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from hundreds of MIT courses visit mitop courseware at align:start position:0% align:start position:0% ocw.mit.edu everybody would like to align:start position:0% ocw.mit.edu everybody would like to align:start position:0% ocw.mit.edu everybody would like to believe that they just simply sort of align:start position:0% believe that they just simply sort of align:start position:0% believe that they just simply sort of think about the subject a little bit and align:start position:0% think about the subject a little bit and align:start position:0% think about the subject a little bit and it'll come to them it's kind of like the align:start position:0% it'll come to them it's kind of like the align:start position:0% it'll come to them it's kind of like the osmosis theory of learning but in fact align:start position:0% osmosis theory of learning but in fact align:start position:0% osmosis theory of learning but in fact the way you get good at something is align:start position:0% the way you get good at something is align:start position:0% the way you get good at something is practicing it's the same for music it's align:start position:0% practicing it's the same for music it's align:start position:0% practicing it's the same for music it's the same for sports it's the same for align:start position:0% the same for sports it's the same for align:start position:0% the same for sports it's the same for academics so the way you get good in align:start position:0% academics so the way you get good in align:start position:0% academics so the way you get good in this subject is to work on the homework align:start position:0% this subject is to work on the homework align:start position:0% this subject is to work on the homework we'll give two kinds of align:start position:0% we'll give two kinds of align:start position:0% we'll give two kinds of homework there will be conventional align:start position:0% homework there will be conventional align:start position:0% homework there will be conventional kinds of problems followed by align:start position:0% kinds of problems followed by align:start position:0% kinds of problems followed by engineering design align:start position:0% engineering design align:start position:0% engineering design problems the conventional kinds of align:start position:0% problems the conventional kinds of align:start position:0% problems the conventional kinds of problems are intended to be the kinds of align:start position:0% problems are intended to be the kinds of align:start position:0% problems are intended to be the kinds of problems align:start position:0% problems align:start position:0% problems that have simple align:start position:0% that have simple align:start position:0% that have simple unambiguous easily checkable align:start position:0% unambiguous easily checkable align:start position:0% unambiguous easily checkable answers so well-defined problem single align:start position:0% answers so well-defined problem single align:start position:0% answers so well-defined problem single unambiguous align:start position:0% unambiguous align:start position:0% unambiguous answer uh the kinds of problems that you align:start position:0% answer uh the kinds of problems that you align:start position:0% answer uh the kinds of problems that you might be expecting to see on an exam for align:start position:0% might be expecting to see on an exam for align:start position:0% might be expecting to see on an exam for example but then we'll also have uh align:start position:0% example but then we'll also have uh align:start position:0% example but then we'll also have uh engineering design problems engineering align:start position:0% engineering design problems engineering align:start position:0% engineering design problems engineering design problems are intended to align:start position:0% design problems are intended to align:start position:0% design problems are intended to be a little more ambitious a little more align:start position:0% be a little more ambitious a little more align:start position:0% be a little more ambitious a little more fun a little harder align:start position:0% fun a little harder align:start position:0% fun a little harder Perhaps Perhaps not completely well align:start position:0% Perhaps Perhaps not completely well align:start position:0% Perhaps Perhaps not completely well specified sort of more like the kind of align:start position:0% specified sort of more like the kind of align:start position:0% specified sort of more like the kind of a problem that your boss might give you align:start position:0% a problem that your boss might give you align:start position:0% a problem that your boss might give you after you've align:start position:0% after you've align:start position:0% after you've graduated so part of the exercise will align:start position:0% graduated so part of the exercise will align:start position:0% graduated so part of the exercise will be figuring out exactly how to convince align:start position:0% be figuring out exactly how to convince align:start position:0% be figuring out exactly how to convince somebody that you got the right answer align:start position:0% somebody that you got the right answer align:start position:0% somebody that you got the right answer so those kinds of problems are often the align:start position:0% so those kinds of problems are often the align:start position:0% so those kinds of problems are often the it will often be the case that numerics align:start position:0% it will often be the case that numerics align:start position:0% it will often be the case that numerics will help you in making an align:start position:0% will help you in making an align:start position:0% will help you in making an argument so it's often will be the case align:start position:0% argument so it's often will be the case align:start position:0% argument so it's often will be the case that we'll ask you to plot something or align:start position:0% that we'll ask you to plot something or align:start position:0% that we'll ask you to plot something or maybe you just decide that a plot is the align:start position:0% maybe you just decide that a plot is the align:start position:0% maybe you just decide that a plot is the most effective way to communicate your align:start position:0% most effective way to communicate your align:start position:0% most effective way to communicate your result we are completely Ambi ambivalent align:start position:0% result we are completely Ambi ambivalent align:start position:0% result we are completely Ambi ambivalent about what programming language we that align:start position:0% about what programming language we that align:start position:0% about what programming language we that you use since uh 601 and 602 are align:start position:0% you use since uh 601 and 602 are align:start position:0% you use since uh 601 and 602 are prerequisites we assume you all know align:start position:0% prerequisites we assume you all know align:start position:0% prerequisites we assume you all know python so all the examples that I give align:start position:0% python so all the examples that I give align:start position:0% python so all the examples that I give in lecture handouts or in homework align:start position:0% in lecture handouts or in homework align:start position:0% in lecture handouts or in homework solutions will be in Python but if you'd align:start position:0% solutions will be in Python but if you'd align:start position:0% solutions will be in Python but if you'd rather use mat lab I just don't care align:start position:0% rather use mat lab I just don't care align:start position:0% rather use mat lab I just don't care okay we're completely ambivalent about align:start position:0% okay we're completely ambivalent about align:start position:0% okay we're completely ambivalent about what programming language you use but it align:start position:0% what programming language you use but it align:start position:0% what programming language you use but it will be important that in some of the align:start position:0% will be important that in some of the align:start position:0% will be important that in some of the problems you'll find that useful to be align:start position:0% problems you'll find that useful to be align:start position:0% problems you'll find that useful to be able to to um generate a numerical kind align:start position:0% able to to um generate a numerical kind align:start position:0% able to to um generate a numerical kind of align:start position:0% of align:start position:0% of solution these are very different kinds align:start position:0% solution these are very different kinds align:start position:0% solution these are very different kinds of problems and to help you with both of align:start position:0% of problems and to help you with both of align:start position:0% of problems and to help you with both of them we have a different idea about align:start position:0% them we have a different idea about align:start position:0% them we have a different idea about each so to help you with the align:start position:0% each so to help you with the align:start position:0% each so to help you with the conventional problems the kind that have align:start position:0% conventional problems the kind that have align:start position:0% conventional problems the kind that have precise align:start position:0% precise align:start position:0% precise easy to Che answers we've developed for align:start position:0% easy to Che answers we've developed for align:start position:0% easy to Che answers we've developed for the first time ever over the summer a align:start position:0% the first time ever over the summer a align:start position:0% the first time ever over the summer a tutor type environment for align:start position:0% tutor type environment for align:start position:0% tutor type environment for 63 so this was intended to be like the align:start position:0% 63 so this was intended to be like the align:start position:0% 63 so this was intended to be like the tutor environment that we use in 601 align:start position:0% tutor environment that we use in 601 align:start position:0% tutor environment that we use in 601 it's one of the more popular aspects of align:start position:0% it's one of the more popular aspects of align:start position:0% it's one of the more popular aspects of 61 the fact that you can put in an align:start position:0% 61 the fact that you can put in an align:start position:0% 61 the fact that you can put in an answer and hit the button that says align:start position:0% answer and hit the button that says align:start position:0% answer and hit the button that says check okay so we're going to have one of align:start position:0% check okay so we're going to have one of align:start position:0% check okay so we're going to have one of those so the idea is going to be that in align:start position:0% those so the idea is going to be that in align:start position:0% those so the idea is going to be that in the case that we ask you a simple align:start position:0% the case that we ask you a simple align:start position:0% the case that we ask you a simple question that has a pre precise answer align:start position:0% question that has a pre precise answer align:start position:0% question that has a pre precise answer we'll use a tutor environment to let you align:start position:0% we'll use a tutor environment to let you align:start position:0% we'll use a tutor environment to let you check your answer to see if you got the align:start position:0% check your answer to see if you got the align:start position:0% check your answer to see if you got the right answer in the case of the align:start position:0% right answer in the case of the align:start position:0% right answer in the case of the engineering design problems that won't align:start position:0% engineering design problems that won't align:start position:0% engineering design problems that won't be the case it will not be the case that align:start position:0% be the case it will not be the case that align:start position:0% be the case it will not be the case that it's very easy at all to think about a align:start position:0% it's very easy at all to think about a align:start position:0% it's very easy at all to think about a computer program that would check align:start position:0% computer program that would check align:start position:0% computer program that would check you so the alternative is going to be align:start position:0% you so the alternative is going to be align:start position:0% you so the alternative is going to be that we will have extended office align:start position:0% that we will have extended office align:start position:0% that we will have extended office hours so we're going to have something align:start position:0% hours so we're going to have something align:start position:0% hours so we're going to have something that I think of as open or Block office align:start position:0% that I think of as open or Block office align:start position:0% that I think of as open or Block office hours Monday and Tuesday afternoon and align:start position:0% hours Monday and Tuesday afternoon and align:start position:0% hours Monday and Tuesday afternoon and early evening we've reserved the align:start position:0% early evening we've reserved the align:start position:0% early evening we've reserved the basement of sta align:start position:0% basement of sta align:start position:0% basement of sta 32044 for use of this class the idea align:start position:0% 32044 for use of this class the idea align:start position:0% 32044 for use of this class the idea being that that's a nice space you can align:start position:0% being that that's a nice space you can align:start position:0% being that that's a nice space you can just come there to work anytime you want align:start position:0% just come there to work anytime you want align:start position:0% just come there to work anytime you want to hopefully because it's align:start position:0% to hopefully because it's align:start position:0% to hopefully because it's nice and hopefully because you all know align:start position:0% nice and hopefully because you all know align:start position:0% nice and hopefully because you all know that there will be other 63 people there align:start position:0% that there will be other 63 people there align:start position:0% that there will be other 63 people there you'll just decide that's a good place align:start position:0% you'll just decide that's a good place align:start position:0% you'll just decide that's a good place to work and that's convenient because in align:start position:0% to work and that's convenient because in align:start position:0% to work and that's convenient because in these engineering design problems if you align:start position:0% these engineering design problems if you align:start position:0% these engineering design problems if you run into something that's unclear to you align:start position:0% run into something that's unclear to you align:start position:0% run into something that's unclear to you we don't really want you to spend an align:start position:0% we don't really want you to spend an align:start position:0% we don't really want you to spend an hour pondering what was being align:start position:0% hour pondering what was being align:start position:0% hour pondering what was being asked so we want to make it easy for you align:start position:0% asked so we want to make it easy for you align:start position:0% asked so we want to make it easy for you to get align:start position:0% to get align:start position:0% to get help and you can get help from peers align:start position:0% help and you can get help from peers align:start position:0% help and you can get help from peers from other students who are there but align:start position:0% from other students who are there but align:start position:0% from other students who are there but we'll also make sure that there's at align:start position:0% we'll also make sure that there's at align:start position:0% we'll also make sure that there's at least one six3 staff member there all align:start position:0% least one six3 staff member there all align:start position:0% least one six3 staff member there all the align:start position:0% the align:start position:0% the time okay so the homeworks will be due align:start position:0% time okay so the homeworks will be due align:start position:0% time okay so the homeworks will be due on Wednesday that's the Strategic value align:start position:0% on Wednesday that's the Strategic value align:start position:0% on Wednesday that's the Strategic value of Mondays and Tuesdays so that you can align:start position:0% of Mondays and Tuesdays so that you can align:start position:0% of Mondays and Tuesdays so that you can uh use that time to get ready for align:start position:0% uh use that time to get ready for align:start position:0% uh use that time to get ready for turning in the homework on Wednesday align:start position:0% turning in the homework on Wednesday align:start position:0% turning in the homework on Wednesday questions or align:start position:0% align:start position:0% comments okay we're required to say align:start position:0% comments okay we're required to say align:start position:0% comments okay we're required to say something about what we mean by a align:start position:0% something about what we mean by a align:start position:0% something about what we mean by a collaboration policy so in this course align:start position:0% collaboration policy so in this course align:start position:0% collaboration policy so in this course we would like you to talk to each other align:start position:0% we would like you to talk to each other align:start position:0% we would like you to talk to each other and help each other figure out what's align:start position:0% and help each other figure out what's align:start position:0% and help each other figure out what's going align:start position:0% align:start position:0% on at the same time we'd like to reward align:start position:0% on at the same time we'd like to reward align:start position:0% on at the same time we'd like to reward you for being on the honest Side by align:start position:0% you for being on the honest Side by align:start position:0% you for being on the honest Side by which I'm going to mean if you sign your align:start position:0% which I'm going to mean if you sign your align:start position:0% which I'm going to mean if you sign your name to something you actually did it align:start position:0% align:start position:0% so we'd like to believe that the align:start position:0% so we'd like to believe that the align:start position:0% so we'd like to believe that the homework that you turn in under your align:start position:0% homework that you turn in under your align:start position:0% homework that you turn in under your name you align:start position:0% name you align:start position:0% name you did you're perfectly welcome to talk to align:start position:0% did you're perfectly welcome to talk to align:start position:0% did you're perfectly welcome to talk to each other to talk to the staff to talk align:start position:0% each other to talk to the staff to talk align:start position:0% each other to talk to the staff to talk to friends who took this course align:start position:0% to friends who took this course align:start position:0% to friends who took this course previously get all the help you like in align:start position:0% previously get all the help you like in align:start position:0% previously get all the help you like in trying to understand the concepts but align:start position:0% trying to understand the concepts but align:start position:0% trying to understand the concepts but when it comes down to writing up your align:start position:0% when it comes down to writing up your align:start position:0% when it comes down to writing up your homework and turning it in we expect align:start position:0% homework and turning it in we expect align:start position:0% homework and turning it in we expect that what you signed your name to is align:start position:0% that what you signed your name to is align:start position:0% that what you signed your name to is something you did if you got especially align:start position:0% something you did if you got especially align:start position:0% something you did if you got especially large amount of help at the conceptual align:start position:0% large amount of help at the conceptual align:start position:0% large amount of help at the conceptual level we we'd like you to acknowledge align:start position:0% level we we'd like you to acknowledge align:start position:0% level we we'd like you to acknowledge that you don't need to acknowledge that align:start position:0% that you don't need to acknowledge that align:start position:0% that you don't need to acknowledge that the ta's helped you we expect that but align:start position:0% the ta's helped you we expect that but align:start position:0% the ta's helped you we expect that but if you collaborated with somebody to align:start position:0% if you collaborated with somebody to align:start position:0% if you collaborated with somebody to understand what the problem was we'd align:start position:0% understand what the problem was we'd align:start position:0% understand what the problem was we'd like you to tell us that just right at align:start position:0% like you to tell us that just right at align:start position:0% like you to tell us that just right at the top collaborated on figuring out the align:start position:0% the top collaborated on figuring out the align:start position:0% the top collaborated on figuring out the concept with so and align:start position:0% concept with so and align:start position:0% concept with so and so but we'd like you to have written align:start position:0% so but we'd like you to have written align:start position:0% so but we'd like you to have written your homework so for example we would align:start position:0% your homework so for example we would align:start position:0% your homework so for example we would like you not to say I copied so and so's align:start position:0% like you not to say I copied so and so's align:start position:0% like you not to say I copied so and so's homework word for word even though align:start position:0% homework word for word even though align:start position:0% homework word for word even though that's an honest statement that's not align:start position:0% that's an honest statement that's not align:start position:0% that's an honest statement that's not something that we're actually looking align:start position:0% something that we're actually looking align:start position:0% something that we're actually looking for okay is that is that clear we want align:start position:0% for okay is that is that clear we want align:start position:0% for okay is that is that clear we want you to work together but we would like align:start position:0% you to work together but we would like align:start position:0% you to work together but we would like you to write up your own align:start position:0% align:start position:0% Solutions okay there's a dead we have align:start position:0% Solutions okay there's a dead we have align:start position:0% Solutions okay there's a dead we have firm align:start position:0% firm align:start position:0% firm deadlines uh the homeworks will always align:start position:0% deadlines uh the homeworks will always align:start position:0% deadlines uh the homeworks will always be do on align:start position:0% be do on align:start position:0% be do on Wednesday you'll be able to slip on one align:start position:0% Wednesday you'll be able to slip on one align:start position:0% Wednesday you'll be able to slip on one without align:start position:0% without align:start position:0% without penalty so if you turn in one late it align:start position:0% penalty so if you turn in one late it align:start position:0% penalty so if you turn in one late it won't change it won't have any effect on align:start position:0% won't change it won't have any effect on align:start position:0% won't change it won't have any effect on your grade whatever if you do twice that align:start position:0% your grade whatever if you do twice that align:start position:0% your grade whatever if you do twice that could have an effect unless you're align:start position:0% could have an effect unless you're align:start position:0% could have an effect unless you're excused by a Dean or an instructor or a align:start position:0% excused by a Dean or an instructor or a align:start position:0% excused by a Dean or an instructor or a medical align:start position:0% medical align:start position:0% medical person uh that submission will count align:start position:0% person uh that submission will count align:start position:0% person uh that submission will count half what it would normally align:start position:0% half what it would normally align:start position:0% half what it would normally count I should have mentioned when I was align:start position:0% count I should have mentioned when I was align:start position:0% count I should have mentioned when I was talking about homework homework one is align:start position:0% talking about homework homework one is align:start position:0% talking about homework homework one is already posted it will be due next align:start position:0% align:start position:0% Wednesday uh the online submission part align:start position:0% Wednesday uh the online submission part align:start position:0% Wednesday uh the online submission part is almost ready to be posted and will be align:start position:0% is almost ready to be posted and will be align:start position:0% is almost ready to be posted and will be posted later this afternoon so right now align:start position:0% posted later this afternoon so right now align:start position:0% posted later this afternoon so right now you can see all the problems but the align:start position:0% you can see all the problems but the align:start position:0% you can see all the problems but the online will become available this align:start position:0% align:start position:0% afternoon okay here's uh3 at a align:start position:0% afternoon okay here's uh3 at a align:start position:0% afternoon okay here's uh3 at a glance it's not too unlike what you align:start position:0% glance it's not too unlike what you align:start position:0% glance it's not too unlike what you might have expected just a couple things align:start position:0% might have expected just a couple things align:start position:0% might have expected just a couple things to point out the order of coverage is align:start position:0% to point out the order of coverage is align:start position:0% to point out the order of coverage is not the same as the order of coverage in align:start position:0% not the same as the order of coverage in align:start position:0% not the same as the order of coverage in oppenheim and align:start position:0% oppenheim and align:start position:0% oppenheim and wiski the reason for doing that is that align:start position:0% wiski the reason for doing that is that align:start position:0% wiski the reason for doing that is that having taught this class for 15 years align:start position:0% having taught this class for 15 years align:start position:0% having taught this class for 15 years years I just think there's some subjects align:start position:0% years I just think there's some subjects align:start position:0% years I just think there's some subjects that are easier starting align:start position:0% that are easier starting align:start position:0% that are easier starting points than align:start position:0% points than align:start position:0% points than others so I'm doing what I think is the align:start position:0% others so I'm doing what I think is the align:start position:0% others so I'm doing what I think is the easiest entry to this align:start position:0% easiest entry to this align:start position:0% easiest entry to this material with the idea that after you've align:start position:0% material with the idea that after you've align:start position:0% material with the idea that after you've got the easy stuff under your belt it's align:start position:0% got the easy stuff under your belt it's align:start position:0% got the easy stuff under your belt it's easier to move on to the more difficult align:start position:0% easier to move on to the more difficult align:start position:0% easier to move on to the more difficult material so the order is not precisely align:start position:0% material so the order is not precisely align:start position:0% material so the order is not precisely the same as that in oppenheim and wilski align:start position:0% the same as that in oppenheim and wilski align:start position:0% the same as that in oppenheim and wilski but there is a map on the website that align:start position:0% but there is a map on the website that align:start position:0% but there is a map on the website that gives you a week-by-week align:start position:0% gives you a week-by-week align:start position:0% gives you a week-by-week breakdown of what part of oppenheim and align:start position:0% breakdown of what part of oppenheim and align:start position:0% breakdown of what part of oppenheim and wil are we working on right align:start position:0% wil are we working on right align:start position:0% wil are we working on right now so Oppenheimer wiski is the align:start position:0% now so Oppenheimer wiski is the align:start position:0% now so Oppenheimer wiski is the recommended text but it's not quite in align:start position:0% recommended text but it's not quite in align:start position:0% recommended text but it's not quite in the regular order of oppenheim and align:start position:0% the regular order of oppenheim and align:start position:0% the regular order of oppenheim and wilsky we will eventually cover exactly align:start position:0% wilsky we will eventually cover exactly align:start position:0% wilsky we will eventually cover exactly the same material that is in that align:start position:0% the same material that is in that align:start position:0% the same material that is in that oppenheim and wilsky was written for align:start position:0% oppenheim and wilsky was written for align:start position:0% oppenheim and wilsky was written for this class we'll cover the same material align:start position:0% this class we'll cover the same material align:start position:0% this class we'll cover the same material just not quite in the same align:start position:0% align:start position:0% order another thing to notice is exams align:start position:0% order another thing to notice is exams align:start position:0% order another thing to notice is exams we'll have three midterm align:start position:0% we'll have three midterm align:start position:0% we'll have three midterm exams and a final the midterm term exams align:start position:0% exams and a final the midterm term exams align:start position:0% exams and a final the midterm term exams are all evening exams 7:30 to 9:30 on align:start position:0% are all evening exams 7:30 to 9:30 on align:start position:0% are all evening exams 7:30 to 9:30 on Wednesday align:start position:0% align:start position:0% evening the idea is that there's align:start position:0% evening the idea is that there's align:start position:0% evening the idea is that there's intended to be a gentle ramp through the align:start position:0% intended to be a gentle ramp through the align:start position:0% intended to be a gentle ramp through the exams exam one is worth 10% exam two is align:start position:0% exams exam one is worth 10% exam two is align:start position:0% exams exam one is worth 10% exam two is worth 15% exam three is worth 20% the align:start position:0% worth 15% exam three is worth 20% the align:start position:0% worth 15% exam three is worth 20% the final exam is worth align:start position:0% final exam is worth align:start position:0% final exam is worth 40% so the idea is that if you have a align:start position:0% 40% so the idea is that if you have a align:start position:0% 40% so the idea is that if you have a mismatch on the way if you don't quite align:start position:0% mismatch on the way if you don't quite align:start position:0% mismatch on the way if you don't quite understand the way we're going to be align:start position:0% understand the way we're going to be align:start position:0% understand the way we're going to be asking questions and so forth there's align:start position:0% asking questions and so forth there's align:start position:0% asking questions and so forth there's not a big penalty for screwing up the align:start position:0% not a big penalty for screwing up the align:start position:0% not a big penalty for screwing up the first exam that's the idea so there's a align:start position:0% first exam that's the idea so there's a align:start position:0% first exam that's the idea so there's a intended to be a gentle ramp um in how align:start position:0% intended to be a gentle ramp um in how align:start position:0% intended to be a gentle ramp um in how much the exams align:start position:0% align:start position:0% count okay and finally I'd like feedback align:start position:0% count okay and finally I'd like feedback align:start position:0% count okay and finally I'd like feedback on what you think is happening so I align:start position:0% on what you think is happening so I align:start position:0% on what you think is happening so I would like to ask for a few volunteers align:start position:0% would like to ask for a few volunteers align:start position:0% would like to ask for a few volunteers few is four or align:start position:0% align:start position:0% fewer who would be interested to meet align:start position:0% fewer who would be interested to meet align:start position:0% fewer who would be interested to meet with me once a week just to tell me what align:start position:0% with me once a week just to tell me what align:start position:0% with me once a week just to tell me what you think align:start position:0% you think align:start position:0% you think think ideally those people would be kind align:start position:0% think ideally those people would be kind align:start position:0% think ideally those people would be kind of outgoing and tell me things that align:start position:0% of outgoing and tell me things that align:start position:0% of outgoing and tell me things that other people think too but if you're align:start position:0% other people think too but if you're align:start position:0% other people think too but if you're completely introverted and want to only align:start position:0% completely introverted and want to only align:start position:0% completely introverted and want to only tell me what you think that's fine align:start position:0% tell me what you think that's fine align:start position:0% tell me what you think that's fine too so uh this is an opportunity for you align:start position:0% too so uh this is an opportunity for you align:start position:0% too so uh this is an opportunity for you to tell the staff to give us feedback on align:start position:0% to tell the staff to give us feedback on align:start position:0% to tell the staff to give us feedback on how things are going we're going too align:start position:0% how things are going we're going too align:start position:0% how things are going we're going too slow we're going too fast we're going align:start position:0% slow we're going too fast we're going align:start position:0% slow we're going too fast we're going too boring we're intensely too align:start position:0% too boring we're intensely too align:start position:0% too boring we're intensely too interesting that kind of stuff align:start position:0% interesting that kind of stuff align:start position:0% interesting that kind of stuff um uh align:start position:0% um uh align:start position:0% um uh and it's a it's an opportunity for you align:start position:0% and it's a it's an opportunity for you align:start position:0% and it's a it's an opportunity for you to understand our perspective like well align:start position:0% to understand our perspective like well align:start position:0% to understand our perspective like well the reason we did that was blah and then align:start position:0% the reason we did that was blah and then align:start position:0% the reason we did that was blah and then you can tell us we but that's not align:start position:0% you can tell us we but that's not align:start position:0% you can tell us we but that's not important so it's an opportunity for um align:start position:0% important so it's an opportunity for um align:start position:0% important so it's an opportunity for um you to convey to align:start position:0% you to convey to align:start position:0% you to convey to us how you think things are going and align:start position:0% us how you think things are going and align:start position:0% us how you think things are going and how you think things should be different align:start position:0% how you think things should be different align:start position:0% how you think things should be different it's also an opportunity for you to align:start position:0% it's also an opportunity for you to align:start position:0% it's also an opportunity for you to learn about teaching so if you think align:start position:0% learn about teaching so if you think align:start position:0% learn about teaching so if you think about teaching as a career it's a good align:start position:0% about teaching as a career it's a good align:start position:0% about teaching as a career it's a good thing to do completely voluntary has align:start position:0% thing to do completely voluntary has align:start position:0% thing to do completely voluntary has absolutely absolutely no effect on your align:start position:0% absolutely absolutely no effect on your align:start position:0% absolutely absolutely no effect on your grade uh if you think you might be align:start position:0% grade uh if you think you might be align:start position:0% grade uh if you think you might be interested it's probably going to meet align:start position:0% interested it's probably going to meet align:start position:0% interested it's probably going to meet on we'll probably meet on Thursday align:start position:0% on we'll probably meet on Thursday align:start position:0% on we'll probably meet on Thursday afternoon but that's negotiable since align:start position:0% afternoon but that's negotiable since align:start position:0% afternoon but that's negotiable since it's only four people and if you are align:start position:0% it's only four people and if you are align:start position:0% it's only four people and if you are interested please send me an email and align:start position:0% interested please send me an email and align:start position:0% interested please send me an email and the first four people who ask they're align:start position:0% the first four people who ask they're align:start position:0% the first four people who ask they're there align:start position:0% align:start position:0% okay that's it on align:start position:0% align:start position:0% Administration are there things you'd align:start position:0% Administration are there things you'd align:start position:0% Administration are there things you'd like to know about course Administration align:start position:0% like to know about course Administration align:start position:0% like to know about course Administration before we go on to technical align:start position:0% align:start position:0% things wonderful okay this course is align:start position:0% things wonderful okay this course is align:start position:0% things wonderful okay this course is about align:start position:0% align:start position:0% systems this is not the first course align:start position:0% systems this is not the first course align:start position:0% systems this is not the first course you've had about align:start position:0% you've had about align:start position:0% you've had about systems you've had lots of courses in align:start position:0% systems you've had lots of courses in align:start position:0% systems you've had lots of courses in systems every Physics course you've ever align:start position:0% systems every Physics course you've ever align:start position:0% systems every Physics course you've ever taken is about align:start position:0% align:start position:0% systems what's different about this align:start position:0% systems what's different about this align:start position:0% systems what's different about this course is the way we think about systems align:start position:0% course is the way we think about systems align:start position:0% course is the way we think about systems we think about systems in a particular align:start position:0% we think about systems in a particular align:start position:0% we think about systems in a particular way that turns out to be extraordinarily align:start position:0% way that turns out to be extraordinarily align:start position:0% way that turns out to be extraordinarily powerful useful shows up all the time align:start position:0% powerful useful shows up all the time align:start position:0% powerful useful shows up all the time and that's what the focus of this course align:start position:0% and that's what the focus of this course align:start position:0% and that's what the focus of this course is the the abstraction that we're going align:start position:0% is the the abstraction that we're going align:start position:0% is the the abstraction that we're going to use is we think about a system as a align:start position:0% to use is we think about a system as a align:start position:0% to use is we think about a system as a thing that has an input and an align:start position:0% thing that has an input and an align:start position:0% thing that has an input and an output given the input and the system align:start position:0% output given the input and the system align:start position:0% output given the input and the system you can compute the align:start position:0% you can compute the align:start position:0% you can compute the output systems have one input and one align:start position:0% output systems have one input and one align:start position:0% output systems have one input and one output it's a very special align:start position:0% output it's a very special align:start position:0% output it's a very special way of thinking about a align:start position:0% way of thinking about a align:start position:0% way of thinking about a system that turns out to be surprisingly align:start position:0% system that turns out to be surprisingly align:start position:0% system that turns out to be surprisingly powerful it's so powerful that that's align:start position:0% powerful it's so powerful that that's align:start position:0% powerful it's so powerful that that's really the thing that we will focus on align:start position:0% really the thing that we will focus on align:start position:0% really the thing that we will focus on in this class so it's the thing that I align:start position:0% in this class so it's the thing that I align:start position:0% in this class so it's the thing that I will call align:start position:0% will call align:start position:0% will call the align:start position:0% the align:start position:0% the align:start position:0% align:start position:0% abstraction that's going to be the theme align:start position:0% abstraction that's going to be the theme align:start position:0% abstraction that's going to be the theme in everything we do that represents align:start position:0% in everything we do that represents align:start position:0% in everything we do that represents entation and the best way to get across align:start position:0% entation and the best way to get across align:start position:0% entation and the best way to get across I think how that's different from other align:start position:0% I think how that's different from other align:start position:0% I think how that's different from other representations that you're already align:start position:0% representations that you're already align:start position:0% representations that you're already quite familiar with is to just look at align:start position:0% quite familiar with is to just look at align:start position:0% quite familiar with is to just look at an example so here's a system that I'm align:start position:0% an example so here's a system that I'm align:start position:0% an example so here's a system that I'm sure you all know all about right this align:start position:0% sure you all know all about right this align:start position:0% sure you all know all about right this is not the first time you've seen this align:start position:0% is not the first time you've seen this align:start position:0% is not the first time you've seen this kind of a align:start position:0% kind of a align:start position:0% kind of a system so you all know how to do this align:start position:0% system so you all know how to do this align:start position:0% system so you all know how to do this free body diagrams f equals ma there's align:start position:0% free body diagrams f equals ma there's align:start position:0% free body diagrams f equals ma there's an enormous number of ways that you can align:start position:0% an enormous number of ways that you can align:start position:0% an enormous number of ways that you can analyze such a system we're going to align:start position:0% analyze such a system we're going to align:start position:0% analyze such a system we're going to take a very special approach where we align:start position:0% take a very special approach where we align:start position:0% take a very special approach where we think about the align:start position:0% think about the align:start position:0% think about the system as having an input and an align:start position:0% align:start position:0% output okay that's a little bit align:start position:0% output okay that's a little bit align:start position:0% output okay that's a little bit arbitrary we'll say a little more in a align:start position:0% arbitrary we'll say a little more in a align:start position:0% arbitrary we'll say a little more in a minute about how arbitrary it is but the align:start position:0% minute about how arbitrary it is but the align:start position:0% minute about how arbitrary it is but the idea is going to be that this system the align:start position:0% idea is going to be that this system the align:start position:0% idea is going to be that this system the mass spring system the Anvil with a align:start position:0% mass spring system the Anvil with a align:start position:0% mass spring system the Anvil with a spring there's an align:start position:0% spring there's an align:start position:0% spring there's an input and there's an output so for align:start position:0% input and there's an output so for align:start position:0% input and there's an output so for example for some particular purpose I align:start position:0% example for some particular purpose I align:start position:0% example for some particular purpose I might want to think that the align:start position:0% might want to think that the align:start position:0% might want to think that the input is the position of my align:start position:0% input is the position of my align:start position:0% input is the position of my hand one reason I might want to think align:start position:0% hand one reason I might want to think align:start position:0% hand one reason I might want to think about that is it is under my control so align:start position:0% about that is it is under my control so align:start position:0% about that is it is under my control so it might be reasonable to think about my align:start position:0% it might be reasonable to think about my align:start position:0% it might be reasonable to think about my hand being the input so for example my align:start position:0% hand being the input so for example my align:start position:0% hand being the input so for example my hand can do align:start position:0% hand can do align:start position:0% hand can do this right so I might think about the align:start position:0% this right so I might think about the align:start position:0% this right so I might think about the position of my hand being the align:start position:0% position of my hand being the align:start position:0% position of my hand being the input so if I did that then what I could align:start position:0% input so if I did that then what I could align:start position:0% input so if I did that then what I could do is characterize the input by some align:start position:0% do is characterize the input by some align:start position:0% do is characterize the input by some waveform a align:start position:0% align:start position:0% signal in order to use the abstraction I align:start position:0% signal in order to use the abstraction I align:start position:0% signal in order to use the abstraction I also have to say what the output is I align:start position:0% also have to say what the output is I align:start position:0% also have to say what the output is I might be interested for example in align:start position:0% might be interested for example in align:start position:0% might be interested for example in knowing the position of the align:start position:0% knowing the position of the align:start position:0% knowing the position of the mass then the output would be the align:start position:0% mass then the output would be the align:start position:0% mass then the output would be the displacement of the align:start position:0% displacement of the align:start position:0% displacement of the mass that assignment of input and output align:start position:0% mass that assignment of input and output align:start position:0% mass that assignment of input and output is somewhat align:start position:0% align:start position:0% arbitrary I get to choose that to make align:start position:0% arbitrary I get to choose that to make align:start position:0% arbitrary I get to choose that to make the problem that I'm interested it in as align:start position:0% the problem that I'm interested it in as align:start position:0% the problem that I'm interested it in as easy as align:start position:0% easy as align:start position:0% easy as possible so for example if I'm thinking align:start position:0% possible so for example if I'm thinking align:start position:0% possible so for example if I'm thinking about this uh mass and spring system align:start position:0% about this uh mass and spring system align:start position:0% about this uh mass and spring system here at using my hand as the input that align:start position:0% here at using my hand as the input that align:start position:0% here at using my hand as the input that seems kind of natural because that's the align:start position:0% seems kind of natural because that's the align:start position:0% seems kind of natural because that's the thing I control thinking about the align:start position:0% thing I control thinking about the align:start position:0% thing I control thinking about the position of the mass as the output that align:start position:0% position of the mass as the output that align:start position:0% position of the mass as the output that seems reasonable because that's the PCT align:start position:0% seems reasonable because that's the PCT align:start position:0% seems reasonable because that's the PCT that's the thing you could take a align:start position:0% that's the thing you could take a align:start position:0% that's the thing you could take a picture of it's the thing you can align:start position:0% picture of it's the thing you can align:start position:0% picture of it's the thing you can observe from where you are now but it's align:start position:0% observe from where you are now but it's align:start position:0% observe from where you are now but it's not unique I might have wanted to think align:start position:0% not unique I might have wanted to think align:start position:0% not unique I might have wanted to think about the force that I'm putting into it align:start position:0% about the force that I'm putting into it align:start position:0% about the force that I'm putting into it as the align:start position:0% as the align:start position:0% as the input I might have wanted to think about align:start position:0% input I might have wanted to think about align:start position:0% input I might have wanted to think about the output being the acceleration of the align:start position:0% the output being the acceleration of the align:start position:0% the output being the acceleration of the mass maybe maybe there's some limit to align:start position:0% mass maybe maybe there's some limit to align:start position:0% mass maybe maybe there's some limit to how much acceleration some Mass can can align:start position:0% how much acceleration some Mass can can align:start position:0% how much acceleration some Mass can can take so you get to choose the input and align:start position:0% take so you get to choose the input and align:start position:0% take so you get to choose the input and the align:start position:0% the align:start position:0% the output but you have to choose an input align:start position:0% output but you have to choose an input align:start position:0% output but you have to choose an input and an output that's part of the align:start position:0% and an output that's part of the align:start position:0% and an output that's part of the abstraction and that's part of the power align:start position:0% abstraction and that's part of the power align:start position:0% abstraction and that's part of the power we'll see later that part of the power align:start position:0% we'll see later that part of the power align:start position:0% we'll see later that part of the power of doing it this way is that you draw align:start position:0% of doing it this way is that you draw align:start position:0% of doing it this way is that you draw attention to the things things of align:start position:0% attention to the things things of align:start position:0% attention to the things things of interest and push into the background align:start position:0% interest and push into the background align:start position:0% interest and push into the background the parts that are not of interest at align:start position:0% the parts that are not of interest at align:start position:0% the parts that are not of interest at least not align:start position:0% align:start position:0% now this kind of an abstraction is align:start position:0% now this kind of an abstraction is align:start position:0% now this kind of an abstraction is extremely align:start position:0% extremely align:start position:0% extremely versatile here's a completely different align:start position:0% versatile here's a completely different align:start position:0% versatile here's a completely different kind of align:start position:0% kind of align:start position:0% kind of problem imagine water flowing into a align:start position:0% problem imagine water flowing into a align:start position:0% problem imagine water flowing into a tank the Tank's leaky so it flows align:start position:0% tank the Tank's leaky so it flows align:start position:0% tank the Tank's leaky so it flows out there's a second tank okay what on align:start position:0% out there's a second tank okay what on align:start position:0% out there's a second tank okay what on Earth could this be this could be maybe align:start position:0% Earth could this be this could be maybe align:start position:0% Earth could this be this could be maybe this is the it's appropriate to today align:start position:0% this is the it's appropriate to today align:start position:0% this is the it's appropriate to today right it's raining or it was raining uh align:start position:0% right it's raining or it was raining uh align:start position:0% right it's raining or it was raining uh so it was pouring when I came in it was align:start position:0% so it was pouring when I came in it was align:start position:0% so it was pouring when I came in it was pouring uh maybe this align:start position:0% pouring uh maybe this align:start position:0% pouring uh maybe this is nature delivering align:start position:0% is nature delivering align:start position:0% is nature delivering rain this is the wuber align:start position:0% rain this is the wuber align:start position:0% rain this is the wuber reservoir from which we get water in align:start position:0% reservoir from which we get water in align:start position:0% reservoir from which we get water in Boston maybe this is the Fresh Pond align:start position:0% Boston maybe this is the Fresh Pond align:start position:0% Boston maybe this is the Fresh Pond Reservoir from which we get water in align:start position:0% Reservoir from which we get water in align:start position:0% Reservoir from which we get water in Cambridge maybe this is is the rate at align:start position:0% Cambridge maybe this is is the rate at align:start position:0% Cambridge maybe this is is the rate at which water goes from wubber align:start position:0% which water goes from wubber align:start position:0% which water goes from wubber to Fresh align:start position:0% to Fresh align:start position:0% to Fresh Pond maybe this is the rate at which align:start position:0% Pond maybe this is the rate at which align:start position:0% Pond maybe this is the rate at which water leaves Fresh Pond and goes into align:start position:0% water leaves Fresh Pond and goes into align:start position:0% water leaves Fresh Pond and goes into consumers align:start position:0% consumers align:start position:0% consumers houses the point is that there's some align:start position:0% houses the point is that there's some align:start position:0% houses the point is that there's some abstraction there's there's some align:start position:0% abstraction there's there's some align:start position:0% abstraction there's there's some physical I didn't say that right there's align:start position:0% physical I didn't say that right there's align:start position:0% physical I didn't say that right there's a physical align:start position:0% a physical align:start position:0% a physical thing rain wuber Wu Reservoir um Fresh align:start position:0% thing rain wuber Wu Reservoir um Fresh align:start position:0% thing rain wuber Wu Reservoir um Fresh Pond Reservoir consumer houses there align:start position:0% Pond Reservoir consumer houses there align:start position:0% Pond Reservoir consumer houses there some physical thing there's some laws of align:start position:0% some physical thing there's some laws of align:start position:0% some physical thing there's some laws of physics that dictate things but we align:start position:0% physics that dictate things but we align:start position:0% physics that dictate things but we ignore or ignore is not quite the right align:start position:0% ignore or ignore is not quite the right align:start position:0% ignore or ignore is not quite the right way to think about it we think about align:start position:0% way to think about it we think about align:start position:0% way to think about it we think about those rules of physics as rules that align:start position:0% those rules of physics as rules that align:start position:0% those rules of physics as rules that govern the input output relationship so align:start position:0% govern the input output relationship so align:start position:0% govern the input output relationship so we think about then the entire system align:start position:0% we think about then the entire system align:start position:0% we think about then the entire system rather than thinking about it as align:start position:0% rather than thinking about it as align:start position:0% rather than thinking about it as reservoirs and rain and water and flow align:start position:0% reservoirs and rain and water and flow align:start position:0% reservoirs and rain and water and flow and all that sort of thing we think align:start position:0% and all that sort of thing we think align:start position:0% and all that sort of thing we think about it as there's a signal align:start position:0% about it as there's a signal align:start position:0% about it as there's a signal in rain align:start position:0% in rain align:start position:0% in rain and there's a signal out water align:start position:0% and there's a signal out water align:start position:0% and there's a signal out water usage and we take all the details of the align:start position:0% usage and we take all the details of the align:start position:0% usage and we take all the details of the system and bury it in this box it's an align:start position:0% system and bury it in this box it's an align:start position:0% system and bury it in this box it's an abstraction it's a way to suppress some align:start position:0% abstraction it's a way to suppress some align:start position:0% abstraction it's a way to suppress some details to highlight align:start position:0% align:start position:0% others okay it's and we use this in huge align:start position:0% others okay it's and we use this in huge align:start position:0% others okay it's and we use this in huge variety of situations we can think about align:start position:0% variety of situations we can think about align:start position:0% variety of situations we can think about a third example here a cell phone cell align:start position:0% a third example here a cell phone cell align:start position:0% a third example here a cell phone cell phone is enormously complicated system align:start position:0% phone is enormously complicated system align:start position:0% phone is enormously complicated system but for the purpose of understanding the align:start position:0% but for the purpose of understanding the align:start position:0% but for the purpose of understanding the input output align:start position:0% input output align:start position:0% input output characteristics if what I really want to align:start position:0% characteristics if what I really want to align:start position:0% characteristics if what I really want to know is how good is the quality of the align:start position:0% know is how good is the quality of the align:start position:0% know is how good is the quality of the audio not that that's even a little bit align:start position:0% audio not that that's even a little bit align:start position:0% audio not that that's even a little bit important for cell phones right cell align:start position:0% important for cell phones right cell align:start position:0% important for cell phones right cell phones are important for texting and align:start position:0% phones are important for texting and align:start position:0% phones are important for texting and taking pictures and have nothing to do align:start position:0% taking pictures and have nothing to do align:start position:0% taking pictures and have nothing to do with voice but in the old days cell align:start position:0% with voice but in the old days cell align:start position:0% with voice but in the old days cell phones had to do with voice and I made align:start position:0% phones had to do with voice and I made align:start position:0% phones had to do with voice and I made this slide back when that was still true align:start position:0% this slide back when that was still true align:start position:0% this slide back when that was still true so I apologize it's out of date um so align:start position:0% so I apologize it's out of date um so align:start position:0% so I apologize it's out of date um so here the important thing of the cell align:start position:0% here the important thing of the cell align:start position:0% here the important thing of the cell phone system was the sound in and the align:start position:0% phone system was the sound in and the align:start position:0% phone system was the sound in and the sound out align:start position:0% sound out align:start position:0% sound out and the idea was to represent the cell align:start position:0% and the idea was to represent the cell align:start position:0% and the idea was to represent the cell phone system by the align:start position:0% phone system by the align:start position:0% phone system by the transformation between how the sound align:start position:0% transformation between how the sound align:start position:0% transformation between how the sound comes in and how the sound comes align:start position:0% comes in and how the sound comes align:start position:0% comes in and how the sound comes out okay so we do this for an enormous align:start position:0% out okay so we do this for an enormous align:start position:0% out okay so we do this for an enormous number of reasons one is that it's align:start position:0% number of reasons one is that it's align:start position:0% number of reasons one is that it's widely applicable you can do this kind align:start position:0% widely applicable you can do this kind align:start position:0% widely applicable you can do this kind of a characterization for systems from align:start position:0% of a characterization for systems from align:start position:0% of a characterization for systems from uh electrical systems mechanical systems align:start position:0% uh electrical systems mechanical systems align:start position:0% uh electrical systems mechanical systems Optical systems acoustic systems align:start position:0% Optical systems acoustic systems align:start position:0% Optical systems acoustic systems biological systems Financial systems all align:start position:0% biological systems Financial systems all align:start position:0% biological systems Financial systems all over the align:start position:0% over the align:start position:0% over the place align:start position:0% place align:start position:0% place that means that this kind of an approach align:start position:0% that means that this kind of an approach align:start position:0% that means that this kind of an approach is very powerful because it's so widely align:start position:0% is very powerful because it's so widely align:start position:0% is very powerful because it's so widely applicable it also means that for align:start position:0% applicable it also means that for align:start position:0% applicable it also means that for example every engineering discipline align:start position:0% example every engineering discipline align:start position:0% example every engineering discipline everywhere has some course like this align:start position:0% everywhere has some course like this align:start position:0% everywhere has some course like this it's just too powerful not to do align:start position:0% it's just too powerful not to do align:start position:0% it's just too powerful not to do it so what regardless of what department align:start position:0% it so what regardless of what department align:start position:0% it so what regardless of what department you were in in the School of Engineering align:start position:0% you were in in the School of Engineering align:start position:0% you were in in the School of Engineering for example there is something like align:start position:0% align:start position:0% this another reason this is interesting align:start position:0% this another reason this is interesting align:start position:0% this another reason this is interesting is that it provides a layer of align:start position:0% is that it provides a layer of align:start position:0% is that it provides a layer of abstraction that lets you focus on align:start position:0% abstraction that lets you focus on align:start position:0% abstraction that lets you focus on certain things so for example let's align:start position:0% certain things so for example let's align:start position:0% certain things so for example let's expand on the cell phone network imagine align:start position:0% expand on the cell phone network imagine align:start position:0% expand on the cell phone network imagine that sound comes into a cell phone well align:start position:0% that sound comes into a cell phone well align:start position:0% that sound comes into a cell phone well the job of the phone is to communicate align:start position:0% the job of the phone is to communicate align:start position:0% the job of the phone is to communicate to a align:start position:0% to a align:start position:0% to a tower and that communication is via some align:start position:0% tower and that communication is via some align:start position:0% tower and that communication is via some electromag magnetic align:start position:0% electromag magnetic align:start position:0% electromag magnetic signal then the job of the tower is to align:start position:0% signal then the job of the tower is to align:start position:0% signal then the job of the tower is to communicate with another align:start position:0% communicate with another align:start position:0% communicate with another tower that happens in all kinds of ways align:start position:0% tower that happens in all kinds of ways align:start position:0% tower that happens in all kinds of ways it could be if optic link it could be a align:start position:0% it could be if optic link it could be a align:start position:0% it could be if optic link it could be a satellite link it could be align:start position:0% satellite link it could be align:start position:0% satellite link it could be electromagnetism it could be lots of align:start position:0% electromagnetism it could be lots of align:start position:0% electromagnetism it could be lots of different things there's a lot of align:start position:0% different things there's a lot of align:start position:0% different things there's a lot of different Technologies for getting align:start position:0% different Technologies for getting align:start position:0% different Technologies for getting Towers to talk to Towers then Towers align:start position:0% Towers to talk to Towers then Towers align:start position:0% Towers to talk to Towers then Towers talk back to cell phones via align:start position:0% talk back to cell phones via align:start position:0% talk back to cell phones via electromagnetic signals just like these align:start position:0% electromagnetic signals just like these align:start position:0% electromagnetic signals just like these and gener and the cell phone eventually align:start position:0% and gener and the cell phone eventually align:start position:0% and gener and the cell phone eventually generates an acoustical align:start position:0% generates an acoustical align:start position:0% generates an acoustical output the idea is that by thinking align:start position:0% output the idea is that by thinking align:start position:0% output the idea is that by thinking about the signals and systems approach align:start position:0% about the signals and systems approach align:start position:0% about the signals and systems approach we have um abstracted away everything align:start position:0% we have um abstracted away everything align:start position:0% we have um abstracted away everything other than the flow of information align:start position:0% other than the flow of information align:start position:0% other than the flow of information it makes it easy to align:start position:0% it makes it easy to align:start position:0% it makes it easy to concentrate to follow the flow of align:start position:0% concentrate to follow the flow of align:start position:0% concentrate to follow the flow of information through a complex align:start position:0% information through a complex align:start position:0% information through a complex system because we have pushed every align:start position:0% system because we have pushed every align:start position:0% system because we have pushed every element into the same framework every align:start position:0% element into the same framework every align:start position:0% element into the same framework every element has an input and align:start position:0% element has an input and align:start position:0% element has an input and output regardless of the underlying align:start position:0% output regardless of the underlying align:start position:0% output regardless of the underlying substrate the analysis is align:start position:0% substrate the analysis is align:start position:0% substrate the analysis is similar that means that components who align:start position:0% similar that means that components who align:start position:0% similar that means that components who that are characterized using the align:start position:0% that are characterized using the align:start position:0% that are characterized using the substraction are easily align:start position:0% substraction are easily align:start position:0% substraction are easily combined so we refer to this as we say align:start position:0% combined so we refer to this as we say align:start position:0% combined so we refer to this as we say that these systems are align:start position:0% that these systems are align:start position:0% that these systems are combinational they're combinational in align:start position:0% combinational they're combinational in align:start position:0% combinational they're combinational in the same sense that python was align:start position:0% the same sense that python was align:start position:0% the same sense that python was combinational in align:start position:0% combinational in align:start position:0% combinational in 601 if you represent things by python align:start position:0% 601 if you represent things by python align:start position:0% 601 if you represent things by python functions it's very easy to combine the align:start position:0% functions it's very easy to combine the align:start position:0% functions it's very easy to combine the functions to have a bigger align:start position:0% functions to have a bigger align:start position:0% functions to have a bigger function whose purpose and details can align:start position:0% function whose purpose and details can align:start position:0% function whose purpose and details can be understood without knowing what was align:start position:0% be understood without knowing what was align:start position:0% be understood without knowing what was was inside every individual align:start position:0% was inside every individual align:start position:0% was inside every individual function similarly here by knowing the align:start position:0% function similarly here by knowing the align:start position:0% function similarly here by knowing the input Behavior the input output behavior align:start position:0% input Behavior the input output behavior align:start position:0% input Behavior the input output behavior of the cell phone the input output align:start position:0% of the cell phone the input output align:start position:0% of the cell phone the input output behavior of the tower the input output align:start position:0% behavior of the tower the input output align:start position:0% behavior of the tower the input output behavior of this Tower is easy to align:start position:0% behavior of this Tower is easy to align:start position:0% behavior of this Tower is easy to compose them and in fact we will spend a align:start position:0% compose them and in fact we will spend a align:start position:0% compose them and in fact we will spend a fair amount of time thinking through the align:start position:0% fair amount of time thinking through the align:start position:0% fair amount of time thinking through the rules of align:start position:0% align:start position:0% combination um so that's kind of an align:start position:0% combination um so that's kind of an align:start position:0% combination um so that's kind of an overview of the most important thing align:start position:0% overview of the most important thing align:start position:0% overview of the most important thing we're going to talk about in this class align:start position:0% we're going to talk about in this class align:start position:0% we're going to talk about in this class which is the 63 abstraction the idea of align:start position:0% which is the 63 abstraction the idea of align:start position:0% which is the 63 abstraction the idea of representing a align:start position:0% representing a align:start position:0% representing a system by the way it transforms an input align:start position:0% system by the way it transforms an input align:start position:0% system by the way it transforms an input into an align:start position:0% into an align:start position:0% into an output so with that kind of overview align:start position:0% output so with that kind of overview align:start position:0% output so with that kind of overview what I want to do next is say a little align:start position:0% what I want to do next is say a little align:start position:0% what I want to do next is say a little more about what are signals and what are align:start position:0% more about what are signals and what are align:start position:0% more about what are signals and what are systems it's the signals and systems align:start position:0% systems it's the signals and systems align:start position:0% systems it's the signals and systems abstraction we're going to need to know align:start position:0% abstraction we're going to need to know align:start position:0% abstraction we're going to need to know some more details about what is a align:start position:0% some more details about what is a align:start position:0% some more details about what is a signal and what is a align:start position:0% signal and what is a align:start position:0% signal and what is a system so basically a signal is just a align:start position:0% system so basically a signal is just a align:start position:0% system so basically a signal is just a mathematical function in all the align:start position:0% mathematical function in all the align:start position:0% mathematical function in all the examples I talked about so align:start position:0% examples I talked about so align:start position:0% examples I talked about so far and in many of the examples that align:start position:0% far and in many of the examples that align:start position:0% far and in many of the examples that we'll talk about throughout the align:start position:0% we'll talk about throughout the align:start position:0% we'll talk about throughout the term the function is a function of align:start position:0% term the function is a function of align:start position:0% term the function is a function of time and the signal can have many align:start position:0% time and the signal can have many align:start position:0% time and the signal can have many different dimensions so for example the align:start position:0% different dimensions so for example the align:start position:0% different dimensions so for example the mass spring system uh evolved as a align:start position:0% mass spring system uh evolved as a align:start position:0% mass spring system uh evolved as a function of time the tank system The align:start position:0% function of time the tank system The align:start position:0% function of time the tank system The Leaky tank system evolved as a function align:start position:0% Leaky tank system evolved as a function align:start position:0% Leaky tank system evolved as a function of time the cell phone system had align:start position:0% of time the cell phone system had align:start position:0% of time the cell phone system had acoustic signals that were functions of align:start position:0% acoustic signals that were functions of align:start position:0% acoustic signals that were functions of time time was the same in each case but align:start position:0% time time was the same in each case but align:start position:0% time time was the same in each case but the dependent variable can be a a align:start position:0% the dependent variable can be a a align:start position:0% the dependent variable can be a a variety of things here it was align:start position:0% variety of things here it was align:start position:0% variety of things here it was position so here it was flow rates meter align:start position:0% position so here it was flow rates meter align:start position:0% position so here it was flow rates meter cubed per second of align:start position:0% cubed per second of align:start position:0% cubed per second of water uh here it was uh perhaps Pascal's align:start position:0% water uh here it was uh perhaps Pascal's align:start position:0% water uh here it was uh perhaps Pascal's some unit of pressure to characterize align:start position:0% some unit of pressure to characterize align:start position:0% some unit of pressure to characterize the acoustic waveform align:start position:0% the acoustic waveform align:start position:0% the acoustic waveform so the point is that signals are align:start position:0% so the point is that signals are align:start position:0% so the point is that signals are generally align:start position:0% generally align:start position:0% generally functions we'll see that just like align:start position:0% functions we'll see that just like align:start position:0% functions we'll see that just like functions in align:start position:0% functions in align:start position:0% functions in mathematics there's a lot more to align:start position:0% mathematics there's a lot more to align:start position:0% mathematics there's a lot more to functions than dependent variables and align:start position:0% functions than dependent variables and align:start position:0% functions than dependent variables and independent variables and in fact that's align:start position:0% independent variables and in fact that's align:start position:0% independent variables and in fact that's going to be a key feature of our align:start position:0% going to be a key feature of our align:start position:0% going to be a key feature of our analysis of signals but that's to come align:start position:0% analysis of signals but that's to come align:start position:0% analysis of signals but that's to come so for the time being the simplest and align:start position:0% so for the time being the simplest and align:start position:0% so for the time being the simplest and complete model for what a signal is is align:start position:0% complete model for what a signal is is align:start position:0% complete model for what a signal is is it's just a function the function align:start position:0% it's just a function the function align:start position:0% it's just a function the function doesn't have to be a onedimensional align:start position:0% doesn't have to be a onedimensional align:start position:0% doesn't have to be a onedimensional function in fact a lot of the align:start position:0% function in fact a lot of the align:start position:0% function in fact a lot of the interesting um applications of signals align:start position:0% interesting um applications of signals align:start position:0% interesting um applications of signals and systems is to look at align:start position:0% and systems is to look at align:start position:0% and systems is to look at multi-dimensional functions so for align:start position:0% multi-dimensional functions so for align:start position:0% multi-dimensional functions so for example my research is in align:start position:0% example my research is in align:start position:0% example my research is in hearing I'm interested to align:start position:0% hearing I'm interested to align:start position:0% hearing I'm interested to study how do the cells respond to sound align:start position:0% study how do the cells respond to sound align:start position:0% study how do the cells respond to sound so that we can understand how broken align:start position:0% so that we can understand how broken align:start position:0% so that we can understand how broken ears work align:start position:0% ears work align:start position:0% ears work differently when you have a when you align:start position:0% differently when you have a when you align:start position:0% differently when you have a when you have a hearing deficit like I do when align:start position:0% have a hearing deficit like I do when align:start position:0% have a hearing deficit like I do when you have a hearing deficit how it what align:start position:0% you have a hearing deficit how it what align:start position:0% you have a hearing deficit how it what is the manifestation of that at a signal align:start position:0% is the manifestation of that at a signal align:start position:0% is the manifestation of that at a signal processing level how do the cells align:start position:0% processing level how do the cells align:start position:0% processing level how do the cells respond differently to people who have align:start position:0% respond differently to people who have align:start position:0% respond differently to people who have impaired hearing from people who have align:start position:0% impaired hearing from people who have align:start position:0% impaired hearing from people who have normal align:start position:0% normal align:start position:0% normal hearing uh in order to study that align:start position:0% hearing uh in order to study that align:start position:0% hearing uh in order to study that problem we take video align:start position:0% problem we take video align:start position:0% problem we take video pictures of the cells at large align:start position:0% pictures of the cells at large align:start position:0% pictures of the cells at large magnifications and watch them wiggle align:start position:0% magnifications and watch them wiggle align:start position:0% magnifications and watch them wiggle when sounds hit align:start position:0% when sounds hit align:start position:0% when sounds hit them so that's a picture processing align:start position:0% them so that's a picture processing align:start position:0% them so that's a picture processing example so the signals are not the align:start position:0% example so the signals are not the align:start position:0% example so the signals are not the independent variable is not just time align:start position:0% independent variable is not just time align:start position:0% independent variable is not just time it's a picture so it might have X and align:start position:0% it's a picture so it might have X and align:start position:0% it's a picture so it might have X and Y in fact the pictures we take are three align:start position:0% Y in fact the pictures we take are three align:start position:0% Y in fact the pictures we take are three dimensional so it has X Y and Z and in align:start position:0% dimensional so it has X Y and Z and in align:start position:0% dimensional so it has X Y and Z and in fact they're four dimensional because align:start position:0% fact they're four dimensional because align:start position:0% fact they're four dimensional because they change with align:start position:0% they change with align:start position:0% they change with time so we do we apply this kind of a align:start position:0% time so we do we apply this kind of a align:start position:0% time so we do we apply this kind of a technique on 4D signals X YZ time but align:start position:0% technique on 4D signals X YZ time but align:start position:0% technique on 4D signals X YZ time but it's still the the property it still has align:start position:0% it's still the the property it still has align:start position:0% it's still the the property it still has a property that the signals of Interest align:start position:0% a property that the signals of Interest align:start position:0% a property that the signals of Interest are align:start position:0% are align:start position:0% are functions align:start position:0% functions align:start position:0% functions we will have an we will be especially align:start position:0% we will have an we will be especially align:start position:0% we will have an we will be especially interested in this align:start position:0% interested in this align:start position:0% interested in this class in two distinct representations of align:start position:0% class in two distinct representations of align:start position:0% class in two distinct representations of signals we will call them CT signals and align:start position:0% signals we will call them CT signals and align:start position:0% signals we will call them CT signals and DT signals CT is continuous time DT is align:start position:0% DT signals CT is continuous time DT is align:start position:0% DT signals CT is continuous time DT is discreet time we're very interested in align:start position:0% discreet time we're very interested in align:start position:0% discreet time we're very interested in that because we're align:start position:0% align:start position:0% Engineers a lot of align:start position:0% Engineers a lot of align:start position:0% Engineers a lot of physics lives in continuous align:start position:0% physics lives in continuous align:start position:0% physics lives in continuous time okay the m spring dashbot system align:start position:0% time okay the m spring dashbot system align:start position:0% time okay the m spring dashbot system the signals were functions of align:start position:0% the signals were functions of align:start position:0% the signals were functions of time time was a continuously varying uh align:start position:0% time time was a continuously varying uh align:start position:0% time time was a continuously varying uh independent align:start position:0% independent align:start position:0% independent variable so uh so the Leaky tank the align:start position:0% variable so uh so the Leaky tank the align:start position:0% variable so uh so the Leaky tank the cell phones those were all systems whose align:start position:0% cell phones those were all systems whose align:start position:0% cell phones those were all systems whose signals evolve in time uh so they're all align:start position:0% signals evolve in time uh so they're all align:start position:0% signals evolve in time uh so they're all continuous you can you can start with a align:start position:0% continuous you can you can start with a align:start position:0% continuous you can you can start with a second and break it in half and get a align:start position:0% second and break it in half and get a align:start position:0% second and break it in half and get a half a second and a half and a half and align:start position:0% half a second and a half and a half and align:start position:0% half a second and a half and a half and a half and there's no limit to how many align:start position:0% a half and there's no limit to how many align:start position:0% a half and there's no limit to how many halves you can take align:start position:0% halves you can take align:start position:0% halves you can take by contrast a lot of the a lot of the align:start position:0% by contrast a lot of the a lot of the align:start position:0% by contrast a lot of the a lot of the systems that we will look at are things align:start position:0% systems that we will look at are things align:start position:0% systems that we will look at are things that are um that evolve in discrete time align:start position:0% that are um that evolve in discrete time align:start position:0% that are um that evolve in discrete time what's your bank account well it only align:start position:0% what's your bank account well it only align:start position:0% what's your bank account well it only gets computed once a align:start position:0% gets computed once a align:start position:0% gets computed once a day so it doesn't make sense to talk align:start position:0% day so it doesn't make sense to talk align:start position:0% day so it doesn't make sense to talk about your bank account at 9:00 am and align:start position:0% about your bank account at 9:00 am and align:start position:0% about your bank account at 9:00 am and and and 2 p.m right the bank account align:start position:0% and and 2 p.m right the bank account align:start position:0% and and 2 p.m right the bank account updates your account the the bank align:start position:0% updates your account the the bank align:start position:0% updates your account the the bank updates your account once a day so it's align:start position:0% updates your account once a day so it's align:start position:0% updates your account once a day so it's something that happens in discret time align:start position:0% something that happens in discret time align:start position:0% something that happens in discret time it's something that happens in all align:start position:0% it's something that happens in all align:start position:0% it's something that happens in all computational align:start position:0% computational align:start position:0% computational systems so computational systems align:start position:0% systems so computational systems align:start position:0% systems so computational systems generally operate on signals that uh align:start position:0% generally operate on signals that uh align:start position:0% generally operate on signals that uh that are functions of discret align:start position:0% that are functions of discret align:start position:0% that are functions of discret time what was the state at Time Zero align:start position:0% time what was the state at Time Zero align:start position:0% time what was the state at Time Zero what was the state at time one what was align:start position:0% what was the state at time one what was align:start position:0% what was the state at time one what was the state at time align:start position:0% the state at time align:start position:0% the state at time two so the system is something that eats align:start position:0% two so the system is something that eats align:start position:0% two so the system is something that eats a discrete time signal and generates a align:start position:0% a discrete time signal and generates a align:start position:0% a discrete time signal and generates a discrete time align:start position:0% discrete time align:start position:0% discrete time output align:start position:0% output align:start position:0% output signal and a unique part of this class align:start position:0% signal and a unique part of this class align:start position:0% signal and a unique part of this class will be converting between the two align:start position:0% will be converting between the two align:start position:0% will be converting between the two representations because as Engineers we align:start position:0% representations because as Engineers we align:start position:0% representations because as Engineers we often want to take build something that align:start position:0% often want to take build something that align:start position:0% often want to take build something that operates in the physical world masses align:start position:0% operates in the physical world masses align:start position:0% operates in the physical world masses and springs and reservoirs and that kind align:start position:0% and springs and reservoirs and that kind align:start position:0% and springs and reservoirs and that kind of align:start position:0% of align:start position:0% of stuff and do the processing align:start position:0% stuff and do the processing align:start position:0% stuff and do the processing computationally ever since the Advent of align:start position:0% computationally ever since the Advent of align:start position:0% computationally ever since the Advent of digital electronics it's just much align:start position:0% digital electronics it's just much align:start position:0% digital electronics it's just much easier digital one it's just much easier align:start position:0% easier digital one it's just much easier align:start position:0% easier digital one it's just much easier to process signals in the in the digital align:start position:0% to process signals in the in the digital align:start position:0% to process signals in the in the digital domain so we will often be interested in align:start position:0% domain so we will often be interested in align:start position:0% domain so we will often be interested in how do you align:start position:0% how do you align:start position:0% how do you represent a signal who naturally lives align:start position:0% represent a signal who naturally lives align:start position:0% represent a signal who naturally lives in the physical world and is therefore align:start position:0% in the physical world and is therefore align:start position:0% in the physical world and is therefore part at whose signals are continuous in align:start position:0% part at whose signals are continuous in align:start position:0% part at whose signals are continuous in nature continuous time how do you align:start position:0% nature continuous time how do you align:start position:0% nature continuous time how do you convert it into a discrete align:start position:0% convert it into a discrete align:start position:0% convert it into a discrete representation so you can crunch it on a align:start position:0% representation so you can crunch it on a align:start position:0% representation so you can crunch it on a computer and in fact how do you go back align:start position:0% computer and in fact how do you go back align:start position:0% computer and in fact how do you go back so for align:start position:0% so for align:start position:0% so for example if we were thinking about align:start position:0% example if we were thinking about align:start position:0% example if we were thinking about processing audio signals we might want align:start position:0% processing audio signals we might want align:start position:0% processing audio signals we might want to think about how we would take a align:start position:0% to think about how we would take a align:start position:0% to think about how we would take a signal of continuous time and turn it align:start position:0% signal of continuous time and turn it align:start position:0% signal of continuous time and turn it into a discrete time align:start position:0% into a discrete time align:start position:0% into a discrete time representation that's precisely what we align:start position:0% representation that's precisely what we align:start position:0% representation that's precisely what we do when we want to record something in align:start position:0% do when we want to record something in align:start position:0% do when we want to record something in mp3 mp3 represents a sound a sound is align:start position:0% mp3 mp3 represents a sound a sound is align:start position:0% mp3 mp3 represents a sound a sound is something that I think of as a signal in align:start position:0% something that I think of as a signal in align:start position:0% something that I think of as a signal in continuous time it's pascals as a align:start position:0% continuous time it's pascals as a align:start position:0% continuous time it's pascals as a function of align:start position:0% function of align:start position:0% function of seconds but we want to represent it by a align:start position:0% seconds but we want to represent it by a align:start position:0% seconds but we want to represent it by a sequence of numbers because it's a align:start position:0% sequence of numbers because it's a align:start position:0% sequence of numbers because it's a sequence of numbers that's easy to store align:start position:0% sequence of numbers that's easy to store align:start position:0% sequence of numbers that's easy to store to communicate align:start position:0% to communicate align:start position:0% to communicate Etc we do the same sort of thing with align:start position:0% Etc we do the same sort of thing with align:start position:0% Etc we do the same sort of thing with images the images of the ear that I align:start position:0% images the images of the ear that I align:start position:0% images the images of the ear that I talked about in my research are things align:start position:0% talked about in my research are things align:start position:0% talked about in my research are things that I think about happening at align:start position:0% that I think about happening at align:start position:0% that I think about happening at continuous align:start position:0% continuous align:start position:0% continuous space every time we look at a smaller align:start position:0% space every time we look at a smaller align:start position:0% space every time we look at a smaller and smaller Dimension we get a different align:start position:0% and smaller Dimension we get a different align:start position:0% and smaller Dimension we get a different answer because the the image has no align:start position:0% answer because the the image has no align:start position:0% answer because the the image has no quantization in align:start position:0% quantization in align:start position:0% quantization in space but when I crunch it I don't want align:start position:0% space but when I crunch it I don't want align:start position:0% space but when I crunch it I don't want to work in something in continuous space align:start position:0% to work in something in continuous space align:start position:0% to work in something in continuous space I want to work in a in a sampled version align:start position:0% I want to work in a in a sampled version align:start position:0% I want to work in a in a sampled version like a JPEG align:start position:0% like a JPEG align:start position:0% like a JPEG version similarly we want to convert align:start position:0% version similarly we want to convert align:start position:0% version similarly we want to convert back I mean it would be useless for the align:start position:0% back I mean it would be useless for the align:start position:0% back I mean it would be useless for the case of the cell phone to convert it align:start position:0% case of the cell phone to convert it align:start position:0% case of the cell phone to convert it into a discrete representation and then align:start position:0% into a discrete representation and then align:start position:0% into a discrete representation and then not be able to hear align:start position:0% not be able to hear align:start position:0% not be able to hear it so we'll think about the inverse align:start position:0% it so we'll think about the inverse align:start position:0% it so we'll think about the inverse process which is reconstruction if you align:start position:0% process which is reconstruction if you align:start position:0% process which is reconstruction if you had a discrete representation of a align:start position:0% had a discrete representation of a align:start position:0% had a discrete representation of a signal how would you turn it align:start position:0% signal how would you turn it align:start position:0% signal how would you turn it into a continuous representation and align:start position:0% into a continuous representation and align:start position:0% into a continuous representation and there's lots of ways people do that align:start position:0% there's lots of ways people do that align:start position:0% there's lots of ways people do that here's a way that we call zero order align:start position:0% here's a way that we call zero order align:start position:0% here's a way that we call zero order hold where you convert every sample into align:start position:0% hold where you convert every sample into align:start position:0% hold where you convert every sample into a corresponding align:start position:0% a corresponding align:start position:0% a corresponding voltage and just hold that voltage until align:start position:0% voltage and just hold that voltage until align:start position:0% voltage and just hold that voltage until the next sample comes along this is the align:start position:0% the next sample comes along this is the align:start position:0% the next sample comes along this is the representation that is most commonly align:start position:0% representation that is most commonly align:start position:0% representation that is most commonly used in things like align:start position:0% align:start position:0% mp3 by contrast we might do something align:start position:0% mp3 by contrast we might do something align:start position:0% mp3 by contrast we might do something cleverer we might linearly extrapolate align:start position:0% cleverer we might linearly extrapolate align:start position:0% cleverer we might linearly extrapolate between align:start position:0% between align:start position:0% between samples that is commonly done in picture align:start position:0% samples that is commonly done in picture align:start position:0% samples that is commonly done in picture processing the reason we use the two align:start position:0% processing the reason we use the two align:start position:0% processing the reason we use the two different schemes is that align:start position:0% different schemes is that align:start position:0% different schemes is that psychophysically the things that are align:start position:0% psychophysically the things that are align:start position:0% psychophysically the things that are important to your eyes and the things align:start position:0% important to your eyes and the things align:start position:0% important to your eyes and the things that are important to your ears are align:start position:0% that are important to your ears are align:start position:0% that are important to your ears are different so you would hear the errors align:start position:0% different so you would hear the errors align:start position:0% different so you would hear the errors more if you did the linear interpolation align:start position:0% more if you did the linear interpolation align:start position:0% more if you did the linear interpolation and you would see the errors align:start position:0% and you would see the errors align:start position:0% and you would see the errors more if you did the other interpolation align:start position:0% more if you did the other interpolation align:start position:0% more if you did the other interpolation and we'll do lots of examples later on align:start position:0% and we'll do lots of examples later on align:start position:0% and we'll do lots of examples later on in the course so you see why that's align:start position:0% align:start position:0% true okay too much talking not enough align:start position:0% true okay too much talking not enough align:start position:0% true okay too much talking not enough thinking so I've told you a lot about align:start position:0% thinking so I've told you a lot about align:start position:0% thinking so I've told you a lot about signals now I'd like you to think about align:start position:0% signals now I'd like you to think about align:start position:0% signals now I'd like you to think about some align:start position:0% some align:start position:0% some signals and so a common thing I'm going align:start position:0% signals and so a common thing I'm going align:start position:0% signals and so a common thing I'm going to do in this class is ask you to think align:start position:0% to do in this class is ask you to think align:start position:0% to do in this class is ask you to think about a question talk to your neighbor align:start position:0% about a question talk to your neighbor align:start position:0% about a question talk to your neighbor come to a consensus and then vote so align:start position:0% come to a consensus and then vote so align:start position:0% come to a consensus and then vote so that's what we're going to do here I'm align:start position:0% that's what we're going to do here I'm align:start position:0% that's what we're going to do here I'm going to play some sounds you're going align:start position:0% going to play some sounds you're going align:start position:0% going to play some sounds you're going to try to figure out what signals were align:start position:0% to try to figure out what signals were align:start position:0% to try to figure out what signals were represented by those sounds but I would align:start position:0% represented by those sounds but I would align:start position:0% represented by those sounds but I would like you to have an opinion before we we align:start position:0% like you to have an opinion before we we align:start position:0% like you to have an opinion before we we communicate the answer because you can align:start position:0% communicate the answer because you can align:start position:0% communicate the answer because you can prove that people get more out of it align:start position:0% prove that people get more out of it align:start position:0% prove that people get more out of it when there's something at stake so I align:start position:0% when there's something at stake so I align:start position:0% when there's something at stake so I would like you to work with a partner so align:start position:0% would like you to work with a partner so align:start position:0% would like you to work with a partner so in order to work with a partner align:start position:0% in order to work with a partner align:start position:0% in order to work with a partner everybody stand up introduce yourself to align:start position:0% everybody stand up introduce yourself to align:start position:0% everybody stand up introduce yourself to your neighbor figure out a good partner align:start position:0% your neighbor figure out a good partner align:start position:0% your neighbor figure out a good partner figure out where they align:start position:0% align:start position:0% live align:start position:0% align:start position:0% okay so what I'm I'm going to do align:start position:0% okay so what I'm I'm going to do align:start position:0% okay so what I'm I'm going to do now is play a computer generated sound align:start position:0% now is play a computer generated sound align:start position:0% now is play a computer generated sound this is a computerated computer align:start position:0% this is a computerated computer align:start position:0% this is a computerated computer generated speech sound made by Bob align:start position:0% generated speech sound made by Bob align:start position:0% generated speech sound made by Bob Donovan who did his PhD thesis figuring align:start position:0% Donovan who did his PhD thesis figuring align:start position:0% Donovan who did his PhD thesis figuring out how to do computer generated align:start position:0% out how to do computer generated align:start position:0% out how to do computer generated speech and then I'm going to play Four align:start position:0% speech and then I'm going to play Four align:start position:0% speech and then I'm going to play Four transformations of that align:start position:0% transformations of that align:start position:0% transformations of that sound and the your task is to identify align:start position:0% sound and the your task is to identify align:start position:0% sound and the your task is to identify what's the mathematical representation align:start position:0% what's the mathematical representation align:start position:0% what's the mathematical representation of the Transformations that I'm telling align:start position:0% of the Transformations that I'm telling align:start position:0% of the Transformations that I'm telling you okay so first the original align:start position:0% you okay so first the original align:start position:0% you okay so first the original first is the signal that Bob Donovan align:start position:0% align:start position:0% MaDee and now I'm going to play a align:start position:0% MaDee and now I'm going to play a align:start position:0% MaDee and now I'm going to play a sequence of four align:start position:0% sequence of four align:start position:0% sequence of four Transformations 1 2 3 four and the align:start position:0% Transformations 1 2 3 four and the align:start position:0% Transformations 1 2 3 four and the question is going to be is this the ma align:start position:0% question is going to be is this the ma align:start position:0% question is going to be is this the ma the correct mathematical representation align:start position:0% the correct mathematical representation align:start position:0% the correct mathematical representation for this sound the sound you just heard align:start position:0% for this sound the sound you just heard align:start position:0% for this sound the sound you just heard was F of t align:start position:0% was F of t align:start position:0% was F of t is the following sound which by align:start position:0% is the following sound which by align:start position:0% is the following sound which by definition is align:start position:0% definition is align:start position:0% definition is F1 the same as F of align:start position:0% align:start position:0% [Music] align:start position:0% align:start position:0% 2T you're allowed to align:start position:0% align:start position:0% talk align:start position:0% align:start position:0% e align:start position:0% align:start position:0% okay remember that answer now is the align:start position:0% okay remember that answer now is the align:start position:0% okay remember that answer now is the following sound minus F align:start position:0% align:start position:0% [Music] align:start position:0% align:start position:0% of align:start position:0% align:start position:0% okay remember that answer is the align:start position:0% okay remember that answer is the align:start position:0% okay remember that answer is the following sound F of align:start position:0% align:start position:0% 2T align:start position:0% align:start position:0% and align:start position:0% and align:start position:0% and finally is the following signal oneir of align:start position:0% finally is the following signal oneir of align:start position:0% finally is the following signal oneir of f of align:start position:0% align:start position:0% te okay conver confer with your align:start position:0% te okay conver confer with your align:start position:0% te okay conver confer with your partner figure out how many are true align:start position:0% partner figure out how many are true align:start position:0% partner figure out how many are true because in 30 seconds I'm going to ask align:start position:0% because in 30 seconds I'm going to ask align:start position:0% because in 30 seconds I'm going to ask you to raise your align:start position:0% align:start position:0% hand align:start position:0% align:start position:0% question align:start position:0% question align:start position:0% question or fft was the original align:start position:0% align:start position:0% signal uh align:start position:0% signal uh align:start position:0% signal uh pressure pressure in the acoustic align:start position:0% pressure pressure in the acoustic align:start position:0% pressure pressure in the acoustic waveform align:start position:0% align:start position:0% that's align:start position:0% align:start position:0% true okay how many of those statements align:start position:0% true okay how many of those statements align:start position:0% true okay how many of those statements are true raise your hand with some align:start position:0% are true raise your hand with some align:start position:0% are true raise your hand with some number of fingers keeping in mind that align:start position:0% number of fingers keeping in mind that align:start position:0% number of fingers keeping in mind that you're voting having collaborated with align:start position:0% you're voting having collaborated with align:start position:0% you're voting having collaborated with your partner so it's not your fault you align:start position:0% your partner so it's not your fault you align:start position:0% your partner so it's not your fault you have a right anyway okay so how many so align:start position:0% have a right anyway okay so how many so align:start position:0% have a right anyway okay so how many so everybody raise your hands I want to align:start position:0% align:start position:0% see okay it's about 90% no less than align:start position:0% see okay it's about 90% no less than align:start position:0% see okay it's about 90% no less than that about 85% correct okay uh so first align:start position:0% that about 85% correct okay uh so first align:start position:0% that about 85% correct okay uh so first one was that F of align:start position:0% one was that F of align:start position:0% one was that F of 2T why do you think that everybody seems align:start position:0% 2T why do you think that everybody seems align:start position:0% 2T why do you think that everybody seems to be saying align:start position:0% to be saying align:start position:0% to be saying yes it was fast it sounded faster so align:start position:0% yes it was fast it sounded faster so align:start position:0% yes it was fast it sounded faster so what's faster mean from a signal point align:start position:0% what's faster mean from a signal point align:start position:0% what's faster mean from a signal point of align:start position:0% align:start position:0% view change the time scales that's a align:start position:0% view change the time scales that's a align:start position:0% view change the time scales that's a very good way of thinking about it so if align:start position:0% very good way of thinking about it so if align:start position:0% very good way of thinking about it so if we thought about if we made a simple align:start position:0% we thought about if we made a simple align:start position:0% we thought about if we made a simple representation for align:start position:0% representation for align:start position:0% representation for f ignore what it really is let's say align:start position:0% f ignore what it really is let's say align:start position:0% f ignore what it really is let's say it's align:start position:0% it's align:start position:0% it's that what would F1 look like in order to align:start position:0% that what would F1 look like in order to align:start position:0% that what would F1 look like in order to make it sound align:start position:0% align:start position:0% faster okay we have people doing this align:start position:0% faster okay we have people doing this align:start position:0% faster okay we have people doing this and we have people doing this and I'm align:start position:0% and we have people doing this and I'm align:start position:0% and we have people doing this and I'm not quite sure what that means so could align:start position:0% not quite sure what that means so could align:start position:0% not quite sure what that means so could somebody be a little more align:start position:0% somebody be a little more align:start position:0% somebody be a little more descriptive hand gestures are fine yes align:start position:0% descriptive hand gestures are fine yes align:start position:0% descriptive hand gestures are fine yes squished squished exactly right so what align:start position:0% squished squished exactly right so what align:start position:0% squished squished exactly right so what we want to have happen align:start position:0% we want to have happen align:start position:0% we want to have happen in order to sound faster we want to have align:start position:0% in order to sound faster we want to have align:start position:0% in order to sound faster we want to have it come out faster so we would like that align:start position:0% it come out faster so we would like that align:start position:0% it come out faster so we would like that to happen to it so it's all over in one align:start position:0% to happen to it so it's all over in one align:start position:0% to happen to it so it's all over in one second if the original was all over in align:start position:0% second if the original was all over in align:start position:0% second if the original was all over in two seconds right that's what faster align:start position:0% two seconds right that's what faster align:start position:0% two seconds right that's what faster means so we would like that to be true align:start position:0% means so we would like that to be true align:start position:0% means so we would like that to be true is that what's going on in that align:start position:0% is that what's going on in that align:start position:0% is that what's going on in that function align:start position:0% function align:start position:0% function yeah okay so that's right second align:start position:0% yeah okay so that's right second align:start position:0% yeah okay so that's right second one align:start position:0% one align:start position:0% one yes align:start position:0% yes align:start position:0% yes and that also makes it higher pitched in align:start position:0% and that also makes it higher pitched in align:start position:0% and that also makes it higher pitched in fact we will talk a lot about this later align:start position:0% fact we will talk a lot about this later align:start position:0% fact we will talk a lot about this later but pitch has to do with a related kind align:start position:0% but pitch has to do with a related kind align:start position:0% but pitch has to do with a related kind of waveform if I thought about a align:start position:0% of waveform if I thought about a align:start position:0% of waveform if I thought about a waveform that was a single align:start position:0% waveform that was a single align:start position:0% waveform that was a single tone that might look like align:start position:0% tone that might look like align:start position:0% tone that might look like this so if you were to play an align:start position:0% this so if you were to play an align:start position:0% this so if you were to play an OBO and play Just A C constantly you get align:start position:0% OBO and play Just A C constantly you get align:start position:0% OBO and play Just A C constantly you get some periodic align:start position:0% some periodic align:start position:0% some periodic waveform if I did the same align:start position:0% waveform if I did the same align:start position:0% waveform if I did the same transformation on that periodic wave align:start position:0% transformation on that periodic wave align:start position:0% transformation on that periodic wave form what would happen to that waveform align:start position:0% form what would happen to that waveform align:start position:0% form what would happen to that waveform what would happen to this if I did align:start position:0% what would happen to this if I did align:start position:0% what would happen to this if I did transformation align:start position:0% transformation align:start position:0% transformation one squish what's squish align:start position:0% one squish what's squish align:start position:0% one squish what's squish being more High higher frequency more align:start position:0% being more High higher frequency more align:start position:0% being more High higher frequency more cycles per second yeah so how you make align:start position:0% cycles per second yeah so how you make align:start position:0% cycles per second yeah so how you make the same go as fast have the same that's align:start position:0% the same go as fast have the same that's align:start position:0% the same go as fast have the same that's very hard in fact I worked on that as a align:start position:0% very hard in fact I worked on that as a align:start position:0% very hard in fact I worked on that as a research project um so one of the align:start position:0% research project um so one of the align:start position:0% research project um so one of the techniques that we use to try to fix align:start position:0% techniques that we use to try to fix align:start position:0% techniques that we use to try to fix people's align:start position:0% people's align:start position:0% people's ears uh it turns out that people who align:start position:0% ears uh it turns out that people who align:start position:0% ears uh it turns out that people who have hearing disorders like align:start position:0% have hearing disorders like align:start position:0% have hearing disorders like mine can more easily understand male align:start position:0% mine can more easily understand male align:start position:0% mine can more easily understand male speech than female speech just because align:start position:0% speech than female speech just because align:start position:0% speech than female speech just because of the shift in frequencies male speech align:start position:0% of the shift in frequencies male speech align:start position:0% of the shift in frequencies male speech is primarily about an octave lower on align:start position:0% is primarily about an octave lower on align:start position:0% is primarily about an octave lower on average than female speech so we tried align:start position:0% average than female speech so we tried align:start position:0% average than female speech so we tried to convert every female into a male align:start position:0% to convert every female into a male align:start position:0% to convert every female into a male speaker and every male into the Jolly align:start position:0% speaker and every male into the Jolly align:start position:0% speaker and every male into the Jolly Green align:start position:0% Green align:start position:0% Green Giant and it worked males sounded like align:start position:0% Giant and it worked males sounded like align:start position:0% Giant and it worked males sounded like the Jolly Green Giant and females align:start position:0% the Jolly Green Giant and females align:start position:0% the Jolly Green Giant and females sounded like males and it didn't help align:start position:0% sounded like males and it didn't help align:start position:0% sounded like males and it didn't help hearing at all and that's primarily align:start position:0% hearing at all and that's primarily align:start position:0% hearing at all and that's primarily because it's a hard problem so but we align:start position:0% because it's a hard problem so but we align:start position:0% because it's a hard problem so but we will say more about that as we go on align:start position:0% will say more about that as we go on align:start position:0% will say more about that as we go on through the course but is a hard align:start position:0% through the course but is a hard align:start position:0% through the course but is a hard problem because you end up having to do align:start position:0% problem because you end up having to do align:start position:0% problem because you end up having to do other than just align:start position:0% other than just align:start position:0% other than just squish align:start position:0% squish align:start position:0% squish okay second one is this the right align:start position:0% okay second one is this the right align:start position:0% okay second one is this the right transformation no why align:start position:0% transformation no why align:start position:0% transformation no why not align:start position:0% align:start position:0% yeah if you flipped over the align:start position:0% yeah if you flipped over the align:start position:0% yeah if you flipped over the amplitude so what it's so flip over the align:start position:0% amplitude so what it's so flip over the align:start position:0% amplitude so what it's so flip over the amplitude you're saying that didn't do align:start position:0% amplitude you're saying that didn't do align:start position:0% amplitude you're saying that didn't do it what what was the transformation if align:start position:0% it what what was the transformation if align:start position:0% it what what was the transformation if it wasn't flip over the amplitude what align:start position:0% it wasn't flip over the amplitude what align:start position:0% it wasn't flip over the amplitude what is it align:start position:0% is it align:start position:0% is it yeah F of minus t it was flipped this align:start position:0% yeah F of minus t it was flipped this align:start position:0% yeah F of minus t it was flipped this way right flipping this way is largely align:start position:0% way right flipping this way is largely align:start position:0% way right flipping this way is largely inaudible if you had a pressure wave align:start position:0% inaudible if you had a pressure wave align:start position:0% inaudible if you had a pressure wave form versus a rare faction waveform align:start position:0% form versus a rare faction waveform align:start position:0% form versus a rare faction waveform positive pressures versus negative align:start position:0% positive pressures versus negative align:start position:0% positive pressures versus negative pressures this is very hard to tell not align:start position:0% pressures this is very hard to tell not align:start position:0% pressures this is very hard to tell not impossible if you're one of those audio align:start position:0% impossible if you're one of those audio align:start position:0% impossible if you're one of those audio types like me you can probably hear the align:start position:0% types like me you can probably hear the align:start position:0% types like me you can probably hear the difference but it's very hard hearing align:start position:0% difference but it's very hard hearing align:start position:0% difference but it's very hard hearing this not align:start position:0% this not align:start position:0% this not hard okay uh how about this align:start position:0% align:start position:0% one obviously wrong because this one was align:start position:0% one obviously wrong because this one was align:start position:0% one obviously wrong because this one was right okay how about this align:start position:0% right okay how about this align:start position:0% right okay how about this one yeah so the idea was that um there align:start position:0% one yeah so the idea was that um there align:start position:0% one yeah so the idea was that um there were two that were correct so the 85% of align:start position:0% were two that were correct so the 85% of align:start position:0% were two that were correct so the 85% of you or so who got two presumably you got align:start position:0% you or so who got two presumably you got align:start position:0% you or so who got two presumably you got the right two right the bet I won't ask align:start position:0% the right two right the bet I won't ask align:start position:0% the right two right the bet I won't ask um okay here's a here's a an image align:start position:0% um okay here's a here's a an image align:start position:0% um okay here's a here's a an image processing align:start position:0% processing align:start position:0% processing question think about this picture of align:start position:0% question think about this picture of align:start position:0% question think about this picture of sta I know it's hard to think about align:start position:0% sta I know it's hard to think about align:start position:0% sta I know it's hard to think about pictures of sta that's align:start position:0% pictures of sta that's align:start position:0% pictures of sta that's okay think about this picture of sta align:start position:0% okay think about this picture of sta align:start position:0% okay think about this picture of sta I've indexed XM Y and I've written that align:start position:0% I've indexed XM Y and I've written that align:start position:0% I've indexed XM Y and I've written that as f a signal that depends on two align:start position:0% as f a signal that depends on two align:start position:0% as f a signal that depends on two independent variables XM align:start position:0% independent variables XM align:start position:0% independent variables XM Y and I've got three Transformations the align:start position:0% Y and I've got three Transformations the align:start position:0% Y and I've got three Transformations the F1 transformation the F2 and the align:start position:0% align:start position:0% F3 the question is is this the right align:start position:0% F3 the question is is this the right align:start position:0% F3 the question is is this the right mathematical align:start position:0% mathematical align:start position:0% mathematical representation for transforming that align:start position:0% representation for transforming that align:start position:0% representation for transforming that picture into this align:start position:0% picture into this align:start position:0% picture into this one is this the one for that one and is align:start position:0% one is this the one for that one and is align:start position:0% one is this the one for that one and is is this a representation for this one is align:start position:0% is this a representation for this one is align:start position:0% is this a representation for this one is this a representation for that one take align:start position:0% this a representation for that one take align:start position:0% this a representation for that one take 30 seconds talk to your partner and and align:start position:0% 30 seconds talk to your partner and and align:start position:0% 30 seconds talk to your partner and and figure out how many of those align:start position:0% figure out how many of those align:start position:0% figure out how many of those Transformations are align:start position:0% align:start position:0% correct align:start position:0% align:start position:0% e align:start position:0% align:start position:0% okay so how many of the Transformations align:start position:0% okay so how many of the Transformations align:start position:0% okay so how many of the Transformations are correct raise your align:start position:0% are correct raise your align:start position:0% are correct raise your hand everybody raise your align:start position:0% hand everybody raise your align:start position:0% hand everybody raise your hand oh align:start position:0% hand oh align:start position:0% hand oh uh it's a zero or three okay just align:start position:0% uh it's a zero or three okay just align:start position:0% uh it's a zero or three okay just checking okay about 85% correct again align:start position:0% checking okay about 85% correct again align:start position:0% checking okay about 85% correct again okay probably a different 85% um so how align:start position:0% okay probably a different 85% um so how align:start position:0% okay probably a different 85% um so how do you think about this problem so align:start position:0% do you think about this problem so align:start position:0% do you think about this problem so there's a variety of ways you could align:start position:0% there's a variety of ways you could align:start position:0% there's a variety of ways you could think about align:start position:0% think about align:start position:0% think about it let me show you a way that focuses on align:start position:0% it let me show you a way that focuses on align:start position:0% it let me show you a way that focuses on an idea called mapping because I think align:start position:0% an idea called mapping because I think align:start position:0% an idea called mapping because I think is a very powerful way so if you think align:start position:0% is a very powerful way so if you think align:start position:0% is a very powerful way so if you think about a map how does the how does the T align:start position:0% about a map how does the how does the T align:start position:0% about a map how does the how does the T variable in the first problem map to the align:start position:0% variable in the first problem map to the align:start position:0% variable in the first problem map to the T variable in the second align:start position:0% T variable in the second align:start position:0% T variable in the second problem so a way you can think about align:start position:0% problem so a way you can think about align:start position:0% problem so a way you can think about this is if this is true for um all X and align:start position:0% this is if this is true for um all X and align:start position:0% this is if this is true for um all X and Y it's true for particular X so let's align:start position:0% Y it's true for particular X so let's align:start position:0% Y it's true for particular X so let's ask is it true for x equals align:start position:0% ask is it true for x equals align:start position:0% ask is it true for x equals z so how does the point x equals 0 map align:start position:0% z so how does the point x equals 0 map align:start position:0% z so how does the point x equals 0 map from one image to another so if you align:start position:0% from one image to another so if you align:start position:0% from one image to another so if you substitute at x equals 0 then F1 of0 so align:start position:0% substitute at x equals 0 then F1 of0 so align:start position:0% substitute at x equals 0 then F1 of0 so if you look at uh this align:start position:0% if you look at uh this align:start position:0% if you look at uh this transformation the claim would be that align:start position:0% transformation the claim would be that align:start position:0% transformation the claim would be that fub1 of0 is the same as F of align:start position:0% fub1 of0 is the same as F of align:start position:0% fub1 of0 is the same as F of zero is that align:start position:0% align:start position:0% true so F1 of align:start position:0% true so F1 of align:start position:0% true so F1 of zero bcts align:start position:0% align:start position:0% sta F of Z bisects align:start position:0% sta F of Z bisects align:start position:0% sta F of Z bisects Theta so yeah that's the right thing align:start position:0% Theta so yeah that's the right thing align:start position:0% Theta so yeah that's the right thing everybody see align:start position:0% everybody see align:start position:0% everybody see that is the statement true for xal align:start position:0% that is the statement true for xal align:start position:0% that is the statement true for xal 250 well F1 of align:start position:0% 250 well F1 of align:start position:0% 250 well F1 of 250 so F1 of align:start position:0% 250 so F1 of align:start position:0% 250 so F1 of 250 is align:start position:0% 250 is align:start position:0% 250 is through uh the right hand side of align:start position:0% through uh the right hand side of align:start position:0% through uh the right hand side of sta sort of where the steel and the align:start position:0% sta sort of where the steel and the align:start position:0% sta sort of where the steel and the bricks come align:start position:0% bricks come align:start position:0% bricks come together is that the same as align:start position:0% together is that the same as align:start position:0% together is that the same as substituting in here if x is 250 then F align:start position:0% substituting in here if x is 250 then F align:start position:0% substituting in here if x is 250 then F of 500 align:start position:0% of 500 align:start position:0% of 500 F of 500 is off the screen so is that align:start position:0% F of 500 is off the screen so is that align:start position:0% F of 500 is off the screen so is that transformation hold at the point xal 250 align:start position:0% transformation hold at the point xal 250 align:start position:0% transformation hold at the point xal 250 no okay it can't be right okay one one align:start position:0% no okay it can't be right okay one one align:start position:0% no okay it can't be right okay one one bad sample proves that it can't be the align:start position:0% bad sample proves that it can't be the align:start position:0% bad sample proves that it can't be the general align:start position:0% general align:start position:0% general transformation okay everybody see align:start position:0% transformation okay everybody see align:start position:0% transformation okay everybody see that similarly for the second align:start position:0% that similarly for the second align:start position:0% that similarly for the second transformation let's just try some align:start position:0% transformation let's just try some align:start position:0% transformation let's just try some points if we try the point xal align:start position:0% points if we try the point xal align:start position:0% points if we try the point xal 0 this says that F2 at location Z should align:start position:0% 0 this says that F2 at location Z should align:start position:0% 0 this says that F2 at location Z should be F at location minus align:start position:0% be F at location minus align:start position:0% be F at location minus 250 is that align:start position:0% align:start position:0% correct so F of zero well that's the align:start position:0% correct so F of zero well that's the align:start position:0% correct so F of zero well that's the left hand side of align:start position:0% left hand side of align:start position:0% left hand side of theta and so F2 of Zer that's the left align:start position:0% theta and so F2 of Zer that's the left align:start position:0% theta and so F2 of Zer that's the left hand side of theta F of 2 * align:start position:0% hand side of theta F of 2 * align:start position:0% hand side of theta F of 2 * 0 minus 250 IS F of 250 minus 250 that's align:start position:0% 0 minus 250 IS F of 250 minus 250 that's align:start position:0% 0 minus 250 IS F of 250 minus 250 that's also the left of sta align:start position:0% also the left of sta align:start position:0% also the left of sta okay so by using that reasoning you can align:start position:0% okay so by using that reasoning you can align:start position:0% okay so by using that reasoning you can go through and see if certain points map align:start position:0% go through and see if certain points map align:start position:0% go through and see if certain points map to the corresponding place where they align:start position:0% to the corresponding place where they align:start position:0% to the corresponding place where they were supposed to go that's the idea of align:start position:0% were supposed to go that's the idea of align:start position:0% were supposed to go that's the idea of mapping and in this particular case if align:start position:0% mapping and in this particular case if align:start position:0% mapping and in this particular case if we Trace two align:start position:0% we Trace two align:start position:0% we Trace two points and the map 2x - 250 is a linear align:start position:0% points and the map 2x - 250 is a linear align:start position:0% points and the map 2x - 250 is a linear function of align:start position:0% function of align:start position:0% function of X so two points determine a straight align:start position:0% X so two points determine a straight align:start position:0% X so two points determine a straight line so after you've determined two two align:start position:0% line so after you've determined two two align:start position:0% line so after you've determined two two points you're align:start position:0% points you're align:start position:0% points you're done seem seem align:start position:0% done seem seem align:start position:0% done seem seem right so similarly you can figure out align:start position:0% right so similarly you can figure out align:start position:0% right so similarly you can figure out that the particular points didn't work align:start position:0% that the particular points didn't work align:start position:0% that the particular points didn't work here and so the answer is that only one align:start position:0% here and so the answer is that only one align:start position:0% here and so the answer is that only one of those Transformations was align:start position:0% align:start position:0% right okay so I spent a fair amount of align:start position:0% right okay so I spent a fair amount of align:start position:0% right okay so I spent a fair amount of time on S signals in the last minute oh align:start position:0% time on S signals in the last minute oh align:start position:0% time on S signals in the last minute oh minute no I'm not going to do that am I align:start position:0% minute no I'm not going to do that am I align:start position:0% minute no I'm not going to do that am I okay I spent a fair amount of time align:start position:0% okay I spent a fair amount of time align:start position:0% okay I spent a fair amount of time talking about signals next align:start position:0% talking about signals next align:start position:0% talking about signals next time we'll talk about align:start position:0% time we'll talk about align:start position:0% time we'll talk about similar ways of thinking about simple align:start position:0% similar ways of thinking about simple align:start position:0% similar ways of thinking about simple systems so have a good align:start position:0% align:start position:0% weekend