text stringlengths 30 4k | source stringlengths 60 201 |
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Recursion Theorem (Sipser
Theorem 6.3):
Let T be a TM that computes a
(possibly partial) 2-argument
function t: Σ* × Σ* → Σ*.
<M>
w
T
t(<M>, w)
Then there is another TM R that
computes the function r: Σ* → Σ*,
where for any w, r(w) = t(<R>, w).
w
R
t(<R>, w)
Applications of the Recursion
Theorem
Applications of... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
w> then accept.
Application 1: AccTM is undecidable
• Suppose for contradiction that D decides AccTM.
• R: On input w:
– Obtain < R >
– Run D on input <R, w>
– Do the opposite of what D does:
• If D accepts <R, w> then reject.
• If D rejects <R, w> then accept.
• Now get a contradiction:
– If R accepts w, then
• D ... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
else), by definition of R.
– If R does not accept 01, then
• D rejects <R> since D is a decider for Acc01TM, so
• R accepts 01 (and everything else), by definition of R.
• Contradiction. So D can’t exist, so Acc01TM is undecidable.
Applications of Recursion Theorem
• Application 3: Using Recursion Theorem to prove
R... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
• Rice’s Theorem: Let P be a nontrivial property of Turing-
recognizable languages. Let MP = { < M > | L(M) ∈ P }.
Then MP is undecidable.
• L(M1) ∈ P, L(M2) ∉ P.
• D decides MP.
• R: On input w:
– Obtain < R >
– Run D on input <R>
– If D accepts <R> then run M2 on input w and do the same thing.
– If D rejects <R> t... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
the same
language }.
– Theorem: MINTM is not Turing-recognizable.
– Note: This doesn’t follow from Rice:
• Requires non-T-recognizability, not just undecidability.
• Besides, it’s not a language property.
– Proof:
• Assume for contradiction that MINTM is Turing-recognizable.
• Then it’s enumerable, say by enumerator ... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
.
w
Q
q(w) = < Pw >
Proof of RT: Special Case
• Lemma: (Sipser Lemma 6.1): There is a
computable function q: Σ* → Σ* such
that, for any string w, q(w) is the
description of a TM Pw that just prints out
w and halts.
w
Q
q(w) = < Pw >
• Now, back to the machine that outputs its own
description…
• Consists of 2 sub-... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
• Claim A ° B outputs its own description, which is < A ° B >.
• Check this…
• A is P<B>, so the output from A to B is <B>:
<B>
A = P<B>
• Substituting B for M in B’s output:
<B>
A = P<B>
B
B
< P<B> ° B >
Combining the Pieces
• A ° B:
A
B
• Claim A ° B outputs its own description, which is < A ° B >.
<B>
A = P<B>
B
< ... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
�* → Σ*,
where for any w, r(w) = t(<R>, w).
• Construct R from:
– The given T, and
– Variants of A and B from the special-
case proof.
<M>
w
T
R
t(<M>, w)
w
t(<R>, w)
Proof of RT: General Case
• R looks like:
A
B
T
• Write this as (A ° B) °1 T
– The °1 means that the output from (A ° B)
connects to the first (top) ... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
B
• Now combine with T, plugging in R for M in T’s input:
<B °1 T >
< R>
A
B
t(<R>,w)
T
w
Combining the Pieces
<B °1 T >
< R>
A
B
t(<R>,w)
T
w
• Thus, R = (A ° B) °1 T, on input w, produces
t(<R>,w), as needed for the Recursion Theorem.
w
R
t(<R>, w)
Next time…
• More on computabilty theory
• Reading:
– "Computin... | https://ocw.mit.edu/courses/6-045j-automata-computability-and-complexity-spring-2011/2fe1aea3ab4dd67aff81f2a04fc010ef_MIT6_045JS11_lec10.pdf |
Examples of Transient RC and RL Circuits.
The Series RLC Circuit
Impulse response of RC Circuit.
Let’s examine the response of the circuit shown on Figure 1. The form of the source
voltage Vs is shown on Figure 2.
R
Vs
C
+
vc
-
Figure 1. RC circuit
Vs
Vp
0
tp
Figure 2.
t
We will investigate the response
vc t
( ... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf |
⎦
⎞
⎟
⎟
⎠
0
t
≤ ≤
tp
(1.3)
When
RC t(cid:21)
the higher order terms may be neglected resulting in
vc t
( )
(cid:17)
Vp
t
RC
0
t
≤ ≤
tp
At the end of the pulse (at
t
tp=
) the voltage becomes
vc t
(
=
tp
)
(cid:17)
Vptp
RC
6.071/22.071 Spring 2006, Chaniotakis and Cory
(1.4)
(1.5)
2
... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf |
2 Volt car
battery. The spark plug is connected actors the inductor and current may flow though it
only if the voltage across the gap of the plug exceeds a very large value (about 20 kV).
+
Vb
-
R
L
+
vL
-
Figure 5
spark
plug
When the switch is closed, the current through the inductor reaches a maximum value of
/V... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf |
The voltage across the coil when the switch is opened is
v L
=
i
∆
t
∆
=
0.01
2.4
1 10
×
−
6
=
24
kV
6.071/22.071 Spring 2006, Chaniotakis and Cory
5
Response of RC circuit driven by a square wave.
Let’s now consider the RC circuit shown on Figure 6(a) driven by a square wave signal of
the for... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf |
⎤
⎥
⎦
+
t
−
RC
⎤
Vp e
⎥
⎥
⎦
(1.13)
(1.14)
Similarly the response during the first part of the second cycle starts with the value of vc
at t=T and evolves towards the value Vp.
If the time constant is small compared to the period of the square wave, the response will
reach the maximum and minimum values of the sq... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf |
9
Second Order Circuits
Series RLC circuit
The circuit shown on Figure 10 is called the series RLC circuit. We will analyze this
circuit in order to determine its transient characteristics once the switch S is closed.
Vs
S
+ vR -
+ vL -
R
L
C
+
vc
-
Figure 10
The equation that describes the response of... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf |
And
2
s
+
R
L
s
+
1
LC
=
0
α=
R
L
2
οω =
1
LC
The characteristic equation becomes
2
s
+
=
οα ω 0
2
+
s
2
The roots of the characteristic equation are
1s
2s
= − +
2
α α ω
ο
−
2
= − −
2
α α ω
ο
−
2
And the homogeneous solution becomes
hvc
=
A e
1
1
s t
2
s t
+
A e
2
The total solution now becomes
vc Vs A e
1
=
+
s... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf |
−
α ω
ο
=
j
2
ω
ο
2
−α In this case the roots s1 and s2 are complex
2
α ω α
−
2
ο
j
s
1
= − +
numbers:
oscillatory behavior
Under Damped System
s
, 2
= − −
2
α ω α
−
j
2
ο
. System exhibits
Important observations for the series RLC circuit.
• As the resistance increases the value of α increases and the system is ... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf |
solution of the form
stAe the characteristic equation is
Where
οω =
1
LC
The two roots are
2
s
2
οω+
=
0
1s
j οω= +
2s
j οω= −
And the solution is a linear combination of
11 s t
A e
and
A e
22 s t
( )
vc t
=
1
A e
oj
ω
t
+
2
A e
t
j
ω−
ο
By using Euler’s relation Equation (1.34) may also be written as
vc t
( )... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf |
(
t
ω ω
ο
ο
)
(1.36)
(1.37)
(1.38)
(1.39)
(1.40)
And the voltage across the inductor is easily determined from KVL or from the element
relation of the inductor
vL L
=
di
dt
vL
= −
= −
vc
Vo
cos(
tω
)o
(1.41)
Figure 13 shows the plots of
vL t and i t
( )
between vc(t) and vL(t) and the 90 degree phase differenc... | https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf |
Image Quality Metrics
• Image quality metrics
• Mutual information (cross-entropy) metric
• Intuitive definition
• Rigorous definition using entropy
• Example: two-point resolution problem
• Example: confocal microscopy
• Square error metric
• Receiver Operator Characteristic (ROC)
• Heterodyne detection
MIT 2.71... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/302f8fb4454a52e44502785c2e046203_imgq.pdf |
2σ
Precision of measurement
=
C
+
...
n1
∑
2
k 1=
ln
+
1
2
ln +
1
µ
2
−
2
t
2σ
µ
k
=
2σ
ln
+
+
1
≈precision
of (t-2)th measurement
E.g. 0.5470839348
these digits worthless
if σ ≈10-5
MIT 2.717
Image quality metrics p-5
2
<
<
−µ
σ
µ
2
2
t
t 1
no... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/302f8fb4454a52e44502785c2e046203_imgq.pdf |
6
Formal definition of cross-entropy (2)
• Fair coin: p(H)=1/2; p(T)=1/2
1
Entropy = −
2
log
2
1
2
−
1
2
log2
=
bit1
1
2
• Unfair coin: p(H)=1/4; p(T)=3/4
1
Entropy = −
4
log2
log
2
1
4
3
4
−
3
=
4
81.0
bits
Maximum entropy ⇔⇔⇔⇔⇔⇔⇔⇔ Maximum uncertainty
Maximum uncertainty
Maximum entropy
MIT 2.717
... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/302f8fb4454a52e44502785c2e046203_imgq.pdf |
hardware
channel
“physical
attributes”
(measurement)
H
detection
g
Formal definition of cross-entropy (5)
object
hardware
channel
“physical
attributes”
(measurement)
f
field
propagation
H
detection
g
Noise adds uncertainty
⇔ eliminates information
adds uncertainty to the measurement wrt the object
elim... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/302f8fb4454a52e44502785c2e046203_imgq.pdf |
2.717
Image quality metrics p-13
Entropy & Differential Entropy
• Discrete objects (can take values among a discrete set of states)
– definition of entropy
(
Entropy = −∑ x p
k
)log 2
(
x p
k
)
k
– unit: 1 bit (=entropy value of a YES/NO question with 50%
uncertainty)
• Continuous objects (can take values fro... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/302f8fb4454a52e44502785c2e046203_imgq.pdf |
metrics p-16
n
+
1
∑
ln
1
2
=
1
k
2
k
2
µ
σ
As noise increases
is lost whenever
• one rank of
σ2
overcomes a new eigenvalue
• the remaining ranks lose precision
H
2σ
Example: two-point resolution
Finite-NA imaging system, unit magnification
Two point-sources
(object)
f A A
x ... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/302f8fb4454a52e44502785c2e046203_imgq.pdf |
��
+
1
2
)
( 1
−
s
2σ
MIT 2.717
Image quality metrics p-19
1
+
2
ln +
1
( 1
+
s
2σ
)
2
IMI vs source separation
MIT 2.717
Image quality metrics p-20
s→ 1
)
(
=
SNR
1
σ
2
s→ 0
IMI for rectangular matrices (1)
H
=
=
H
underdetermined
underdetermined
(more unknowns than
... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/302f8fb4454a52e44502785c2e046203_imgq.pdf |
Image quality metrics p-23
Confocal microscope
Small pinhole:
Depth resolution
Light efficiency
Large pinhole:
Depth resolution
Light efficiency
virtual slice
pi
nhole
object
beam
splitter
Intensity
detector
MIT 2.717
Image quality metrics p-24
Depth “resolution” vs. noise
NA=0.2
Object structure:
p... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/302f8fb4454a52e44502785c2e046203_imgq.pdf |
result
of
inversion
– e.g. pseudoinverse minimizes MSQ in an overdetermined problem
– obvious problem: most of the time, we don’t know what f is!
– more when we deal with Wiener filters and regularization
• • Receiver Operator Cha
Receiver Operator Charracteacterriisstictic
– measures the performance of a c... | https://ocw.mit.edu/courses/2-717j-optical-engineering-spring-2002/302f8fb4454a52e44502785c2e046203_imgq.pdf |
6.231: DYNAMIC PROGRAMMING
LECTURE 1
LECTURE OUTLINE
Problem Formulation
Examples
The Basic Problem
Significance of Feedback
•
•
•
•
1 DP AS AN OPTIMIZATION METHODOLOGY
Generic optimization problem:
•
min g(u)
u∈U
where u is the optimization/decision variable, g(u)
is the cost function, and U is the constraint set
•
Cat... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/304cf17d604e774625dae63810777264_MIT6_231F15_Lec1.pdf |
Stock at Period k
xk
Inventory System
Stock at Period k + 1
xk + 1 = xk + uk - wk
Cost of Period k
r(xk) + cuk
Stock ordered at
Period k
uk
Discrete-time system
xk+1 = fk(xk, uk, wk) = xk + uk
wk
−
Cost function that is additive over time
•
•
N −1
E
gN (xN ) +
(
k=0
X
N −1
gk(xk, uk, wk)
)
= E
cuk + r(xk + uk
(
k=0
X (... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/304cf17d604e774625dae63810777264_MIT6_231F15_Lec1.pdf |
CCA
CCD
CD
ABC
CCD
ACB
CBD
ACD
CDB
CAB
CBD
CAD
CDB
CBC
CCB
CCD
CAB
CAD
CDA
CDA
CAB
6 STOCHASTIC FINITE-STATE PROBLEMS
•
Example: Find two-game chess match strategy
Timid play draws with prob. pd > 0 and loses
pd. Bold play wins with prob. pw <
•
with prob. 1
−
1/2 and loses with prob. 1
pw
−
0.5-0.5
pd
0 - 0
1 - pd
pw
... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/304cf17d604e774625dae63810777264_MIT6_231F15_Lec1.pdf |
uk) of wk
∈
Expected cost of π starting at x0 is
•
•
•
Jπ(x0) = E
gN (xN ) +
(
N −1
k=0
X
gk(xk, µk(xk), wk)
)
Optimal cost function
J ∗(x0) = min Jπ(x0)
π
Optimal policy π∗ satisfies
•
•
Jπ∗ (x0) = J ∗(x0)
When produced by DP, π∗ is independent of x0.
8 SIGNIFICANCE OF FEEDBACK
•
Open-loop versus closed-loop policies
w... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/304cf17d604e774625dae63810777264_MIT6_231F15_Lec1.pdf |
Horizon Problems - Simple (Vol. 1, Ch.
−
−
•
7, 3 lectures)
********************************************
•
−
Infinite Horizon Problems - Advanced (Vol. 2)
Chs. 1, 2: Discounted problems - Computa-
tional methods (3 lectures)
Ch. 3: Stochastic shortest path problems (2
lectures)
Chs. 6, 7: Approximate DP (6 lectures)
−
−... | https://ocw.mit.edu/courses/6-231-dynamic-programming-and-stochastic-control-fall-2015/304cf17d604e774625dae63810777264_MIT6_231F15_Lec1.pdf |
6.776
High Speed Communication Circuits
Lecture 2
Transceiver Architectures
Massachusetts Institute of Technology
February 3, 2005
Copyright © 2005 by H.-S. Lee and M. H. Perrott
Transceivers for Amplitude Modulation
H.-S. Lee & M.H. Perrott
MIT OCW
Amplitude Modulation Review
Transmitter Output
0
x(t)
y(t)
2cos(2πfo... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/30624026937064d06c6438a7a2b5197a_lec2.pdf |
AM Transmitter
Balanced
modulator
-90o phase
shifter
Balanced
modulator
Power
Amp
+
RF
Filter
QAM
H.-S. Lee & M.H. Perrott
MIT OCW
SSB Transmitter I
(cid:131) Phase-shift SSB Modulator
Balanced
modulator
-90o phase
shifter
-90o phase
shifter
Balanced
modulator
Power
Amp
+
RF
Filter
SSB
(cid:131) Sideband removal depen... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/30624026937064d06c6438a7a2b5197a_lec2.pdf |
Crystal (piezo)
Earpiece
(cid:131) Applicable only to standard AM signals (DC shifted baseband)
(cid:131) No active component: very simple and cheap
(cid:131) Low sensitivity: only strong stations can be tuned in
(cid:131) Poor selectivity (single RF filter)
(cid:131) Low baseband output power: can only drive high effi... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/30624026937064d06c6438a7a2b5197a_lec2.pdf |
& M.H. Perrott
MIT OCW
Super-regeneration Receiver
R1
-R2
Envelope
Detector
Audio
out
Quench
(cid:131) Quench circuit is either an oscillator (quenching at
regular intervals) or amplitude detector (quenches
when predetermined amplitude is reached)
(cid:131) Large effective RF gain can be achieved by a single
stage ... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/30624026937064d06c6438a7a2b5197a_lec2.pdf |
Perrott
MIT OCW
Superheterodyne Receiver Spectra
H.-S. Lee & M.H. Perrott
MIT OCW
Image Rejection in Superheterodyne Receivers
(cid:131) Key Point: image signal at equidistance from flo
converts to the same IF band
(cid:131) The RF filter must remove image! (image reject filter)
(cid:131) Want high IF frequency for e... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/30624026937064d06c6438a7a2b5197a_lec2.pdf |
Extractor or
LO
LPF
Audio Freq.
(AF) Amplifier
(cid:131) Mixes RF signal with the carrier frequency down
directly to baseband: no image to reject
H.-S. Lee & M.H. Perrott
MIT OCW
Homodyne Receiver Cont’d
(cid:131) No local oscillator if pilot carrier is present – carrier
extracted from the transmitted signal (carri... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/30624026937064d06c6438a7a2b5197a_lec2.pdf |
and high selectivity in the
baseband processing circuit
(cid:131) Channel filtering is typically performed by DSP
(cid:131) No image to reject
(cid:131) Time-varying DC offset due to local oscillator leakage is
an important issue
(cid:131) DC offset can be larger than signal and saturate
baseband circuits
H.-S. Lee ... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/30624026937064d06c6438a7a2b5197a_lec2.pdf |
-
4&1
I-Phase
(cid:131) Similar to Weaver SSB generator (P.S. #1)
(cid:131) Image rejection by phase relationship – no passive
Figure by MIT OCW.
components
(cid:131) Image rejection limited by amplitude and phase
matching of 6 mixers!
H.-S. Lee & M.H. Perrott
MIT OCW
Low-IF Receiver
RF Filter
RF Amplifier
(LNA)
Qua... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/30624026937064d06c6438a7a2b5197a_lec2.pdf |
’s (Albert Jerng’s VCO’s)
1.8V
1.8V
L
L
Vtune
L
Vtune
L
NMOS VCO
PMOS VCO
H.-S. Lee & M.H. Perrott
Figure by MIT OCW.
MIT OCW
PLL-Based Frequency Modulation
Phase
detector
Loop
filter
VCO
(cid:131) The loop bandwidth must be lower than the lowest
signal frequency
(cid:131) The center frequency is precisely maintaine... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/30624026937064d06c6438a7a2b5197a_lec2.pdf |
H.-S. Lee & M.H. Perrott
MIT OCW
FM Demodulator Using Phase-Locked Loop
Limiter
Phase
detector
Loop
filter
VCO
(cid:131) VCO input voltage is used as output
(cid:131) VCO is in feedback loop: input output characteristic is
the inverse of VCO function (thus f-to-v conversion).
H.-S. Lee & M.H. Perrott
MIT OCW | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/30624026937064d06c6438a7a2b5197a_lec2.pdf |
Lecture 7
PN Junction and MOS Electrostatics(IV)
MetalOxideSemiconductor Structure (contd.)
Outline
1. Overview of MOS electrostatics under bias
2. Depletion regime
3. Flatband
4. Accumulation regime
5. Threshold
6.
Inversion regime
Reading Assignment:
Howe and Sodini, Chapter 3, Sections 3.8-3.9
6.012 Sprin... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/30d115908b75764ef7ebe79517ac5e8a_MIT6_012S09_lec07.pdf |
example:
Depletion region thickness:
xd (VGB ) =
εεεεs
1+
Cox
2C 2 (φφφφB + VGB)
εεεεsqNa
ox
− 1
Potential drop across semiconductor SCR:
VB (VGB ) =
qN x 2
d
a
2ε s
Surface potential
φφφφ(0) = φφφφp + VB(VGB)
Potential drop across oxide:
Vox (VGB ) =
qN a x d t ox
ε ox
6.012 Spri... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/30d115908b75764ef7ebe79517ac5e8a_MIT6_012S09_lec07.pdf |
) kT
Solve for φ(0) at VGB = VT:
φφφφ( 0 )
VGB =VT
=
kT
q
• ln
n ( 0 )
n i VGB = VT
=
kT
q
• ln
N
a = − φφφφp
ni
Hence:
VB (VT ) = −2φφφφp
6.012 Spring 2009
Lecture 7
9
Computation of threshold voltage (contd.)
Second, compute potential drop in oxide at threshold.
Obtain xd(VT) using relationship betwe... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/30d115908b75764ef7ebe79517ac5e8a_MIT6_012S09_lec07.pdf |
↑↑. The higher the doping, the more
voltage required to produce n(0) = Na
• If C ox ↑↑↑↑ (tox ↓↓↓↓) ⇒⇒⇒⇒ VT ↓↓↓↓. The thinner the oxide, the
less voltage dropped across the oxide.
6.012 Spring 2009
Lecture 7
11
6. Inversion
What happens for VGB > VT?
More electrons at Si/SiO2 interface than acceptors
⇒⇒⇒⇒ inv... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/30d115908b75764ef7ebe79517ac5e8a_MIT6_012S09_lec07.pdf |
inv.) ≈ VB(VT ) = −2φφφφP
6.012 Spring 2009
Lecture 7
14
ChargeControl Relation (contd..)
• All extra voltage beyond VT used to increase
inversion charge Qn. Think of it as capacitor:
– Top plate: metal gate
– Bottom plate: inversion layer
Q = CV
⇒
QN = −Cox (VGB − VT )
Coul/cm2
for VGB > VT
Existence of... | https://ocw.mit.edu/courses/6-012-microelectronic-devices-and-circuits-spring-2009/30d115908b75764ef7ebe79517ac5e8a_MIT6_012S09_lec07.pdf |
s
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1
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2
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3
2
3
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5
-15
4
1
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-5
2
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10
The s... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/313d12c426c44380ccd838dff3151690_MIT15_093J_F09_lec10.pdf |
3
10
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MIT OpenCourseWare
http://ocw.mit.edu
15.093J / 6.255J Optimization Methods
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/313d12c426c44380ccd838dff3151690_MIT15_093J_F09_lec10.pdf |
Advanced System Architecture
ESD.342/EECS 6.883
2006
• Goals of this course:
• Gain an understanding of system architecture
• Learn existing theoretical and analytical methods
• Compare systems in different domains and
understand what influences their architectures
• Apply/extend existing theory in case studies
Adv Sy... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/31440ed9b876651c3287b430ba75f77e_lec1.pdf |
assignments and exercises to
learn to use the software
• Case study project with periodic reports in class
• Class overheads, assigned reading, and optional
reading posted on class website.
Adv Sys Arch intro
8/24/2006
© Daniel E Whitney
5
Grading Formula
• 15% in-class participation (especially reading
connections... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/31440ed9b876651c3287b430ba75f77e_lec1.pdf |
through these different lenses.
Adv Sys Arch intro
8/24/2006
© Daniel E Whitney
9
A “Perfect” Theory of Architecture
Would Permit Us To:
• Measure
• Characterize
• Understand at a fundamental level
• Design, operate, evaluate, improve
• Predict future behavior
Adv Sys Arch intro
8/24/2006
© Daniel E Whitney
10
A Def... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/31440ed9b876651c3287b430ba75f77e_lec1.pdf |
4 (see assignment 1)
Adv Sys Arch intro
8/24/2006
© Daniel E Whitney
14
Some Things Do Not Have Architectures
with Internal Structure
• Random Networks
• Perfect gases
• Crowds of people
• Their behavior can still be analyzed and often
forms a baseline for comparison to things that do
have architectures with signif... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/31440ed9b876651c3287b430ba75f77e_lec1.pdf |
Overtly designed
– Can be an
architect
– A design strategy
is practical
– Products, product
families
– Cars, airplanes
– Bell System
– Organizations
– Centrally-planned
economies
•
Infrastructures
– Architect not common
– Protocols and standards
are crucial
– Design strategy may or
may not be practical
– May be d... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/31440ed9b876651c3287b430ba75f77e_lec1.pdf |
9
Comments on Typologies:
Attributes of Effective Classification
Standards for Taxonomy
•
– Collectively Exhaustive and Mutually Exclusive
– Internally Homogeneous
– Stability
– Understandable Representation and Naming
• None of the approaches really fulfill these criteria. Interestingly (more
later in course), no c... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/31440ed9b876651c3287b430ba75f77e_lec1.pdf |
they be used?
• Assuming we know what functions, performance, and
ilities we want, what methods can be used to create a
suitable architecture?
• Assuming we know what architecture we want, what are
the most effective ways of influencing the architecture of
complex, evolving engineering systems?
Adv Sys Arch intro
8... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/31440ed9b876651c3287b430ba75f77e_lec1.pdf |
2.997 Decision-Making in Large-Scale Systems
MIT, Spring 2004
March 8
Handout #13
Lecture Note 10
1 Value Function Approximation
DP problems are centered around the cost-to-go function J ⁄ or the Q-factor Q⁄. In certain problems, such as
linear-quadratic-Gaussian systems, J ⁄ exhibits some structure which allows for it... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
current falling piece. More speciflcally, we have b(i; j) = 1, if
position (i; j) of the board is fllled, and b(i; j) = 0 otherwise.
If there are p difierent types of pieces, and the board has dimension n £ m, the number of states is on the
order of p £ 2n£m, which grows exponentially with n and m.
Since exact solution of... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
‚(u);
(1)
where U is the set of all possible policies. In principle, we could solve (1) by enumerating all policies and
choosing the one with the smallest value of ‚(u); however, note that the number of policies is exponential
in the number of states | we have jYj = jAjjSj; if there is no special structure to U, this p... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
efiective approximation. First, we need to choose a
parameterization ~J that can closely approximate the desired cost-to-go function. In this respect, a suitable
choice requires some practical experience or theoretical analysis that provides rough information on the shape
of the function to be approximated. \Regularitie... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
easy to see that a neural network represents a function ~J(x; r), where x is the input and r is the set of
weights in each of the perceptrons. Recall that we are interested in representing a function J ⁄(x) as ~J(x; r),
for some set of weights r. Part of the appeal of neural networks is that they can be e–ciently train... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
partitioning of the state space as a tree or using adaptive methods for choosing the
partitions, for instance.
4
(cid:15)
(cid:15)
(cid:16)
(cid:16)
(cid:17)
(cid:17)
(cid:18)
(cid:18)
(cid:19)
(cid:19)
(cid:20)
(cid:20)
2.3 Features
A special case of state space partitioning consists of mapping states to features, an... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/3154a1a393c870d0128e5a3e62579a8b_lec_10_v1.pdf |
130
RICHARD B. MELROSE
18. Solutions to (some of) the problems
Solution 18.1 (To Problem 10). (by Matjaˇz Konvalinka).
Since the topology on N, inherited from R, is discrete, a set is com
C)→
pact if and only if it is finite. If a sequence {xn} (i.e. a function N
is in C0(N) if and only if for any � > 0 there exist... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
�
y
�→
xnyn
is linear and bounded (
�x�0, the mapping
�
|
∞
=1n
n=1
xnyn| ≤ ∞
=1n
�
|xn||yn|
≤ �x�0 �y�1) by
Φ : l1
�−→ c∗
0
defined by
�
�→ y
x
�→
�
∞
�
xnyn
n=1
is a (linear) welldefined mapping with norm at most 1. In fact, Φ is
an isometry because if |xj| = �x�0 then Φ(x)(ej)| = 1 where ej is
the j... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
H(x)ϕ
�
∞
i
�
ϕ
(x) dx = i(0 − ϕ(0)) = −iδ(ϕ),
we get DxH = Cδ for C = −i.
0
LECTURE NOTES FOR 18.155, FALL 2004
131
Solution 18.3 (To Problem 40). (Matjaˇz Konvalinka) Let us prove this
in the case where n = 1. Define (for b = 0)
U (x) = u(b) − u(x) − (b − x)u�(x) − . . . −
(b − x)k−1
(k − 1)!
u
(k−1)(x);
t... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
decomposition is u(x) = p(x) + v(x) for
p(x) = u(0) + u�(0)x +
u��(0)
2
2
x + . . . +
u(k−1)(0)
(k − 1)!
x k−1 +
u(k)(0)
k!
k
x ,
v(x) = u(x) − p(x) =
u(k)(ζ) − u(k)(0)
k!
k
x
for ζ between 0 and x, and since u(k) is continuous, (u(x) − p(x))/xk
tends to 0 as x tends to 0.
The proof for general n is n... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
p (od degree
k), and we have
vx(1)
|k
|
x
so it is bounded by a positive combination of terms of the form
wx (ζx) − wx (0)
k!|x|k
u(x) − p(x)
|k
|
x
=
=
,
(k)
(k)
�
�
�
�
∂x l1
∂lu
1 ∂x l2
2 · · ·
li
∂x i
(ζxx) −
∂lu
1 ∂x l2
2 · · ·
∂x l1
li
∂x i
(0)
�
�
�
�
with l1 + . . . + li = k and 0 < ζx < 1... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
welldefined, coincides with f on Sn−1, and is continuous:
if M is
the maximum of |g| on Sn−1, and � > 0 is given, then f (x) < � for
|
|
x < �/M.
|
|
Solution 18.5. (partly Matjaˇz Konvalinka)
For any ϕ ∈ S(R)
∞
�
|
−∞
ϕ(x)dx| ≤
|
ϕ(x) dx ≤ sup((1+x
|
we have
�
∞
−∞
�
�
∞
2
|
|
) ϕ(x) )
|
(1+
x|
|
2)−1dx
... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
)
�
LECTURE NOTES FOR 18.155, FALL 2004
133
so it suffices to show that xkAψ is bounded for any k as |x| → ±∞.
Since ψ(t) − cφ(t) ≤ Ckt−k−1 in t ≥ 1 it follows from (18.2) that
x kAψ(x) ≤ Cx k
|
|
∞
t−k−1dt ≤ C �, k > 1, in x > 1.
�
x
A similar estimate as x → −∞ follows from (18.1). Now, A is clearly
linear, a... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
�
Rn
�ξ�2(m+m� )
2 dξ < ∞.
|
|
u
�
�ξ�2(m+m�)|�|2 dξ =
u
�ξ�2m� (1 + ξ1
2 + . . . + ξ2
n)m|u|
�
2 dξ =
But that is true since
�
Rn
�
=
Rn
�ξ�2m�
⎛
⎝
�
|α|≤m
Cαξ2α
�
Rn
⎞
⎠ |�|2 dξ =
u
�
|α|≤m
Cα
��
Rn
�
�ξ�2m� ξ2α u
|�|2 dξ
u = ξ�m� Dαu is in L2(Rn) (note that u ∈ H m(Rn)
�
� (Rn), α ≤ m). The... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
j
Then obviously we have 1 =
χEj v. Then �x� is bounded by
Ej, 1 ≤ j ≤ n, we have
�
x�
xj|
|
≤
(1 + n|xj|2)1/2
|xj|
= �
n + 1/|xj
�1/2 ≤ (2n)1/2 ,
2
|
134
RICHARD B. MELROSE
n xjwj for wj ∈ L2(Rn).
j=1
= xjwj for wj ∈ L2(Rn). But that means that �x�v = w0 +
� x�vj
so �
If u is in L2(Rn) then �u ∈ L2(Rn), a... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
=
Solution 18.9. It is equivalent to ask when �ξ�mδ�0 is in L2(Rn). Since
−∞
i.−
δ�0(ψ) = δ0(ψ�) = ψ�(0) =
ψ(x) dx = 1(ψ),
�
Rn
ξ�
2m has a finite integral over
this is equivalent to finding m such that �
Rn . One option is to write �ξ� = (1 + r2)1/2 in spherical coordinates,
and to recall that the Jacobian of sph... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
1, . . . , n, vi ∈ S; (Rn) have Fourier transforms ξ−n−1χAi.
=
and for i
|ξi|
> c�ξ� on the support of v�i for each i = 1, . . . , n, each term
Since
{
ξ; ξi = supj |ξj , ξ
| | ≥ 1}
{ | |
|
|
i
LECTURE NOTES FOR 18.155, FALL 2004
135
is in H m for any m < 1 + n/2 so, by the Sobolev embedding theorem,
each vi ∈ C... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/316b53904ce106dc33393334695469c8_solution_prob.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.014 Calculus with Theory
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-014-calculus-with-theory-fall-2010/316f0420829f20de483bbc975159dea1_MIT18_014F10_ChBnotes.pdf |
The Challenges of Delivering
The Challenges of Delivering
Content on the Internet
Content on the Internet
Tom Leighton
Tom Leighton
Chief Scientist
Chief Scientist
Akamai Technologies
Akamai Technologies
Outline
Outline
How the Web Works
How the Web Works
Services
Akamai’s Services
Akamai’s
Technology Overview
Techn... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
Bottleneck Implications
Slow downloads
•• Slow downloads
Content must traverse multiple backbones and long distances
-- Content must traverse multiple backbones and long distances
Unreliable performance
•• Unreliable performance
Content may be blocked by congestion or backbone
-- Content may be blocked by congestion o... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
Results
Results
Keynote
Web Site Performance
Typical Improvement with Akamai
5
1
y
a
M
n
o
o
N
6
1
y
a
M
n
o
o
N
7
1
y
a
M
n
o
o
N
8
1
y
a
M
n
o
o
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9
1
y
a
M
n
o
o
N
0
2
y
a
M
n
o
o
N
1
2
y
a
M
n
o
o
N
2
2
y
a
M
n
o
o
N
3
2
y
a
M
n
o
o
N
4
2
y
a
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n
o
o
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5
2
y
a
M
n
o
o
N
6
2
y
a
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o
o
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7
2
y
a
M
n
o
o
N
Web ob... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
functionality of ordinary conference calls
Akamai Forum:: enables businesses to
enables businesses to
Webcasts
produce live, interactive Webcasts
produce live, interactive
•• Akamai Conference
•• Akamai Forum
Akamai Forum
Akamai Forum
No special
No special
client software
client software
Demand
Live or On--Deman... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
amai’s
storage management service
•• ACSACS:: storage management service
that persistently stores content delivered to
that persistently stores content delivered to
network
Akamai’s network
end users via Akamai’s
end users via
a comprehensive
Digital Parcel Service:: a comprehensive
•• Digital Parcel Service
dig... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
0 Minutes
10
1
Browser’s
Cache
2
9
3
8
OS
Downloading www.xyz.com
Downloading www.xyz.com
Akamai’s EdgeSuite
with with Akamai’s EdgeSuite
WWW.XYZ.COM
WWW.XYZ.COM
11
22
DNS
66
55
77
33
User enters www.xyz.com
•• User enters www.xyz.com
Browser requests IP
•• Browser requests IP
address for www.xyz.com
address for ... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
.net
20.20.123.55
11
12 a212.g.akamai.net
.net Root
(InterNIC)
Akamai High-Level DNS Servers
30.30.123.5 13
Akamai Low-Level DNS Servers
End User
16
1
Local Name
Server
3
14
Browser’s
Cache
2
15
OS
DNS Maps & Time--ToTo--LiveLive
DNS Maps & Time
Maps created using
•• Maps created using
info on:
info on:
Internet c... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
also breaks
also breaks
performance
performance
bottlenecks when
bottlenecks when
distributed across
distributed across
12,000 servers
12,000 servers
Used as an API to
•• Used as an API to
party
third--party
third
applications on
applications on
Akamai’s network
Akamai’s network
<html>
<asi version = “1.0... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
Satellite
Uplink
Encoding
Entry
Point
1 2 33 44
1 2
X X X X
1 2 3 4
1 2 3 4
x
11 22 3 43 4
1 2 3 4
1 2 3 4
Top-level
reflectors
Regions
Outline
Outline
How the Web Works
How the Web Works
Services
Akamai’s Services
Akamai’s
Technology Overview
Technology Overview
Technological Challenges
Technological Challenges
T... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
data mining
time monitoring of system for NOCC with
•• RealReal--time monitoring of system for NOCC with
meaningful alerts and performance metrics
meaningful alerts and performance metrics
time SQL queries to the system
Support for real--time SQL queries to the system
•• Support for real
Technological Challenges
Tec... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
we called
Akamai. Tuesday night we
were Akamaized and
instantly 6-10 times faster.”
Craig Maccubbin
CTO of BET.com
BET.com Akamaized 90% of
Each Web Page with FreeFlow:
• Improved site performance (6-10 times)
• Quadrupled page view capacity
• Postponed 2nd data center build out
• Preserved graphic-rich page design... | https://ocw.mit.edu/courses/18-996-topics-in-theoretical-computer-science-internet-research-problems-spring-2002/3170cbaa7807c4e266f79bcb58e6b646_lec1present.pdf |
L1: 6.111 Course Overview
L1: 6.111 Course Overview
Acknowledgements:
Materials in this lecture are courtesy of the following sources and are used with
permission.
Rex Min
J. Rabaey, A. Chandrakasan, B. Nikolic. Digital Integrated Circuits: A Design Perspective.
Prentice Hall/Pearson, 2003.
L1: 6.111 Spring 2006
Intro... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
(cid:134) Prior digital design experience is NOT Required
(cid:134) 6.004 is not a prerequisite!
(cid:129) Take 6.004 before 6.111 or
(cid:129) Take 6.004 after 6.111 or
(cid:129) Take both in the same term
(cid:134) Must have basic background in circuit theory
(cid:134) Some basic material might be a review for those ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
) Must emphasize digital concepts, but inclusion of analog interfaces
(e.g., data converters, sensors or motors) common and often
desirable
(cid:134) Proposal Conference
(cid:134) Design Review(s)
(cid:132) Design presentation in class (% of the final grade for the in-
class presentation)
(cid:132) Top projects will ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
should be joint with individual authors specified for each section
(cid:122) Copy anything you want (with attribution) for your project report
L1: 6.111 Spring 2006
Introductory Digital Systems Laboratory
6
The First Computer
The First Computer
Photograph of the
Babbage Difference Engine.
Image removed due to
copyri... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
132) Shannon’s 1937 MIT Master’s Thesis introduces the world to
binary digital electronics
L1: 6.111 Spring 2006
Introductory Digital Systems Laboratory
9
Evolution of Digital Electronics
Evolution of Digital Electronics
Vacuum Tubes
Transistors
VLSI Circuits
ENIAC, 1946
First Transistor
Bell Labs, 1948
4004, 1971
U... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
(cid:132) Labs & Design project
(cid:132) Product specs
algorithm selection,
flowcharts, etc.
Behavioral Description
software-like
programming
(cid:132) Algorithms, RTL, etc.
(cid:132) Flowcharts
(cid:132) State transition diagrams
HDL Description
(cid:132) Verilog code
(cid:132) VHDL code
automated synthesis
Hardware ... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
:132) Dataflow Level
(cid:134) The flow of data through components is specified based on the idea of how
data is processed
(cid:132) Gate Level
(cid:134) Specified as wiring between logic gates
(cid:134) Not practical for large examples
(cid:132) Switch Level
(cid:134) Description in terms of switching (modeling a tr... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
Embedded Digital System
Embedded Digital System
Analog Inputs
(sensors, audio,
video, tablet)
A/D
digitize
synchronize
Digital Inputs
(peripherals,
buses, switches)
Sync.
Memory
D/A
Control
Data
Processing
Analog
Outputs
(actuators, motors,
multimedia)
Digital
Outputs
(peripherals,
buses, lights)
(cid:132) Digit... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
are important
components of digital design
L1: 6.111 Spring 2006
Introductory Digital Systems Laboratory
19
The Inverter: Voltage Transfer Characteristic
The Inverter: Voltage Transfer Characteristic
IN
OUT
Digital circuits perform operations on logical (or Boolean) variables
A logical variable is a mathematical abst... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
of inverters
v0
v1
v2
v3
v4
v5
v6
out
v3
v1
f (v)
Simulated response
5
3
1
)
t
l
o
V
(
V
fin v(v)
v0
v1
v 2
v2
v0
2 1
0
in
2
4
6
8
10
t (nsec)
| Voltage gain | should be > 1 between logic states
L1: 6.111 Spring 2006
Introductory Digital Systems Laboratory
23
Lab Hours, Equipment, Computers
Lab Hours, Equipment, Com... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
use ‘setup 6.111’- ‘setup 6.111’ sources /mit/6.111/.attachrc
which attaches 6.111-nfs and sources /mit/6.111-nfs/.attachrc which
sets up your path and environment variables, etc.
L1: 6.111 Spring 2006
Introductory Digital Systems Laboratory
24
The 6.111 Lab
The 6.111 Lab
Courtesy of Tony Rinaldo. Used with permis... | https://ocw.mit.edu/courses/6-111-introductory-digital-systems-laboratory-spring-2006/317da3eef4612e543559758c871d25d8_l1_overview.pdf |
Determinants 2. Area and Volume
Area and volume interpretation of the determinant:
(1)
±
�
� a
1 a
2
�
b2
b1
�
�
�
�
�
= area of parallelogram with edges A = (a1, a2), B = (b1, b2).
B
θ
A
(2)
�
�
� a1 a2 a3 �
�
�
b3 � = volume of parallelepiped with edges row-vectors A, B, C.
± � b1
b2
�
�
c3 �... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/317eb93dbd9e833fbe51dfc6b2d3e034_MIT18_02SC_MNotes_d2.pdf |
=
= A · B ′ ,
= a1b2 − a2b1
′ ,
by the above observations
b
1
B
b
2
by the geometric definition of dot product
by the formula for B ′
This proves the area interpretation (1) if A and B have the position shown. If their positions
are reversed, then the area is the same, but the sign of the determinant is chan... | https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/317eb93dbd9e833fbe51dfc6b2d3e034_MIT18_02SC_MNotes_d2.pdf |
D-Lab
Spring 2010
1 Today in class:
• Review of the Design Process
• Design for Manufacture
– No Spare Parts
• Books Assignment
• Readings
2
D-Lab
Design for Manufacture
3 DfM Definition:
Adapting a design to make it more
easily manufactured and to reduce
its manufacturing costs.
4 DfM Definition:
To g... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/31bc1da7c60661b5848601aa2f9eba07_MITEC_720JS10_lec12.pdf |
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