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Fourier transform | What is the frequency transfer function when $x(t) = e^{-3t}u(t)$ and $y(t) = 2u(t)[e^{-t}-e^{-4t}]$ | https://dsp.stackexchange.com/questions/38018/what-is-the-frequency-transfer-function-when-xt-e-3tut-and-yt-2u | <p>What I did was take the fourier transform of both $x(t)$ and $y(t)$ and then divided $Y(j\omega)/X(j\omega)$. So
$$Y(j\omega) = \frac{2}{1+j\omega}-\frac{2}{4+j\omega}\quad\text{and}\quad X(j\omega) = \frac{1}{3+j\omega}$$ </p>
<p>However, the answer is of the form: $\displaystyle \frac{A_1}{1+j\omega}+\frac{A_2}{... | <p>Your answer is correct; it's just in a different form than the given answer. In order to get your answer in the given form, do the following:</p>
<ol>
<li>rewrite $Y(j\omega)$ by combining the two terms: $Y(j\omega)=\displaystyle\frac{N(j\omega)}{(j\omega+1)(j\omega+4)}$, where $N(j\omega)$ is a (very simple) polyn... | 34 |
Fourier transform | Significance of an impulse in the frequency domain | https://dsp.stackexchange.com/questions/44735/significance-of-an-impulse-in-the-frequency-domain | <p>I know that $X(f)$ gives the amplitude associated with the frequency component $f$ of a signal $x(t)$.</p>
<p>Now, a sinusoidal signal in time $x(t) = A \cos (2 \pi f_0 t)$, has a Fourier transform $X(f) = \frac{A}{2}[\delta(f-f_0) + \delta(f+f_0) ]$.</p>
<p>My question is that the Dirac Delta funcion tends to $\i... | <p>The Dirac delta is not strictly a function but a distribution. The Dirac delta is such that $\delta(x)=0 \ \forall x\neq0$ and it has to meet the following restriction:</p>
<p>$$\int_{-\infty}^{\infty} \delta(x) \ \mathrm{d}x = 1$$</p>
<p>This means that the unit impulse must integrate $1$ over all the real number... | 35 |
Fourier transform | Mean Square Error and Gibbs oscillations | https://dsp.stackexchange.com/questions/52088/mean-square-error-and-gibbs-oscillations | <p>While studying the convergence of Fourier transform, I got to know two conditions. </p>
<ul>
<li>$$\sum_{n=-\infty}^{\infty}|x(n)|<\infty$$</li>
<li>$$\sum|x(n)|^{2} \leq [\sum|x(n)|]^{2}$$</li>
</ul>
<p>While I was reading the text, I found this paragraph quite confusing. I didn't understood this. </p>
<block... | <p>The first condition mentioned in your question is absolute summability, which is sufficient for the discrete-time Fourier transform (DTFT) to exist. In this case, the sum given by the DTFT of a sequence converges uniformly. The other condition you probably mean is square summability:</p>
<p><span class="math-contai... | 36 |
Fourier transform | Is the following property true? | https://dsp.stackexchange.com/questions/53581/is-the-following-property-true | <p>I was looking at a solution of a Fourier Transform question and following property was used, if:
<span class="math-container">$$ x(t)\rightarrow X(jw) $$</span>
then:</p>
<p><span class="math-container">$$ e^{jw_ot}x(t)\rightarrow X(j(w-w_0)) $$</span>
<span class="math-contain... | <p>For <span class="math-container">$\cos$</span>, assuming <span class="math-container">$\omega_0$</span> is real, <a href="https://en.wikipedia.org/wiki/Fourier_transform#Functional_relationships,_one-dimensional" rel="nofollow noreferrer">the identity is</a>:
<span class="math-container">$$
x(t) \cos(\omega_0)t = \f... | 37 |
Fourier transform | What is a correct way to find or "guess" a kernel which transforms an image into another image using Fourier Transformations? | https://dsp.stackexchange.com/questions/55468/what-is-a-correct-way-to-find-or-guess-a-kernel-which-transforms-an-image-into | <p>Assuming I have two images, apple and orange; also assuming a filter kernel that transforms an apple image into an orange image possibly exists, how would some series of Fourier Transformations (and other spectral operations) get me a filter kernel? Is this possible? If possible, can it be immune to rotations/scalin... | 38 | |
Fourier transform | Applying frequency-domain filters on a centered Fourier transform | https://dsp.stackexchange.com/questions/56160/applying-frequency-domain-filters-on-a-centered-fourier-transform | <p>I understand why we shift the Fourier transform such that the 0-frequency is centered for visualization. In the shifted DFT(u,v) of an M*N 2-dimensional image,</p>
<ul>
<li>the top-left corner of the 4th quadrant is (0,0) frequency or (low u, low v)</li>
<li>the bottom-left corner of the 1st quadrant, (M-1,0) or (h... | <p>Note that an FFT of strictly real data is conjugate symmetric. </p>
<p>For a length M FFT of strictly real data, the data in FFT result bin M-1 has the same magnitude as in bin 1 (for a low frequency), but complex conjugated. So to maintain symmetry after filtering, which is necessary for the IFFT-ed image to rem... | 39 |
Fourier transform | Fourier transform and anti-trasform--identity missing | https://dsp.stackexchange.com/questions/59436/fourier-transform-and-anti-trasform-identity-missing | <p>I have a very silly doubt:</p>
<p>If we define the power spectral density:</p>
<p>S(f)=<span class="math-container">$\frac{1}{2\pi}\int exp(-i\tau2\pi f)r(\tau)d\tau$</span> (1)</p>
<p>where <span class="math-container">$r(\tau)$</span> is the correlation coefficient.</p>
<p>If we do the Fourier anti-transform, ... | <p>Different ways of showing it, depends where you start. Are you willing to accept that the Fourier transform of <span class="math-container">$\delta(t)$</span> is <span class="math-container">$1$</span> and vice versa (i.e., <span class="math-container">$\int_{-\infty}^\infty {\color{red}1} \cdot {\rm e}^{\jmath 2\pi... | 40 |
Fourier transform | If the cosine function is periodic, why does it have a Fourier Transform? | https://dsp.stackexchange.com/questions/60763/if-the-cosine-function-is-periodic-why-does-it-have-a-fourier-transform | <p>As far as I understand Fourier Transforms are for non-periodic signals and Fourier Series for periodic signals.</p>
<p>So why is it we can take the Fourier Transform of a cosine when it is a periodic function, assuming the above paragraph is correct?</p>
| <p>Indeed there are two things you have to know.</p>
<p>First, it can be shown that the continuous-time Fourier transform can be obtained from the continuous-time Fourier series by letting the period <span class="math-container">$T$</span> go to infinity. </p>
<p>Second, formally speaking the Fourier transform integr... | 41 |
Fourier transform | Why is a circular mask appropriate for Fourier filtering rectangular images? | https://dsp.stackexchange.com/questions/61184/why-is-a-circular-mask-appropriate-for-fourier-filtering-rectangular-images | <p>Suppose I apply 2D DFT to an image with dimensions <span class="math-container">$H{\times}W$</span> where <span class="math-container">$H \neq W$</span>, then shift the DC component to the center. Why does a circular mask capture the lowest frequency components, i.e. why is it not an ellipse given that the image is ... | <p>For simplicity, let's not do any shifting and only consider non-negative frequencies.</p>
<p>Let's assume that the horizontal and vertical image dimensions are even integers <span class="math-container">$W$</span> and <span class="math-container">$H$</span>. Looking at the output of a <span class="math-container">$H... | 42 |
Fourier transform | Fourier antitransform using scaling property? | https://dsp.stackexchange.com/questions/62520/fourier-antitransform-using-scaling-property | <p>I'm trying to calculate the antitransform of:</p>
<p><span class="math-container">$\frac{1}{2\cdot(1+5w)^2}$</span></p>
<p>Now I know the antitransform of <span class="math-container">$\frac{1}{(1+5w)^2} = t \cdot e^{-5t} u(t) $</span></p>
<p>But in this case I got that divided by 2. I assumed I had to use the s... | <p>If <span class="math-container">$h(t)$</span> is the inverse Fourier transform of <span class="math-container">$H(\omega)$</span>, then by linearity the inverse Fourier transform of <span class="math-container">$aH(\omega)$</span> is simply <span class="math-container">$ah(t)$</span>. This has nothing to do with the... | 43 |
Fourier transform | Fourier Transform of an acceleration signal containing engine orders | https://dsp.stackexchange.com/questions/63148/fourier-transform-of-an-acceleration-signal-containing-engine-orders | <p>I am trying to understand how to evaluate this equation in the context of acceleration data which contain engine orders</p>
<p><span class="math-container">$a^{f_{e}^{crit}}(f)=\sum_{o}^{K}A^{o,f_{e}^{crit}}\mathscr{F}(cos(2\pi \cdot f_{e}^{crit} \cdot o \cdot t))$</span></p>
<p><span class="math-container">$a^{f... | 44 | |
Fourier transform | Fourier transform of $\sum_{n=-\infty}^\infty(-1)^n\delta(t-nT_0)$ | https://dsp.stackexchange.com/questions/63194/fourier-transform-of-sum-n-infty-infty-1n-deltat-nt-0 | <p>Given <span class="math-container">$x(t)$</span> and <span class="math-container">$h(t)=\sum_{n=-\infty}^\infty(-1)^n\delta(t-nT_0)$</span>, I have to compute <span class="math-container">$Y(f)$</span>, where <span class="math-container">$y(t)=x(t)h(t)$</span>. I have thought about using that, in this case, <span cl... | <p><strong>HINT:</strong></p>
<p>Note that the given <span class="math-container">$h(t)$</span> can be written as</p>
<p><span class="math-container">$$h(t)=g(t)-g(t-T_0)\tag{1}$$</span></p>
<p>with some <span class="math-container">$g(t)$</span> the Fourier transform <span class="math-container">$G(f)$</span> of wh... | 45 |
Fourier transform | why does the spectral envelope of human speech not change w.r.t. pitch when taking a Fourier transform? | https://dsp.stackexchange.com/questions/63517/why-does-the-spectral-envelope-of-human-speech-not-change-w-r-t-pitch-when-taki | <p>In the context of speech recognition (recognizing individual speech sounds), the pitch of a certain person can change at different times. </p>
<p>Excerpt from Statistical Signal Processing by Steven Kay: </p>
<blockquote>
<p>This is a natural variability due to the nature of human speech. The
spectral envelop... | <p>The spectral envelope, which determines where the formant frequencies are, is determined by the shape of the mouth, tongue, lips, and nasal coupling. That is independent of the pitch, which is dependent on the tension or stiffness of the vocal cords.</p>
| 46 |
Fourier transform | Anti-Aliasing and the Fourier Transform, Gonzalez Digital Image Processing | https://dsp.stackexchange.com/questions/67554/anti-aliasing-and-the-fourier-transform-gonzalez-digital-image-processing | <p>In Gonzalez book Digital Image Processing, section 4.34 (third edition), he writes:</p>
<blockquote>
<p>Unfortunately, except for some special cases mentioned blow, aliasing
is always present in sampled signals because, even if the original
sampled function is band-limited, infinite frequency components are
introduc... | <p>Note that in order to obtain the Fourier transform of the windowed time domain signal <span class="math-container">$f(t)h(t)$</span>, you need to convolve the Fourier transforms of <span class="math-container">$f(t)$</span> and <span class="math-container">$h(t)$</span>. We know that the Fourier transform of <span c... | 47 |
Fourier transform | Fourier Transform of Impulse Train | https://dsp.stackexchange.com/questions/34146/fourier-transform-of-impulse-train | <p>Why is the fourier transform of impulse train a impulse train? Is there a intuitive reason behind it?</p>
| <p>Intuition can sometimes be misleading. But here are some ideas that might help one move towards creating a mental picture.</p>
<p>An infinitely long pure sinewave in the time domain (consisting of just one frequency FT or DFT basis function) will be a single impulse in the frequency domain.</p>
<p>Distort the sin... | 48 |
Fourier transform | Fourier transform as the integral of a parameter multiplied by an homogeneous wave | https://dsp.stackexchange.com/questions/73087/fourier-transform-as-the-integral-of-a-parameter-multiplied-by-an-homogeneous-wa | <p>Can a Fourier transform in space be interpreted as the integral of a parameter multiplied by an homogeneous wave <span class="math-container">$\sigma$</span>?</p>
<p>where <span class="math-container">$\sigma$</span> is:</p>
<p><span class="math-container">$\sigma$</span>=<span class="math-container">$e^{-ikx}$</spa... | <p>given your definition, and the definition of the Fourier transfer:</p>
<p>yes. That is literally the definition:</p>
<p><span class="math-container">$$\int_{\mathbb R} s(x) e^{i2\pi f x} \mathrm dx$$</span></p>
<p>By comparison, with <span class="math-container">$k= 2\pi f$</span>, you get the Fourier transform. Eve... | 49 |
Fourier transform | Techniques to deriving DTFTs | https://dsp.stackexchange.com/questions/3369/techniques-to-deriving-dtfts | <p>Are there general techniques to derive DTFTs? Given a bandlimited function $x(t)$, how do I find</p>
<p>$$X(\omega)=\sum_{n=-\infty}^\infty x[n]e^{-i\omega n}$$</p>
<p>Generally, it is easier to derive the continuous transform (never mind the constants):</p>
<p>$$X(f)=\int_{-\infty}^{\infty}x(t)e^{-i \omega t}\, ... | <p>As an example, the rules of series can work. A typical example is the $x[n] = a^n u[n]$ where $|a| < 1$. For instance, say we want to find the DTFT of the signal $x[n] = 0.9^n u[n]$. Then,</p>
<p>$$X(e^{j\omega}) = \sum_{n = -\infty}^{\infty} 0.9^n u[n] e^{-j\omega n} \\
= \sum_{n = 0}^{\infty} (0.9 e^{-j\omega}... | 50 |
Fourier transform | Getting bpm of song with fft | https://dsp.stackexchange.com/questions/14717/getting-bpm-of-song-with-fft | <p>I would like to get the bpm of a song analyzing the spectrum of the volume.
Doing a fft what I get is a peak at the origin and of course that can't be the frequency corresponding to the bpm, so I do the following:</p>
<p>$\overline{h} = h - \frac{1}{l}\sum_0^l h$</p>
<p>where $h$ is the fft of the volume and $l$ i... | <p>Unless your FFT is very large you're not going to get much resolution in the range of interest since each FFT bin holds Fs/N Hz of spectrum. ( Fs = sample rate, N=FFT size).</p>
<p>I have successfully got very accurate BPM values by using two FFT's in series and then picking the biggest peak in the region of intere... | 51 |
Fourier transform | Sine of frequency 0 contains sines of all frequencies at once in it | https://dsp.stackexchange.com/questions/15333/sine-of-frequency-0-contains-sines-of-all-frequencies-at-once-in-it | <p>You know that a sine corresponds to a pulse by J.Fourier transform. The lower is the frequency, the closer is the pulse to the origin. A constant signal is a sine (or cosine, that may be important) of frequency 0. It is a pulse in the origin. This is ok, since <a href="https://dsp.stackexchange.com/questions/9842">w... | <p>You are confusing time domain and frequency domain. A constant time domain function (a sine with frequency 0, if you like) corresponds to a delta impulse at the origin (i.e., frequency zero!) in the <em>frequency domain</em>. A delta impulse at frequency zero is zero for all other frequencies $\omega\neq 0$. Consequ... | 52 |
Fourier transform | What is the difference between multiplying a delta and a step versus convolving a delta and a step? | https://dsp.stackexchange.com/questions/20418/what-is-the-difference-between-multiplying-a-delta-and-a-step-versus-convolving | <p>Seems both will produce another step. there is no difference? Thanks</p>
| <p>First of all you need to see whether you are performing these operations for a continous time signal or discrete time signal.</p>
<p>Sampling theorem says that multiplication of a signal $x(t).\delta(t)$=$x(0).\delta(t)$ provided $x(t)$ is continous at $t=0$. But here in your question $x(t)$ is a unit step function... | 53 |
Fourier transform | GSP as an extenstion of DSP | https://dsp.stackexchange.com/questions/68291/gsp-as-an-extenstion-of-dsp | <p>I am a PhD. in pure mathematics. </p>
<ol>
<li>Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-based Fourier analysis theory.</li>
<li>I am reading the interesting <a href="http://www.eusipco2016.org... | 54 | |
Fourier transform | How to get a non-equally spaced FFT back into the time domain | https://dsp.stackexchange.com/questions/74937/how-to-get-a-non-equally-spaced-fft-back-into-the-time-domain | <p>I have a signal that I STFT and then filter using an ERB spaced filterbank. At some point after this I want to get the signal back into the time domain, how can I go about this? Using a standard iSTFT function won't work because it assumed linearly spaced frequency bins, AFAIK? I've put a code snippet below.</p>
<p>... | <p>The short answer is that you can solve a least squares problem with the input signal as the decision variables.</p>
<h3>About Nonequispaced FFT</h3>
<p>There is a <a href="https://stackoverflow.com/questions/67350588/example-python-nfft-fourier-transform-issues-with-signal-reconstruction-normal">related question in ... | 55 |
Fourier transform | What frequencies are present in the Fourier transform of the Dirac impulse? | https://dsp.stackexchange.com/questions/51085/what-frequencies-are-present-in-the-fourier-transform-of-the-dirac-impulse | <p>When I do the Fourier transform of the Dirac impulse I get a pure sinusoid (or complex exponential, however you wanna call it) but I read in several places that all frequencies are present in the dirac impulse and all of them with the same amplitude. How is this possible? Am I wrong when I perform the transform?</p>... | <p>A Dirac impulse <span class="math-container">$x(t)=\delta(t-d)$</span> has the continuous-time Fourier transform <span class="math-container">$X(\Omega)$</span> of
<span class="math-container">$$\mathcal{F}\{\delta(t-d) \} = 1 e^{-j\Omega d} $$</span></p>
<p>whose <strong>magnitude</strong> is <span class="math-cont... | 56 |
Fourier transform | Solving Fouriertransform exercises without explicitly doing the transform | https://dsp.stackexchange.com/questions/76059/solving-fouriertransform-exercises-without-explicitly-doing-the-transform | <p>Hey there in the signal processing course I am studying there is an excercise that reads:</p>
<p>The sequence <span class="math-container">$x(n)$</span> is given <span class="math-container">$x(n)=\{-1\quad2\quad \underline{-3}\quad 2\quad -1\}$</span> and the fouriertransform <span class="math-container">$X(\omega)... | <p>Some key take-aways /properties to know about the Fourier Transform will help reveal the answers. There is a method to the madness in extracting some key take-aways and high level understanding of what the Fourier Transform represents that makes this exercise useful.</p>
<p>The line under the 3 indicates the assumed... | 57 |
Fourier transform | Can the magnitude of a discrete-time Fourier transform be negative? | https://dsp.stackexchange.com/questions/79374/can-the-magnitude-of-a-discrete-time-fourier-transform-be-negative | <p>Consider the discrete-time system
<span class="math-container">$$
H(z) = 1 + z^{-1} + z^{-2} + z^{-3}
$$</span>
To obtain the magnitude of the discrete-time Fourier transform, I substitute <span class="math-container">$z = e^{j\omega}$</span> to get
<span class="math-container">\begin{align}
H(\omega) &= 1 + e^{... | <p>You can't conclude that the magnitude response is <span class="math-container">$r(\omega)$</span> and the phase response is <span class="math-container">$\theta(\omega)$</span>.</p>
<p>Note that <span class="math-container">$e^{j\pi}=-1$</span>, at the frequencies that <span class="math-container">$r(\omega) < 0$... | 58 |
Fourier transform | Where did I make the mistake in the Fourier transform? | https://dsp.stackexchange.com/questions/79568/where-did-i-make-the-mistake-in-the-fourier-transform | <p><span class="math-container">$$ \begin{align}
X(f) & = \int_{-\infty}^{\infty} x(t)e^{-j2\pi ft}dt & \\
& = \int_{-\infty}^{\infty} x(t)\left(e^{-j2\pi} \right)^{ft}dt & \;\;\mathrm{where}\; e^{-j2\pi}=1 \\
& = \int_{-\infty}^{\infty} x(t) (1)^{ft}\; dt, &\;\;\mathrm{but}\ \; 1^{ft}... | <p><a href="https://en.wikipedia.org/wiki/Exponentiation#Complex_exponents_with_a_positive_real_base" rel="nofollow noreferrer">See this Wikipedia article.</a></p>
<p>You can only multiply exponents if they are real. Let's look at the example of <span class="math-container">$f\cdot t = 1/4$</span></p>
<p><span class="m... | 59 |
Fourier transform | Fourier Transform of the signum function, using the integral property | https://dsp.stackexchange.com/questions/80888/fourier-transform-of-the-signum-function-using-the-integral-property | <p>Cheers, I am trying to find the fourier transform of the signum function, which is</p>
<p><span class="math-container">$$ \operatorname{sgn}(t) \triangleq \begin{cases}
1 \qquad & t>0 \\
0 \qquad & t=0 \\
-1 \qquad & t<0 \\
\end{cases} $$</span></p>
<p>I rewrite this as:</p>
<p><span class="math-co... | <p>By taking the derivative you loose all information about the DC value of the original signal. Any signal</p>
<p><span class="math-container">$$x(t) = 2\cdot u(t)- a$$</span></p>
<p>has the same derivative, regardless of what <span class="math-container">$a$</span> is. So you do have to calculate the DC value by hand... | 60 |
Fourier transform | Positive or negative sign on Fourier transform formula | https://dsp.stackexchange.com/questions/26221/positive-or-negative-sign-on-fourier-transform-formula | <p>I have seen both the formula of Fourier transform with positive and negative sign on exponential as $$ X(\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt$$ and $$ X(\omega)=\int_{-\infty}^{\infty} x(t)e^{j\omega t}dt$$ I am confused which one is the correct formula. I also solved for Fourier transform by taking... | <p>The definition with the negative in the exponent is the accepted definition of the Fourier transform... however, this is an arbitrary choice. It could just as easily be defined with $e^{jw}$ and the inverse transform with $e^{-jw}$. </p>
| 61 |
Fourier transform | Approximation of a complex-valued function of real variable | https://dsp.stackexchange.com/questions/81900/approximation-of-a-complex-valued-function-of-real-variable | <p>I have a problem to approximate a complex-valued function of a real argument. In other words, how to find a function’s analytic form in the complex domain if the sets of values of a function of z=x+yi type and arguments n are given.
Thus, if the sequences of real arguments (n) and complex-valued functions (z=x+yi) a... | <p>The analytic signal is given as:</p>
<p><span class="math-container">$$x_a(t) = x(t) + j \hat x(t)$$</span></p>
<p>Where</p>
<p><span class="math-container">$x_a(t)$</span> is the complex analytic signal</p>
<p><span class="math-container">$x(t)$</span> is a real signal</p>
<p><span class="math-container">$\hat x(t)... | 62 |
Fourier transform | is $y(t) = (x(t))^2$ non-linear and time-invariant system? | https://dsp.stackexchange.com/questions/82269/is-yt-xt2-non-linear-and-time-invariant-system | <p>i was able to show that it is not linear but for time-invariant I am not sure.
<span class="math-container">$y(t) = (x(t))^2$</span></p>
<p>Let <span class="math-container">$y(t)$</span> be the output corresponding to the input <span class="math-container">$x(t).$</span> Let <span class="math-container">$x_T(t) = (x... | <p>The system defined through the transfer function expression:
<span class="math-container">$$
y(t)=(x(t))^2
$$</span>
is <em>Time Invariant</em>, since the time is explicitely not included in the expression, besides <span class="math-container">$x(t)$</span>, and <span class="math-container">$y(t)$</span>.</p>
<p>Als... | 63 |
Fourier transform | Spectrally flat binary sequence | https://dsp.stackexchange.com/questions/72894/spectrally-flat-binary-sequence | <p>I'm trying to construct a binary sequence of length <span class="math-container">$2^n$</span>. This sequence will be converted to a square signal of <span class="math-container">$\pm 1$</span>, where 0 produces <span class="math-container">$-1$</span> and 1 produces <span class="math-container">$1$</span>. I want th... | <p>Golay Complementary Sequences are spectrally flat. See <a href="https://www.isg.rhul.ac.uk/%7Ekp/golaysurvey.pdf" rel="nofollow noreferrer">https://www.isg.rhul.ac.uk/~kp/golaysurvey.pdf</a> or
<a href="https://www.sfu.ca/%7Ejed/Papers/Davis%20Jedwab.%20Golay%20Reed-Muller.%201999.pdf" rel="nofollow noreferrer">ht... | 64 |
Fourier transform | Can you quickly find the inverse Fourier Transform using the duality property? | https://dsp.stackexchange.com/questions/79526/can-you-quickly-find-the-inverse-fourier-transform-using-the-duality-property | <p>Cheers, in an exercise of mine I reach the point that I have to find the <span class="math-container">$F^{-1}\{Λ(ω)\}$</span> (where <span class="math-container">$Λ(ω)$</span> is the triangle function, with <span class="math-container">$1-|ω|$</span> for <span class="math-container">$|ω| \leq 1 $</span> and 0 elsewh... | <p>You know a Fourier transform pair</p>
<p><span class="math-container">$$x(t)\Longleftrightarrow X(\omega)\tag{1}$$</span></p>
<p>with</p>
<p><span class="math-container">$$\mathcal{F}\big\{x(t)\big\}=X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt\tag{2}$$</span></p>
<p>and</p>
<p><span class="math-container">... | 65 |
Fourier transform | Recover Fourier Transform of flipped signal from the FFT of orignal signal | https://dsp.stackexchange.com/questions/82725/recover-fourier-transform-of-flipped-signal-from-the-fft-of-orignal-signal | <p>I trying to recover the Fourier Transform of a flipped signal directly from the Fourier transform of the original signal.</p>
<p>More precisly, let <code>s</code> be a random signal:</p>
<pre><code>s = np.random.randn(n)
</code></pre>
<p>Let <code>s1_fft</code> and <code>s2_fft</code> the Fourier Transform of the si... | <p>We can start with a the simple DFT relationship of the time reversal, i.e.</p>
<p>If <span class="math-container">$ \mathcal{F} (x[n]) = X[k] $</span>, then <span class="math-container">$ \mathcal{F} (x[-n]) = X'[k] $</span>, where <span class="math-container">$'$</span> denotes complex conjugation.</p>
<p>Now f... | 66 |
Fourier transform | Interpretation of complex time-domain signal resulting from time-shift property of Fourier transform | https://dsp.stackexchange.com/questions/83217/interpretation-of-complex-time-domain-signal-resulting-from-time-shift-property | <p>I am currently working on simulating RF transmissions for beamforming and other applications in Matlab.</p>
<p>One of the fundamental properties that I need to simulate is signal propagation delay due to transmission distance. This can either be done by generating the signal <span class="math-container">$s(t-\tau)$<... | <p>It is nothing about the numerical precision in your case, the main reason is the fractional delay. We know that phase shift of DFT corresponds to a <strong>circular shift</strong> in time domain. The DFT of <span class="math-container">$x[n]$</span> is
<span class="math-container">$$
X[k] = \sum_{n=0}^{N-1} x[n] e^{... | 67 |
Fourier transform | What is the meaning of $Ta_k$ of fourier series or transform? | https://dsp.stackexchange.com/questions/9050/what-is-the-meaning-of-ta-k-of-fourier-series-or-transform | <p>What is the meaning of $Ta_k$ of fourier series or transform?
I am taking a course on signal and systems.</p>
<p>In 286 page of my textbook, it says that as T becomes arbitrarily large the original periodic square wave approaches a rectangular pulse.
Also it says that all that remains in the time domain is an aperi... | <p>The idea is that a <a href="http://en.wikipedia.org/wiki/Fourier_series" rel="nofollow">Fourier series</a> is only defined for periodic signals. In the discussion in the linked slides, the author is considering a rectangular pulse train with period $T$. That is, a pulse of width $2S$ repeats periodically with a spac... | 68 |
Fourier transform | Calculate maximum of filter kernel | https://dsp.stackexchange.com/questions/9124/calculate-maximum-of-filter-kernel | <p>I'm sure there must be an easy way to do this, but given the Fourier transform of an isotropic filter kernel, $\hat{f}(\mathbf{u}) = \mathcal{F}f(\mathbf{z})$, can one calculate the value of the kernel at $\mathbf{z} = 0$?</p>
| <p>Since $$f(\mathbf{z})=\int_{\mathbf{R}^n}\hat{f}(\mathbf{u})e^{2\pi i\mathbf{z}\cdot\mathbf{u}}\;d\mathbf{u}$$</p>
<p>$$f(\mathbf{0})=\int_{\mathbf{R}^n}\hat{f}(\mathbf{u})\;d\mathbf{u}$$</p>
<p>So you simply integrate (or sum in the discrete case) over $\hat{f}(\mathbf{u})$. </p>
| 69 |
Fourier transform | Frequency Axis problem in a DTFT | https://dsp.stackexchange.com/questions/13930/frequency-axis-problem-in-a-dtft | <p>I have a doubt related to calculating the Discrete Time Fourier Transform (DTFT) by hand. Specifically in how calculate the frequency axis of the spectrum. My signal has N values and was sampled at FS Hz, the spectrum would have N entries too (where N/2 values are a mirror of the other half). The maximum representab... | <p>Generally I like to compose my sinusoids using this format:</p>
<pre><code>x = cos(2*pi*f/fs*(0:num_samps-1))
</code></pre>
<p>Depending on the FFT routine you will use it will provide either the onesided or twosided result. Calculating the domain of the frequency axis is as follows. It should be noted that num_sa... | 70 |
Fourier transform | Discrete Fourier Transform by hand | https://dsp.stackexchange.com/questions/18461/discrete-fourier-transform-by-hand | <p>I have an assignment where I'm given the DFT of a sequence $x[n]$ as $X[k]=\{4,3,2,1,0,1,2,3\}$ and also
$$y[n] = \left\{ \begin{array}[cc]
xx[n/2] & \text{if n is even} \\
0 & \text{otherwise}
\end{array} \right. $$</p>
<p>and I'm supposed to find and sketch the DFT of $y[n]$.</p>
<p>So $y[n] = \{x[0], 0,... | <p>If you use the DFT formula, you get:<br>
$$
Y[k] = \sum_{n=0}^{2N-1}y[n]e^{\frac{-2\pi k n}{2N}}
$$
Now, substituting the definition of $y[n]$ you get:<br>
$$
Y[k] = \sum_{n=0}^{N-1}x[n]e^{\frac{-2\pi k (2n)}{2N}} = \sum_{n=0}^{N-1}x[n]e^{\frac{-2\pi k n}{N}}
$$<br>
So, for $0\leq k < N$ you get that
$$ Y[k] = X... | 71 |
Fourier transform | Inverse Fourier transform of complex exponential with frequency dependent shift | https://dsp.stackexchange.com/questions/74642/inverse-fourier-transform-of-complex-exponential-with-frequency-dependent-shift | <p>In the case of a constant delay <span class="math-container">$\tau$</span>, we have the following equality:</p>
<p><span class="math-container">$$\begin{align}\mathcal{F^{-1}}\left\{e^{-j\omega \tau}\right\}=\delta(t-\tau)\end{align}$$</span></p>
<p>If the delay is frequency dependent <span class="math-container">$\... | <p>In the general case: no.</p>
<p>For the (inverse) Fourier transform of a function to be composed of a sum of countable diracs (i.e. to be discrete), the function needs to be periodic.</p>
<p>Your <span class="math-container">$e^{-j\omega\tau(f)}$</span> is not periodic (it <em>can</em> be periodic, if <span class="m... | 72 |
Fourier transform | Vector parameters in uncountably infinite-dimensional spaces | https://dsp.stackexchange.com/questions/43154/vector-parameters-in-uncountably-infinite-dimensional-spaces | <p>My question was, in an uncountably infinite-dimensional vector spaces, how to represent a vector by a list of parameters, as we do in finite-dimensional spaces? I was assuming that if we can not express a vector as a list of discrete parameters, we have a big issue...but during the writting up of this question, it s... | 73 | |
Fourier transform | In Fourier transforms, can momentum space be analogized to frequency, and position space be analogized to wavelength? | https://dsp.stackexchange.com/questions/157/in-fourier-transforms-can-momentum-space-be-analogized-to-frequency-and-positi | <p>We know that in quantum mechanics, momentum space is the fourier transform of position space (and vice versa)</p>
<p>And also, in time-series analysis, that frequency (of cycles) is the fourier transform of the distribution of all cycle lengths.</p>
<p>What about electromagnetic radiation? Is the distribution of f... | <p>The notions of position and momentum are not fundamental to the uncertainty principle, but the fact that position and momentum are analogous to instantaneous time and instantaneous frequency is. There is no necessity to translate the spatial domain of an image and its fourier representation in terms of position and ... | 74 |
Fourier transform | What happens if we change the limits of integral in Fourier transform? | https://dsp.stackexchange.com/questions/6282/what-happens-if-we-change-the-limits-of-integral-in-fourier-transform | <p>By definition of Fourier transform</p>
<p>$$X(\omega)=\int_{-\infty}^\infty x(t) e^{-j\omega t} dt $$</p>
<p>Now what will happen to the answer of transform for example in case of $x(t)= \cos(\omega_0 t)$ if limit is $0$ to $A$ instead of $-\infty$ to $\infty$? </p>
<p>For $x(t)=\cos(\omega_0 t)$ its fourier tr... | <p>Yes, it will affect the answer. What you're suggesting is known as the short-time Fourier transform. In the sinusoidal case that you proposed, you will observe spectral leakage, as the truncation of the integral limits is equivalent to multiplication of the sinusoid by a rectangular window function. This multiplicat... | 75 |
Fourier transform | Unit Impulse funciton FT | https://dsp.stackexchange.com/questions/17675/unit-impulse-funciton-ft | <p>How is FT of $ \delta $(t) equal to 1 ? Normal FT gives the result $\infty$. Can someone please explain? I did the normal integration and substituted the limits. </p>
<p>Is it because $ \delta $(t) is a unit impulse function so as it's height is large it's width is very small so no matter what the FT will always b... | <p>If follows directly from the definition of the Dirac delta distribution. It is defined so that</p>
<p>$$\int_\mathbb{R} \delta(x) f(x) dx := f(0)$$</p>
<p>for any test function $f(x)$. In other words, the Dirac distribution is the generator of the linear functional that extracts a single function value.</p>
<p>Wi... | 76 |
Fourier transform | Multi-Time Window FFT | https://dsp.stackexchange.com/questions/27165/multi-time-window-fft | <p>One can achieve better resolution results by taking FFT of different sizes of the input signal. FFT size decreases as frequency increases, i.e. longer FFT length for lower frequencies and shorter FFT length for higher frequencies. I have tried to find papers on this topic but did not find any so far. Rational Acoust... | <p>If your goal is to plot a magnitude spectrum with the frequency axis on a log scale, but with a roughly even visual resolution along that axis, then a single FFT might provide too low a density of plot points (without interpolation) at low frequencies, and more plot points than can be plotted on a line (without aver... | 77 |
Fourier transform | Question about ramp filter used in filtered backprojection | https://dsp.stackexchange.com/questions/34424/question-about-ramp-filter-used-in-filtered-backprojection | <p>Question is this.
First, a ramp filter (in frequency domain) is defined by $H(Q)=|Q|$.
What are the responses of a ramp filter to
(1) a constant function $f(r)=c$ and (2) a sinusoid function $f(r)=\sin(wr)$?
What does the response mean? Following is my work. </p>
<p>My work: </p>
<ol>
<li><p>First, take fourier... | <p>To see if your math is correct, it is useful to first understand what is, in general, the effect of a filter on a signal, and then see if you can predict what the theoretical result should look like.</p>
<p>If a filter has frequency response $H(Q)$, this means that its response to an input $e^{j2\pi Q_0 r}$ is the ... | 78 |
Fourier transform | How to apply an FFT | https://dsp.stackexchange.com/questions/41870/how-to-apply-an-fft | <p>Okay, round 2.</p>
<p>The issue I am having with implementing FFT is that different implementations require passing as arguments different types of data. From the WAV file you obtain samples of the amplitude recorded at the sample rate. </p>
<p>As an example, the NAudio library takes an array of complex numbers as... | <p>The data section of a WAVE file is often an array of 16-bit signed integers. You may need to convert each element of that array from an integer into a floating-point double, and put that converted value into the real component of an element of a complex array, in order to use many common floating point FFTs.</p>
<... | 79 |
Fourier transform | Understanding the meaning of amplitude in FFT | https://dsp.stackexchange.com/questions/41988/understanding-the-meaning-of-amplitude-in-fft | <p>I am recording data with a magnetometer of the background magnetic field in a building. I have applied the FFT algorithm to the data in order to look for the frequencies that appear in it. I would like to use this in order to identify (or at least make an educated guess) of the sources of the disturbances that I obs... | <p>The continuous-time Fourier transform of a function <span class="math-container">$f(t)$</span> is in essence an integration of <span class="math-container">$f(t)$</span> multiplied with a complex exponential kernel:
<span class="math-container">$$ F(\omega)=\int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt \tag{1}$$</s... | 80 |
Fourier transform | What happens with signal in frequency spectrum when it is time shifted in time spectrum? | https://dsp.stackexchange.com/questions/17009/what-happens-with-signal-in-frequency-spectrum-when-it-is-time-shifted-in-time-s | <p>I have some trouble to understand what is going on with signal in frequency spectrum when it is time shifted in time spectrum.</p>
<p>I am hoping that somebody will help me to understand that.</p>
<p>Thanks you very much.</p>
| <p>Each frequency in the FT of a time shifted waveform is rotated in phase by an amount proportional to the frequency and proportional to the amount of time shift.</p>
<p>If you delay a pure sinusoid by 25% of its period, it's phase referenced to any fixed point in time will change by pi/2 radians. Delay a slightly h... | 81 |
Fourier transform | Why is signum function used to calculate Fourier transform of unit step function | https://dsp.stackexchange.com/questions/26406/why-is-signum-function-used-to-calculate-fourier-transform-of-unit-step-function | <p>I read in a standard textbook that the Fourier transform of unit impulse function is calculated with the help of approximations and signum function as the integration of unit impulse does not converge. What's so special about signum function that it is used to calculate Fourier transform? I tried to find out an appr... | <p>If somebody you trust told you that the Fourier transform of the sign function is given by</p>
<p>$$\mathcal{F}\{\text{sgn}(t)\}=\frac{2}{j\omega}\tag{1}$$</p>
<p>you could of course use this information to compute the Fourier transform of the unit step $u(t)$. Using</p>
<p>$$u(t)=\frac12(1+\text{sgn}(t))\tag{2}$... | 82 |
Fourier transform | How the Fourier transform of a cosine signal is existed? | https://dsp.stackexchange.com/questions/33785/how-the-fourier-transform-of-a-cosine-signal-is-existed | <p>As I know, if an aperiodic continuous-time signal be absolutely integrable, i.e. </p>
<p>$$\int\limits_{-\infty}^\infty \vert x(t) \vert \ dt \ < \ \infty $$</p>
<p>its Fourier transform is existed. </p>
<p>Also, the Fourier transform of $\cos(\omega_0 t)$ is $\pi(\delta(\omeg... | <p>The Fourier Transform is defined only for functions that are absolutely integrable or in other words $f\in\mathcal{L}_1$, where $\mathcal{L}_1$ is the set of all absolutely integrable functions. </p>
<p>If you want to be mathematically rigorous, then you should assume that the Fourier transform of the cosine does n... | 83 |
Fourier transform | How to recover $f(t)$ from Fourier Transform of its absolute value $\mathcal{F}|f(t)|$? | https://dsp.stackexchange.com/questions/34782/how-to-recover-ft-from-fourier-transform-of-its-absolute-value-mathcalf | <p>Let the Fourier Transform of a real signal, $f(t)$, be $\mathcal{F}(\omega)$. And the FT of the absolute value of the same signal, $|f(t)|$, be $\mathcal{F}(u)$. </p>
<p>Can $\mathcal{F}(w)$ be recovered from $\mathcal{F}(u)$?</p>
<p>For instance, the FT of $a \cdot \cos(ft)$ returns a spectrum in which the freque... | <p>I recently <a href="https://dsp.stackexchange.com/questions/34373/pilot-tone-frequency-doubling/34374#comment65236_34374">was pointed to</a> a very nice trick by Robert Bristow Johnson which possibly applies here too to demonstrate this "inability" of recovery. I thought I'd share it here, in addition to the accepte... | 84 |
Fourier transform | Spectrum of windowed version of original continuous signal | https://dsp.stackexchange.com/questions/37548/spectrum-of-windowed-version-of-original-continuous-signal | <p>Suppose we have the complex signal $x(t)= \exp(j\omega_0 t)$. Using the properties of Fourier transform we can prove its CTFT is Dirac $\delta$ function. </p>
<p>If any one ask me about the spectrum of $x(t)$ "Does $x(t)$ has continuous spectrum or discrete spectrum", my answer will be "The spectrum of $x(t)$ is di... | <p>If you don't understand the difference between the <a href="https://en.wikipedia.org/wiki/Fourier_transform" rel="nofollow noreferrer">Continuous Time Fourier Transform</a> (CTFT), the <a href="https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform" rel="nofollow noreferrer">Discrete Time Fourier Transform</a... | 85 |
Fourier transform | Fourier transform of cosine to the power of 3 | https://dsp.stackexchange.com/questions/6038/fourier-transform-of-cosine-to-the-power-of-3 | <p>How can I find the Fourier transform of</p>
<p>$$ f(x) = ( \cos(x) )^3$$</p>
<p>I know that for $ g(x) = \cos(x) $</p>
<p>$$\mathcal F \Big\{ g(x) \Big\} = \mathcal F \Big\{ \cos(x) \Big\} = \pi \Big [ \delta(w-\pi / 2) + \delta(w+\pi / 2) \Big ]$$</p>
<p>But using this pair of Fourier transform how to obtain th... | <p>One way would be to use the <a href="http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formula" rel="noreferrer">power-reduction trigonometric identity</a>:</p>
<p>$$
\cos^3(x) = \frac{3 \cos(x) + \cos(3x)}{4}
$$</p>
<p>Due to the linearity property of the Fourier transform, you can tra... | 86 |
Fourier transform | Why does the periodic signal in time always give a discrete frequency spectrum | https://dsp.stackexchange.com/questions/17060/why-does-the-periodic-signal-in-time-always-give-a-discrete-frequency-spectrum | <p>I would like to know, why does the periodic signal in time always give a discrete frequency spectrum in FT?</p>
<p>I know the equations, but I simply dont understand why is it so.</p>
<p>Thanks!</p>
| <p>Here's an intuitive explanation if the convolution theorem is taken for granted:</p>
<p>Since the time-domain signal is periodic, one can say that it can be built by "copying and pasting" the same block of signal every period: your periodic signal can be expressed as a little block of signal (spanning one period) <... | 87 |
Fourier transform | Filtering and Fourier Transforming, does the order matter? | https://dsp.stackexchange.com/questions/22076/filtering-and-fourier-transforming-does-the-order-matter | <p>I have a signal $x(t)$. I want to find the Fourier Transform of it, $X(f)$, and then extract a narrow frequency range from $X(f)$ by use of a Band Pass Filter (BPF) in frequency domain.</p>
<p>Can I instead filter $x(t)$ by using a BPF in time domain and then find the Fourier Transform of the filtered signal?</p>
... | <p>There may be some slight differences due to the band-pass filtering of FFT results in the frequency domain being a circular convolution (with some wrap-around artifacts) rather than a pure linear convolution.</p>
<p>If you filter first, starting in time before your FFT window, any windowing artifacts from any out-o... | 88 |
Fourier transform | Repeated Fourier transform - what happens? | https://dsp.stackexchange.com/questions/31285/repeated-fourier-transform-what-happens | <p>I have a Fourier transformable complex function that is a function of independent real variable a. Now I take the Fourier transform of it, giving me a complex function of real variable b. Now I treat the resulting function as if it is in the original domain of a and again take Fourier transform of it - in stead of... | <p>if you define the continuous Fourier Transform in a <a href="https://en.wikipedia.org/wiki/Unitary_operator" rel="nofollow">unitary</a> manner, my preferred unitary definition is</p>
<p>$$ X(f) \triangleq \mathscr{F}\{ x(t) \} = \int\limits_{-\infty}^{+\infty} x(t) \, e^{-j 2 \pi f t} \ dt $$</p>
<p>$$ x(t) \trian... | 89 |
Fourier transform | Instantaneous frequency vs fourier frequency | https://dsp.stackexchange.com/questions/19469/instantaneous-frequency-vs-fourier-frequency | <p>Lets consider a pure sine signal at $\nu$ that is chopped using square pulses (like a burst mode on signal generators). My understanding is that instantaneous frequency is $\nu$ when oscillations are ON and 0 when they are OFF. On the other hand fourier spectrum is constant over time and contains also other frequenc... | <p>Yes, your understanding ist correct. Instantaneous frequency is the time derivative of the sine argument. As Robert mentions in his answer, this argument is not defined where there is no sine (or complex exponential) function but I think its reasonable to consider it a sine with amplitude zero and constant argument.... | 90 |
Fourier transform | Fourier Transform negative amplitude meaning | https://dsp.stackexchange.com/questions/52406/fourier-transform-negative-amplitude-meaning | <p>I am reading this example <a href="http://www.thefouriertransform.com/pairs/truncatedCosine.php" rel="nofollow noreferrer">http://www.thefouriertransform.com/pairs/truncatedCosine.php</a></p>
<p>What does it mean to have some of the frequency components be negative in its amplitude ? I am not talking about the nega... | <p>The fft returns complex values, to get the amplitude you need to take the abs( ). The real and imaginary portion tell you about the signals phase. Remember the fft is changing the basis by projecting your signal onto a complex sinusoid:
<span class="math-container">$$e^{i \omega t} = \cos(\omega t) + i \sin(\omega t... | 91 |
Fourier transform | Fourier transform is an isomorphism...but we don’t get when each frequency appears? | https://dsp.stackexchange.com/questions/62491/fourier-transform-is-an-isomorphism-but-we-don-t-get-when-each-frequency-appea | <p>Statistician here who wants to get some DSP knowledge for time series analysis.</p>
<p>I’ve known for years that if we hit a function with a Fourier transform, we have an inverse Fourier transform that will recover the original function. However, doesn’t the interpretation of the Fourier transform in the frequency ... | <p>It's true that taking the Fourier transform will leave you without any (visible) information on time and vice versa, but of course you don't lose any information, you just represent it in a way such that in one domain you only see time information, and in the other you only see frequency information.</p>
<p>Take as... | 92 |
Fourier transform | Applying duality property to fourier transform of unit step function | https://dsp.stackexchange.com/questions/56388/applying-duality-property-to-fourier-transform-of-unit-step-function | <p>For Continuous time aperiodic signals, the duality property of Continuous Time Fourier Transform (CTFT) is following</p>
<p><span class="math-container">$$\mathscr{F}\Big\{x(t)\Big\} = X(f), \qquad\text{then} \quad \mathscr{F}\Big\{X(t)\Big\} = x(-f)$$</span></p>
<p>Now we know while Dirichlet conditions are not s... | <p>The result you got is correct and it is also expected according to the two first formulas in your question. If <span class="math-container">$X(f)$</span> is the Fourier transform of <span class="math-container">$x(t)$</span>, then the Fourier transform of <span class="math-container">$X(t)$</span> equals <span class... | 93 |
Fourier transform | Fourier transform of a damped cosine wave with a linear frequency chirp | https://dsp.stackexchange.com/questions/59160/fourier-transform-of-a-damped-cosine-wave-with-a-linear-frequency-chirp | <p>I want to take the Fourier transform of the following transient signal,
<span class="math-container">$$f(t) = e^{-t/\tau} \cos((\omega_0 + m t)t)$$</span>, where <span class="math-container">$m$</span> is some gradient parameter in units of <span class="math-container">$\rm{Hz}/s$</span>.
I thought this would be qui... | <p>For Fourier Transform of the LFM chirp portion - you use the Principle of Stationary Phase (POSP). The POSP essentially says the main contribution in the Fourier Integral comes from the portion of where the derivative of the phase is zero - it assumes that the integral of the oscillating components cancel themselves... | 94 |
Fourier transform | Fourier transform of discrete time unit step function | https://dsp.stackexchange.com/questions/61903/fourier-transform-of-discrete-time-unit-step-function | <p>To obtain fourier transform of u[n],
<code>u[n] - u[n-1] = delta[n]</code> , taking fourier transform of both sides of the equation results in :
<code>U(w) - exp(-jw) U(w) = 1</code> , hence :
<code>U(w) = 1/(1-exp(-jw))</code> which is wrong and the right answer has an extra term.
Which step is wrong in this poss... | <p>The DFT of a unit step response is
<span class="math-container">$$U(\omega) = \frac{1}{1 - e^{-j \omega}} + \pi \delta(\omega)$$</span>
Applying the shift property as you did will give:
<span class="math-container">$$\mathcal{F}(u[n] - u[n-1]) = U(\omega) - U(\omega)e^{-j \omega} = \frac{1}{1 - e^{-j \omega}} + \pi ... | 95 |
Fourier transform | Time shift and Phase Examples | https://dsp.stackexchange.com/questions/82329/time-shift-and-phase-examples | <p>Given are two cosines according to the following formula
<span class="math-container">$x_i(t) = cos(2\pi f_i t)$</span>
with <span class="math-container">$f_1 = 1Hz$</span> , <span class="math-container">$f_2 = 2Hz$</span> and <span class="math-container">$f_3 = 3Hz$</span> .</p>
<p>The two cosines are delayed by <s... | <p>Any signal <span class="math-container">$x(t)$</span> can be time-shifted: simply calculate <span class="math-container">$x(t + \Delta t)$</span>.</p>
<p>A sinusoid can also be phase-shifted. Consider the cosine signal with phase <span class="math-container">$\phi$</span>: <span class="math-container">$$x(t) = \cos(... | 96 |
Fourier transform | complex numbers and fourier transform | https://dsp.stackexchange.com/questions/59468/complex-numbers-and-fourier-transform | <p>Is it possible to define a scaling property for fourier transform when the scale factor is complex?
Usually the scaling factor is real. What happen when a scaling factor is complex? </p>
| <p>there are issues. given this convention for the continuous Fourier transform (and inverse)</p>
<p><span class="math-container">$$ \mathscr{F} \Big\{ x(t) \Big\} \triangleq X(f) \triangleq \int\limits_{-\infty}^{+\infty} x(t) e^{-j 2 \pi f t} \ \mathrm{d}t $$</span></p>
<p><span class="math-container">$$ \mathscr{... | 97 |
Fourier transform | Windowing function for Inverse Fourier Transform | https://dsp.stackexchange.com/questions/70813/windowing-function-for-inverse-fourier-transform | <p>It is a common practice to apply windowing function, such as Hann or Hamming, to a time domain signal before FFT, in order to reduce spectral leakage. Often, we do 1) Windowing, 2) FFT, 3) frequency domain processing, such as filtering, then 4) Inverse FFT. My questions are: before inverse FFT, do we need to apply a... | 98 | |
Fourier transform | FFT of a stretched vector | https://dsp.stackexchange.com/questions/74933/fft-of-a-stretched-vector | <p>Lets say I have a small size vector x=[a b c d]. Now I stretch this vector 3 times and I got x3=[a a a b b b c c c d d d]. What would be the relation between fft(x) and fft(x3)?</p>
| <p>Conceptually you should split this into two steps</p>
<ol>
<li>Up-sample by a factor of 3, i.e. x = [a 0 0 b 0 0 ...]. This results in a 3 times periodic repetition of the spectrum.</li>
<li>Convolve with a rectangular pulse of length three, i.e. h = [1 1 1]. This crates time stretched sequence you want. Convolution... | 99 |
Laplace transform | Inverse Laplace transform of two-sided and one-sided Laplace transform | https://dsp.stackexchange.com/questions/54855/inverse-laplace-transform-of-two-sided-and-one-sided-laplace-transform | <p>As I read in <a href="https://en.wikipedia.org/wiki/Laplace_transform" rel="noreferrer">Wikipedia</a>, there are two types of Laplace transforms</p>
<ul>
<li><p>One-sided Laplace transform: <span class="math-container">$F(s) = \int_{0}^\infty e^{-st} f(t) dt$</span></p></li>
<li><p>Two-sided Laplace transform: <spa... | <p>The inversion formula is the same for both types of transforms:</p>
<p><span class="math-container">$$f(t)=\frac{1}{2\pi j}\int_{\alpha-j\infty}^{\alpha+j\infty}F(s)e^{st}ds\tag{1}$$</span></p>
<p>The difference is in the choice of the constant <span class="math-container">$\alpha$</span>. The line <span class="ma... | 100 |
Laplace transform | Is the Laplace transform redundant? | https://dsp.stackexchange.com/questions/26146/is-the-laplace-transform-redundant | <p>The Laplace transform is a generalization of the Fourier transform since the Fourier transform is the Laplace transform for $s = j\omega$ (i.e. $s$ is a pure imaginary number = zero real part of $s$).</p>
<blockquote>
<p>Reminder:</p>
<p>Fourier transform: $X(\omega) = \int x(t) e^{-j\omega t} dt$</p>
<... | <p>The Fourier and the Laplace transform obviously have many things in common. However, there are cases where only one of them can be used, or where it's more convenient to use one or the other.</p>
<p>First of all, even though in the definitions you simply replace $s$ by $j\omega$ or vice versa to go from one transfo... | 101 |
Laplace transform | Finding Laplace Transform without ROC | https://dsp.stackexchange.com/questions/27369/finding-laplace-transform-without-roc | <p>While studying Laplace Transform i found that region of convergence (ROC) is important because for some problems we have same Laplace Transform but different ROC helps us to take correct inverse Laplace Transform.
So, Now i am practicing inverse Laplace Transform problem i found almost every probelm to find $x(t)$ i... | <p>Strictly speaking you can't because without specifying the ROC, the inverse Laplace transform is generally not unique. However, in many contexts there is the implicit assumption of causality of the corresponding time function (i.e., $x(t)=0$ for $t<0$), which is equivalent to stating that the ROC is a right half-... | 102 |
Laplace transform | From Fourier transform to Laplace Transform | https://dsp.stackexchange.com/questions/56171/from-fourier-transform-to-laplace-transform | <p>It's well known that you can estimate the Fourier Transform <span class="math-container">$X(f)$</span> of a signal <span class="math-container">$x(t)$</span> via its Laplace Transform <span class="math-container">$X(s)$</span>, just by setting <span class="math-container">$s = j2\pi f$</span> to the latter, as long ... | <p>You need to distinguish three cases:</p>
<ol>
<li><p>There are Dirac impulses in the expression for the Fourier transform. In this case you can't just replace <span class="math-container">$j\omega$</span> by <span class="math-container">$s$</span> to obtain the Laplace transform. The Laplace transform might not exi... | 103 |
Laplace transform | Why the unilateral Laplace transform? | https://dsp.stackexchange.com/questions/61733/why-the-unilateral-laplace-transform | <p>Why is the Laplace transform commonly taught as the unilateral Laplace transform?</p>
<p>I mean, for the Fourier transform, we commonly have the bilateral transform... if the signal is 0 for <span class="math-container">$t<0$</span>, then it turns into a unilateral Fourier transform. Why not have this same conve... | <p>The widespread use of the unilateral Laplace transform reflects the fact that in practice we often deal with causal systems and signals that have a defined starting time (usually chosen as <span class="math-container">$t_0=0$</span>).</p>
<p>The Fourier transform is mainly used for analyzing ideal signals and syste... | 104 |
Laplace transform | Laplace Transform and Inverse laplace Transform for 2D images python code available? | https://dsp.stackexchange.com/questions/93330/laplace-transform-and-inverse-laplace-transform-for-2d-images-python-code-availa | <p>I am wondering if there is any implementation of Laplace Transform and Inverse Laplace Transform available for 2D data (i.e., images). For example, a batch of <code>N</code> input sequence of <code>D</code> can be reshaped into a 2D image with width of <code>W</code> and height of <code>H</code> and then 2D FFT is a... | 105 | |
Laplace transform | Laplace transform of derivative | https://dsp.stackexchange.com/questions/82749/laplace-transform-of-derivative | <p>Here is a short proof that Laplace Transform of <span class="math-container">$x'(t)$</span> is Laplace transform of <span class="math-container">$x(t)$</span> multiplied by s:</p>
<p><a href="https://i.sstatic.net/HMI1l.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/HMI1l.png" alt="enter image descri... | <p>The existence of <span class="math-container">$\mathcal L\{x*\delta'\}=\mathcal L\{x\}\cdot\mathcal L\{\delta'\}$</span> requires the same subexponential behaviour from <span class="math-container">$x$</span> (if <span class="math-container">$\lim_{|t|\to\infty}xe^{-st} \ne 0$</span>, then <span class="math-containe... | 106 |
Laplace transform | Laplace transform of averaging operator | https://dsp.stackexchange.com/questions/36106/laplace-transform-of-averaging-operator | <p>I am studying dc-dc converter now. I got a problem with Laplace transform of the averaging operator as in the image below.</p>
<p>Can anyone help me derive the Laplace transform result $G_{av}(s)$ as in the image?</p>
<p><a href="https://i.sstatic.net/VBXLQ.png" rel="nofollow noreferrer"><img src="https://i.sstati... | <p>Here's the outline of the argument, feel free to fill in the details.</p>
<p>The averaging operator is like a convolution with a "square" pulse of height $1/T_s$ supported on the interval $[-T_s/2, T_s/2]$. </p>
<p>You can express the square pulse as a sum of two heaviside step functions. </p>
<p>Finally, recall ... | 107 |
Laplace transform | Confusion in basics of Laplace Transform | https://dsp.stackexchange.com/questions/27179/confusion-in-basics-of-laplace-transform | <p>I have few confusions while starting Laplace Transform. So far I have studied, Fourier series and Fourier Transform. The basic difference which I found from different books is Fourier Transform is only considered the imaginary part whereas the Laplace transform considers both real and imaginary for general values.<b... | <p>The unilateral Laplace transform is used for analyzing causal linear time-invariant systems, which have an impulse response $h(t)$ that is zero for $t<0$. The unilateral Laplace transform can be used to solve initial value problems, due to the correspondence</p>
<p>$$x'(t)\Longleftrightarrow sX(s)-x(0)$$</p>
<p... | 108 |
Laplace transform | Intuitive interpretation of Laplace transform | https://dsp.stackexchange.com/questions/11008/intuitive-interpretation-of-laplace-transform | <p>So I am getting to grasps with Fourier transforms. Intuitively now I definately understand what it does and will soon follow some classes on the mathematics (so the actual subject). But then I go on reading about the laplace transform and there I kind of lose it. What is the moment of a signal? Why is the fourier tr... | <p>If you have an understanding of Fourier transforms then you probably already have a conceptual model of transforming signals into the frequency domain. The Laplace transform provides an alternative frequency domain representation of the signal - usually referred to as the "S domain" to differentiate it from other fr... | 109 |
Laplace transform | Basic difference between Fourier transform and laplace transform? | https://dsp.stackexchange.com/questions/58413/basic-difference-between-fourier-transform-and-laplace-transform | <p>I have read few links about difference between Fourier transform and Laplace transform but still not satisfied</p>
<p>Please correct me if i am wrong
Simply put, the main difference between Fourier transform and Laplace transform is that real part is set to zero in Fourier transform while real part is non zero in l... | <p>Fourier transform is an <strong>intuitive</strong> tool that's a bridge between domain of physics and mathematics, as it quantitatively describes the periodic content of the signals and also frequency response characterisation of systems that occur in physical (and engineering) applications. The use of frequencies i... | 110 |
Laplace transform | Questions related to Laplace Transform | https://dsp.stackexchange.com/questions/27189/questions-related-to-laplace-transform | <p>While studying Laplace transform, I also some questions which I want to understand: </p>
<p>a) We used to say that Laplace transform include both real and imaginary part whereas in Fourier transform we only have imaginary part. But when we have to say about convergence we also choose Real part to be either >0 or ... | <p>The Fourier transform is the Laplace transform along the imaginary axis in the complex plane.</p>
<p>The convergence of the Laplace transform ignores the complex part, as the imaginary part breaks down the signal into sinusoids: which are bounded, and so have no effect on the convergence.</p>
| 111 |
Laplace transform | Confusions regarding differences between Fourier transform & Laplace transform? | https://dsp.stackexchange.com/questions/79569/confusions-regarding-differences-between-fourier-transform-laplace-transform | <p>Although this topic has already been addressed in multiple popular questions of SE but i have few confusions in this regard</p>
<p>Number 1)</p>
<p>Link of question
<a href="https://electronics.stackexchange.com/questions/86489/relation-and-difference-between-fourier-laplace-and-z-transforms">https://electronics.sta... | <p>Concerning your first question, both, the Laplace and the Fourier transform, are frequency domain representations of a function or signal. In the Fourier transform we deal with a real-valued frequency variable <span class="math-container">$\omega$</span>, whereas in the Laplace transform we have a generally complex-... | 112 |
Laplace transform | Laplace transform plot isn't right | https://dsp.stackexchange.com/questions/75543/laplace-transform-plot-isnt-right | <p>I'm trying to plot the Laplace transform of a function. Here's my MatLab script</p>
<pre><code>clear
clc
syms t
L = 100;
sigma=(-10:0.1:(10-0.1));
omega = (-L/2:L/2-1)*(2*pi*0.1);
x = sin(2 * pi * t);
X_symbolic = laplace(x);
X = matlabFunction(X_symbolic);
result = [];
for j=1:length(omega)
resultCol = []... | 113 | |
Laplace transform | Unilateral Laplace Transform's Differentiation Property | https://dsp.stackexchange.com/questions/74093/unilateral-laplace-transforms-differentiation-property | <p>I've read in numerous places that the unilateral laplace transform is extermely useful in solving differential equations with initial conditions based on the differentiation property of the unilateral transform:</p>
<p><span class="math-container">$\mathscr{L}{f′(t)}=sF(s)−f(0_−)$</span></p>
<p>What i don't understa... | <p>You need to look at the derivation of that property. Integration by parts gives</p>
<p><span class="math-container">$$\begin{align}\mathcal{L}\{f'(t)\}&=\int_{0^-}^{\infty}f'(t)e^{-st}dt\\&=f(t)e^{-st}\Big|_{0^-}^{\infty}+s\underbrace{\int_{0^-}^{\infty}f(t)e^{-st}dt}_{F(s)}\\&=\lim_{t\to\infty}f(t)e^{-s... | 114 |
Laplace transform | Conversion from laplace transform to z-transform | https://dsp.stackexchange.com/questions/14483/conversion-from-laplace-transform-to-z-transform | <p>I would like to know if</p>
<p>$$
\text {Z-Transform ( } G(s)H(s) \text{ )} = \text {Z-Transform (}G(s) \text{)} \text { Z-Transform (} H(s) \text{) } = G(z)H(z)
$$</p>
<p>where G(s), H(s) are the Laplace transform representations of g and h, and G(z) and H(z) are the Z-transform representation of g and h.</p>
... | <p>Your question makes no sense. Z transform is performed on a discrete signal/series.</p>
<p>Since $H(s)$ is a continuous function, you can't just calculate a Z-transform of $H(s)$ without first sampling it, to make it discrete. Also, it doesn't make much sense to do a time->spectrum transform (such as a Z-transform)... | 115 |
Laplace transform | confused about time shifting property of Laplace Transform | https://dsp.stackexchange.com/questions/54400/confused-about-time-shifting-property-of-laplace-transform | <p>In book signals and systems 2 edition a question is given which is as follows:</p>
<p><span class="math-container">$$ x(t)=e^{-3(t+1)}u(t+1) $$</span></p>
<p>and we are asked to find the unilateral Laplace Transform of the signal. The method that is given in the solution manual is as follows:</p>
<p>Usi... | <p>If you have written the function correctly then its Laplace transform could be found very similary to your first example:</p>
<p>Given
<span class="math-container">$$x(t) = e^{-2 t} u(t-1)$$</span>
its Laplace transform could be found as follows. First denote the signal</p>
<p><span class="math-container">$$x_0(t... | 116 |
Laplace transform | Inverse Laplace transform Using Inversion Formula | https://dsp.stackexchange.com/questions/30701/inverse-laplace-transform-using-inversion-formula | <blockquote>
<p>Use the complex inversion formula to calculate the inverse Laplace transform $f(t)$ of the following Laplace transform:
$$F_L (s) = \frac{1}{(s+2)(s^2 +4)}.$$
When the region of convergence is:
\begin{align}(1)& \quad Re(s)<-2;\\(2)&\quad -2<Re(s)<0;\\(3)&\quad Re(s)>0.\... | <p>In engineering practice, the complex inversion integral is hardly ever used. As an engineer, you will almost exclusively need to invert rational functions, and this can be done by partial fraction expansion and elementary inversions. So first I'll show you how to obtain the inverse Laplace transform by partial fract... | 117 |
Laplace transform | Confusion regarding Laplace transform calculation in MATLAB | https://dsp.stackexchange.com/questions/83278/confusion-regarding-laplace-transform-calculation-in-matlab | <p>I am trying to learn about Laplace transform and especially about ROC and i found out on <a href="http://jntuhsd.in/uploads/programmes/Module15_LT_13.01_.2017_.PDF" rel="nofollow noreferrer">this weblink.</a></p>
<p>I have also attached a snapshot of this link and highlighted where it is being said that although the... | <p>The <a href="https://www.mathworks.com/help/symbolic/sym.laplace.html" rel="nofollow noreferrer">Matlab implementation</a> of the Laplace transform computes the <em>uni-lateral</em> (one-sided) Laplace transform:</p>
<p><a href="https://i.sstatic.net/F19tu.png" rel="nofollow noreferrer"><img src="https://i.sstatic.n... | 118 |
Laplace transform | Confusion in proof of Inverse Laplace Transform | https://dsp.stackexchange.com/questions/27288/confusion-in-proof-of-inverse-laplace-transform | <p>For the proof of inverse Laplace transform, we change the integral from $\omega$ to $s$. I want to know the reason why we need to change the integral?
<a href="https://i.sstatic.net/qjRzE.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qjRzE.png" alt="enter image description here"></a></p>
| <p>To summarize the discussion:</p>
<ul>
<li><p>The usual substitution is $s = \sigma + j \omega$ where $\sigma$ is the real part of the $s$ variable and $\omega$ is the imaginary part.</p></li>
<li><p>The equation in the image is for the <strong>Fourier</strong> transform, not the <strong>Laplace</strong> transform. ... | 119 |
Laplace transform | Time Setting of $z$ and Laplace Transforms | https://dsp.stackexchange.com/questions/56821/time-setting-of-z-and-laplace-transforms | <p>I'm aware that the z-transform and the Laplace Transform have an analogous relationship but I want to be doubly-sure that the z-transform only works in discrete-time and that the Laplace transform only works in the continuous time settings?</p>
<p>Thanks.</p>
| <p>In typical (or all) applications, the Laplace Transform is used for continuous time systems and the z-Transform for discrete systems. However, the Laplace Transform for Discrete Time Systems certainly exists but would be more complicated than necessary to solve. The z transform exists as a mathematical simplificatio... | 120 |
Laplace transform | How to compute Laplace Transform in Python? | https://dsp.stackexchange.com/questions/66428/how-to-compute-laplace-transform-in-python | <p>I am trying to do practicals for signal processing where I need to Laplace Transform a function. Used 'fft' of numpy before. Nothing of Laplace is found in the documentation. Do we have any other alternative?</p>
<p>Please go through the notebook to understand the problem (would love to get suggestions/contribution... | <p>Given the approach started in the OP's Github code I have this suggestion:</p>
<p>Observe that the unilateral Laplace Transform given as:</p>
<p><span class="math-container">$$X(s) = \int_0^\infty x(t)e^{-st}dt$$</span></p>
<p>Is just the Fourier Transform of a causal function with a weighting exponential:</p>
<p><s... | 121 |
Laplace transform | $y(0)$ terms in the Laplace transform | https://dsp.stackexchange.com/questions/94361/y0-terms-in-the-laplace-transform | <p>When taking the Laplace transform (in my case, for building a transfer function) of a signal <span class="math-container">$y(n)$</span> the substitution below is often made directly:</p>
<p><span class="math-container">$$\mathscr{L} \big\{ y^{(n)}(t) \big\} = s^n \mathscr{L} \big\{ y(t) \big\}$$</span></p>
<p>But th... | <p>Linear systems, by definition, obey the the superposition principle. As such, depending on what you're doing, the initial conditions can be irrelevant.</p>
<p>If you want to know what the overall response is, then including the initial conditions is required because they make up part of the overall response. For exa... | 122 |
Laplace transform | Why we take Laplace Transform of functions which converged using Fourier Transform | https://dsp.stackexchange.com/questions/27230/why-we-take-laplace-transform-of-functions-which-converged-using-fourier-transfo | <p>There are several functions for which we know that Fourier Transform will exist but still we calculate its Laplace Transform. Can I know the reason why we need to take Laplace transform for which we know its convergence?</p>
<p>Thanks</p>
| <p>There is a large class of functions for which both the Fourier transform and the Laplace transform exist, and for which one can be obtained from the other by setting $s=j\omega$. (Note that even when both exist, the latter need not be the case). So for this class of functions, obtaining the Laplace transform from th... | 123 |
Laplace transform | How is causality in Laplace transform related to Fourier transform? | https://dsp.stackexchange.com/questions/95015/how-is-causality-in-laplace-transform-related-to-fourier-transform | <ol>
<li><p>Taking the Laplace transform of a system given by a differential equation yields its transfer function <span class="math-container">$H(s)$</span>. The region of convergence of the causal impulse response of the system lies right of the most right pole in the complex plane. Suppose the system is stable. Then... | <p>If we are given a function <span class="math-container">$H(s)$</span> and we're told that it is a Laplace transform, then there are usually many possible corresponding time-domain functions. Let's assume that <span class="math-container">$H(s)$</span> is rational (so we can talk about poles), then the possible regio... | 124 |
Laplace transform | Confusion in initial condition of differential equation using Laplace transform transform | https://dsp.stackexchange.com/questions/69667/confusion-in-initial-condition-of-differential-equation-using-laplace-transform | <p>I'm confused in solving linear constant coefficients differential equations (LCCDEs) by Laplace transform if initial conditions are given at time</p>
<ol>
<li>just before <span class="math-container">$t=0$</span></li>
<li>just after <span class="math-container">$t=0$</span></li>
<li>exactly at <span class="math-cont... | <p>Initial conditions are always given at <span class="math-container">$t=0^-$</span>, because they define the state of the system <em>before</em> any input is applied, and - by definition - the input is applied at <span class="math-container">$t=0$</span>. The state at <span class="math-container">$t=0^+$</span> is de... | 125 |
Laplace transform | Bilateral Laplace transform and existence of Fourier transform | https://dsp.stackexchange.com/questions/50462/bilateral-laplace-transform-and-existence-of-fourier-transform | <p>I was reading from Athanosios Papoulis' "The Fourier integral and its applications." and they referenced the bilateral Laplace transform and Fourier Transform as:</p>
<p>$$F(p)=\int_{-\infty}^{\infty}e^{-pt}f(t)dt$$
$$F(\omega)=\int_{-\infty}^{\infty}e^{-j\omega t}f(t)dt$$</p>
<p>and stability indicates that the r... | <p>The bilateral Laplace transform converges in a vertical strip $a<\text{Re}\{p\}<b$, called the region of convergence (ROC). Compare this to the bilateral $\mathcal{Z}$-transform which converges in an annulus centered at the origin of the complex plane: $r_1<|z|<r_2$. For causal signals we have $b=\infty$... | 126 |
Laplace transform | Causal Signal - Fourier Transform or Laplace Transform | https://dsp.stackexchange.com/questions/40201/causal-signal-fourier-transform-or-laplace-transform | <p>I am dealing with a physics problem which is related to signal processing. The problem requires me to calculate the instantaneous force acting on a body which depends on some physical parameter $x$. Assume that $x(t)$ is periodic in time for the moment. Since $x(t)$ is periodic, then it can be expanded as a Fourier ... | <p>What you want is an all-pass filter with frequency response</p>
<p>$$H(\omega)=e^{j\phi(\omega)}\tag{1}$$</p>
<p>where $\phi(\omega)$ is the desired phase shift (and $j$ is how we denote the imaginary unit over here). This system is called an all-pass filter because clearly $|H(\omega)|=1$ holds.</p>
<p>The type ... | 127 |
Laplace transform | What are the advantages of Laplace Transform vs Fourier Transform in signal theory? | https://dsp.stackexchange.com/questions/45910/what-are-the-advantages-of-laplace-transform-vs-fourier-transform-in-signal-theo | <p>What are the advantages of Laplace Transform vs Fourier Transform in signal theory?</p>
| <p>Laplace transforms can capture the transient behaviors of systems. Fourier transforms only capture the steady state behavior. Of course, Laplace transforms also require you to think in complex frequency spaces, which can be a bit awkward, and operate using algebraic formula rather than simply numbers.</p>
<p>If y... | 128 |
Laplace transform | What are the advantages and disadvantages of Laplace transform over Z transform? | https://dsp.stackexchange.com/questions/31384/what-are-the-advantages-and-disadvantages-of-laplace-transform-over-z-transform | <p>Laplace transform for continuous signal $x(t)$ is given by</p>
<p>$$
X(s) = \int\limits_{-\infty}^{+\infty} x(t) e^{-s t} dt. \quad (1)
$$</p>
<p>Z-transform for discrete signal $x(n)$ is given by</p>
<p>$$ X(z) = \sum\limits_{n=-\infty}^{+\infty} x[n] z^{-n}. \quad (2)$$</p>
<p>I can say that only difference... | <p>Both transforms are equivalent tools, but the Laplace transform is used for continuous-time signals, whereas the $\mathcal{Z}$-transform is used for discrete-time signals (i.e, sequences).</p>
<p>You can see that they are equivalent by using the continuous-time representation of a discrete-time signal, and then app... | 129 |
Laplace transform | Laplace Transform of $-e^{-at}u(-t)$ | https://dsp.stackexchange.com/questions/27287/laplace-transform-of-e-atu-t | <p>I have found a problem in applying Laplace Transform to $-e^{-at}u(-t)$
I am doing these steps:</p>
<p>$$ = - \int_{-\infty}^{+\infty} e^{-at}u(-t) e^{-st}dt$$
$$ = - \int_{-\infty}^{0} e^{-at} e^{-st}dt$$
$$ = - \int_{-\infty}^{0} e^{-(a+s)t}dt$$
$$ = - [-\frac{1}{a+s} e^{-(a+s)t}]|_{-\infty}^{0}$$
$$ = - [-\fra... | 130 | |
Laplace transform | Laplace Transform: zeros and corresponding impulse response $h(t)$ | https://dsp.stackexchange.com/questions/71611/laplace-transform-zeros-and-corresponding-impulse-response-ht | <h2>Poles and the impulse response</h2>
<p>If our impulse response is in the form :</p>
<p><span class="math-container">$$h(t) = e^{-\sigma_0 t}\cos(\omega_0 t) \, u(t)$$</span></p>
<p>(where <span class="math-container">$u(t)$</span> is the unit step function)</p>
<p>And its Laplace transform is :</p>
<p><span class="... | <p>For you first question you can use the <a href="https://mathworld.wolfram.com/DeltaFunction.html" rel="nofollow noreferrer">following</a></p>
<p><span class="math-container">$$
\int_{-\infty}^{\infty} \delta (t-a)\,f(t)\,dt = f(a),
$$</span></p>
<p>with <span class="math-container">$f(t)$</span> any function. In you... | 131 |
Laplace transform | Laplace transform : integral vs poles & zeros | https://dsp.stackexchange.com/questions/71560/laplace-transform-integral-vs-poles-zeros | <p>If Laplace transform is expressed as :</p>
<p><span class="math-container">$$\int_{-\infty}^{+\infty} h(t)e^{-st}dt $$</span></p>
<p>with :</p>
<p><span class="math-container">$$s = \sigma + j\omega$$</span></p>
<p>and <span class="math-container">$h(t)$</span> an impulse response expressed as :</p>
<p><span class="... | <p>The original post has been updated to add information of why the integral diverges or has some finite complex value.</p>
<p>Figure 32.5 (original question) can't be understood (especially "b. Exact cancellation") if we consider :</p>
<p><span class="math-container">$$
h(t) = e^{-\sigma_0t}\cos{\ome... | 132 |
Laplace transform | What is the inverse Laplace transform of squared denominator term? | https://dsp.stackexchange.com/questions/60655/what-is-the-inverse-laplace-transform-of-squared-denominator-term | <p>Referring to the image below, what would the inverse Laplace transform be? I can't seem to find any tables that include this case.</p>
<p><a href="https://i.sstatic.net/OcDj5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OcDj5.png" alt="enter image description here"></a></p>
| <p>This is quite straightforward to solve. Either just use <a href="https://en.wikipedia.org/wiki/Laplace_transform#Table_of_selected_Laplace_transforms" rel="nofollow noreferrer">this table</a> where you can directly find the corresponding result, or "derive" it yourself with very basic knowledge of the Laplace transf... | 133 |
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