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Question: <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}+\... | https://dsp.stackexchange.com/questions/38018/what-is-the-frequency-transfer-function-when-xt-e-3tut-and-yt-2u |
Question: <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 te... | https://dsp.stackexchange.com/questions/44735/significance-of-an-impulse-in-the-frequency-domain |
Question: <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. </... | https://dsp.stackexchange.com/questions/52088/mean-square-error-and-gibbs-oscillations |
Question: <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="ma... | https://dsp.stackexchange.com/questions/53581/is-the-following-property-true |
Question: <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 rotati... | https://dsp.stackexchange.com/questions/55468/what-is-a-correct-way-to-find-or-guess-a-kernel-which-transforms-an-image-into |
Question: <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-... | https://dsp.stackexchange.com/questions/56160/applying-frequency-domain-filters-on-a-centered-fourier-transform |
Question: <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-t... | https://dsp.stackexchange.com/questions/59436/fourier-transform-and-anti-trasform-identity-missing |
Question: <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>
Answer: <p>Indeed there are two things you have to... | https://dsp.stackexchange.com/questions/60763/if-the-cosine-function-is-periodic-why-does-it-have-a-fourier-transform |
Question: <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... | https://dsp.stackexchange.com/questions/61184/why-is-a-circular-mask-appropriate-for-fourier-filtering-rectangular-images |
Question: <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) $</span></p>
<p>But in this case I got that divided by 2. I assumed I had to... | https://dsp.stackexchange.com/questions/62520/fourier-antitransform-using-scaling-property |
Question: <p>I am trying to understand how to evaluate this equation in the context of acceleration data which contain engine orders</p>
<p><span class="math-container">$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))$</span></p>
<p><span class="math-contai... | https://dsp.stackexchange.com/questions/63148/fourier-transform-of-an-acceleration-signal-containing-engine-orders |
Question: <p>Given <span class="math-container">$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)$</span>. I have thought about using that, in this case... | https://dsp.stackexchange.com/questions/63194/fourier-transform-of-sum-n-infty-infty-1n-deltat-nt-0 |
Question: <p>In the context of speech recognition (recognizing individual speech sounds), the pitch of a certain person can change at different times. </p>
<p>Excerpt from Statistical Signal Processing by Steven Kay: </p>
<blockquote>
<p>This is a natural variability due to the nature of human speech. The
spectr... | https://dsp.stackexchange.com/questions/63517/why-does-the-spectral-envelope-of-human-speech-not-change-w-r-t-pitch-when-taki |
Question: <p>In Gonzalez book Digital Image Processing, section 4.34 (third edition), he writes:</p>
<blockquote>
<p>Unfortunately, except for some special cases mentioned blow, aliasing
is always present in sampled signals because, even if the original
sampled function is band-limited, infinite frequency components ar... | https://dsp.stackexchange.com/questions/67554/anti-aliasing-and-the-fourier-transform-gonzalez-digital-image-processing |
Question: <p>Why is the fourier transform of impulse train a impulse train? Is there a intuitive reason behind it?</p>
Answer: <p>Intuition can sometimes be misleading. But here are some ideas that might help one move towards creating a mental picture.</p>
<p>An infinitely long pure sinewave in the time domain (cons... | https://dsp.stackexchange.com/questions/34146/fourier-transform-of-impulse-train |
Question: <p>Can a Fourier transform in space be interpreted as the integral of a parameter multiplied by an homogeneous wave <span class="math-container">$\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^{-... | https://dsp.stackexchange.com/questions/73087/fourier-transform-as-the-integral-of-a-parameter-multiplied-by-an-homogeneous-wa |
Question: <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 \o... | https://dsp.stackexchange.com/questions/3369/techniques-to-deriving-dtfts |
Question: <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... | https://dsp.stackexchange.com/questions/14717/getting-bpm-of-song-with-fft |
Question: <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/questio... | https://dsp.stackexchange.com/questions/15333/sine-of-frequency-0-contains-sines-of-all-frequencies-at-once-in-it |
Question: <p>Seems both will produce another step. there is no difference? Thanks</p>
Answer: <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)$ ... | https://dsp.stackexchange.com/questions/20418/what-is-the-difference-between-multiplying-a-delta-and-a-step-versus-convolving |
Question: <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.eusip... | https://dsp.stackexchange.com/questions/68291/gsp-as-an-extenstion-of-dsp |
Question: <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 belo... | https://dsp.stackexchange.com/questions/74937/how-to-get-a-non-equally-spaced-fft-back-into-the-time-domain |
Question: <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 tran... | https://dsp.stackexchange.com/questions/51085/what-frequencies-are-present-in-the-fourier-transform-of-the-dirac-impulse |
Question: <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">... | https://dsp.stackexchange.com/questions/76059/solving-fouriertransform-exercises-without-explicitly-doing-the-transform |
Question: <p>Consider the discrete-time system
<span class="math-container">$$
H(z) = 1 + z^{-1} + z^{-2} + z^{-3}
$$</span>
To obtain the magnitude of the discrete-time Fourier transform, I substitute <span class="math-container">$z = e^{j\omega}$</span> to get
<span class="math-container">\begin{align}
H(\omega) &... | https://dsp.stackexchange.com/questions/79374/can-the-magnitude-of-a-discrete-time-fourier-transform-be-negative |
Question: <p><span class="math-container">$$ \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}\... | https://dsp.stackexchange.com/questions/79568/where-did-i-make-the-mistake-in-the-fourier-transform |
Question: <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 clas... | https://dsp.stackexchange.com/questions/80888/fourier-transform-of-the-signum-function-using-the-integral-property |
Question: <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... | https://dsp.stackexchange.com/questions/26221/positive-or-negative-sign-on-fourier-transform-formula |
Question: <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 ... | https://dsp.stackexchange.com/questions/81900/approximation-of-a-complex-valued-function-of-real-variable |
Question: <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... | https://dsp.stackexchange.com/questions/82269/is-yt-xt2-non-linear-and-time-invariant-system |
Question: <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>.... | https://dsp.stackexchange.com/questions/72894/spectrally-flat-binary-sequence |
Question: <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 $</span> an... | https://dsp.stackexchange.com/questions/79526/can-you-quickly-find-the-inverse-fourier-transform-using-the-duality-property |
Question: <p>I trying to recover the Fourier Transform of a flipped signal directly from the Fourier transform of the original signal.</p>
<p>More precisly, let <code>s</code> be a random signal:</p>
<pre><code>s = np.random.randn(n)
</code></pre>
<p>Let <code>s1_fft</code> and <code>s2_fft</code> the Fourier Transform... | https://dsp.stackexchange.com/questions/82725/recover-fourier-transform-of-flipped-signal-from-the-fft-of-orignal-signal |
Question: <p>I am currently working on simulating RF transmissions for beamforming and other applications in Matlab.</p>
<p>One of the fundamental properties that I need to simulate is signal propagation delay due to transmission distance. This can either be done by generating the signal <span class="math-container">$s... | https://dsp.stackexchange.com/questions/83217/interpretation-of-complex-time-domain-signal-resulting-from-time-shift-property |
Question: <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 i... | https://dsp.stackexchange.com/questions/9050/what-is-the-meaning-of-ta-k-of-fourier-series-or-transform |
Question: <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>
Answer: <p>Since $$f(\mathbf{z})=\int_{\mathbf{R}^n}\hat{f}(\mathbf{u})e^{2\pi ... | https://dsp.stackexchange.com/questions/9124/calculate-maximum-of-filter-kernel |
Question: <p>I have a doubt related to calculating the Discrete Time Fourier Transform (DTFT) by hand. Specifically in how calculate the frequency axis of the spectrum. My signal has N values and was sampled at FS Hz, the spectrum would have N entries too (where N/2 values are a mirror of the other half). The maximum r... | https://dsp.stackexchange.com/questions/13930/frequency-axis-problem-in-a-dtft |
Question: <p>I have an assignment where I'm given the DFT of a sequence $x[n]$ as $X[k]=\{4,3,2,1,0,1,2,3\}$ and also
$$y[n] = \left\{ \begin{array}[cc]
xx[n/2] & \text{if n is even} \\
0 & \text{otherwise}
\end{array} \right. $$</p>
<p>and I'm supposed to find and sketch the DFT of $y[n]$.</p>
<p>So $y[n] = ... | https://dsp.stackexchange.com/questions/18461/discrete-fourier-transform-by-hand |
Question: <p>In the case of a constant delay <span class="math-container">$\tau$</span>, we have the following equality:</p>
<p><span class="math-container">$$\begin{align}\mathcal{F^{-1}}\left\{e^{-j\omega \tau}\right\}=\delta(t-\tau)\end{align}$$</span></p>
<p>If the delay is frequency dependent <span class="math-con... | https://dsp.stackexchange.com/questions/74642/inverse-fourier-transform-of-complex-exponential-with-frequency-dependent-shift |
Question: <p>My question was, in an uncountably infinite-dimensional vector spaces, how to represent a vector by a list of parameters, as we do in finite-dimensional spaces? I was assuming that if we can not express a vector as a list of discrete parameters, we have a big issue...but during the writting up of this ques... | https://dsp.stackexchange.com/questions/43154/vector-parameters-in-uncountably-infinite-dimensional-spaces |
Question: <p>We know that in quantum mechanics, momentum space is the fourier transform of position space (and vice versa)</p>
<p>And also, in time-series analysis, that frequency (of cycles) is the fourier transform of the distribution of all cycle lengths.</p>
<p>What about electromagnetic radiation? Is the distrib... | https://dsp.stackexchange.com/questions/157/in-fourier-transforms-can-momentum-space-be-analogized-to-frequency-and-positi |
Question: <p>By definition of Fourier transform</p>
<p>$$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 ... | https://dsp.stackexchange.com/questions/6282/what-happens-if-we-change-the-limits-of-integral-in-fourier-transform |
Question: <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 wil... | https://dsp.stackexchange.com/questions/17675/unit-impulse-funciton-ft |
Question: <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. Ratio... | https://dsp.stackexchange.com/questions/27165/multi-time-window-fft |
Question: <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, ta... | https://dsp.stackexchange.com/questions/34424/question-about-ramp-filter-used-in-filtered-backprojection |
Question: <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 ... | https://dsp.stackexchange.com/questions/41870/how-to-apply-an-fft |
Question: <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 ... | https://dsp.stackexchange.com/questions/41988/understanding-the-meaning-of-amplitude-in-fft |
Question: <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>
Answer: <p>Each frequency in the FT of a time shifted waveform is rotated in p... | https://dsp.stackexchange.com/questions/17009/what-happens-with-signal-in-frequency-spectrum-when-it-is-time-shifted-in-time-s |
Question: <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 o... | https://dsp.stackexchange.com/questions/26406/why-is-signum-function-used-to-calculate-fourier-transform-of-unit-step-function |
Question: <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 $$</p>
<p>its Fourier transform is existed. </p>
<p>Also, the Fourier transform of $\cos(\omega_0 t)$ is $\pi(\d... | https://dsp.stackexchange.com/questions/33785/how-the-fourier-transform-of-a-cosine-signal-is-existed |
Question: <p>Let the Fourier Transform of a real signal, $f(t)$, be $\mathcal{F}(\omega)$. And the FT of the absolute value of the same signal, $|f(t)|$, be $\mathcal{F}(u)$. </p>
<p>Can $\mathcal{F}(w)$ be recovered from $\mathcal{F}(u)$?</p>
<p>For instance, the FT of $a \cdot \cos(ft)$ returns a spectrum in which ... | https://dsp.stackexchange.com/questions/34782/how-to-recover-ft-from-fourier-transform-of-its-absolute-value-mathcalf |
Question: <p>Suppose we have the complex signal $x(t)= \exp(j\omega_0 t)$. Using the properties of Fourier transform we can prove its CTFT is Dirac $\delta$ function. </p>
<p>If any one ask me about the spectrum of $x(t)$ "Does $x(t)$ has continuous spectrum or discrete spectrum", my answer will be "The spectrum of $x... | https://dsp.stackexchange.com/questions/37548/spectrum-of-windowed-version-of-original-continuous-signal |
Question: <p>How can I find the Fourier transform of</p>
<p>$$ 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... | https://dsp.stackexchange.com/questions/6038/fourier-transform-of-cosine-to-the-power-of-3 |
Question: <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>
Answer: <p>Here's an intuitive explanation if the convolution theorem is taken for granted:</p>
<p>Since... | https://dsp.stackexchange.com/questions/17060/why-does-the-periodic-signal-in-time-always-give-a-discrete-frequency-spectrum |
Question: <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 sig... | https://dsp.stackexchange.com/questions/22076/filtering-and-fourier-transforming-does-the-order-matter |
Question: <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 - i... | https://dsp.stackexchange.com/questions/31285/repeated-fourier-transform-what-happens |
Question: <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 othe... | https://dsp.stackexchange.com/questions/19469/instantaneous-frequency-vs-fourier-frequency |
Question: <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 abou... | https://dsp.stackexchange.com/questions/52406/fourier-transform-negative-amplitude-meaning |
Question: <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 ... | https://dsp.stackexchange.com/questions/62491/fourier-transform-is-an-isomorphism-but-we-don-t-get-when-each-frequency-appea |
Question: <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)$$</span></p>
<p>Now we know while Dirichlet conditions... | https://dsp.stackexchange.com/questions/56388/applying-duality-property-to-fourier-transform-of-unit-step-function |
Question: <p>I want to take the Fourier transform of the following transient signal,
<span class="math-container">$$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 wo... | https://dsp.stackexchange.com/questions/59160/fourier-transform-of-a-damped-cosine-wave-with-a-linear-frequency-chirp |
Question: <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... | https://dsp.stackexchange.com/questions/61903/fourier-transform-of-discrete-time-unit-step-function |
Question: <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 del... | https://dsp.stackexchange.com/questions/82329/time-shift-and-phase-examples |
Question: <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>
Answer: <p>there are issues. given this convention for the continuous Fourier transform (and inverse)</p>
<p><span c... | https://dsp.stackexchange.com/questions/59468/complex-numbers-and-fourier-transform |
Question: <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 ... | https://dsp.stackexchange.com/questions/70813/windowing-function-for-inverse-fourier-transform |
Question: <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>
Answer: <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 ... | https://dsp.stackexchange.com/questions/74933/fft-of-a-stretched-vector |
Question: <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 trans... | https://dsp.stackexchange.com/questions/54855/inverse-laplace-transform-of-two-sided-and-one-sided-laplace-transform |
Question: <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$<... | https://dsp.stackexchange.com/questions/26146/is-the-laplace-transform-redundant |
Question: <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 fin... | https://dsp.stackexchange.com/questions/27369/finding-laplace-transform-without-roc |
Question: <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... | https://dsp.stackexchange.com/questions/56171/from-fourier-transform-to-laplace-transform |
Question: <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 ... | https://dsp.stackexchange.com/questions/61733/why-the-unilateral-laplace-transform |
Question: <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 2... | https://dsp.stackexchange.com/questions/93330/laplace-transform-and-inverse-laplace-transform-for-2d-images-python-code-availa |
Question: <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 im... | https://dsp.stackexchange.com/questions/82749/laplace-transform-of-derivative |
Question: <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:... | https://dsp.stackexchange.com/questions/36106/laplace-transform-of-averaging-operator |
Question: <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... | https://dsp.stackexchange.com/questions/27179/confusion-in-basics-of-laplace-transform |
Question: <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 ... | https://dsp.stackexchange.com/questions/11008/intuitive-interpretation-of-laplace-transform |
Question: <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... | https://dsp.stackexchange.com/questions/58413/basic-difference-between-fourier-transform-and-laplace-transform |
Question: <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 eith... | https://dsp.stackexchange.com/questions/27189/questions-related-to-laplace-transform |
Question: <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://elect... | https://dsp.stackexchange.com/questions/79569/confusions-regarding-differences-between-fourier-transform-laplace-transform |
Question: <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)
resu... | https://dsp.stackexchange.com/questions/75543/laplace-transform-plot-isnt-right |
Question: <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'... | https://dsp.stackexchange.com/questions/74093/unilateral-laplace-transforms-differentiation-property |
Question: <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)
$$</p>
<p>where G(s), H(s) are the Laplace transform representations of g and h, and G(z) and H(z) are the Z-transform representation of g a... | https://dsp.stackexchange.com/questions/14483/conversion-from-laplace-transform-to-z-transform |
Question: <p>In book signals and systems 2 edition a question is given which is as follows:</p>
<p><span class="math-container">$$ x(t)=e^{-3(t+1)}u(t+1) $$</span></p>
<p>and we are asked to find the unilateral Laplace Transform of the signal. The method that is given in the solution manual is as follows:</... | https://dsp.stackexchange.com/questions/54400/confused-about-time-shifting-property-of-laplace-transform |
Question: <blockquote>
<p>Use the complex inversion formula to calculate the inverse Laplace transform $f(t)$ of the following Laplace transform:
$$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... | https://dsp.stackexchange.com/questions/30701/inverse-laplace-transform-using-inversion-formula |
Question: <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 al... | https://dsp.stackexchange.com/questions/83278/confusion-regarding-laplace-transform-calculation-in-matlab |
Question: <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>
... | https://dsp.stackexchange.com/questions/27288/confusion-in-proof-of-inverse-laplace-transform |
Question: <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>
Answer: <p>In typical (or all) applications, ... | https://dsp.stackexchange.com/questions/56821/time-setting-of-z-and-laplace-transforms |
Question: <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/co... | https://dsp.stackexchange.com/questions/66428/how-to-compute-laplace-transform-in-python |
Question: <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\}$$</span></p>... | https://dsp.stackexchange.com/questions/94361/y0-terms-in-the-laplace-transform |
Question: <p>There are several functions for which we know that Fourier Transform will exist but still we calculate its Laplace Transform. Can I know the reason why we need to take Laplace transform for which we know its convergence?</p>
<p>Thanks</p>
Answer: <p>There is a large class of functions for which both the ... | https://dsp.stackexchange.com/questions/27230/why-we-take-laplace-transform-of-functions-which-converged-using-fourier-transfo |
Question: <ol>
<li><p>Taking the Laplace transform of a system given by a differential equation yields its transfer function <span class="math-container">$H(s)$</span>. The region of convergence of the causal impulse response of the system lies right of the most right pole in the complex plane. Suppose the system is st... | https://dsp.stackexchange.com/questions/95015/how-is-causality-in-laplace-transform-related-to-fourier-transform |
Question: <p>I'm confused in solving linear constant coefficients differential equations (LCCDEs) by Laplace transform if initial conditions are given at time</p>
<ol>
<li>just before <span class="math-container">$t=0$</span></li>
<li>just after <span class="math-container">$t=0$</span></li>
<li>exactly at <span class=... | https://dsp.stackexchange.com/questions/69667/confusion-in-initial-condition-of-differential-equation-using-laplace-transform |
Question: <p>I was reading from Athanosios Papoulis' "The Fourier integral and its applications." and they referenced the bilateral Laplace transform and Fourier Transform as:</p>
<p>$$F(p)=\int_{-\infty}^{\infty}e^{-pt}f(t)dt$$
$$F(\omega)=\int_{-\infty}^{\infty}e^{-j\omega t}f(t)dt$$</p>
<p>and stability indicates ... | https://dsp.stackexchange.com/questions/50462/bilateral-laplace-transform-and-existence-of-fourier-transform |
Question: <p>I am dealing with a physics problem which is related to signal processing. The problem requires me to calculate the instantaneous force acting on a body which depends on some physical parameter $x$. Assume that $x(t)$ is periodic in time for the moment. Since $x(t)$ is periodic, then it can be expanded as ... | https://dsp.stackexchange.com/questions/40201/causal-signal-fourier-transform-or-laplace-transform |
Question: <p>What are the advantages of Laplace Transform vs Fourier Transform in signal theory?</p>
Answer: <p>Laplace transforms can capture the transient behaviors of systems. Fourier transforms only capture the steady state behavior. Of course, Laplace transforms also require you to think in complex frequency sp... | https://dsp.stackexchange.com/questions/45910/what-are-the-advantages-of-laplace-transform-vs-fourier-transform-in-signal-theo |
Question: <p>Laplace transform for continuous signal $x(t)$ is given by</p>
<p>$$
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)$$</p>
<p>I can say that only ... | https://dsp.stackexchange.com/questions/31384/what-are-the-advantages-and-disadvantages-of-laplace-transform-over-z-transform |
Question: <p>I have found a problem in applying Laplace Transform to $-e^{-at}u(-t)$
I am doing these steps:</p>
<p>$$ = - \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}$$
$$ ... | https://dsp.stackexchange.com/questions/27287/laplace-transform-of-e-atu-t |
Question: <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><sp... | https://dsp.stackexchange.com/questions/71611/laplace-transform-zeros-and-corresponding-impulse-response-ht |
Question: <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><sp... | https://dsp.stackexchange.com/questions/71560/laplace-transform-integral-vs-poles-zeros |
Question: <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>
Answer: <p>T... | https://dsp.stackexchange.com/questions/60655/what-is-the-inverse-laplace-transform-of-squared-denominator-term |
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