Questions tagged [fourier-transform]

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum.

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31 views

Frequency Domain Distribution

I have a complex signal in the time domain normally distributed. What will be its distribution in the frequency domain? I assumed since the frequency domain is a linear transformation the distribution ...
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Does Short-time Fourier transform impact quality of signal?

I am playing with spleeter to separate voice from an audio signal. They go to the frequency domain by computing the Short-time Fourier transform to create a spectrogram. As there are some ...
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2answers
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Meaning of negative frequencies in Baseband non sinusoidal, non periodic signal

I can understand the meaning of negative frequencies in a sine or cosine signal, since by using Euler's identities, you have two complex phasors moving to different directions, which when added, give ...
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169 views

Fourier transform of the magnitude of the fourier transformed signal

I've come to realize that the Fourier transform of an already Fourier transformed signal gives the time-reversal signal. $$\mathcal F(\mathcal F(x(t)))=x(−t)$$ ref 1,ref 2 However, my question is, if ...
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1answer
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Behaviour of integrator at steady state

I wanted to calculate response of integrator of sinusoidal input at steady state via these two methods as mention in image but these two methods give two different answers at steady state, so where ...
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RMS of signal vs average amplitude

I am trying to estimate the average amplitude of some signal with frequency 6 Hz, sampled at ~300 Hz. See figures for a part of the signal and its dft calculated using matlab. I estimate the average ...
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2answers
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Frequencies of complex exponentials in Discrete Fourier Transform

What I understand is that using DFT, we are representing a given discrete signal using a basis of complex exponentials of different harmonic frequencies. If I am taking 16 point DFT of a signal ...
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2answers
19 views

MRI K-space to image: why track frequencies in two dimensions?

While trying to better understand how an MRI goes from k-space to an image, I came across this wonderful website that explains how you would represent an image as a collection of rows of pixels where ...
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Names of system functions in frequency domain

I was just trying to refresh my systems theory known from long ago, and I realized that I had forgetten the name of the basic functions. Specifically, what are the names of these functions ...
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Showing error energy goes to zero

Let $$\hat{x}[k] = \frac{1}{2\pi}\int_{-W}^{W}X(e^{j\omega})e^{j\omega k}d\omega,\label{ift}\tag1$$ where $$X(e^{j\omega}) = \sum_{n=-\infty}^{+\infty} x[n]e^{-j\omega n}\label{dft}\tag2$$ Also, $$d[k]...
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1answer
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Steady state variance of a stochastic differential equation - relation between the frequency and time domains

Consider a stochastic differential equation: $$ dx(t) = a x(t)dt + b y(t)dt \quad (1) $$ where $y(t)$ is a stochastic process satisfying $\langle y(t)y(t')\rangle = \delta(t-t')$. We will assume that ...
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Conjugate symmetry of the DFT of real-valued sequences

I have read about Fourier transformation that real signals are "mirrored" in the real and negative halves of the Fourier transform because of the nature of the Fourier transform. For ...
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Understanding index transformation in derivation of Fourier transform for sampling rate reduction

Was going over some notes regarding deriving fourier transform equation for Sampling Rate Reduction. Reference to Notes from below link https://ocw.mit.edu/courses/electrical-engineering-and-computer-...
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1answer
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Auto Tune Shifting

So i have an algorithm that gets the fundamental frequency of windows, and can shift the pitch of each window as well (by a factor of x). For example. if x is 2 and the f0 is 440, the resulting f0 ...
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1answer
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Spectral analysis of a cross correlation function [duplicate]

So it's widely known that the Cross Correlation of 2 signals helps us in figuring out the time delay in those signals by analyzing the peak of the correlation coefficient in the time domain. For ...
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Fourier transform of an integrator filter

I have to find the Fourier transform , and $y(t)$ of an $ x(t) = e^{- \frac {t}{T} } u(t) $ that passes into a integrator filter. I know that $ Y(f) = X(f) H(f) $ so I first calculate the Fourier ...
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Does it make sense to apply convolution in the frequency dimension of a fourier transformed signal?

What I have is more likely a theoretical question. Since I am not from a signal processing background it is hard for me to grasp the issue in using convolution in the frequency dimension of a Fourier ...
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2answers
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Duality Property for DFT

I was watching a youtube video for the duality property for continuous time Fourier transforms, which shows that if Fourier transform of $x(t)$ is $X(\omega)$ then Fourier transform of $X(t)$ is $2\pi ...
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What is the gradient of fft?

I have a time-series of length N generated by the following equation: ...
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1answer
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Fourier Transform of a PSD and response of a PSD input

Does a Fourier Transform of a white noise exist? If so, what is its general form? It is possible to compute the fourier transform of a Power Spectral Density? My problem in detail. I have to compute ...
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How to interpret the window size and FFT size while performing FFT? [closed]

I have a discrete signal of length 98,000 samples and I am supposed to a FFT of the discrete signal with certain window size and FFT size. Can someone help me explain what is meant by the window and ...
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2answers
62 views

Proof of fourier transformation of multiplication of two signals

I've been trying to find a proof of the following, but still I m unable to proof it, can someone help me? $$ ℱ[x(t)g(t)] = \frac{1}{2\pi} [X(\omega)*G(\omega)] $$
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Understanding zero-phase filters

I'm reading the book "Digital Image Processing" by Gonzalez and Woods and I'm wondering how their definition of zero-phase-shift filter is equivalent to the one given here. "Digital Image Processing" ...
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1answer
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DFT symmetry vs DFT duality in Richard Lyons' “Understanding DSP”

I am reading Richard lyons, understanding dsp, chap 3. Article 3.2 is about property of dft symmetry but any where in this chapter, i am unable to find discussion about dft duality property I want ...
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Find a LTI system such that $\mathcal{T}\{\frac{\sin t}{t}\} = \frac{\sin 2t}{t}$

Let $$x(t) = \frac{\sin t}{t} \qquad\text{and}\qquad y(t) = \frac{\sin 2t}{t}$$ Is it possible to find a LTI system such that $\mathcal{T}\{x(t)\} = y(t)$? If not, what's the reason for ...
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Zero-padding or Interpolation in 3D FFT

I'm trying to perform a FFT of a 3D regular grid and then compute the bin average (in spherical shell bins) of the Fourier transformed grid. The problem is that the resulted vector is very noisy as I'...
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Calculating Transfer function numerator and denominator from the rationalfit model

I have a frequency response data called 'AC_data' which is a vector of complex numbers (real and imaginary part) at different frequency points. I have calculated a rationalfit model for the AC_data ...
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Wavelets vs Fourier Transforms

It's supposed that computing a spectrogram for a signal is faster using wavelets. However, wavelets need to be applied to the signal for every time step and frequency, giving the implementation a time ...
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2answers
75 views

Impulse response of a time scaling system

Assume a bandlimited signal $X(t)$. Given that the output for this signal is $X(t/2)$, what will be the impulse response $h(t)$ of such a system? \begin{array}{l} X( \omega ) \ =\ \int ^{\infty }_{-\...
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3answers
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What is the DFT of a binomial filter?

I hope this question is not too simple, I just started learning digital image processing. The 1D binomial filter of size 2 is defined by $B_2 = \frac{1}{4}\begin{bmatrix}1 & 2 & 1\end{bmatrix}$...
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1answer
48 views

Fourier transform of unit step

I was reading pdf by caltech and in one of its section, Fourier transform of Unit step signal is calculated but I am confused, how this can be possible if region of convergence for Laplace transform ($...
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2answers
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Why can we have non-integer frequency bins in FFT?

I am studying DFT/FFT and I'm very confused about one thing. I read online that the frequencies we can sample with DFT must be integer (Why does the frequency in the DFT have to be an integer?). Later ...
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1answer
68 views

Discrete-time Fourier transform of $\frac{\cos(\frac{n\pi} 6)}{(n+3)\pi}$

Find the Discrete-time Fourier transform of $\frac{\cos(\frac{n\pi} 6)}{(n+3)\pi}$ I thought of making it to be a sinc, but at the bottom there is $n+3$ and if I replace $n+3$ then I don’t know how ...
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finding FL and FH from continuous time signal

I have a continuous signal $x_{a}$ which is defined as $X_{a}(F)=0$ for $|F|>B$. Now if I multiplied the continuous signal $x_{a}$ by $cos6\pi Bt$. Then the fourier transform of the signal can be ...
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1answer
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Confusion regarding DFT calculation?

I am studying Richard lyon chap3 , article "Understanding the DFT equation " But i am bit confused how x(n) is calculated specially x(0) and x(1) because apparently x(n) is calculated by plugging ...
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1answer
56 views

Pitch shifting with bins creating large feedback

I am attempting to create a pitch correction algorithm. I started by performing a test. The test goes as such: Get WAV file Split it into bins of size n (512 in my case) Shift each bin by 2 semitones ...
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How to identify the frequencies of periodic peak signals in a noisy time series?

Suppose to have two time series with peak signals at different frequencies, like these two: ...
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34 views

Construct IFFT from only the major frequencies of an input signal?

I have data which when plotted looks like the green line in the below image. The data is from the walking-motion of a reinforcement-learning-model in a simulation and therefore the values do not ...
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2answers
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DFT without knowing the exact value of sampling rate?

I have a sequence of (real) numbers that represent the magnitude of a certain natural event. I know that the samples are equispaced in time, but not the exact value of the spacings. So does that mean ...
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5answers
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Positive and negative frequencies in DFT due to frequency folding, or due to negatively indexed frequencies?

When I look for the cause of the mirroring of frequencies in DFT output, I get two types of explanations: The first one which says the frequencies are mirrored because of the complex exponential ...
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1answer
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Cepstrum of white gaussian noise

What are the statistics of the cepstrum of gaussian white noise? \begin{align}\newcommand{\Nfft}{ {N_{\mathrm{FFT}} }}\DeclareMathOperator{\FFT}{FFT}\DeclareMathOperator{\IFFT}{IFFT} x_i &\sim \...
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64 point FFT Hamming window interpolated to 720-point sequences

Can someone here explain the meaning of, 64 point FFT Hamming window interpolated to 720-point sequences and its inverse? 720 point is from 0 to $2 \pi$ or (0 to 720 degree). I get the 64 point ...
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1answer
49 views

GSP as an extenstion of DSP

I am a PhD. in pure mathematics. Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-...
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27 views

GSP as an extenstion of DSP

I am a PhD. in pure mathematics. Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-...
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0answers
59 views

Is rotation of a Fourier transform the same thing as Fourier transform of a rotation?

I'm working on an image processing problem and wondering if DFT(rotation(image)) == rotation(DFT(image)) (1). My final goal is to apply rotations in the Fourier domain then do an inverse Fourier ...
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$2\pi$ Periodicity is not working for me for Fourier of Discrete Time Signal

please help me find the error in the following counter example. Consider we take sinus with period of $2\pi$. We sample it many time, and more than 3. We make convolution with rectangle of height 1 ...
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Calculating DCT in reversed vector

I'm doing an exercise in which I need to show that the DCT of $\tilde{x} = (x_{N-1}, x_{N-2}, ..., x_1, x_0) $, with $\tilde x_m = x_{N-m-1}$, is equal to $ \tilde{X}_k = (-1)^{k}X_{k}$, but I have ...
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1answer
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Anti-Aliasing and the Fourier Transform, Gonzalez Digital Image Processing

In Gonzalez book Digital Image Processing, section 4.34 (third edition), he writes: Unfortunately, except for some special cases mentioned blow, aliasing is always present in sampled signals because, ...
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1answer
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time scaling and shifting of cosine in Fourier transform

I've met some problems when calculating the Fourier transform of $\cos(at+b)$. I want to use the shifting and scaling properties to solve this problem. First, when I look up in the book and some ...
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1answer
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Scipy.signal noise with rfft compared to fft

I'm trying to get the fourier transform of a signal with real values, however the results I get with rfft are noiser than those with fft. I wrote the following code: ...

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