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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Fast Hartley Transform Implementation in MATLAB

I want to implement Fast Hartley Transform (Specifically Discrete Hartley Transform) in a script file in MATLAB. Does anyone know have a reference implementation of this in MATLAB or another language ...
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559 views

Instantaneous frequency vs fourier frequency [closed]

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 ...
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FFT for a single frequency

I was looking for a more efficient way of finding the magnitude and phase of a signal at a certain frequency without performing an FFT because it produces more information than I need and I came ...
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Online DFT Algorithm

I have a discrete audio stream $x$ that needs to be processed in real-time. Specifically, as the each new sample is received, I would like to compute a Fourier transform of the last $n$ samples of the ...
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398 views

Cross-correlation, sharp peak at 0?

First of all, I have to stress that I am not a professional of coding, no more than a professional of signal processing. I am a chemist that happen to be working on a project involving both. So in ...
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217 views

Fourier transform 4 times = original function (2D and higher)

The Signal Processing SE post linked below shows how the Fourier Transform applied 4 times to a 1D function returns the original function, i.e. F{ F{ F{ F{ g(x) } } } } = g(x) Link to 1D case: ...
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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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Expression to represent alternating pulse train?

I have a question regarding an alternating pulse train signal, $p(t)$, and a system which looks like this: Immediately, I'm supposed to be able to extrapolate that $$ P_t = P_1(t)-P_1(t-\Delta) $$ ...
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FFT method input argument have to be $2^n$ ?

Does FFT method input argument have to be power of 2, i.e, $2^n$ I just realized there are many algorithm for FFT implementation,...
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Fourier transform with periodicity at the harmonic frequency

Let's suppose I have a signal $F(t)$ that is periodic, with two periodicities $P1$ and $P2$, with $P1>P2$ Suppose that I know the values of the two periodicities. Using the Fast Fourier transform ...
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IDFT of $Y[k]=2X[k]$ for even $k$

If the 16-point DFTs of $x[n]$ and $y[n]$ are given as $Y[k]=\begin{cases}2X[k], & k=0,2,4,...,14 \\ 0, & k=1,3,5,...,15\end{cases}$, where $x[n],y[n]=0, \forall n<0, n>15$, how can I ...
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Significance of modular arithmetic in DFT?

In what ways does modular arithmetic plays a part in DFT? Why is it a so integral part of DFT?
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486 views

What is the role of complex exponential?

What is the role of complex exponential $ e^{jθ} $ in Fourier Transform? Is it different in the continuous and in discrete time domain?
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Fourier Transform of Morlet wavelet Function?

As you know the Morlet wavelet function is given by: $$\frac{1}{\sqrt{\pi f_b}}e^{\frac{-t^2}{f_b}}e^{j2\pi f_c}$$ The Fourier transform of this equation is: $e^{-\pi^2 f_b(f-f_c)^2}$ (is it right)? ...
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Can I study continuous time Fourier Transform and treat the rest as special cases

Say I learned the theoretical result of continuous time Fourier transform. And I want to extends some results(say "convolution rule") to Lapace transform, Z transform, DTFT, DFT, Fourier sequence ...
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568 views

Fourier Transform of Kernel Density Estimation - Convolution Theorem?

I am reading this paper about density estimation (Appendix A), where the authors apply a Fourier transform to the estimated probability density (the $X_j$ are a sample of $N$ data points drawn ...
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523 views

How to choose a phase for the deconvolution of an autocorrelation?

Say I have a function, $C=C\left(x\right)$, whose fourier transform is denoted by $c=c\left(k\right)$, i.e. $C\left(x\right)=\sum_{k=-\infty}^{\infty}c\left(k\right)\chi\left(x\right)$, where $\chi\...
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What is the time-integration property in the Fourier series analysis?

In the continuous Fourier series properties for a periodic continuous-time signal, we have time-integration property: $$ \int_{-\infty}^t x(\alpha)d\alpha \leftrightarrow \frac{a_k}{jk\omega_0} $$ ...
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What does the exponential term in the Fourier transform mean?

We know that Fourier transform $F(\omega)$ of function $f(t)$ is summation from $-\infty$ to $+\infty$ product of $f(t)$ and $e^{-j \omega t}$: $$ F(\omega) = \int\limits_{-\infty}^{+\infty} f(t) \ e^...
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122 views

FFT - second and further divides and conquers - need help

​ ​Hello, I would like to ask you for help in understanding Fast Fourier Transform. Most articles about FFT describe a simple DFT example with N=8 number of samples. They divide it on half, to evens ...
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Why is a circular mask appropriate for Fourier filtering rectangular images?

Suppose I apply 2D DFT to an image with dimensions $H{\times}W$ where $H \neq W$, then shift the DC component to the center. Why does a circular mask capture the lowest frequency components, i.e. why ...
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Implementing rotation in frequency domain and map it back to spatial domain

Please consider the following small example: ...
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182 views

Hilbert transform from analytic signal

Show that the Hilbert transform of $h(t) = m(t) \cos(2 \pi \nu_c t)$ is $$\hat{h} (t) = m(t) \sin(2 \pi \nu_c t),$$ where $m(t)$ is a real valued, band-limited function (i.e. we have Fourier ...
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How do I interpret the result of a Fourier Transform?

For example, I entered the following "equation" into Wolfram|Alpha: FourierTransform[Piecewise[{{sin[t],t > 0 and t < 2*pi}}, 0], t, \[Omega]] So as to ...
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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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Create an aliased image in FD

Is there an accurate way to create an aliased image from the Fourier transform of the original image? in other words, i have the Fourier coefficients of an image and i want to make down-sample in ...
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How to interpret output of matched filter with complex input?

I have implemented a matched filter based on the Fourier Transform approach. In the real numbers domain that means that I use as the coefficients of my filter (B) the inverted time-samples of the ...
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186 views

Properties of Spectral Transformations - Allocation (decomposition into even and odd part)

I am trying to understand the Allocation property of Spectral Transformations. I can't. I know that every function can be separated into an even part and into an odd part. My problem is ...
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Discrete and Continuous Signals

I am new to the signals area and have been reading through a lot and have some questions. Suppose I have a saved audio file on my computer, .wav file. I can view the time domain of the signal by ...
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883 views

How to Match 2 Signals which Are a Shifted and Scaled Version of Each Other

I have 2 signals S1 and S2 that contain the same information, but S2 is shifted and scaled compared to S1 by an unknown amount (but small; eg shift would be of the order of 1-10 samples). What is the ...
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360 views

Fourier transform of certain noisy function

So, I have a noisy signal in time domain, $f(t) = t \eta(t)$ where $\eta(t)$ is white noise with variance $\sigma$ and mean zero, and that it has the property $\langle \eta(t)\eta(t') \rangle = \sigma^...
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What is the correct solution for Fourier transform of unit step signal?

The unit step signal defined as $$ u[n]= \lbrace 1; n>=0; \\ \qquad0; n<0 \rbrace $$ has three possible solutions for its Fourier domain representation depending on the type of ...
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Hilbert transform pair proof

I am looking for the proof that the Hilbert transform of $\displaystyle\frac{\sin(at)}{at}$ is given by $$\frac{\sin^2(at/2)}{at/2}.$$ How do we prove this? This is a $\operatorname{sinc}(at)$ ...
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679 views

Why are edges in spatial images represented as edges in their Fourier transform image?

Here is a well-known image and its Fourier Transform (magnitude). If I understand correctly the theory behind the FFT, each pixel in the FFT image represents a certain 2D sine wave with frequency ...
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444 views

Why is the second half of the FFT negative frequencies

There are lots of questions here about what the negative frequencies in fft mean, but I'm confused on why the second half of the fft calculation is the correct calculation for a negative frequency. ...
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Prove the dirac delta contains all frequencies

I'm looking for a mathematical proof that the dirac delta contains all frequencies. I just read in a text book that the frequency spectrum of a dirac is just a horizontal line of amplitude 1, whereas ...
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304 views

Which Approach Is Better for Decomposing an Image into High Frequency and Low Frequency Components?

Which approach is better or there is mathematical justification for using Bilater filter and Fourier Transform to decompose a image into High Frequency and Low Frequency Component. Both Bilateral ...
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Phase difference measurement of a signal sampled with two different sampling frequencies

I am working on phase interferometry for locating a transmitter. The direction of arrival of an incident wave can be estimated from the phase difference caused by the antenna separation as shown ...
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Autocorrelation to diagnose faults

I'm attending a very practical course on signals and i have some doubts, i hope to receive answers in layman terms. 1) My prof said i can use the autocorrelation of the output of a process to ...
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Why doesn't this complex multiplication in the frequency domain produce my expected phase shift?

I know how to change the phase of a complex number by multiplying by $\cos \theta + i \sin \theta$. And I understand that the phase of a sine wave is reflected in its Fourier transform. So, I am ...
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Simple FFT filtering vs. e.g. butterworth filtering

I am currently working on functional connectivity analysis of EEG, and need to bandpass filter my data into different frequency bands (Delta, Theta, Alpha and Beta). An important thing is that the ...
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161 views

One sided frequency spectrum (Matlab vs. Origin)

There are a lot of queries on fft frequency all over the web. I guess the following point not discussed anywhere explicitly. Hope someone can provide an insight here. If we have and even number of ...
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Is there a closed form expression for main-lobe width increase given a window?

We know that when we window a signal, we increase the main-lobe width. Let 'main-lobe-width' here be the null-to-null bandwidth of the main lobe. Let us further more say that the main-lobe width of a ...
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470 views

Why are the basis functions for DFT so?

When you get a DFT of a signal, you use the basis functions as: $e^{-j2\pi kn/N}$ Why is it so? Why don't we use the conjugate, $e^{j2\pi kn/N}$, or any other function?
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Can I use Fourier transforms instead of Laplace transforms (analyzing RC circuit)?

I don't study electrical engineering or something related but I was assigned a problem on transfer functions, impulse responses, and in general, everything related to this post. (Specifically, I'm ...
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How do I understand Fourier descriptors more visually and intuitively?

I read the book Image Processing, Vision and Machine Vision and find the concept Fourier descriptors hard to understand, although literally its derivation is somewhat reasonable. Can anyone give me a ...
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I have a pressure signal and want to do SPL analysis on it

Signal I have an acoustic signal from a Ffowcs Williams Hawkings CFD analysis and would like to convert it to the frequency domain and see the SPL and OASPL. I know I need to use fft() but I am ...
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Why look at power spectral density for stochastic processes?

I have been told that for deterministic signals, it makes sense to look at their respective Fourier transforms/spectra. For stochastic processes on the other hand, I am supposed to work with power ...
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convergence of Fourier transform of $e^{-t}\sin(2\pi ft)u(t)$

As you see Fourier transform function is being divergent for the first statement but it seems to converge. What is my fault? $$ \begin{align} \int\limits_{-\infty }^{+\infty }{{{e}^{-t}}\sin(2\pi ...