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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Circular wrapping of an asymmetric function (in DFT calculations)

Convoluting a signal (using discrete FT) for a given interval [a, b] with a Gaussian can be done by circular wrapping as shown in Numerical Recipes. I found a shortcut for circular wrapping so that ...
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Processes/Transforms involved to get brainwave data from raw EEG? (Autocorrelation confusion)

Not clear on what the autocorrelation function of raw EEG means physically why can't you take the FT of a the EEG itself and get frequency data? With BCI & basic electrode setups you can ...
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RMS in frequency domain using Plancherel theorem

I have an acceleration measurement in x, y and z direction, i.e. three vectors each of length 3000. $x = [x_{1}, x_{2}, x_{3},...,x_{3000}]$ $y = [y_{1}, y_{2}, y_{3},...,y_{3000}]$ $z = [z_{1}, z_{2},...
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Discrete Wavelet Transform Time Series

My problem is to cluster some time series together. But due to a huge length I was interested in using some methods to reduce the dimensionality. I was thinking of Discrete Wavelet Transform since the ...
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How do I calculate the inverse Fourier transform of this function [closed]

How do I compute this integral to find the inverse Fourier transform
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Deconvolution of sidelobes in a point spread function?

It seems that most deconvolution algorithms mainly handle the main lobe of a point spread function (PSF) and assume that sidelobes can be safely neglected. For a direct algorithm trying to perform a ...
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4answers
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What is the correct gain of an RRC Filter?

Breaking my brain all morning with this reading previous questions and googling ... I have made an RRC filter from the equation on wikipedia. It works fine and I compared it to commpy library in ...
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3answers
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Summation limits in DFT

Assume a discrete time signal $(x_n)$ is given. Some texts define the DFT as $$ X[k] = \sum_{n=-N}^N x_n\exp\left(\frac{-2\pi j k n}{N}\right) $$ while others define it as $$X[k] = \sum_{n=0}^{N-1} ...
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Phase rotation without Fourier Transform

Is there any known math equation or other method to perform phase rotation on signal without decomposing it to frequency domain? In frequency domain it is obvious. You just need to perform phase shift ...
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1answer
33 views

Does it matter if the scaling term is in the DFT versus the inverse DFT?

From Wikipedia, the equation for the 1D DFT is From a separate source, the equation for the 2D DFT is Notice how the 2D DFT definition features a scaling term while the first definition does not. Is ...
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Derivation of Inverse Fourier transform from forward Fourier transform

Consider the Fourier pairs: $$\psi(x,t) \stackrel{\mathrm{FT}}{\longleftrightarrow} \Psi(k,t)$$ $$\text{If } \quad \quad\Psi(k,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \psi(x,t) e^{-ikx} \, dx \...
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DTFT Pairs confusion

When I am in the DT Fourier Domain, and I want to come back to the time domain, which pair do I use? Asking because both pairs have the exact same "form" in the Fourier domain, and that is ...
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1answer
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How to accurately compute the Winger-Ville Distribution of an exponential

I am using MATLAB for this question so hopefully you can help me out. I am trying to compute the Wigner-Ville distribution (WVD) of a sinusoidal signal defined as \begin{equation} x(t) = e^{-i\omega_0 ...
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Find corresponding wavenumber from FFT - Python

I have a set of data taken from a high speed camera. I've done some image processing which results in getting a pixel location at each frame. This oscillates with time and so I have performed an FFT ...
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1answer
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How do I find the Energy Density Function of $g(t)$ if i am not given an input or impulse response?

$$g(t)=\frac{12a}{t^2+a^2}$$ I need to find the Energy Density Function of the signal, but everywhere I look has an input and an impulse response. Does anyone know how to solve this. Would I just take ...
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Unit impulse fourier transform confusion

I just saw this question. What is the correct solution for Fourier transform of unit step signal? But i cant understand how the 3d form is equal to the 1st one since de 1st one is purely imaginary and ...
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Fourier transform as the integral of a parameter multiplied by an homogeneous wave

Can a Fourier transform in space be interpreted as the integral of a parameter multiplied by an homogeneous wave $\sigma$? where $\sigma$ is: $\sigma$=$e^{-ikx}$ Are there papers or book that ...
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2answers
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Difficulty with a Fourier Transform

What would be the best way to take the Fourier transform of $$ f(t)\cdot \cos\big(\pi(t-1)\big) $$ I'm aware that when you take the Fourier Transform of $\cos(kt)$ you get two impulse at the location ...
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How do I perform 2D Fourier domain multiplication if the filter mask doesn't match the image size?

Let's say I have an image that is 512 x 512 pixels. I've been tasked with creating two ideal half-band low-pass filters that will filter the image. The first filter is 8 x 8, and the second one is 16 ...
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How to find inverse Fourier transform of summ of delta functions?

I am practicing for my exam that I have this semester and I stumbled upon this one. How can i find inverse Fourier transform given: $$ X(j\omega) = \sum_{k=-\infty}^{\infty}\delta(\omega-2k+1) $$
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how to compute the signal passing from the low pass filter?

I am currently trying to solve this question. Let $x[n]=\cos(\frac{\pi}{2}n)$ and $h[n]=\frac{1}{5}\text{sinc}(\frac{n}{5})$. Compute the convolution $y[n]=x[n]∗h[n],$ and write the value of $y[5].$ ...
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How do you reduce $H\left(e^{j\frac{\pi}{2}}\right)$ further according to a textbook solution

I want to know how I could get from the first line to the second. I've been trying to figure it out for a while with no luck. Thank you in advance! \begin{align} H\left(e^{j0.5\pi}\right) &= \frac{...
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Finding set of orthogonal basis functions for composite signals

I've been given a list of 5 composite signals, where each is composed of 10 sinusoids of different frequencies. For instance, the first composite signal $S_1$ is given by $$ S_1 = \sum_{i=1}^{10} A_i \...
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What is the requirement to reconstruct a spatial domain signal if we sample in the frequency domain?

I've come across an interesting question with regarding to signal reconstruction. The sampling theorem states that a signal in the time or spatial domain must be sampled at twice a rate twice the ...
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How to find Integration of sinc function and then it's infinite summation?

I was doing this problem,i got answers of (a) and (b) by going following way, first i computed F.T. of x(t) as follows which in turn will be used to estimate it's bandwidth which will come as 2Hz and ...
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Continuous vs. Discrete Fourier Transform for representing a physical phenomenon

This question is related to the differences between discrete and continuous Fourier Transform equations. Although I've found some good explanations about the difference in this forum, I cannot see how ...
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Why do we not encode frequency in 3 dimensions in MRI?

If I understand correctly, to select a particular slice (of an object, during an MRI scan) that is orthogonal to the direction of the main magnetic field applied, we apply a magnetic field that varies ...
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Interpreting scipy spectrum from audio signal

I have recorded some audio and would like to obtain a spectrum from it. As the audio is a real valued signal, I figure that I could perform a FFT with some window and obtain a spectrum for that period....
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63 views

Spectrally flat binary sequence

I'm trying to construct a binary sequence of length $2^n$. This sequence will be converted to a square signal of $\pm 1$, where 0 produces $-1$ and 1 produces $1$. I want the resultant signal to be as ...
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Removing frequencies and reconstructing signal

I am trying to implement this section of this paper: https://arxiv.org/abs/2012.15846 The rPPG signal is the spatial average of green value per frame over a video. Head orientation signals are a ...
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The difference between tensorflow stft and scipy stft

I'm trying to wrap my head around stft, and checking the docs, tf.signal.stft and ...
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Why the transfer function is equal to the output in this case

In this description of transfer functions on the z-plane (image linked), I'm confused by equation 1.49, which says that $H(f)=v_{out}(f)$ when $v_{in}(f)= 1 * e^{j 2 \pi (f/f_s)}$. (For another matter ...
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1answer
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The magnitude spectrum of a sharpening filter

I'm trying to derive an expression for the magnitude spectrum of the following sharpening filter. $$ g(m,n) = \delta(m,n)+\lambda (\delta(m,n) - h(m,n)) $$ where $\lambda$ is some positive constant ...
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1answer
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Recovering DTFT from Z-transform

The relationship between the Z-transform and DTFT can be expressed like: $$ H(e^{j \omega}) = H(z)|_{z = e^{j \omega}}$$ Graphically, evaluating the Z-transform on the unit circle is shown as sweeping ...
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Approximate using DFT phase shifting property

I have a discrete signal $x(n)$ having $N$ samples with DFT $X(n)$. Here $N$ is large say $N=600$. Let the samples of $x(n)$ be, $x(n) = \big[x(0), x(1), x(2), ..., x(N-1)\big]$. But if suppose ...
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Issue with demonstrating amplitude modulation of sinusoid using MatLab

I am trying to demonstrate Conventional and DSB-SC modulation of a signal $ \sin(2\pi1000t) $ using MatLab. My carrier signal is $ \sin(2\pi f_ct)$ where $f_c = 100\text{ MHz}$. Now according to ...
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The Fourier transform of sinusoids' products with possible other components

I know that in general it transforms the signal from the time to the frequency field but these specific cases seem pretty demanding. Do I calculate each part separately and then just leave them with ...
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1answer
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Why is the Fourier Series a special case of the Fourier Transform and not the other way around?

I was reading a text book on the frequency domains in signal processing and my understanding is that the Fourier Transform considers signals that are a-periodic in time while the Fourier Series ...
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Spectral decomposition of the DC part of the signal

The title of the question might be misleading, but it shows how clueless I am about this problem. I have a time series of some quantity $X(t)$. I can calculate autocorrelation of this quantity $Y(\tau)...
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1answer
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Importance of Phase in FFT of an image

While processing digital images in the Fourier domain, mostly we exploit the amplitude and not the phase. This could be because the amplitude is much more structured and the amplitude spectrum reveals ...
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How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?

Let $H(z)$ be the rational system function of an LTI system. How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?
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multiplication of a function with a Fourier-transformed equals to Fourier-transformed with a function

I already showed b item using the fact that it is $h\left(0\right)=\int \:f\left(t\right)g\left(0-t\right)dt$ I struggle a lot of hours trying to find the trick in item C. Can anyone help please ?
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Convert measured spectrum to impulse response

I am still learning, so please bear with me: My audio measurement tool exports the measured spectrum of the DUT in the following format: Frequency [Hz] Magnitude [dB] Phase [degree] It is stored in ...
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How to compose a Discrete Prolate Spheroid (DPSS) dictionary?

I have a model of signal as $$ Y=AX + N $$ where $Y$ is received data in a linear array, $A$ is steering matrix, $X$ is data of sources and $N$ is noise. If $A$ has the form of $A=\exp(\alpha \sin(\...
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Why does alpha of 0.5 in detrended fluctuation analysis indicate randomness?

I'm trying to get an intuitive understanding of the different coefficients in detrended fluctuation analysis (DFA). It is used to detect fractal patterns in time series and it yields a coefficient, ...
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Difference between $\tt fft$ and $\tt dftmtx$ in MATLAB

I have the following MATLAB code: ...
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64 views

Normalization factor in the convolution theorem

Maybe it's a trivial question, but I couldn't find any explanation for this. According to the convolution theorem, in the continues case we add normalization factor, i.e. $$ \mathcal F\left\{h\star g\...
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Derivative of cosine at Nyquist

is negative sine, or zero, which is trouble; the imaginary DFT basis (sine) is likewise zero. Is there a way to meaningfully define the derivative of cosine when sampled at $f_s/2$ (such that it's not ...
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1answer
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Inverse Fourier transform of $\frac{\frac{2}{1+2i\pi f}-\frac{2}{4+2i\pi f}}{\frac{1}{1+2i\pi f}+\frac{1}{3+2i\pi f}}$

I wanted to calculated the inverse fourier transform of the transfer function : \begin{align} H(f) &= \frac{\frac{2}{1+2i\pi f}-\frac{2}{4+2i\pi f}}{\frac{1}{1+2i\pi f}+\frac{1}{3+2i\pi f}}\\ &...
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what does frequencies in non periodic signals mean?

What do the frequencies in the a Fourier transform of a non-periodic signal mean physically? Are there another definition of frequency that doesn't include the FT?

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