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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What is the significance of the linear phase in the phase response of an M-point moving average filter?

I have plotted the magnitude and the phase responses of an M-point moving average filter, the following are the plots when M = 10: We can observe that corresponding to every lobe in the magnitude ...
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1answer
40 views

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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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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Deriving the integration property of the Fourier Transform

I want to derive the property of the Fourier Transform that states that if $X(j\omega) = \mathcal{F} (x(t))$ then $$\mathcal{F} \left( \int_{-\infty}^{t} x(\tau) \mathrm{d} \tau \right) = \frac{1}{j\...
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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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1answer
45 views

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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1answer
25 views

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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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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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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Why is the continuous time Fourier series of DC signal an impulse?

In case of continuous time Fourier transform(CTFT), I can easily calculate the Fourier transform of DC signal by using Fourier duality or inverse CTFT. But I don't know how to calculate the continuous ...
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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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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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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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Understanding the meaning of amplitude in FFT

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 ...
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Phase information of a signal when creating a spectrogram

Sorry for the uninformed question but I need some help understanding this. I'm trying to understand what phase information is and it would be very helpful if someone could correct my understanding of ...
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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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1answer
70 views

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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How does one interpret an element of the “transfer matrix” used to calculate frequency domain granger causality (via VAR models)?

I am attempting to gain a better mathematical understanding for how autoregressive models can be used to infer frequency-domain granger causality. All freq. domain measures of causality that utilize ...
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1answer
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Bluestein's algorithm to evaluate the DFT from $f_o$ to $f_o + k\Delta_F$

Briefly, the convolution between $x(nT) e^{-j2\pi f_o nT} e^{-j \pi \Delta_F Tn^2}$ and $c(nT) = e^{j \pi \Delta_F T n^2}$ multiplied $e^{j \pi \Delta_F T k^2}$ allows me to find the DFT $X(f_k = f_o +...
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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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Non Equispaced / Non Uniform DFT Bandwidth

I need to construct Fourier transform of non-equispaced data. That is, I have signal $s(t)$, $t\in[0,T]$ sampled at non-equispaced points $t_k$, $k=0...N-1$ with sample values $s_k = s(t_k)$. For ...
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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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Why Is the Total Time Equal to $ N \cdot {T}_{s} $ and Not $ \left( N - 1 \right) \cdot {T}_{s} $ In the Context of DFT?

In the definitions of the DFT DFT $$ X(j)=\sum_{k=0}^{N-1} x(k) \exp \left(-i 2 \pi\left(\frac{j}{N}\right) k\right) $$ Let us say, if we have $10$ points, $N=10$, each sampled at $0.2$ seconds, why ...
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1answer
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Fourier transform of time division

I know that Fourier transform of $t^n f(t)= i^n \frac{d}{d\omega^n} F(\omega)$. But does this work when $n<0$? Is there any direct relation to compute the Fourier transform of $\frac{f(t)}{t}$?
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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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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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Why does Hilbert filter distorts the shape of the signal?

If all the harmonics composing a the signal are shifted by the same amount, this would be the same as sampling later or earlier in time. I think. Take a simple pulse train as an example. If Fourier ...
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606 views

What does it mean for a function to have frequencies?

In a lecture, my professor mentioned that $\cos$ has two frequencies. I see that using the inverse Euler's formula we can express $\cos$ as a some linear combination of complex exponentials, each with ...
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507 views

Covariance between real and imaginary parts of Fourier transform of a stationary time series

Since Fourier transform of a random stationary process in time (in the case of existence) is not necessarily real, my question is what is the relation between the covariance of real and imaginary ...
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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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203 views

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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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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Zero Padding of FFT

There are many question related to the zero padding a time domain signal to get more frequency bins after performing Fourier transform. As I understand this process is equivalent to trigonometric ...
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749 views

FFT of input length 1536

Does anyone knows can i find a FFT of 1536 length input. Its a specification given in 3gpp Lte and we need a transform of 1536 input size which is neither a power of any number i would say. I just ...
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Derive DTFT of $x[2n]$

If the DTFT of discrete sequence $x[n]$ is $X(e^{j\omega})$, what is the DTFT of $g[n] = x[2n]$? I see the textbook answer is \begin{align*} G(e^{j\omega}) &= \frac{1}{2} \left( X(e^{j\omega/2}...
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Difference between CTFT and DTFT?

I have tried to read different articles but still confused in difference between continuous time Fourier transform and discrete time Fourier transform?
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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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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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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 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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Anyone explain to me this video?

I was watching a video in time 24:48 I would like to know where you got the value .9 (1.14z + .941) and 1.0232 + .757 Does anyone explain how he got those numbers?
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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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What does the frequency axis of a Power Spectral Density mean?

I have never really understood what the frequency axis meant when we plot the Power Spectral Density(PSD). Does it correspond to frequency as we get after we take the Fourier Transform of a time ...

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