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Questions tagged [dft]

The Discrete Fourier Transform (DFT) is a mapping between a finite set of discrete points in a (primal) domain (time, space) and the dual frequency domain. DFT requires an input sequence which is discrete, such as a sampling from an analogue audio signal.

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Zero Padding in Implementing FFT from scratch

I'm trying to implement an FFT algorithm from scratch. I'm using the recursive algorithm where if N is a power of 2, then I have M = N/2. The algorithm is divided into even and odd parts and I have ...
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Strided unpadding energy relationship?

If x1 = x0[::2] is unaliased subsampling, then $E(x_1) = E(x_0)/2$, which can be proven via Parseval's theorem. For same $x_0, x_1$, however, ...
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Phase extraction from Fourier transform

Is it possible in principle to correctly extract the phase from Fourier transform? I just tried to do so using Python, here some attempts: ...
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FFT of a gaussian signal in Python

I've been trying to get the FFT of a gaussian in Python. When I use the following parameters, the FFT goes hand in hand with the theoretical FT of the gaussian, but if I increase $\sigma$ they rapidly ...
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Can anyone explain how dft works as a filter bank?

When we take the fft of input signal, the fft formulas say us to down convert the 2pik/N frequency content of input signal and sum one period interval. This gives us a just one complex number,not an ...
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Bandwidth visualization in frequency domain

Consider some signal in frequency domain: the maximum length of which corresponds to the half of the original signal ($N/2$), here $N=32$. It is known that the bandwidth of each sample is $2/N$, so ...
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What is going wrong with the plot of 2D spatial spectrum at a specific frequency?

I've a set of 09 sensors in the following arrangement and the script for the sensor positions as follows: ...
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1 answer
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Lowering Spectral Resolution of FFT

I find myself in the position of having to lower the FFT resolution. Basically I have a signal of length M and I would like to make an FFT with N<M frequency bins. I cannot simply make several FFT'...
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1 vote
1 answer
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What Does "Reduced Modulo N" mean in this context?

I am trying to understand a piece of notation used in several papers, the simplest/shortest of which is this paper by Crochiere. The equation in question is Equation 7 on the second page: $x_m(sR) = ...
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2 votes
1 answer
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Sparse signal FFT

Say I had a time domain signal $x[k]$ wich is sparse: $\log(N)^2$ nonzero samples and the fourier transform has only a very (very!) small number of high frequency components. Are there any techniques ...
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Confusion Understanding the mathematical expression of duality property of dft?

Duality Property for DFT Above dsp.se question provides good understanding about dft duality property but i am having difficulty understanding its mathematical expression because on Google when i try ...
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Expression for DFT of linear 2D ramp

One dimensional solution is in “Expression for discrete fourier transform of linear ramp“ I need two-dimensional for image processing. We have function f(x,y) = $a_1 \cdot x + a_2 \cdot y$, My ...
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Removing sawtooth wave from DFT transform [duplicate]

We have samples $x_n$ and sawtooth wave $s_n$ with period N (especially if N=$2^k$) $s_n = c \cdot n$ for n=0...N-1 where c is constant. What is formula for coefficients of DFT this wave? I want ...
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Inferring shape of information signal from its DFT?

I came across this question recently, and I am very confused by (b)(ii). b(i) gives $x_n$ = [0.5, 0, -0.5, 0]. My approach to (ii) was to recognise that $X_m$ represents the frequency content of the ...
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1 answer
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Is there other basis possible for DFT?

As I understand, the DFT of a signal $x$ is a representation of this signal in the basis $$ \{ e^{j2\pi kn/N} \}_{k = 0, 1, \dots, N-1}$$ Is it possible to form a base of such discrete complex ...
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1 answer
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Best way to get the bin amplitudes of an audio DFT "normalized" 0-1

I'm working on a spectral processing plugin that's predicated on allowing the user to manipulate per-bin values in an STFT algorithm. I normalize the bin amplitudes by diving them by $N$ (number of ...
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1 vote
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Entropy Loss in Linear Filters

I've been trying to figure out the result for the entropy loss/entropy gain in linear systems derived in "The Mathematical Theory of Communication" by Claude Shannon. Claude Shannon states ...
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2 answers
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Question about radix-2 DIT FFT and sampling theorem

I'm currently reading this article and trying hard to understand it. According to the article, DFT is as follows: $$ X_k=\sum_{n=0}^{N-1}x_ne^{-\frac{2\pi i}{N}nk} \\\text{where k is integer and its ...
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DFT Graph has too many oscillations? [closed]

I'm very new to the DFT, and am working on it for a HS project. I've taken the DFT of my instrument playing (the original waveform looking like this: But after plotting the DFT onto Desmos, I get a ...
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4 votes
3 answers
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Inverse DFT on the first half time domain

Assume to have c[] representing N DFT coefficients. The complex-valued signal of N samples in the time domain is computed by ...
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1 answer
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What means `crop` in FFT calculation?

In soapy power manual: Crop: -o PERCENT, --overlap PERCENT percent of overlap when frequency hopping (incompatible with -k) -k PERCENT, --crop PERCENT percent of crop when frequency hopping (...
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correct fourier transform of time series starting with different start/end times

I have lots of time series where I want to analyse some periodic signal that occurs beside the signal I wanted to measure (and I didn't expect). The idea is that there's always a peak at a certain ...
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2 answers
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What is edge-effect in discrete convolution?

My professor threw in the random term and never explained what it means cus he's been going on strike for 3 weeks straight.
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1 answer
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Understanding spectral leakage in a pink noise dominated signal

I've been reading about power spectral density estimation based on the DFT, about spectral leakage, windowing functions and the Welch method. I've recorded a signal that's supposed to be pretty much ...
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2 answers
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Understanding $\mathsf{FFT}$ of $x(t):=\cos(2\pi f_{0}t)$

I have been buzzed about this issue for more than an hour. I have been tasked to execute a simple DTFT task using MATLAB for the signal $x(t):=\cos(2\pi f_{0}t)$. Assuming we are sampling at a rate of ...
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What metrics should I use to describe the difference between two magnitude responses in octave band

Suppose that I have two frequency responses $H_1(k)$ and $H_2(k)$, I want to describe the difference or MSE between them in each octave band. The background is that I have a target frequency response $...
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5 votes
1 answer
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Sinc Based Multi Dimension Signal Resampling on the Fourier Spectrum (DFT)

As a generalization of the following questions: The Proper Way to Do Sinc Downsampling (DFT Downsampling) for Uniformly Sampled Discrete Signals with Finite Number of Samples. The Proper Way to Do ...
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1 answer
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Intuition of odd and even complex conjugate symmetry definition of DFT/DTFT so that $X(e^{j w})=X_{e}\left(e^{j w }\right)+X_{o}\left(e^{j w}\right)$

I have been reading through my courses DSP slides and came across something which was not really taught in detail. You can look up here for reference, it is stated almost identical. Given the ...
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4 votes
1 answer
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Applying 2D Sinc Interpolation for Downsampling in the Fourier Domain (DFT / FFT)

Related to The Proper Way to Do Sinc Downsampling (DFT Downsampling) for Uniformly Sampled Discrete Signals with Finite Number of Samples, how can one apply Sinc downsampling in the DFT / FFT domain ...
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1 vote
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Relation between DFT filter bank and sliding DFT

I am reading the book 'spectral audio signal processing'. It says when $n=LN-1$ for any integer $L$, the sliding DFT $$X_n(k)=\sum_{m=0}^{N-1}x(n+m)e^{-j2\pi mk/N}$$ coincides with the DFT filter bank....
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2 votes
2 answers
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How can I check if a signal its periodic from the graph of FFT?

x is a vector of length 1000 that contains the samples of the signal; n is equal to 16 that its the number of bits of each sample; fa=256 Hz (sampling frequency); <...
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4 votes
1 answer
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Applying 2D Sinc Interpolation for Upsampling in the Fourier Domain (DFT / FFT)

Related to The Proper Way to Do Sinc Upsampling (DFT Upsampling) for Uniformly Sampled Discrete Signals with Finite Number of Samples, how can one apply Sinc upsampling in the DFT / FFT domain for a ...
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What is the magnitude and phase of the wave represents in the k-space or Fourier space?

#Code from https://stackoverflow.com/questions/70768384/right-method-for-finding-2-d-spatial-spectrum-from-cross-spectral-densities I've an array of seismic sensors (say, N=34). Every sensor collect ...
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1 vote
2 answers
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How to know if filter is lowpass or highpass?

I was trying to solve the following question: Calculate the DFT of the given filter impulse response $h(n,m)$. Based on the result, determine if the given filter is a high-pass or a low-pass filter. ...
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5 votes
1 answer
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Algorithm that enlarges the image to a resolution of $2N \times 2N$ using DFT operations

I'm trying to solve the following question: Given an image at a resolution of $N \times N$. Describe an algorithm that enlarges the image to a resolution of $2N \times 2N$ using DFT operations. As I ...
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2 answers
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Understanding the result of the fft algorithm

Understanding the result of the fft algorithm. I need help understanding the FFT calculation results. Recently, I have been interested in signal analysis, so I have created and understood fft ...
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0 answers
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Calculating the DFT of the image after using a filter

Studying for my finals in Image Processing course. Trying to solve the following question: Let $h$ be a filter that replaces each pixel value with the average of it's 8 neighbors. Let $f$ be a ...
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1 answer
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Fourier Transform of an 2D image and associated units

I have the following rather simple problem and unfortunately I am not getting forward. Imagine a simple 2D image with pixels and a unique value for each pixel of the image. For example, let the image ...
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6 votes
7 answers
937 views

Why is frequency resolution dependent on the number of samples? (need for intuition)

I know the DFT, I agree with the formula and everything, but I don't get the intuition on the link between frequency resolution and number of samples. Like, why would I get a higher frequency ...
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0 votes
1 answer
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Why are the negative frequencies of the DFT symmetrically reflected at the nyquist to the positive frequencies?

I was playing around with plotting DFT and realized that the negative frequencies are symmetric to the positive frequencies reflected at the nyquist. Plot shown for the signal $f(x) = \cos(\frac{\pi}{...
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1 vote
4 answers
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Why is the nyquist frequency at $\frac{N}{2}$ (or $\lfloor \frac{N}{2}\rfloor$) for the DFT and what is the value for $X[k_{N/2-1}]$

For the definition of the DFT we have $X[k] = \sum\limits_{n=0}^{N-1}x[n]\exp(- \frac{2 \pi i \cdot n}{N} k)$ Let's say for simplification that $N$ is even. Then $k_{N/2-1} = \frac{N}{2}$ is ...
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0 votes
1 answer
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How to calculate the DFT for this sum of cosine's in the form $\sum A_i \cos(\omega_i n + \phi_i)$ for fixed $N$

I am stuck trying to calculate the DFT for a given $N$ Given the signal $x[n] = \cos(\frac{\pi}{2}n - \frac{\pi}{2})+2 \cos(\pi n + \frac{\pi}{2})$ and $N = 4$ I tried to calculate the DFT $X[k] = \...
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1 answer
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DFT algorithm output meanning [duplicate]

The result of calculating the amplitude of the audio through the DFT algorithm above is It came out as below. ...
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1 vote
1 answer
527 views

Why is the arithmetic mean the same as the DC component of its fourier transform?

When we define $$\overline{\left|x\right|} = \frac1T\int_0^T x(t) dt$$ as the arithmetic mean of a signal we can see that it is the same as its dc component in the fourier transform. Why is this the ...
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4 votes
2 answers
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"DFT". Understanding the formula $e^{-i2\pi k}$ $k$ is a real number

I am studying about fast Fourier transform. Assuming that $x_0$, $x_1, \ldots, x_{n-1}$ are complex numbers, the DFT is defined as follows. $$f_j = \sum\limits_{k=0}^{n-1} x_k e^{-\frac{2\pi i}{n}jk},\...
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2 votes
1 answer
154 views

How to interpret a 1D-DFT of an image with a sinus grating/gradient compared to its 2D-DFT outcome?

For getting a better and more intuitive understanding on how the 2D-DFT works I was playing around with sinus gratings in grayscale. I tried to compute the 1D-FFT first and compare it with the 2D-DFT ...
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0 votes
1 answer
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Why does the 2D-DFT of a sinus gradient not show energy along the diagonal straigh lines and only vertical/horizontal from the diagonal point?

I have been experimenting a little bit with simple examples of the 2D DFT to get a better sense for it's interpretation. For this purpose I have been using sinus gratings with the following code: <...
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1 vote
1 answer
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Scaling of FFT2 magnitude in image-processing

I got the following code: ...
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overlap and save method) relation zero padding with double size FIR filter coefficient

i'm doing overlap and save method at frequency domain to do this, i added one block back and forth at filter and signal respectly The problem is my filter is bessel function and that is look like this ...
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1 answer
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iFFT and extracting dw

I have a trivial fourier transform question. I have a correlation function, C(t), with complex components in the time-domain, and dt. I would like it in the frequency domain, C(w), like from ...
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