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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Inverse Fourier Transform of $\omega ^2$ in $[-\omega _0,\omega _0]$

I've been learning about signals for a while now, and I'm just starting to learn about Continuous time Fourier transforms. In this particular case, we were asked to get the inverse Fourier Transform ...
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1 answer
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What should be the size of my FFT values for speed,acceleration,..?

I am not from electrical eng. or physics background, so a layman explanation would be appreciated. I work with sensor data (accelerometer) from wearable device, collected for few hours. I take few ...
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Inverse of wavelet transform modulus gives poor results

I just want to understand, why is the result of my wavelet(?) transform so bad. For $0\le i< k$, where I set $k$ to $10$, I split the signal in blocks of length $s_i:=2^{i+2}$, overlapping by $s/2$...
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How to process a signal in frequency domain then convert it bakc to time domain?

Maybe a example of coupling effect in frequency modulated continuous (FMCW) radar will be helpful to better demonstrate my problem. Let us consider a FMCW radar system. We have a complex transmitted ...
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3 votes
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Does the Kramer-Kronig relations apply to this example $f(t) =\left(1-t^2\right)^4\cdot\theta(1-t^2)$?

Does the Kramer-Kronig relations apply to this example $f(t) =\left(1-t^2\right)^4\cdot\theta(1-t^2)$? with $\theta(t)$ is the Heaviside step function. I made a detailed related question here with ...
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Convolution of two functions using FFTW

I'm trying to perform a discrete convolution of two functions, $f(x) = 1$ and $g(x) = \exp(-x)$ of length nsize using FFTW. I have followed the procedure for zero-...
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Subsampling in frequency domain? Effect of sampling rate on spectrum?

Given a sequence $$ x[n] = [0, 1, 2, 3, 4, 5, 6, 7] $$ and its subsampling (by e.g. factor of 2) $$ x_\text{sub}[n] = [0, 2, 4, 6] $$ are $x_\text{sub}$ and $x$ related in spectrum? That is, given $X =...
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Basic audio denoising in the frequency domain using minimum statistics?

I'm trying to do some example elementary denoising of the audio signal. Let's say input is speech with constant traffic background noise. First I calculated block-based overlap-add Fourier transform (...
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1 answer
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Can DFT magnitude be used to identify repeating patterns in an Image?

Given the DFT magnitude vector of an 1-D image, I want to understand if we can calculate the size and pitch of repeating patterns in the image. Is this possible? I took a few test images and ...
2 votes
1 answer
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What is the Modulation Transfer Function of the filter $ [1, 1, 1] $?

I am trying to compute the MTF (modulation transfer function - Fourier Transform) of the following filter: $ [1, 1, 1] $. Here are my steps: $$ X_k = \frac{1}{N} \sum_{n=0}^{2} e^{-\frac{2 \pi i kn}{3}...
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Spectrum of squared signal? Why is spectrum bandlimited for squaring modulus of a bandlimited signal?

Approximately bandlimiting, or maybe exactly if x is exactly - example: rfft(abs(x)**2)[210] ...
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Why we use Fourier transform rather than Fourier series in signal processing?

In real world, signals are always finite, which means that any aperiodic signal can be periodic signal by repeating themselves. Then, why we don't just use Fourier series for those finite aperiodic ...
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2 answers
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How to sample a complex function?

We know that we can sample a real function such as g(x) = sin(2pi*f*x) with a (approximate) critical sampling rate expressed as ...
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Can we calculate the iFFT of a vector multiplied by 1 and -1 with less computational complexity?

I have the ifft result of vector x, I need to get the ifft of vector multiplied point-wise with a vector containing 1 and -1 which is symmetry and mirror around the center with less computational ...
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Do the direct multiplication of a matrix with vector and divided matrix with part of vector have the same complexity?

According to my understanding the FFT operation for a vector whose length is $N$ has a complexity of $ \frac{N}{2}\log_{2}{N}$ complex multiplication and $ N\log_{2}{N}$ complex addition. I was ...
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Determining minimum window length for DTFT [duplicate]

Let $x(t) = \cos(2\pi\times15t)+\cos(2\pi\times22t)+\cos(2\pi\times35t)+\cos(2\pi\times42t)$ and $\forall t\in\mathbb{R} :w(t) = 1$. We sample $x(t)$ and $w(t)$ with $F_s = 92 \ \text{Hz}$. So we ...
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2D Fourier transform normalization/standardization for machine learning

I am training deep learning models (i.e., CNNs, convolutional deep neural nets) on Fourier transformed images, i.e. the neural net receives as input a 2-channel (real and imaginary) tensor of shape e....
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1 answer
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FFT vs Gabor Transform

I am quite new to signal processing, so I would need some help regarding a project. I am interested in increasing the bass of a song from Spotify. At first I was thinking that applying a FFT to it to ...
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2 votes
1 answer
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Can finite Multi-Resolution Analysis satisfy the Littlewood-Paley Criterion (unity partition)?

Suppose we are working with a multi resolution analysis (MRA) of $L^2(\mathbb{R})$ and let $\phi$ be the corresponding scaling function and $\psi$ the derived wavelets. Using standard notation with $\...
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A mathematical justification of discontinuity detection using STFT

I'm trying to detect rapid changes in a one-dimensional signal say $[0,1]\ni t \mapsto f(t) \in [-1,1]$. By rapid changes, I mean corner points, edges, or sharp transitions at a point for example the ...
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2 votes
1 answer
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Time shifting a signal in python

I am currently modeling FMCW (linear FM) and SFCW (stepped FM) chirps in python. For my project, I need to simulate those signals as a transmit chirp and a received one, scattered from point targets ...
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Computing the DFT: how is the number of operations reduced by splitting the signal into even and odd parts?

On page 137 of "Understanding Digital Signal Processing" by R.G. Lyons I found that if I separate the standard DFT form: $$X(m)=\sum_{n=0}^{N-1}x(n)e^{-j2\pi nm/N}\tag{4-11}$$ by odd and ...
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"There's no ideal lowpass filter" - really?

Sinc is $\propto 1/t$. If $x(t)$ is bounded, then there exists $t: |x(t)/t| < \epsilon_M$, where $\epsilon_M$ is machine epsilon. If $x(t)$ is also time-limited, it also means there's $\tau$ such ...
1 vote
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What is the meaning of pixel 'value' when interpreting as inverse-fourier transform?

I’m working on a image signal project using C++ with DFT, IDEF. I major in physics and have lots of experience dealing with 1d fourier transform.. HOWEVER, 2d dft is really not intuitive. I studied a ...
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Algorithm or a tool to compare two spectrogram outputs for unit testing purposes

I am looking for a good algorithm or a tool to compare two spectrogram outputs for unit testing. I can visually confirm the outputs are similar but I would like to automate this process. The basic ...
1 vote
1 answer
54 views

Do amplitude shift keying modulation occupies a large portion of the spectrum?

In the PySDR online course, to the question "Why can’t we directly transmit the ethernet signal [directly in the antenna]", one answer is the following : Square waves take an excessive ...
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Discrete Cosine Transform (DCT) as the limit of Principal Component Analysis (PCA)

On the Wikipedia article about Discrete cosine transform it is said: For strongly correlated Markov processes, the DCT can approach the compaction efficiency of the Karhunen-Loève transform (which is ...
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1 vote
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Why does MP3 use modified DCT?

I read on Wikipedia here that the MP3 standard does use a modified discrete cosine transform. My question is: why does it use a modified transform and not the original DCT (like the JPEG standard)? ...
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Matlab's FFT wrong?

In my previous question, I tried to implement the DFT. Now I want to compare it to Matlab's FFT My input signal is: x(n) = n-2>=0 Here are the results: As you ...
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Why cant DFT be implemented this way?

I am trying to implement the discrete Fourier transform.Here is my code in MATLAB: ...
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Trapezoidal signal frequency spectrum

I don't understand why the frequency spectrum of a trapezoidal signal looks like this. I was expecting either a $\mathrm{sinc}$ or a $\mathrm{sinc}^2$, but it seems to be a combination of both.
1 vote
2 answers
143 views

Converting a triangle from the frequency domain to the time domain

I’ve been given a triangular signal that looks like this: $$ X^{F}(\omega) = (2 -|\omega|) \cdot W_{[-2,2]}(\omega)$$ (this is just my interpretation of the signal from a picture I’ll add). I was ...
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1 answer
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Discrete Fourier Transform and Incomplete Fourier Series

I'm working on a paper where I'm collecting sound pressure data from a chord's wave and trying to create a frequency spectrum to find the individual frequencies that make up the chord. However, I can'...
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How to derive CWT inverse equation from "Nonorthogonal wavelet transformation for reconstructing gravitational wave signals"

Can someone assist with a complete prove of equation 11, given in the paper (https://arxiv.org/abs/2201.01526) ? It is a method used for performing inverse fourier transform (or signal reconstruction),...
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Signal Reconstruction Using Scipy.signal.cwt

Can someone explain to me how I can reconstruct a signal using the scipy.signal.morlet2? The codes in the link only allows one to do a fourier transform using the morlet wavelet, but there is no such ...
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3 votes
1 answer
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How to get DFT spectral leakage from convolution theorem?

I have an issue, where my numerically calculated leakage from a DFT of a simple cosine does not match the theoretical prediction from the convolution theorem. I will try to present the example using ...
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1 vote
3 answers
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What is theorem under this formula?

I'm new to DSP. As I reading the textbook, I cannot understand the formula $X_{s}(f)=\frac{1}{T}\sum_{n=-\infty}^{\infty} X(f-nf_{s})$. Could you please give me some keywords so I can learn the ...
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2 votes
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Reconstruction of a Ricker Wavelet using inverse discrete fourier transform - signal cut in a half?

I am new here and new to DSP, so maybe my question is really basic. I have the formula for the Ricker wavelet (Mexican Hat) in frequency-domain and I wish to do an inverse Fourier transform to recover ...
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2 answers
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Is square of signal more recoverable than signal itself?

Let $x[n]$ be aliased sampling of real-valued $x(t)$ over $t_0 \leq t \leq t_1$. Can $|x(t)|^2$ be recovered more accurately than $x(t)$, over $t_0 \leq t \leq t_1$? If so, how? For $|x[n]|^2$, ...
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Periodogram giving frequencies beyond Nyquist frequency? [duplicate]

I have a complex-valued signal $q(t)$ which is sampled at $F_s = 24$ Hz. I am using the matlab periodogram function to plot its power spectral density. Here are the matlab commands I use: ...
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Can we control the minimum of a continuous signal $x$ when some Fourier coefficients are constant?

Let us fix a sequence of real numbers $\{a_k\}_{k=-n}^n$ and $\gamma\in \mathbb{R}$. Is there any $2\pi$-periodic continuous signal $x :\mathbb{R}\to \mathbb{R}$ such that the following points ...
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Range-Time Radar Data Processing

I am learning to how it is working this data processing. But I am little bit a confused. There are different ways to obtain this range-time table. As you can see the picture, obtaining the range-fft ...
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Sampling frequency to use with irregular signal

I have experimental data where I collected data points over time and my "signal" looks oscillatory in nature (y-axis values are more or less constrained in a y-axis bandwidth), but the curve ...
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1 answer
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How can one infer the input signal $x[n]$ from the output $y[n]$ of an LTI system with known Gain and Phase Response

I have the gain response of an amplifier and its phase response curves, in an appropriate frequency range. I also have a set of output (from the system) discrete data $y[n]$. How would one go about ...
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Can we control the maximum norm of a continuous signal whose finitely many Fourier coefficients are fixed?

Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous signals $x:\mathbb{R}\to \mathbb{R}$. Fix $n\in \mathbb{N}$ and put $$\Lambda_n=\{y\in C_{2\pi}: \mathcal{F}(y)[k]=0 ~\...
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Shifting the function and its effect on spectogram i.e. Gabor transform

I was reading about Gabor transform and how it allows to localize frequency with time. So it stands to reason that for this to be successful any shift in the function f, should result only in shift in ...
3 votes
3 answers
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How to understand the basis sinusoids of 3D FFT?

I understand intuitively the concept of Fourier transforms of 1D signals and 2D images. In the case of a 1D signal, an FFT gives the relative contribution of sinusoids with different frequencies and ...
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1 answer
84 views

Numerically observe the rotation property of the Fourier transform

I would like to numerically observe the rotation property of the Fourier transform. I believe that it is not possible since the rotation property is for the continuous transform and the DFT introduces ...
1 vote
1 answer
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Reconstructing an undersampled signal by cutting off at the signal's maximum frequency

Assume a (continuous) band-limited signal $f$, that is, a signal for which $F(s) = 0$ for all $\lvert s \rvert > p / 2$. If the signal is sampled with frequency $p$, we can reconstruct it by ...
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Relationship between different 2D FFT/Fourier domain window sizes?

hope everyone is enjoying their holidays. I'm a PhD researcher and I have an idea to filter an image using FFTs but I would like to try capture information at different scales using multiple window ...

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