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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If the cosine function is periodic, why does it have a Fourier Transform?

As far as I understand Fourier Transforms are for non-periodic signals and Fourier Series for periodic signals. So why is it we can take the Fourier Transform of a cosine when it is a periodic ...
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How do I make sense of the cosine wave having Fourier Transform coefficients which have infinite magnitude?

To illustrate my question better, consider the Fourier Transform of an aperiodic (as a periodic cosine wave has a Fourier Transform not Fourier Series) cosine wave $$f(x) = \begin{cases} \cos(2\pi ...
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How does minimum-latency partitioned convolution reverb work when you receive input samples in chunks, rather than one at a time?

I'm writing a reverb system where I receive an input block of samples 480 elements long, do some operation on them, and pass the block on to the next effect. I've been reading up on partitioned ...
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How to find frequency band after folding using single channel ADC

I want to to explore some signal processing techniques that is used to identify frequency band using FFT folding Following sequence is used in implementation Input->ADC->FFT->Frequency detection If ...
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DTFT of even and odd samples

Here to find DTFT of $h(2n)$ they have scaled omega, while in RHS to find DTFT $x(2n+1)$ they didn't, why is that?
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Is there a software to interactively stretch parts of a signal in the time domain (input as array)

I have a collection of audio signals as well as arrays related to the frequency spectrums of other audio sources. The audio signals are 2D arrays (of floats) as well. I need to interactively stretch ...
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Inverse Discrete Time Fourier Transform of $1$

$\textrm{DTFT}(\delta[n]) =1$, but $\textrm{IDTFT(1)} = \frac{\sin(\pi n)}{\pi n}$. Why it is not equal to the unit impulse $\delta[n]$?
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What determines peaks in FFT?

I ran FFT on three audio files and found that the results for some have more peaks than the other. Could anyone give me any conceptual explanation as to what determines these peaks? Below are plots of ...
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Why zero padding the 2-d DFT interpolates images in spatial domain?

I was applying different image interpolation techniques and I came know to about interpolation in frequency domain. In this technique we first take 2d DFT of an image, padd it with zeros and take the ...
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Why the spectral coherence is unity for all frequencies between single-frequency time series and itself

In the example below, I am plotting the coherence between time series and itself. The time series do has one frequency.The coherence magnitude was one for all frequencies. I wonder why it is not zero ...
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How to find a Matched Filter Transfer Function from large signal sample

Lets say I have a system where I have a small sample of a signal with no noise $\hat{x}(t)$ and a lot of a similar signal with noise $y(t) = \hat{x}(t) + n(t)$, and from $\hat{x}(t)$ I want to create ...
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Signal processing using numpy python

To process a .wav audio file with numpy (using fast Fourier transform algorithm). I want to process an audio signal at a particular interval with a sampling frequency 44100hz and sampling rate of 20ms ...
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Fourier Spectra : Significance of the Negative Amplitude [duplicate]

For example, for an aperiodic gate pulse, the Fourier Transforms for the continuous time case is a sinc function, while the discrete time case gives a sine over sine periodic kind of a function. In ...
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Is there a version of Welch's method that doesn't look for power?

Welch's method splits a time signal, $x(n)$ into $M$ periodograms $P_m$, $P_{x_m,M }(k) = \frac{1}{M}|F_k(x_m)|^2$ and averages them to give the Power Spectral Density (PSD), $S_{x}(k) = \frac{1}{K}...
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How do I decide which frequencies are signal and which are noise?

I have an arbitrary recorded digial signal, on which I have run a Fourier transform. I'm not sure what conventions are on a case like this, but I have 1024 frequency bins. Second bin is the highest ...
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Fourier Transform: interpretation of continuous spectrum at specific frequencies

B. P. Lathi in his book "Principles of Linear Systems and Signals" mentions in the Fourier Transform: When $x(t)$ is periodic, the spectrum is discrete, and $x(t)$ can be expressed as a sum of ...
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Aliasing when interpolating with DFT?

I'm coming from an understanding of the continuous-time Fourier Transform, and the effects of doing a DFT and the inverse DFT are mysterious to me. I have created a noiseless signal as: ...
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How to search for irregular signals: Fourier, DWT or k-means?

See my notebook here I want to search for irregular time signals in a data set of ~3 500 000 time signals. By eyeballing I have found hundreds of flat and oscillating signals, but just a few that are ...
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What is Imaginary in Fourier transform?

How to plot graph of $e^{-t}$ in frequency domain. What would be the axis? If its Fourier transform is $1 /(1+j\omega)$, then how can we plot imaginary on frequency domain (amplitude vs frequency ...
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Explain Auto-Tune in a simple way

I have to do a presentation about Auto-Tune and its relation to the Fourier transform. What is a good explanation on how does Auto-Tune work?
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Impulse response of a 3x3 PSF - how to find analytical expression for fourier transform of a 3x3 matrix?

I have a filter $\mu[n_1, n_2]$ with taps: $$ (1/8) (1/4) (1/8)$$ $$ (1/4) (1/2) (1/4)$$ $$ (1/8) (1/4) (1/8)$$ How do I find an analytical expression for $\hat\mu(w_1, w_2) $? Since it looks so ...
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STFT on time varying signal with good time and frequency resolution

I am trying to determine the main frequency of a noisy signal that varies in frequency over time. Ideally I want to detect changes in the frequency as rapidly as possible - say 50Hz update rate, but I ...
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Is there any practical application for performing a double Fourier transform? …or an inverse Fourier transform on a time-domain input?

In mathematics you can take the double derivative, or double integral of a function. There are many cases where performing a double derivative models a practical real-world situation, like finding the ...
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Two versions of Constant Q Transform (CQT) doesn't match each other?

To my knowledge, there's two major CQT papers, the one by Brown in 1991, and the one by Schorkhuber in 2010. The 2010 paper claims to be a more computationally efficient implementation of the 1992 ...
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How can I relate my amplitude from a FTT to the actual signal?

Sorry for disturb you guys, I've been playing with this the last days. I am computing signals of a wave produced with a wavemaker. Cause I have several sensors (wave gauges) I am sensing the same wave ...
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Determining LTI system response to a scale change

We know that for an LTI system, if $y(t)$ is the output for $x(t)$ then the response for $x(t-2)$ will be $y(t-2)$ and so on. But my question is what will be the system response for the input $x(-2t)$...
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How to reconstruct a sound from magnitude spectrogram?

I have an audio magnitude spectrogram but I don't have the phase, try to randomize the phases of each container and then make a reverse fourier, but only pure noise is heard How can I reconstruct the ...
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Non Circulant Translation Using Fourier Transform

The translation property of Fourier Transform (FT) for a two dimensional image $f$ is as $$ f(x-x_0, y - y_0) = F(u, v)e^{-j2\pi(ux_0/M+vy_0/N)} $$ Using this equation, the following code (in Matlab) ...
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Sinc interpolation in spatial domain

I have tried to perform sinc interpolation (in 1D) with the following Matlab code: ...
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how to reconstruct an phase information from the magnitude spectrogram

I need to recreate the phase of a spectogram of magnitude and when inverse fourier, that the sound is understandable and not pure noise Observe these softwares https://photosounder.com/ http://...
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Is Fourier series a sampled version of Fourier transform?

I recently learned about dtft and how dft/dfs is the sampled version of dtft. I was wondering if Fourier series is also obtainable by sampling Fourier transform? I am a noob in the subject so sorry if ...
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Estimation / Reconstruction of an Image from Its Missing Data 2D DFT

Given the 2D DFT of an image i.e. a NxM matrix of complex numbers, with some missing lines (or even partial lines), considering we have zeros in the missing positions. Any suggestions for an ...
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Is the whole data needed for the fast fourier transform with an ongoing signal?

I sample an signal with around 840k values a second, to check the spectrum around <400kHz. But i can't save the whole data as it is just to much for my microcontroller. I know that for the ...
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Fourier transform in Matlab and hermitian symmetry

According to the conjugate symmetry property of Fourier transform, shouldn't the following command not return 1 (=true): ...
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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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What effect does rotation in the spatial domain has on phase in Fourier transforms?

More precisely, let's say I apply a 45 degrees rotation to an image (in the spatial domain) say, in Matlab : Ir=imrotate(myImage,45,'crop'); FT_I=fft2(I); In the ...
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Continuity and its relationship with asymptotic spectral decay

The asymptotic decay of the magnitude of the Fourier transform of a function appears always to be determined by its continuity properties as follows, with examples given in Fig. 1: Continuous ...
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Approximating Lorentzian Fourier transform with FFT

I have a Lorentzian frequency distribution $F(w) = \frac{1+iz}{1+z^2}$ Where $z = \frac{w-\Omega}{R}$ With $\Omega$ being the peak frequency and R the decay constant. I know that analytically ...
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“The Fourier transform cannot measure two phases at the same frequency.” Why not?

I have read that the Fourier transform cannot distinguish components with the same frequency but different phase. For example, in Mathoverflow, or xrayphysics, where I got the title of my question ...
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Is the FFT magnitudes of multiple audio sources additive?

Is the FFT magnitude of two audio signals when played together the sum of their individual FFT magnitudes across frequency bins? Note I'm talking only about the magnitude, ignoring phase. For ...
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Continuous-time vs Discrete-time Fourier transform

case 1) to calculate the Fourier transform of discrete-time signal(sampled signal) we use Discrete-time Fourier transform. but my question is: case 2) if I consider that discrete-time signal as ...
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Non-Linear, Non-Stationary spectral analysis methods! When and where?

I have been reading about non-linear non-stationary signal analysis methods and it seems to do this type of analysis the go-to method is the Empirical Mode Decomposition (EMD), then Hilbert Transform (...
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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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How do I explain a complex exponential intuitively?

What is a complex exponential, explained intuitively? How do I explain to an adolescent a complex exponential function?
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fft function in R vs spec function from 'seewave' package? They don't give a similar frequency spectra

I understand that spec will give me the frequency and the corresponding amplitude of that component, whereas fft will compute the DFT of the signal and throw the complex numbers for each component ...
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Characterization of transfer functions with no local peaks

Assume that you are giving an arbitrary amplitude frequency response $A(\omega)=|H(j\omega)|$ Is there a characterization that ensures that $A$ is monotone? i.e, $A$ has a global maximum at $\omega=0$...
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Removing striped noise from an image

I was just wondering if anyone could explain to me the approach one would take to removing striped noise from the fourier domain of an image. I was reading an article about MRI image from 1 just ...
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Graph Fourier transform: the adjoint notation for the eigenbasis matrix

I already asked this question here but there is no response. I'd like to ask this question in signal processing domain. It is well-known that for a real symmetric matrix $L$ (here, graph ...
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Relation between $X(f)$ and $S_x(f)$

We know that for a signal $x(t)$, it is related to $R_x(\tau)$ as, $$R_x(\tau) = \int_{-\infty}^{\infty}x(t)x(t-\tau)dt$$. $\\$We also know that $$R_x(\tau) \rightleftharpoons S_x(f)$$ $\\$How do we ...
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FFT and Power Spectrum Normalization

On many websites, including MathWorks, it was suggested to normalize the fft spectrum (MATLAB or numpy) by dividing it by the total number of samples ($N$). For a sinusoidal signal, for example: $$x(...