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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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(...
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Convolution Theorem: Hamming Window on a Time Series and Fourier Domain

If we have a set of time series data, y, consisting of 100 data points. One can apply a N (odd) Hamming window as a weighted moving average to decrease the noise. Say, if we choose 7 point Hamming ...
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Does convolution violate duality of a Fourier Transform

Duality says that if $$ x(t) \iff X(\omega) $$ then $$ X(t) \iff 2\pi \cdot x(-\omega) $$ Given that $$ f(t)\circledast g(t) \iff F(\omega)G(\omega) $$ By duality, shouldn't that imply that $$ f(t)...
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What is the significance of y-axis in Lomb-Scargle Periodogram?

What does the y-axis of Lomb-Scargle periodogram represent? Periodogram obtained by Fourier transform contains information about the original data's amplitude, but the so-called "Lomb-Scargle power" ...
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What is the effect on Hilbert transform of signal after multiplication by sinusoid?

Question : Find Hilbert transform of $[u(t-a)-u(t-b)]\cos2\pi f_{0}t\\\\$ such that $\\0<a<b$ my attempt: we know Hilbert transform of $[u(t-a)-u(t-b)]\xrightarrow{\mathcal H} \dfrac{1}{\pi}\...
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Frequency Domain Interpolation: Convolution with Sinc Function

I am reading a paper, and I came across the following definition of sinc interpolation. Warning. I don't have a strong background in signal processing. Also, I have no clue what that bar on $\bar{F}$ ...
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How to group similar patterns together that are shifted along the x axis?

I have a set of time series plots that are shifted along the x-axis. I am looking for an algorithm using which I can group them together. My end goal is to average them. While reviewing the ...
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complex numbers and fourier transform

Is it possible to define a scaling property for fourier transform when the scale factor is complex? Usually the scaling factor is real. What happen when a scaling factor is complex?
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Fourier transform and anti-trasform--identity missing

I have a very silly doubt: If we define the power spectral density: S(f)=$\frac{1}{2\pi}\int exp(-i\tau2\pi f)r(\tau)d\tau$ (1) where $r(\tau)$ is the correlation coefficient. If we do the Fourier ...
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Do not understand spectrogram of signal created from many sines with different frequencies

I get this spectrogram From the following code: ...
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Is fft2 in MATLAB unitary? Some differences happen

I meet a problem when implementing fft2 in MATLAB. The question is I try to simulate the realistic measurements $Y = |FCXF^H|^2$ - the intensity of Fourier domain of object $X$, where $F$ denotes ...
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Proof that DFT does not require more than N points

I'm trying to show how the discrete Fourier transform (DFT) arises from the equation for the continuous-time Fourier Transform. I've run into an interesting caveat which I can't seem to find an ...
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PSD of modulated signal

I know this question has been previously asked just saw it and tried to do myself but reaching an expression which is different from expression of PSD which I remember for modulated signal Actual ...
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Filter interferce within an image using fouire transform in matlab

I am currently reading Fundamentals of Digital Image Processing : A practical approach with Matlab and on page 129 They demonstrate a way to filter an image with strip kinda of noise. Now ...
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DFT of an audio signal stored as a multi-dimensional array

I'm using the Python scipy library to get the data of a particular .wav file into array format. Now, I'd like to find the Discrete Fourier Transform of that signal. The formula for the DFT is, $X_k = ...
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One sided frequency spectrum (Matlab vs. Origin)

There are a lot of queries on fft frequency all over the web. I guess the following point not discussed anywhere explicitly. Hope someone can provide an insight here. If we have and even number of ...
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A question about Fourier transformation

Hello, this is my first time actually asking in stackexchange. I am a computer engineering student and currently i am doing a linear system course (i don't really know how this is equivalent in ...
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Calculate phase lag between two signals with perturbed frequencies

This type of question has been asked quite a few times on this forum and others now, but I still haven't found a satisfactory answer to my problem. Given an input signal: $$x_1(t)=\cos\big(2\pi ft\...
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Apodization in the Fourier (frequency) domain on discrete experimental data

Let us assume we had time domain signal as a raw data R (Window 1) and we wish to perform the deconvolution process on R using another set of raw data G (window 2). This is accomplished dividing FT of ...
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Shannon-Nyquist theorem reconstruct 1Hz sine wave from 2 samples

Lets say I want to set the minimal sampling rate to reconstruct a 1Hz sine wave, according to the Nyquist-Shannon theorem that states that the maximum recoverable frequency is Fs/2 i.e. we must sample ...
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Can the Hanning Window be represented in the time domain

I know the Hanning window, $w_{Hann}(n) = 1 - \cos \left(2\pi \frac{n}{N} \right)$, is typically applied to discrete Fourier transforms. $N$ being the total number of points in a transient signal that ...
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Fourier transform of a damped cosine wave with a linear frequency chirp

I want to take the Fourier transform of the following transient signal, $$f(t) = e^{-t/\tau} \cos((\omega_0 + m t)t)$$, where $m$ is some gradient parameter in units of $\rm{Hz}/s$. I thought this ...
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How to find fundamental frequency of two signals?

I am facing difficulty with finding fundamental frequency of signals I mean by fundamental frequency=(1/Time period) Correct me if I am wrong consider two continuous time signals with Time period ...
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How do I obtain the fourier series coefficients for a signal obtained by multiplication of two signals of different frequency?

What i assume here is that LCM of time periods of the two taken signals exist that is signals periods are not like pi/2 and 1 but are rather like 1 and 2 (just an example) I am given fourier series ...
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Is it possible to extract peak locations in the time domain using help from fourier/wavelet analysis?

The signal I'm studying has fundamental frequencies of 20 and 60 cycles per minute (shown in the Periodogram graph). It is straight forward to extract the peaks in the time domain belonging to the 20 ...
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38 views

Fourier transforms and time shift

There is probably something trivial behind this, but I am missing something. I need to create a stationary random time series data v(t) which is the the sum of another time series u(t) and u(t) with a ...
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Detecting changes in signal due to server delay

I'm currently working on an application that can be used to determine when a signal changes due to a server delay. Essentially, I have an API that is used to output data to a UI. However, for reasons ...
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Reconstructed output mismatch for LTI system

I have a system with measured input (u) and output (y). I assume that this is an linear time-invariant (LTI) system and I want ...
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Image Brightness on 2D Fourier Transformation

What effect does the brightness have on the frequency spectrum of 2D Fourier Transformation? Example. Suppose we have a gray image and calculate 2D Fourier Transformation. Then we increase the ...
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What's the significance of lomb-scargle power?

What is the significance of Lomb-scargle power (y-axis)? I have two data sets. For each plot, above plot is Lomb-scargle periodogram of the lower plot (original data). The first data set has an ...
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Nyquist Frequency on semi-unevenly sampled data

I have a data set that has 'kind of' constant sampling rate - it switches between 1 min and 2 min. About 70% of the times, samples are taken every 1 minute, and about 30%, samples are taken every 2 ...
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Unit of Energy Spectral Density

The continuous-time Fourier Transform (CTFT) of a signal $x(t)$ (with unit $unit$) is: $$X(\omega)=\int_{-\infty}^{\infty} x(t)e^{-i\omega t}dt$$ which should be in $unit\cdot sec$ or $\frac{unit}{...
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Phase response of $H(f)=e^{-j2{\pi}ft_0}$

Given is the impulse response: $$h(t)=\delta(t-t_0)$$. I calculated $$H(f)=e^{-j2 \pi ft_0}=|H(f)|\cdot e^{j\varphi(f)}$$. Now, the magnitude response of $H(f)$ is: $$\begin{align} |H(f)| &=\...
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Python FFT outptut

I have a (real) array of data and am trying to analyze its frequency components. I've been using NumPy's FFT routines, but I realized there is something I don't quite understand: why does the output ...
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How to normalize PSD to get the same magnitude as FFT peak

I am trying to use FFT and power spectra density estimation with python (np.fft.fftand scipy.signal.periodogram). And trying to ...
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Symmetry in Lomb-Scargle transformation

I'm observing weird symmetry and repeating pattern on my unevenly sampled time series data after Lomb-Scargle transformation. I used astropy lomb-scargle. ...
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Correlation/anticorrelation as a function of frequency

I am tracking the values of two fluctuating quantities as a function of time and am trying to analyze possible correlations between the two as a function of frequency. The application is experimental ...
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Effect of Nyquist frequency on Fourier transformed data

Upper plot is the original data's plot, and the bottom plot is Fourier transformed data. For the bottom plot, x-axis is the frequency and y-axis is the amplitude. I don't understand the weird behavior ...
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fftshift in MATLAB with even number of data points in double sided spectrum

I have a question with reference to this Table. With even N, the frequency axis extremes should be $\pm$Fs/2, where Fs is the sampling frequency. However in the array we have only one value ...
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Frequency Axis of Discrete Fourier Transform (DFT) with Odd Number of Data Points

I am trying to understand the logic behind making a frequency axis in DFT. I am using for time based light absorbance. When we have even number of data points (N= even integer), collected over a ...
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Fourier Transform within a certain Limit

I want to evaluate fourier transform within a certain limit in MATLAB,the expression of which is $$X(f) = \int_{1}^{4}{x(t)e^{-i2\pi ft}}\,dt$$ I have to find value of the above expression ...
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Is there any way that we can perform speech recognition without using Fourier transforms?

I am trying to research about speech recognition and why everyone uses Fourier transforms in going about the topic. I know that we get information related to the frequency of each sound uttered which ...
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Is there a generalized method to get the input with the given output and the impulse response?

$y(t)=y(t+12), y(t) = x(t) \ast h(t)$ The continuous time signal output $y(t)$ is a periodic square wave, 50% duty cycle pulse. The impulse response is a box function.($A = 1, T = 2$) By using ...
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inverse discrete fourier transfor with plain python

I am trying to calculate inverse discrete fourier transform for an array of signals. I am using the following formula: And my python code looks as follow. ...
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Plot Frequency Spectrum of Binary Sequence in Matlab

I am new to Matlab and I am trying to implement a section of a published paper, the basic idea of the part that i am implementing is to show the frequency spectrum of camera aperture. The shutter ...
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Basic difference between Fourier transform and laplace transform? [duplicate]

I have read few links about difference between Fourier transform and Laplace transform but still not satisfied Please correct me if i am wrong Simply put, the main difference between Fourier ...
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Why does subbing $s = j\omega$ into the Laplace transform of a cosine wave yield a purely imaginary result?

The Laplace transform of a cosine starting at $t=0$ is given by $$F(s) = \frac{s}{s^2 + \omega_0^2}$$ If I sub in $s = j\omega$, I get the Fourier transform of a cosine starting at $t=0$: $$F(j\...
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2-d circularly symmetric low-pass filter

For a square pixel grid, the ideal 2-d low-pass filter with a horizontal and a vertical cut-off angular frequency $\omega_c$ in radians has an impulse response (kernel) $h_{\small\square}(x, y)$ that ...