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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Use samples of Fourier transform as DFT?
Consider the LTI system given by: $H(z) = 1 - \frac{1}{2}z^{-1}+\frac{3}{4}z^{-2}$
Let $x[n] = (\frac{1}{2})^nu[n]$ be the input to the system. We want to find the output for $n = 0,1,...,N_a$, using ...
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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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Getting different spectrum from velocity, and position data using Omega arithmetic
I am solving a very long problem and one part of it requires me solving an ODE and computing FFT of the resultant data. Essentially I have a differential equation$\frac{dz}{dt}$ for velocity, from ...
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Real Fast Fourier Transformation (FFT) in 2D changes with changing the axes order
I am using Python to calculate the real FFT for a 2D array. I found that the real FFT function does not return an array with the same size as the input array, rather, it returns an array with the same ...
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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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Phase difference measurement of a signal sampled with two different sampling frequencies
I am working on phase interferometry for locating a transmitter. The direction of arrival of an incident wave can be estimated from the phase difference caused by the antenna separation as shown
...
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Limitations in Backing Out a Transfer Function
Suppose you have an LTI system for which the (complex) frequency response $H(j\omega)$ has been measured in some frequency window $[\omega_1,\omega_2]$.
Now imagine that you want to provide an input $...
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Why does an anti-symmetric function has zero amplitude at the center of an even length window
I am performing FFT on a real odd function and the resultant transform has zero amplitude in the last bin. Essentially if Y= rfft(X), then Y[-1] is always zero. I stumbled on this answer which says
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Formula for PSD across an axis of a 2D output
Consider a 2D stationary input $e(x,y)$ and a 2D real convolution function $h(x,y)$. Let $S=h*e$ be the result of the convolution of $e$ by $h$.
If needed, we may assume $e$ is isotropic (spectrum ...
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FFT giving a huge magnitude of first frequency and pretty much zero after that
So I have a (visually) very noisy time series signal and I have applied the fast Fourier transform using numpy's fft function. I am wondering why I am seeing the ...
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I want to invert my fourier transform components to waves again
Hi I am using R to analyze some data
I basically did fft(data) and got a vector of complex numbers but from that now I want to remove certain harmonics from my actual wave but how do I convert one of ...
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Mathematically Inclined Signal and Systems / Signal Processing Book Recommendations
I'm an electronics engineering student with high inclination to analysis and pure mathematics. I was just wondering if there was any book ( or any resource ) that treats signal and systems and signal ...
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Does constant Q transform have linearity property in the transformed domain?
With FFT, I sometimes take advantage of the fact that I can pre-calculate a signals Fourier Transform, and then you can add the noise in the spectral domain:
$$ \mathscr{F} (x[n] + h[n]) = \mathscr{F}...
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redundancy of sin and cos waves with real data
I have the following question. Isn't it true that when applying a fourier transform to a real function (i.e. computing a characteristic function for a density), we only ever need one of the two waves: ...
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Difference between Fourier-Transform and FFT of rectangular pulse
I'm trying to find a link between the Fourier-Transformation of aperiodic Signals and the FFT of them.
So to start with a basic example, let's take a rectangular pulse with width 0.1s and amplitude of ...
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371 views
inverse discrete fourier transform with plain python
I am trying to calculate inverse discrete fourier transform for an array of signals.
I am using the following formula:
$$ x[n] = \tfrac1N \sum\limits_{k=0}^{N-1} X[k] \, e^{j 2 \pi k n/N} $$
And my ...
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LTI system output given input and frequency response
The question I'm trying to understand is as follows: A linear time-invariant continuous-time system has the frequency response function $$H(\omega)=\frac{1}{j\omega+1} $$
Compute the output response $...
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Fourier transform of a damped cosine wave with a linear frequency chirp
I want to take the Fourier transform of the following transient signal,
$$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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Does spectral leakage include negative frequencies?
When choosing a window function, window duration, and/or transmission frequency (assuming sampling rate satisfies Nyquist), one may want to understand what sort of spectral leakage would occur at a ...
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Spherical equivalent of Nyquist frequency
Let $\phi$ be a scalar function defined on the surface of a sphere. I have samples of $\phi$ at various locations on the sphere. I want to apply a spherical harmonic transform. I know that $\phi$ is '...
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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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Fourier transformed acceleration from Fourier Transform of velocity and position using Omega arithmetic
Let's say the position of an object is given by simple sine function. By elementary calculus, I can calculate the acceleration in the time domain and find its Fourier transform.
I can also calculate ...
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Difference between discrete time fourier transform and discrete fourier transform
I have read many articles about DTFT and DFT but am not able to discern the difference between the two except for a few visible things like DTFT goes till infinity while DFT is only till N-1. Can ...
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Is a Fourier transform a sound way to analyse a transient signal?
I am currently working on a project that involves analysing transient signals from sensors.
While not actually part of the analysis itself, I discussed it with the team, and they are using an fft to ...
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Impulse response of a transfer function
Lets say we have a transfer function from an LTI-system that goes as follows, and we want the impulse response for it:
$$\frac{10\cdot e^{\left(-\mathrm{i}\right)\cdot \pi\cdot f\cdot 8}}{5+\mathrm{i}\...
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Fast Hartley Transform Implementation in MATLAB
I want to implement Fast Hartley Transform (Specifically Discrete Hartley Transform) in a script file in MATLAB. Does anyone know have a reference implementation of this in MATLAB or another language ...
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Inverse Short Time Fourier Transform algorithm described in words
I'm trying to conceptually understand what is happening when the forward and inverse Short Time Fourier Transforms (STFT) are applied to a discrete time-domain signal. I've found the classic paper by ...
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Finding n Amplitudes by DFT, what is correct normalization
Please forgive me if this has already been asked.
Let us assume an example with $x(t) = \sum_{i=1}^N A_i \sin(2\pi f_i t) $ under a given sampling frequency $f_s$, frequencies $\omega_i$ and ...
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Convolution of a non-symmetrical window function by a cosine signal in the frequency domain
A have a time signal:
The associated DFT spectrum of this signal:
The time signal can be considered as a non-symmetrical rectangular window function multiplied by a cosine signal with a frequency $...
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1answer
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Fourier transform of discrete time unit step function
To obtain fourier transform of u[n],
u[n] - u[n-1] = delta[n] , taking fourier transform of both sides of the equation results in :
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turn circular convolution into linear convolution by zero padding: A special case
We know that, multiplying a kernel and signal spectrum in Fourier domain will lead to a circular convolution and not a linear convolution, so in order to it become linear convolution we must zero pad ...
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Understanding Fourier Transforms in abstract math terms
I am having a hard time implementing a method that computes Fourier transform coefficients for the complex form using the trapezoid rule.
I have floated questions in the math and stackoverflow ...
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solution on the time domain becomes “periodic” after the inverse fourier transform
I was trying to solve european option pricing problem using Conv method (introduced by Lord in 2008 https://pdfs.semanticscholar.org/0632/460bd50b2151f74ac40028df4cc60e73a884.pdf).
The final step of ...
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2answers
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The Fourier transform of a damped cosine and the units of the result
If I take a simple transient voltage signal of the form
$$f(t) = V_p e^{-t/\tau} \cos(\omega_0 t)$$
and take the Fourier transform in the normal way
$$F(\omega) = \frac{V_p}{\sqrt{2 \pi}} \int^{+\...
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Analytically determine a PSD from a transient function
This question is related to a series of questions I have asked about the units of PSD and ESDs. I include it as a separate question as it may have worth in isolation.
As I understand it to compute ...
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What does the frequency axis of a Power Spectral Density mean?
I have never really understood what the frequency axis meant when we plot the Power Spectral Density(PSD).
Does it correspond to frequency as we get after we take the Fourier Transform of a time ...
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Bin sizes for non-uniform discrete Fourier transforms
For a non-uniform discrete Fourier transforms, do the specified frequencies – i.e., $f_k$ in
– refer to the midpoint of the bin or the lower bound? I read the answer here, but that stated that ...
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What is the form of the spectral derivative in the all-positive-frequency notation in DFT?
The Discrete Fourier Transform (DFT) of a function $u:[0,2\pi] \to \mathbb R$ sampled over $N$ equidistant points $\theta_j = 2\pi j/N,\, j = 0, \dots, N-1,$ is defined by
$$
\tilde U_k = \frac1N \...
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Transfer function and deconvolution
Forewords
This question is about methodology references and numerical application. I am posting on Signal Processing because I think this question belong to this place. I am new to the stack, feel ...
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Why do we need the power spectral density?
Since the power spectral density is just the squared of the fourier transform, why is it useful ? Can't I just replace every solution that requires the psd with the fourier transform ?
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Why the unilateral Laplace transform?
Why is the Laplace transform commonly taught as the unilateral Laplace transform?
I mean, for the Fourier transform, we commonly have the bilateral transform... if the signal is 0 for $t<0$, then ...
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1answer
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Palmprint Identification - Why do we align the images before we use the Fourier Transform?
I have been reading the paper PALMPRINT IDENTIFICATION BY FOURIER TRANSFORM by WENXIN LI, DAVID ZHANG and ZHUOQUN XU about identifying persons based on an image of their palm, one version of the paper ...
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How to perform spectral inversion in the frequency domain to convert a low-pass filter into a high-pass filter?
To convert a linear-phase FIR low-pass filter into a high-pass filter with the same cut-off frequency, we can invert the sign of the low-pass filter's impulse response $h(n)$ and then add one to the ...
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DTFT based Frequency Sampling
H($e^{jw}$)= 1, |w| < $\pi/2$ and 0, $\pi/2$ <= |w| <= $\pi$
I took M equally spaced frequencies from 0 to $2\pi$. If we assume h[n] to be causal, $H(e^{jw})$ should have some phase and it'...
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Implementation of Fourier Domain Denoising with Hard Threshold
I just tried the Fourier denoising method with a hard threshold and my code is as follow:
...
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Fastest implementation of fft in C++?
I have a MATLAB program that uses fft and ifft a lot. Now I want to translate it to C++ for production. I used OpenCV but I ...
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Multitaper F statistics
I'm having some problems interpreting the F-statistics output from multitaper analysis. To illustrate, the following code-snip in R performs multitaper analysis on the same sine-frequency but with ...
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Scaling factor for comparing spectrums obtained via Continuous Fourier Transform and Discrete Fourier Transformation?
Essentially I am trying to calculate the Bremsstrahlung spectrum numerically for magnetized plasma and want to compare the resultant spectra with the standard textbook spectrum formula for ...
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1answer
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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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How to visualize high-pass filtered images?
When I try visualizing a high-pass filtered image, all I see is gray, similar to the middle subfigure in the attached figure from a related paper. The authors claim they normalize the image to have ...
