Questions tagged [fourier-transform]

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2D fourier transform

I an new to image processing and I have some questions I want to understand: What is the meaning of the imaginary and real part of the 2d fourier transform? What can it tell me about the image it ...
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Fast calculations using fast-Fourier convolution

Consider an array $X$ with shape $H \times W$. Let $Y$ be the other array of the same shape and $Z$ is an array of shape $h \times w$. We want to construct an array $R$ of shape $(H-h + 1, W-w+ 1)$ by ...
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Fourier Transform of t/(t^2+1)^2

Can you help me with what steps to follow to find the Fourier transform of t/(t^2+1)^2. I know it will be solved on the lines of time-frequency duality but cannot seem to get the exact steps for it. ...
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Is there a good analogy between spectral representation of a signal and graph theory?

I am working on some time series problems where the Fourier representation of the signal in the frequency domain is also important. I am wondering if there is any connection between time series ...
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Use of FFT in order to multiply two positive integers of $n$ digits

We're given two positive integers $t$ and $s$ of length $n$ in binary representation. Suppose we divide the numbers into ${n \over k}$ blocks of size $k=\lg n$ using FFT algorithm. Suppose ...
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Expected Behavior of FFT

I am debugging an FFT. My FFT uses complex numbers in polar form, $(r,\theta)$. I am trying to establish that it works, with a unit test. My real values are all correct, after a polynomial ...
312 views

Cost of solving a matrix equation using the FFT

I am trying to calculate $$V = (H^TH+I)^{-1} U$$ where $H\in\mathbb{R}^{m\times m}$ is a circulant convolution matrix corresponding to a convolution kernel $h$, and $U\in\mathbb{R}^{m\times n}$. ...
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Calculating force between n points placed on the x-axis

There are $n$ charges placed on the x-axis at points $1,2,3,\dots,n$. We need to calculate the force on each charge by every other charge. I need the exact force. Charges can be arbitrary. Inputs are ...
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Why does a MP3 encoder use a fast Fourier transform before applying the psychoacoustic model?

Karlheinz Brandenburg depicts a MP3 encoder like this: Source: MP3 and AAC Explained I marked the FFT as I'm not quite sure why it is actually necessary to perform one. Why can't the psychoacoustic ...
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Speed up minimizing quadratic function by FFT

I'm trying to understand the following excerpt from a paper: Subproblem 1: computing $S$. The $S$ estimation subproblem corresponds to minimizing  \sum_{p}(S_p - I_p)^2 + \beta((\partial_xS_p -...
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How does the Stockham FFT work?

I'm trying to implement a Fourier transform in an OpenGL shader. I found some literature that explains the general principles, but is a bit sparse on some details. And those details are seriously ...
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Compare phase and magnitude spectrum results of 2 Images

I have read that from Fourier transform we obtain magnitude and phase spectrum. The magnitude spectrum tells you how strong are the harmonics in a image and the phase spectrum tells where this ...
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Morgenstern proof for FFT lower bound

I looked at my notes from a class about fast forier transform , and the professor proved in class theorem thanks to Morgenstern , first he defined linear algorithm as a algorithm that inly uses ...
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Pruned FFT runtime

Pruned fast Fourier transforms compute only a specified subset of the result indices in faster time, although sometimes with a slower implementation constant (because FFT is generally so optimized). ...
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What is the reason that is FFT multiplication slower than other methods for small N?

I've seen plenty of statements in papers and on websites that Fast Fourier Transform-based multiplication algorithms are slower than other multiplication algorithms for relatively small input size N, ...
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How to evaluate all derivatives of a polynomial at a point with FFT? [closed]

I found this problem: Evaluating all derivatives of a polynomial at a point Given a polynomial A(x) of degree-bound n, its tth derivative is defined by From the coefficient ...
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Show how to do FFT by hand

Say you have two polynomials: $3 + x$ and $2x^2 + 2$. I'm trying to understand how FFT helps us multiply these two polynomials. However, I can't find any worked out examples. Can someone show me how ...
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(Slightly) faster simulation of quantum Fourier transform

Suppose I want to write a classical software simulator of a quantum circuit with $N$ qubits. When it comes time to simulate the quantum Fourier transform I can evaluate all $2^N$ states to determine ...
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How can you see which points in the spectrum is from which pixel in the original image?

Take the image and spectrum below. If I look at the spectrum, it just look like noise.... How to make sense of it intuitively? Image: Frequency spectrum of image (using Fourier Transform):
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What the difference between the Fourier Transform of an image and an image histogram?

Consider this small picture of a sunflower, and its histogram: What would the Fourier transform of the first picture look like? Is there any relationship between the histogram and the Fourier ...
426 views

Intuitive way to understand the triangle spectrum?

Image on the top is in the time domain, image on the bottom is in the frequency domain. Why do we see -2T and 2T on image of the time domain and why do we see -1/2T and 1/2T of the image in the ...
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How to distinguish between the different frequency domains?

Sometimes the terms 'Fourier domain', 'complex frequency domain', 'Frequency domain' and 's domain' are used interchangeably. Take these answers here for example. Can you really use them ...
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FFT-less $O(n\log n)$ algorithm for pairwise sums

Suppose we are given $n$ distinct integers $a_1, a_2, \dots, a_n$, such that $0 \le a_i \le kn$ for some constant $k \gt 0$, and for all $i$. We are interested in finding the counts of all the ...