# Questions tagged [fourier-transform]

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### How can I reduce the complexity of an inverse DFT where I have a uniform frequency series being evaluated at non-uniform target points?

I have implemented an N-dimensional Non-Uniform Discrete Fourier Transform (in this case it's specifically an inverse NUDFT) using PyTorch. My goal with this implementation is to have a function which ...
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### How does each element in the output array of a DFT correspond to a specific frequency?

I have a basic understanding of the Fourier Transform, though I'm trying to use it in a program and I'm confused on the specifics. Based on source code I can find online, the DFT takes a set of ...
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3 votes
2 answers
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### Check for common element in two arrays using FFT

My task asks me to check whether there is a common element in two sets $\{x_1,x_2,...,x_n\}$, $\{y_1,y_2,...,y_n\}$ with $x_i,y_i\in\mathbb{N}$ using the Fast Fourier Transform (FFT). (I'm aware that ...
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1 vote
1 answer
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### Difference between the Magnitude and the Phase Spectrum of the Fourier Transform?

What is the difference between the Magnitude and the Phase Spectrum of the Fourier Transform?
1 vote
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### Understanding the recursive fast Fourier Transform Algorithm from CLRS

Consider the following recursive fast Fourier transform algorithm from CLRS: I believe that I understand this algorithm correctly: you split the input coefficients into the odd and even terms, ...
1 vote
1 answer
74 views

### What are 'Butterfly Combinations' and why are they called this way?

In some scientific work describing Discrete Fourier Transform-algorithms, such as the well-known Cooley-Tukey algorithm, I came across the term 'Butterfly operations' and 'Butterfly combinations', ...
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2 votes
1 answer
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### What's the connection between the two "Fast Walsh Transform"?

First Let's take a look at the convolution $\displaystyle C _ { i } = \sum _ { j \oplus k = i } A _ { j } * B _ { k }$, and the $\oplus$represents any boolean operation. And we are able to evaluate $C$...
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### Need help implementing an algorithm to solve roots of a transcendental equation

I'm trying to implement this algorithm but I'm having problems reproducing the exemple that it gives a solution to. The general method that I tried is: Make a grid $\theta \in [0,2\pi)$, with say N=...
2 votes
0 answers
65 views

### generalized translation operator on graph [closed]

David IShuman in " vertix-frequency analysis on graph" claims that,"we generalize one of the most important signal processing tools – windowed Fourier analysis – to the graph setting and When we apply ...
3 votes
1 answer
2k views

### Complexity of polynomial interpolation

Polynomial Interpolation in the general case is $O(n^2)$ time complexity, but it can be done better in particular situations. For instance, when the polynomial can be evaluated at the complex roots of ...
1 vote
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### What is the runtime of Quantum Fourier Sampling

I have seen estimates for the runtime of Shor's Algorithm,, which relies on Quantum Fourier Transformations. What is the runtime of those transformations themselves? Either big-O or more accurate ...
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2 votes
1 answer
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### What is the significance of the imaginary parts in the output of an Inverse Fourier Transform? Can I avoid them?

Starting with an image, I have taken its fourier transform and then modified it, and finally taken the inverse fourier transform of the modified fourier transform to get my resulting image. In my ...
1 vote
1 answer
92 views

### Efficiently compute parallel matrix-vector product for block vectors with FFTs?

Assume I have $P$ processors, each having a different vector $v_p$ of size $N$, $p=1, ..., P$. I learned in this question/answer that for computing the matrix-vector product $$w = (E\otimes I_N)v$$ ...
3 votes
1 answer
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4 answers
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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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2 votes
1 answer
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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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1 vote
1 answer
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• 131
3 votes
2 answers
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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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5 votes
1 answer
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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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2 votes
1 answer
289 views

### 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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1 answer
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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 ...
3 votes
1 answer
231 views

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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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1 vote
1 answer
75 views

### 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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3 votes
1 answer
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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 ...
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0 votes
1 answer
532 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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2 votes
1 answer
311 views

### 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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18 votes
1 answer
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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 ...
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5 votes
9 answers
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### Explaining why FFT is faster than DFT for the general public?

How would you explain why the Fast Fourier Transform is faster than the Discrete Fourier Transform, if you had to give a presentation about it for the general (non-mathematical) public?
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1 answer
537 views

### Confusion related to time complexity of fast fourier transform

I have this confusion related to the time complexity of FFT. I was reading this book related to Design and Analysis of Algorithms and I came across FFT. It says that lets say I have a polynomial of ...
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