# Questions tagged [fourier-transform]

The Fourier transform is important in mathematics, engineering, and the physical sciences. It is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines. The Fourier Transform shows that any waveform can be re-written as the sum of sinusoidal functions. It is named after French mathematician Joseph Fourier.

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### Deriving a formula for Discrete Cosine Transform from Discrete Fourier Transform

I'm trying to derive a formula for Discrete Cosine Transform (DCT) from Discrete Fourier Transform (DFT). I've been trying with Euler's formula $e^{ix} = \cos(x) + i\sin(x)$ and double angle formulas. ...
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### What should be the size of my FFT values for speed,acceleration,..?

I am not from electrical eng. or physics background, so a layman explanation would be appreciated. I work with sensor data (accelerometer) from wearable device, collected for few hours. I take few ...
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### Term by term Fourier Transform of Taylor Series expansion

Taylor series of $\frac{\sin(x)}{x}$ is given by $1 - \frac{x^2}{6} + \frac{x^4}{120} + ...$ We know that the Fourier Transform of this sinc function is a scaled Rect function. Why is it incorrect to ...
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### Proof of Riemann-Lebesgue lemma. What is the dominated function in this case?

I'm reading a proof of Riemann-Lebesgue lemma (about Fourier transform) on wikipedia https://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma This uses DCT but I wonder what is the dominated ...
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### Time-domain functions $f(t)$ whose Fourier Transform is $f(w)$

The Fourier Transform of a Gaussian is another Gaussian with $w$ in place of $t$. Are there other functions whose Fourier Transform results in the same function as the original time-domain function (...
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### Generalised Fourier Transform for Spherical Bessel Functions

I'm trying to find to solve the following PDE: $$\nabla ^2\phi(\vec{x}) +k^2 \phi(\vec{x})=0$$ In spherical coordinates and then use the Generalised Fourier Transform to extract 2 linearly independent ...
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### Laplace transform of right/left derivative?

Assume the derivative of a function $f$ does not exist everywhere, let's say that it exists everywhere except on a countable set, and that it is continuous between each two successive points of this ...
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### Calculating dot product of fourier amplitude spectra through quadratic form in signal space?

Background I have a signal $X \in \mathbb{R}^d$. Let's define the following terms: $F_X \in \mathbb{C}^d$ is the discrete Fourier transform of $X$. Each element is a complex number with a Fourier ...
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### How to transform a linear system?

Consider the following system $$y_i=\sum_{|j-i|\leq k} x_{j}$$ for some $k< \lfloor (n-1)/2\rfloor$, and with indexes in the $\mathbb{Z}/n\mathbb{Z}$ ring. Essentially, $y_i$ is the sum of $x_i$ ...
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I want to find the function $W(x)$ from the following optimization problem: \begin{equation} \textrm{min} \left(I(x) - \int_{-\infty}^{\infty} W(x_0) d(x-x_0) dx_0\right)^2 \end{equation} \begin{...
Consider the one-dimensional wave equation: $$\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi(x, t) = \frac{\partial^2}{\partial x^2}\psi(x, t).$$ We can take the Fourier transform of $\psi(x,t)$ ...