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 ...
- 2,688
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Poisson summation from $n=1$ to $\infty $
If we accept this as valid Proof of Poisson summation
$\displaystyle \sum_{n=-\infty}^{\infty}\hat {f}(t) dt= \sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}f (t)e^{-2 i \pi n t} dt= \int_{-\infty}^...
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Discrete Fourier Transform of repeating sequences
Based mostly on W. Briggs and V.E. Henson’s “The DFT : An owner’s manual to the Discrete Fourier Transform” chapter 3 and question 57 on rarified and repeated sequences, I would like to find the ...
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If $f(x,t) = g(u)$, how do I relate $\iint{fdxdt}$ to $\int{gdu}$?
I'm learning about Fourier Transforms in the context of travelling waves on a dispersive medium, and my textbook sort of handwaves a simplification in which
$$ \textbf{F}(k,\omega) = \int_{-\infty}^{\...
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Problem with equation $u_{xx}+4u_{yy}=0$
Given task:
Find a solution to the boundary value problem in the domain $\left \{ x>0, y>0 \right \}$ in the class of bounded functions using the Fourier transform
$$u_{xx}+4u_{yy}=0, u|_{x=0}=\...
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Construct Circulant matrix from 2D convolution integral
I have a known function $d(x,y) = \frac{k}{\sqrt{x^2+y^2}+k}$ where ($k = $constant). I also have the 2D convolution integral:
\begin{equation}
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} w(x_0,y_0)...
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Why is considering $\delta(\mu)$ as Fourier transform for $f(t) = 1$ rational?
We know for a function $f(t)$, Fourier transform is defined as:
\begin{equation}
F(\mu) = \int_{-\infty}^{\infty} f(t) e^{-i2\pi \mu t} dt
\end{equation}
I saw the Fourier transformation for $f(t) = 1$...
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Integrating the exponential of a fractional order complex polynomial
This post potentially has two questions related to the following integral:
$$
\int_{-\infty}^{\infty}e^{i(ax-bx^{s})}dx
$$
where $1<s\leq2,$ $a\in\mathbb{R}$ and $b>0$.
The first question: is ...
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PDE fourier series and fourier transform periodicity [closed]
can someone help with this question? Much appreciated for your time and effort!
Consider v(x, y, t) satisfying the PDE
PDE
We know that u,U,Q have a period 2 in y. In addition, we assume |u| → 0 as |x|...
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Under this condition, is $\mathcal Ff$ in $L^1$?
Let me denote Fourier transform by $\mathcal F.$
Proposition
Let $f:\mathbb R\to \mathbb C$ is twice differentiable and suppose $f,f',f''\in L^1\cap C$ and $f(x), f'(x)\to 0$ as $|x|\to\infty.$ Then, ...
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What is the domain and codomain of Fourier inverse transform $\mathcal F^{-1}$?
The Fourier transform $\mathcal F$ is defined by
$$Ff(\xi)=\int_{-\infty}^\infty f(t)e^{-i\xi t}dt$$
Generally, the domain of $\mathcal F$ is $L^1$ since otherwise the integral isn't defined, and the ...
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Fourier transform on time integral
We define the Fourier transformation as
$$
\int_{\mathbb{R}} e^{-iwt}dt=\hat{f}(w)
$$
If we apply the Fourier transformation on
$$
\int_{\Omega}\int_0^{\infty}f(x,t)dtdx
$$
by definition is
$$
\int_{\...
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Use of Fubini's Theorem in Papa Rudin's Holomorphic Fourier Transforms
I am starting to read on chapter 19, Holomorphic Fourier Transforms from Real and Complex Analysis by Walter Rudin. In the first page of that chapter I came across the function $$f(z) = \int_0^\infty ...
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Probabilistic interpretation of Fourier Transform
The fourier transform is given as below :
$$F(\omega) = \frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty} f(t) e^{-i \omega t} \mathrm{d}t$$
Now $$F(\omega) = \frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty} ...
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Mellin transform defined for function on group $(\Bbb R^+,\times)$ but integration domain is over semigroup
I came across this question: Mellin transform defined for function on group $(\mathbb{R}^{+}, \times)$.
Let $\varphi_x: s\mapsto x^s $ be a group isomorphism from $(\Bbb R,+)$ to $(\Bbb R^+,\times).$
...
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What does the comma mean in the mathematical expression? [closed]
While reading the Wikipedia page on Fourier-transform spectroscopy, in the section "Extracting the spectrum", I found the following
$$ I(p) = \bigg(p, \int_0^\infty I(p, v) \, {\rm d} v\bigg)...
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Fourier transform of $x \in \mathbb{R} \mapsto (1+x^2)^{-\alpha}$
I hope that the answer is not somewhere in the forum. I searched without success.
I would like to compute the fourier transform of $f:x \in \mathbb{R} \mapsto (1+x^2)^{-\alpha}$ where $\alpha > 1/2$...
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How to understand the analog frequency and the discrete-time frequency in digital signal processing?
In disgital signal processing, we can analyze the sampling of continuous-time signals. We shall come to this understanding: the analog frequency $\Omega$ times the sampling period $T$, is the discrete-...
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Equivalence condition using fourier transforms
Consider the following question
Let $f$ be a function such that $\int_{-\infty}^{+\infty}|f(x)|^2 d x<\infty$. Prove that $\int_{-\infty}^{+\infty} f(x+k) \overline{f(x+l)} d x=0$ for all integers ...
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What is the origin of $ \hat{F} = \int_{-\infty}^{\infty} f(x) e^{-i2 \pi \omega x} \: dx$ ??
The formula for computing the coefficients of Fourier series is:
$$ C_{k} = \frac{1}{2L} \int_{-L}^{L} f(x) e^{-i2 \pi \omega_{k} x} \: dx$$
where $\omega_{k} = \frac{k}{2L}$.
I understand that we can ...
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Find the complex Fourier transform of $f(x) = \dfrac{1}{1 + x^2 + x^4}$.
How do I find the complex Fourier transform of $f(x) = \dfrac{1}{1 + x^2 + x^4}$?
I know that the complex Fourier transform is $\hat{f}(k) = \displaystyle \int_{-\infty}^{\infty} f(x)e^{-ikx}\ dx$.
...
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Deriving the Fourier Transform of Unit Step Function...
I was trying to derive the fourier transform of the step function explicitly, and i came up with doing the same reasonements we can see in the askers' attempts in their respective questions:
...
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Bounding $\widehat{G_{m+1}}\ast\widehat{H_m}(0)$ when $\frac{1}{2}\leq\widehat{H_m}(0)\leq\frac{3}{2}$ and $H_m,G_{m+1}$ are smooth over $\mathbb{T}$
To put this question into proper context, what I am asking is related to the construction of smooth function $H_m$ over the torus $\mathbb{T}$ such that
$$\left|\widehat{H_m}(k)\right|\leq C\log(\left|...
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Logarithmic-Fourier type integral transform
I'm working with transforms of the kind $$\int_{-\infty}^\infty f(x)e^{it\log(x)} dx \qquad (t\in \mathbb{R}),$$
where $f \in L^1(\mathbb{R})$ and you have fixed a branch of the complex logarithm in $\...
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Defining Fourier Transfrom using Banach Algebra from $L^1(\mathbb R)^n \to L^1(\mathbb R)^n$
Context
A way to define the Fourier transform on $L^1$ is by finding the sets of all homomorphism $\left\{ \phi \right\}$ from $L^1(\mathbb R) \to L^1(\mathbb R)$ where in the domain the product is ...
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Infinite sum of positive semidefinite tempered distributions
Suppose that we have a sequence on non-positive functions $\{f_i(x)\}\in C({\mathbb R})$ such that $f(x)=\sum_{i}f_i(x)$ has a moderate growth rate, e.g. polynomial. Let us denote the corresponding ...
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Why does it hold $\operatorname E\left[\left|\int g\right|^2\right]=\int\operatorname E[|g|^2]$ here?
Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be Lebesgue integrable and vanishing outside $[0,1)^d$. Moreover, let $k\in\mathbb N$ and $x_1,\ldots,x_k\in[0,1)^d$. Let $w_1,\ldots,w_k\ge0$ with $\...
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3D Fourier transform on the Single Particle Green's Function
I'm having some troubles understanding how to use the 3D Fourier transform on this particular example:
$$(-\nu ^2 + \nabla ^2) G(\vec{x}) = \delta ^3(\vec{x})$$
Applying a 3D Fourier Transform:
$$(-\...
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Fourier transform of exponentially decaying sinusoid
Calculate $$f(w) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-iwx} {\rm d} x$$ where $$f(x) = \begin{cases} e^{-x} \sin(x), & x \geq 0 \\ 0, & x < 0\end{cases}$$
My solution ...
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What's the fourier transform convolution of $f(t) = \sin(t) \cdot \exp(-t)$ and $g(t) = \cos(t) \cdot \exp(-t)$? [closed]
In finance studies, maybe is necessary use Fourier Transform and convolution applications. In the book Introductory Mathematical Analysis for Quantitative Finance, at chapter 12 is proposed the ...
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Is the approximation of the integral of $f$ by $\sum_{i=1}^kw_if(x_i)$ exact assuming $f$ has no high frequency components?
Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be Lebesgue integrable and vanishing outside $[0,1)^d$. Moreover, let $k\in\mathbb N$ and $x_1,\ldots,x_k\in[0,1)^d$. Let $w_1,\ldots,w_k\ge0$ with $\...
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How fourier transform of a tempered distribution could be a function (Homogeneous Sobolev space)?
I met the following definition of homogeneous Sobolev space:
$\dot{H^s}(\mathbb{R}^n) = \left\{ f \in \mathcal{S}': \hat{f} \in L_{loc}^1(\mathbb{R}^n) \,\,\, \text{and} \,\,\, || f||_{\dot{H^s}} < ...
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Finding the relation between Laplace and the CTFT
Let us take a system where the input is $V_{i}(t)$ and output is $V_{o}(t)$ and the impulse response of the system be $I(t)$ where $t$ represents time domain and $w$ be frequency. we get $\tag{1}V_{o}(...
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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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Fourier Transform of $\frac1{\cosh^2(x)}$.
How do you take the Fourier transform of
$f(x)= \dfrac{1}{\cosh^2(x)}$, using contour integration? (Where the complex Fourier transform of $f(x)$ is defined to be $\hat{f}(k) = \displaystyle \int_{-\...
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Bijection between complex valued functions on the unit circle and on the real line
Let $\mathbb{T}$ be the unit circle in the complex plane, that is $\mathbb{T}=\{z\in\mathbb{C}: |z| = 1\}$. I have read that there is a bijection between the class of complex-valued functions on $\...
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Computing Sobolev norm via discrete cosine transform
While reading a paper, I faced an identity about the relationship between the Sobolev norm and the Discrete Cosine Transform(DCT) which states
if $f \in H^s(\Omega)$ is a non-periodic function, and $\...
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The Fourier transform of $p(r)$
We are given the function $p(r)$ as:
$$p(r)=\int_{-\infty}^{\infty}\frac{1}{2i}k(S(k)-1)e^{ikr}dk$$
with $S(k)$ being an even function. What is the Fourier transform $P(k)$ of $p(r)$?
I have no idea ...
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How many phase-shifted samples are required to eliminate the harmonics of a complex wave?
Let $w$ be a complex periodic wave with an infinite number of harmonics. What is the required number of samples $N$, phase-shifted incrementally by equal steps of $2\pi/N$, in order to eliminate ...
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Computing Fourier Transform of Triangle Function
I am trying to compute the Fourier transform of the triangle function
\begin{equation}
f(x) =
\begin{cases}
1 -| x | & \text{if}~~ | x |\leq 1 \\
0 & \text{if}~~ | x | > 1
\end{cases}
\...
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Spectral represenation of a Monte Carlo estimator and variance bound
Let $\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$, $d\in\mathbb N$, $\Lambda\in\mathcal B(\mathbb R)^{\otimes d}$ with $\lambda^{\otimes d}(\Lambda)\in(0,\infty)$ and $f:\Lambda\to\...
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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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Constrained Optimization using FFT to find function
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{...
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Partial Differential Equations: Fourier Transform in Space and Time?
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 $\psi(x,t)$ ...
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