Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [fourier-transform]

continuous Fourier transform, discrete Fourier transform (DFT), discrete cosine transform (DCT), discrete sine transform (DST)

0
votes
1answer
11 views

Is a smooth function with compact support in $\mathbb{R}$ can be written as the convolution of two square integrable functions?

Suppose h is smooth compact supported function on $\mathbb{R}$. How to show that there exist $f, g \in L^2(\mathbb{R}, m)$, where $m$ is the usual Lebsegue measure, such that $h = f * g $, where $*$ ...
0
votes
1answer
25 views

$\lVert f \rVert _{H^{s}(\mathbb{R}^d )} \leq C \lVert f \rVert_{L^{1} (\mathbb{R}^d )}$ cannot hold at $ s= -d/2$.

How do I show that $\lVert f \rVert _{H^{s}(\mathbb{R}^d )} \leq C \lVert f \rVert_{L^{1} (\mathbb{R}^d )}$ cannot hold at $ s= -d/2$ ? The definition is, $\lVert f \rVert_{H^{s}} = \lVert \langle \...
0
votes
0answers
14 views

Inverse Fourier transform (in frequency) of a rectangular pulse

This is my first question on here so I'm new to the formatting and all so please be indulgent :) I have an exam question where I am given a function H(f) that is a rectangular pulse between -fc and ...
1
vote
0answers
30 views

Compound Binomial-Exponential: Closed form for the PDF?

Setup: Consider the random variable $Y_N$ derived from $$Y_N = \sum_{i=0}^N X_i$$ where $N$ is a random variable with distribution $p_n = {M \choose n} p^n q^{M-n}$ (binomial i.e. $M$ Bernoulli ...
-1
votes
0answers
27 views

Why is a dummy variable needed when proving Parseval's theorem?

As far as I'm aware, in order to prove Parseval's theorem: $$\frac{1}{2\pi} \int_{-\infty}^{\infty} \mid y(x) \mid^2 = \int_{-\infty}^{\infty} \mid \widetilde y(k)\mid^2 dk$$ The following is done: ...
0
votes
0answers
9 views

Spectrogram approximation.

Say we have a discrete signal S, we derive another one, which is four-times the length S_hat = INTERPOLATE(S, mode = "linear") and we have STFT = short time fourier transformation(filter length = 512, ...
1
vote
0answers
40 views

Closed from of $\int_0^{\infty} \frac{e^{iax}}{x^{n}+1}dx$?

I've been trying to find the general form of a certain group of integrals of the form$$I(a,n)=\int_0^{\infty} \frac{e^{iax}}{x^{n}+1}dx$$ I know that the real part of $I(a,2)$ can be calculated using ...
0
votes
0answers
21 views

Signal Processing and the Fourier Transform

I'm working on this problem where I need to find the Fourier Transform of $$ f(t)\approx f_P=\sum_{k\epsilon\ \mathbb{Z}}[\sum_{n=-N}^{N}\widehat {(f_{k})}[n]e^{2\pi int/T}]X_{[kT,(k+1)T]}(t) $$ ...
0
votes
1answer
18 views

What is the Inverse Fourier Transform [on hold]

what is the inverse fourier transform for: $(6w^2 + 20 )/( w^4 + 7w^2 + 12)$ I tried to simplify it and look it up from the table but it didn't work, can someone give me a hint?
0
votes
1answer
33 views

Weak derivative

I really need help with this exercise: Let $f \in L_2 (\mathbb{R})$. Show the equivalence of the following statements: (a) $f \in H_1 (\mathbb{R})$. (b) The function $\xi \mapsto \xi \hat{f}(\xi) \...
0
votes
0answers
20 views

Fourier transform of given function

$ f(x,t) = exp{\dfrac{(-t^2-x^2)}{2a}}*cos(ct-bx) $ Find $ \hat{f}(k,\omega) $ using fourier transform.
1
vote
0answers
24 views

Integrability of the Fourier transform in Sobolev space

I believe that the statement below is a standard fact but I haven't figured out yet: Suppose $f\in L^{1}(\mathbb{R}^{n})$ has integrable partial derivatives of order $n+1$ and $D^{\alpha}f\in L^{1}(...
3
votes
0answers
29 views

Convolution operator on $\ell^\infty_{\mathbb{Z}}$

Let $P_t \colon \ell^\infty_\mathbb{Z} \to \ell^\infty_\mathbb{Z}$ the map defined by $$ \widehat{P_t(f)}(\xi) =e^{-t \sin \pi \xi}\hat{f}(\xi) $$ where $f \in \ell^\infty_{\mathbb{Z}}$, where $$ \hat{...
0
votes
0answers
19 views

Problem on computing the Fourier transform of the Gaussian

The problem sounds like this. Show that $s\to\int_\mathbb{R}e^{-(x+is)^2}dx$ is constant wrt $s\in\mathbb{R}.$ Then use this fact to shot that $\mathcal{F}(e^{-a|x|^2})=e^{-\frac{|x|^2}{a}}$ ...
0
votes
0answers
11 views

How does the following hold?

For a vector valued function $f(t)\in{L_{2}}~[0,\infty)$ with Laplace transform of $f(t)$ being $F(s)$, how to show that the following holds: $\dfrac{1}{2\pi}\int_{-\infty}^{\infty}|F(j\omega)|^2~d\...
0
votes
0answers
31 views

Effect of $\sin^{-1}$ on $ f(x) = \frac{\sin(ax+b)}{\sin(cx+d)}$

I am working on some maths and I would like to find an analytical solution/closed form solution for my formula. It's the real part of the k-th bin of an N-points DFT of $cos(\frac{2\pi}{c}(1+x))$. $$ ...
1
vote
0answers
14 views

Orthogonality of integer shifts and sum of fourier transforms

A function $\psi \in L_2(\mathbb{R})$ is orthogonal to all integer shifts of a function $\varphi \in L_2(\mathbb{R})$ if and if only $$\sum_{k\in \mathbb{z}} \hat{\varphi}(\xi+k)\overline{\hat{\psi}(\...
0
votes
0answers
23 views

Fourier transform of $f(t)\cos(at)$ given fourier transform of $f(t)$

When the fourier transform of $f(t)$ is $F(w)$, how can I find the fourier transform of $f(t)\cos(at)$? Do I need to use some of fourier transform properties?
2
votes
1answer
37 views

How to minimize sum of matrix-convolutions?

Given $A$, what should be B so that $\lVert I \circledast A - I \circledast B \rVert _2$ is minimal for any $I$? $I \in \mathbb{R}^{20x20}, A \in \mathbb{R}^{5x5}, B \in \mathbb{R}^{3x3}. $ Note ...
0
votes
0answers
13 views

Discrete Fourier Transforms, showing the transform of a product of a series

If $a=(a_{1}, \ldots,a_{n})$. Define $F_{a}( \lambda)= n^{-1/2} \sum\limits_{t=1}^{n} a_{t}e^{-it \lambda}$. Let $\lbrace x_{1}, \ldots,x_{n} \rbrace$ and $\lbrace y_{1},\ldots,y_{n}\rbrace$ be real ...
0
votes
1answer
31 views

Strict Inequality for Fourier Coefficients

I've been trying to solve this inequality but I only get the obvious part which is the $\leq$ part. I need the $<$. The problem is the following: Given a subset $A\subset [0,1)$ of measure not $0$,...
-1
votes
1answer
36 views

Fourier transformation of $e^{-ax^2}$

Find the Fourier transformation of $e^{-ax^2}$, for any constant $a>0$. In general, $$\int \frac{1}{\sqrt{(2\pi \sigma^2)}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} e^{-iλx }dx = e^{-i\muλ-\frac{1}{2}\sigma^...
1
vote
2answers
30 views

Why is $\int_{-1}^1 (1-|t|) \cdot e^{-i \omega t} dt$ equal to $2 \cdot \int_0^1 (1-t) \cdot \cos{\omega t}\, dt$

I have a task where I should calculate the fourier transform of $$ \Delta(t) = \begin{cases} (1-|t|)& |t| \le 1 \\ 0 & |t| > 1 \end{cases} $$ The solution says, that $$ \int_{-1}^1 (1-|t|)...
0
votes
1answer
34 views

How can I calculate Fourier transform of this 3D function? [closed]

Calculate Fourier transform of function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ defined by this formula $$f(\mathbf{x})=\frac{1}{1+|\mathbf{x}|^2}.$$
0
votes
0answers
14 views

How can I solve this linear partial differential equation of 2 variables with Fourier transform?

For $x\in \mathbb{R}$ solve using Fourier transform $$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}-\gamma u,$$ where $k, \gamma$ are positive constants and $u(x,t)|_{t=0}=f(x).$ ...
2
votes
1answer
25 views

Inverse Fourier transform with a $\delta$-function integrand

I'm self-studying math and trying to find the inverse Fourier transform of $\frac{4+w^2}{1+w^2}(4\pi * (\delta(w-2)+\delta(w+2)))$ Based on wolframalpha, the result is $32/5\sqrt{2\pi}\cos(2t)$. But ...
0
votes
0answers
21 views

Informations about Fourier Transform for a Python project (sound manipulation)

I have a project in Python with a friend where we want to manipulate music sound, and we'll need Fourier Transforms, so I made research online and wanted to know if I understand correctly the concepts....
1
vote
1answer
105 views

Fourier Transform - Duality formula. What are the necessary conditions?

I've come across two contradicting statements which I'd be glad if you could help me resolve: Theorem: if $f\left( x \right)$ is continuous and absolutely integrable ($\int\limits_{ - \infty }^\infty ...
-1
votes
0answers
29 views

Inverse Fourier Function

$X(\omega)=\cos(4\omega)$ I started the problem this way. $$\frac{1}{2\pi}\int_{-\infty}^\infty F(\omega)e^{i\omega t} d\omega $$ $$\frac{1}{2\pi}\int_{-\infty}^\infty \cos (4\omega)e^{i\omega t} d\...
0
votes
1answer
39 views

Solving ODE with Fourier Transform: $u(x) - u''(x) = x^2$

Solve: $$u(x) - u''(x) = x^2$$ You can use: $$ \mathcal{F} \{ e^{-a|x|} \} = \frac{2a}{a^2 + s^2} $$ I am new to Fourier transforms. I understand that limits have to be used but don't know how to ...
0
votes
0answers
10 views

Sinusoidal decomposition of signal

I have some data of periodic nature. The curve seems to be slightly irregular, and it makes sense to consider it as the sum of two or more different sinusoidals. I'm asking for a source or tools ...
1
vote
1answer
37 views

Show that $\hat{f}$ is continuous and $\hat{f(\xi)}\rightarrow 0 $ as $|\xi|\rightarrow 0$

Suppose $f$ continuous and of moderate decrease. Denote the Fourier Transform of $f$ by $\hat{f}$. I need to prove that $\hat{f}$ is continuous and $\hat{f(\xi)}\rightarrow 0 $ as $|\xi|\rightarrow 0$....
0
votes
1answer
34 views

How to calculate this $\sum\limits_{n=0}^{\infty}\frac{n}{2^n}e^{jwn}$

How to calculate this $\sum\limits_{n=0}^{\infty}\frac{n}{2^n}e^{-jwn}$,because it is not geometric progression,so i can't know how to solve it,can anyone help me?
1
vote
1answer
31 views

Derivative of Fourier transform using residue theorem

If we define the fourier transform of f as $$\hat{f}(\omega) = \frac{1}{\sqrt{2}}\int_{-\infty}^\infty f(x) e^{-i\omega x} dx$$ then if f is differentiable, and the integrals for $\hat{f}$ and $\...
1
vote
0answers
27 views

Fourier and Mellin transforms of Hilbert Transform

I am reading Hilbert transform recently and meet two questions. The book I am reading is Debnath and Bhatta "Integral Transforms and Their Applications". If we define the Hilbert transform on the ...
-1
votes
0answers
18 views

Fourier Transform of a funtion not defined for all real numbers

I am suposed to solve a partial differential equation using Fourier Transform. At some point I need to find the Fourier Transform of the funtion: $$ U(x,0)= 100 \quad \text{if}\quad1 \le x \le 1 \\ ...
0
votes
0answers
12 views

integral equation and Fourier transform of “almost” the convolution

I am facing an integral equation where one of the terms looks like this: $$ V(t) = \int_t^{+\infty} K(x-t) \cdot V(x) \, dx $$ where $$K(x) = N(\frac{-b-a\sigma^2}{\sigma \sqrt{x}}) - d^{-2a}N(\frac{...
0
votes
1answer
42 views

Finding inverse Fourier transform of $\frac{1}{(1+iw)^2}$

Find the inverse Fourier transform of the function $$ \frac{1}{(1+iw)^2} $$ So I know the inverse is given by the integral $$ \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty \frac{e^{iwx}}{(1+iw)^2}dw $...
0
votes
2answers
23 views

How to calculate 8-point FFT of data by hand

Given the following data: Two period sine; samples = [0, 1, 0, -1, 0, 1, 0, -1]; I am asked to calculate the FFT of the sampled data to find the complex coefficients. I don't necessarily want the ...
0
votes
1answer
22 views

Adaptive knot selection for B-spline fitting.

When fitting a B-spline for regression purposes I've seen a lot of cases where knots are fixed uniformly ,but in some situations this could lead to poor estimations because the behaviour of the curve ...
0
votes
1answer
22 views

Can this Fourier transformed function be transformed into partial fraction?

Hi I'm self learning stochastic process, I've come across a problem and found $$H=\frac{2+j(2w)}{(1-w^2)+j(2w)} \\ |H(iw)|^2=\frac{2^2+(2w)^2}{(1-w^2)^2+(2w)^2} \\$$ In attempt to find the power ...
1
vote
0answers
47 views

Limit of sin(x) as x approaches infinity

This question comes from Fourier Transforms, specifically the evaluation of $\mathcal{F}(e^{2\pi iat})$. Normally I see this derived by first finding the Inverse FT of a delta function, i.e. \begin{...
0
votes
0answers
8 views

Discrete Hartley Transform when N=8

Hi I am currently studying FHT Algorithm but have a hard time to grasp it. I want to calculate FHT when N=8. http://mathworld.wolfram.com/HartleyTransform.html thanks to (19) of the upper link, the ...
-1
votes
0answers
28 views

Lighthill's Proof of Fourier's Inversion Theorem

In Lighthill's "Introduction to Fourier Analysis and Generalised Functions," page 16 Theorem 4, he claims "If $g(y)$ is the F.T. of a good function $f(x)$, then $f(y)$ is the F.T. of $g(-x)$. He ...
3
votes
1answer
25 views

Fourier Transform of $\cos x$ on finite interval

Let $f(x)=\cos x$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$, and $0$ everywhere else. Then $\tilde f(\omega)$ becomes $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos (x) e^{-i \omega x} dx=\frac{2\cos(\...
0
votes
0answers
15 views

$\hat{f}\in L^1\implies \hat{g}\in L^1$ ($g(t)=e^{it^2y}f(t)$)

Let $f\in L^{1}(\mathbb R)$ such that its Fourier transform is also in $L^1(\mathbb R).$ Define $$g(t)=e^{it^2y}f(t), t\in \mathbb R$$ and fixed $0\neq y \in \mathbb R.$ Question: Can we say that ...
4
votes
1answer
61 views

Is the Fourier transform of a continuous function necessarily in $L^{\infty}$?

This is a followup to this earlier question. Given a (let's say continuous) function $f:\mathbb{R} \to \mathbb{C}$ such that the Fourier Transform $$\widehat{f}(y)=\int_{\mathbb{R}} f(x)e^{-2\pi i xy} ...
2
votes
1answer
73 views

Does the existence of all the Fourier transforms imply $f \in L^1$?

It is trivial that for $f \in L^1$ and $y \in \mathbb{R}$ the integral $$\int_{\mathbb{R}} f(x)e^{-2\pi i xy} dx$$ converges. I was just wondering whether the converse is true: Given a (let's say ...
2
votes
1answer
89 views

Is the mentioned method appropriate to solve $\int_{-\infty}^{\infty}\frac{\sin x}{x}\, dx$? [duplicate]

The integral is, $$I=\int_{-\infty}^{\infty}\frac{\sin x}{x}\, dx$$ I know the answer would be $\pi$ and I know how to solve this using Feynman's method and Fourier transform. However I was trying ...
0
votes
0answers
17 views

Is the inverse Fourier transform of a radial function radial?

Let $f: \Bbb{R}^d \to \Bbb{R}$ be a rapidly decreasing function in $\Bbb{R}^d$. Concerning Fourier transforms in $\Bbb{R}^d$, we define $\hat{f}(\xi) = \int_{\Bbb{R}^d} f(x) e^{-2 \pi i \xi \cdot x}dx ...