Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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5
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1answer
58 views

How to manipulate this functions to an identity involving the Riemann zeta function

The identity I want to prove is the following (from Stein's book, an introduction to Fourier analysis): $$\pi^{-s/2} \Gamma(s/2) \zeta (s)=\frac{1}{2} \int_{0}^{\infty}t^{\frac{s}{2}-1}(v(t)-1)dt$$ ...
2
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0answers
22 views

Why wavelets based transmitter/receiver digital signal processing aren't common? [migrated]

I have seen this thread: Difference between Fourier transform and Wavelets AFAIK there is no common usage of wavelets in the real-time DSP world (excluding image and video processing). I am curious ...
-3
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0answers
35 views

Complex Fourier integral of cosht [closed]

Hi please help me with finding complex exponential Fourier integral for $f(t) = \begin{cases}cosht & |t|<p \\ 0 & |t| > p \end{cases}$
1
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0answers
32 views

Fourier transform of the Cosine function with Phase Shift?

How can i calculate the Fourier transform of a delayed cosine? I haven't found anywhere how to do that. This is my attempt in hoping for a way to find it without using the definition: $$ x(t) = ...
0
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0answers
18 views

Converting Walsh coefficients to values of a function

I assume I know the Walsh coefficients of a function f: $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2}$. Is there any efficient possibility to get the values of the function f ?
1
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1answer
41 views

Difficulties of convergence of the partial sums of the Fourier inversion formula

Define the Fourier inversion formula over $\mathbb{R}^n$ by $$ f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\hat{f}d\xi $$ where $\hat{f}$ is the Fourier transform over $\mathbb{R}^n$ $$ ...
0
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0answers
17 views

Finding $a_0$ for the function $s(t)=1-e^{-2t}$.

I am working on multiple Fourier series questions about the function $s(t)=1-e^{-2t}$. How do I find a naught as in $a_0=\dfrac 1T\displaystyle \int\limits_{t_0}^{t_0+T}s(t)\,dt$, when $T = 3$? ...
0
votes
1answer
38 views

How to derive the complex Fourier series of $s(t) = 1-e^{-2t}$? [closed]

I have the periodic function $s(t)=1-e^{-2t}$. I am required to derive the complex Fourier series of $s(t)$. I have some knowledge of Fourier series but not enough to know if I am doing it correctly. ...
1
vote
1answer
96 views

integration concerning Fourier transform of homogeneous kernel(of degree 0)

Let $m\in C^\infty(\mathbb{R}^d-\{0\})$ be homogeneous of degree zero and has mean zero on the sphere$(S^{d-1})$. Then $m$ defines a tempered distribution and $\partial_j^dm\in S'(\mathbb{R}^d)$ is ...
0
votes
1answer
497 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
0
votes
1answer
22 views

Connection between autocovariances and Fourier series of a continous function.

Let $f(w)$ be a continuous function of period $2 \pi$ then it's Fourier series is $$f(w) = \sum_{k = 0}^j \left(a_k \cos(kw) + b_k \sin(kw)\right)$$ I wrote that the autocovariances $\gamma(k)$ (of ...
2
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0answers
13 views

What are the statistics of the discrete Fourier transform of a Bernoilli process?

The problem I would like to understand the statistics of the discrete Fourier transform of a sequence of uncorrelated events $\{x_n\}$ each of which takes the value $\pm1$ with probability $1/2$. In ...
6
votes
2answers
212 views

Why are square functions important in analysis?

I have been reading through chapter 1 of E.M. Stein's textbook Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. In chapter 1, Stein discusses the relationship ...
1
vote
1answer
296 views

How to convert FFT magnitude of square wave to dBm?

I wish to convert the FFT magnitude of square wave into dBm. I use FFT to covert voltage of square wave to a complex number, then i absolute the complex number into magnitude. Then i divide the ...
0
votes
1answer
29 views

in Maxima how to get Fourier transform identity ($\mathcal{F}^{-1}(\mathcal{F}(f)) = f$)?

I'm currently trying to figure out something related to signal processing and wanted to draft Maxima for the task. As a starting point I tried to make Maxima evaluate the identity ...
1
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2answers
88 views

Looking for a source: Fourier inversion of $f \in L^1$

Is there a book where I can find a thorough proof of the following assertion? Let $f \in L^1(\mathbb{R}^d)$ be continuous at zero and $\hat{f}\ge0$. Then $\hat{f} \in L^1(\mathbb{R}^d)$ and ...
2
votes
2answers
114 views

A condition for $\hat f$ to be integrable [duplicate]

Let $f \in L^1 (\mathbb R^n)$. Suppose that $f$ is continuous at zero and that the fourier transform $\hat f$ of $f$ is non-negative. Does this imply that $\hat f \in L^1$ (and hence, by the inversion ...
1
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2answers
273 views

Locally Compact Group with Haar Measure

Suppose $G$ is a locally compact abelian group, with Haar Measure $\mu$, then is $\mu(E)=\mu(E^{-1})$ for all subsets $E$ of $G$? I have seen that this is true for all Borel subsets of $G$, but I am ...
0
votes
0answers
17 views

Prooving that multiply by exponent in time domain yields a frequency shift in frequency domain using convolution.

im trying to proove that $F[x(t)e^{-jat}] = X(w-a)$ using convolution. using the convolution property i know i should get a convolution of $F(x(t))$ and $F(e^{-jat})$ So: $$ F[x(t)e^{-jat}]= 1/2\pi ...
1
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0answers
27 views

Diagnalization of block matrix with circulat blocks

I have the following Matrix $A = \begin{pmatrix} X \\ Y \end{pmatrix}$ Where X, and Y are circulant Matrices. I want to diaganlize $AA^T$. I tried the following: $AA^T = \begin{pmatrix} ...
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0answers
22 views

Fourier methods and a conductor bar

I was doing this question bellow: I tried: Could you help me in the 3 (second Picture) and how to solve the problem?
3
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1answer
31 views

Detrending sine waves accurately

I am doing some data analysis where I look at electricity demand over the course of a day, but need to separate the intra-day (constant and periodic) components from daily changes (assumed linear). At ...
1
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1answer
34 views

Methods for solving definite trig. integrals?

I am studying Fourier series and there is a lot of integration going on, specifically with trigonometric functions involved. When solving for the Fourier coefficients, often times, the definite ...
0
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2answers
31 views

How to do Fourier transform for these 2 questions?

I don't get certain of parts of these two questions 1) I'm trying to do the Fourier transform of: $$f(x) = \, xe^{-x^2} $$ In the problem it said to use: $$F \, (e^{-tx^2}) = ...
1
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0answers
37 views

Cosine Fourier series solution of semi-major axis nonlinear integral equaton

Consider an integral equation $$ \frac{1}{z(t)}=f(t)+\alpha\int_0 ^\infty \cos(ts)z(s)\,ds $$ I am required to solve for $z(t)$. I approached this problem by considering the integral on right hand ...
0
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0answers
20 views

Compute phase-shifted variant of a real-valued function

I'm trying to compute a phase-shifted (by angle $\phi_0$) version of a general real-valued function $f(x)$. I realize that the phase shift is convenient to perform in frequency domain, so first I ...
0
votes
1answer
59 views

confused with the FFT output

I am taking some sensor output and doing fft on it. how to get the exact frequencies from the complex output? my understanding is that bin frequencies and the input frequencies are different. Please ...
4
votes
1answer
302 views

A condition on Fourier transforms that implies absolute continuity

Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
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32 views

Fourier transform of an inverse function.

If for a given function $f(x)$, the Fourier transform is $\hat{f}(p)$; Is there a way to find the Fourier transform of $f(x)^{-1}$ in terms of $\hat{f}(p)$?
1
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1answer
54 views

Condition that Fourier inversion formula holds

Recently I am reading Stein's Complex Analysis but I haven't read the book Fourier Analysis before. So I do not have any knowledge about Fourier Transformation. In Chapter 4.3 (p.121), it tells that ...
0
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0answers
23 views

Convolution and space-time Fourier transform

I have a general function $u(x,y,z,t)$. Then, (1) what would be the space-time Fourier transform of $$G \star \frac{\partial^n u}{ \partial t^n }$$ and (2) would the relation $$G \star ...
1
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2answers
29 views

Fourier series on general interval $[a,b]$

Currently I'm studying Fourier series and the first thing I've read is the definition of the series for a function $f : [-\pi,\pi]\to \mathbb{R}$. In that case the Fourier series is ...
1
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0answers
12 views

when Wiener amalgam space is a subset of Lebesgue space?

Let $X=\mathcal{F}L^{p}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{p}(\mathbb R)\},$ and $\|f\|_{X}= \|\hat{f}\|_{L^{p}}.$ In the definition of Wiener amalgam spaces $W(X, L^p)$, I am taking ...
3
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3answers
57 views

Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
1
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1answer
39 views

how to disprove uniform convergence

I've been asked to check the uniform convergence of the following function sequence on the real line: $$ f_{N}(t)=\sum_{n=-N}^{n=N}\sin(n) \,\frac{\sin(\pi t-\pi n)}{\pi t-\pi n} $$ It is asked in a ...
1
vote
1answer
32 views

Is there a difference between cosine and sine transform?

Surely both should work with the same set of functions. Why is only cosine transform used in JPEG? Why not sine? It seems that using fourier transform rather than cosine transform would result in ...
0
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0answers
37 views

Is there a way to find an in-between 1D FFT with one less point?

Given a discrete 1D signal $f(n)$ over the support $-N/2 \leq n \leq N/2$, where $N$ is even, and given an arbitrary scalar value $\alpha$, the definition of 1-D fractional Fourier transform (FrFT) ...
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0answers
19 views

Proof about fundamental frequency for periodic functions

I know that in the Fourier series expansion for $f$, we have $$f(t) = \sum_{n=-\infty}^{\infty} c_n \exp\left(\frac{2\pi int}{T}\right)$$ where the lowest frequency term (ignoring the constant) has ...
0
votes
1answer
30 views

Integral (Fourier transform) of Heaviside radial function in 3D

I am trying to calculate the following integral: $ \int \frac{d k_x d k_y d k_z}{(2 \pi)^3} \left[ \exp( - \frac{(k_x^2 + k_y^2 + k_z^2) \sigma^2}{2}) + \frac{1}{2} H(\sqrt{k_x^2 + k_y^2 + k_z^2} - ...
0
votes
1answer
43 views

Range of Fourier Tranform 0n $L_1(\mathbb{R})$ is dense in $C_0(\mathbb{R})$

I want to prove it through the hint given in the notes available online(link provided below). It says first prove that if $f\in C_c^2(\mathbb{R})$, then $\hat{f}\in L_1(\mathbb{R})$; and hence ...
0
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0answers
46 views

Approximation by sinc functions in L2

I wish to find the best approximation in $ L_{2} (\Re )$ of $f(x)=\frac{sin(ax)}{ax}$ for $0<a<\pi$ and for $a>\pi$ , Using the system of sinc functions: $$g_{n}(x)=sinc(\pi x-\pi n) = ...
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0answers
55 views

Extending an identity for the Dirac delta function

The identity $$x^p \; \delta^{(n)}(x) = (-1)^p \frac{n!}{(n-p)!} \; \delta^{(n-p)}(x)$$ can easily be derived from the generalized Leibnitz formula for $n$ and $p$ positive integers: $$\int \; x^p ...
2
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0answers
26 views

Fourier transform of a constant - Is it a function of distance only?

I have the following power spectrum $P(k)$, function of the modulus $k$ of the vector $\vec{k}$ in Fourier space: $$P(k) = \begin{cases} P_0 \exp (-\frac{k^2 \sigma^2}{2}) & \text{for} \; k \leq ...
2
votes
1answer
40 views

Why this formula doesn't work for $n=1$?

I've been studying Fourier series and in trying to compute the Fourier series for the function $f: (-\pi,\pi)\to \mathbb{R}$ given by $f(x)=|\sin x|$ I've found something quite strange that I'm not ...
0
votes
1answer
63 views

About PDE problem

Let $ \, u(x,t) : \Bbb{R}\times(0,\infty) \rightarrow\Bbb{R}\, $ be $ \,C^2 \,$ such that $$ {\partial u\over \partial t} (x,t)-{\partial^2 u \over \partial x^2}(x,t)+x^2 ...
5
votes
0answers
47 views

For $f \in C_c^\infty(\mathbb{R})$, does $\hat{f}(k)\sum_{j=0}^n \frac{(-k^2)^j}{j!}$ converge to $\hat{f}(k)e^{-k^2}$ in $L^2(\mathbb{R})$.

As the title states: For $f \in C_c^\infty(\mathbb{R})$, does $\hat{f}(k)\sum_{j=0}^n \frac{(-k^2)^j}{j!}$ converge to $\hat{f}(k)e^{-k^2}$ in $L^2(\mathbb{R})$ where $C_c^\infty(\mathbb{R})$ is the ...
0
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0answers
35 views

Find the Fourier transform of the given memory function in the limit volume $V\rightarrow\infty$

The memory function is given by, \begin{equation} \mu (t)=(8\pi e^{2}/3V)\sum_{\vec{k}}|f_{\vec{k}}|^{2}\cos (ckt) \end{equation} where $V$ is the volume, $f_{\vec{k}}$ is the form factor. In this ...
2
votes
0answers
32 views

Another equivalent characterization of Schwartz function?

Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that $$ \sup_{x\in\mathbb{R}^n}\left||x|^k\Delta^{p}\psi(x)\right|<\infty $$ for all ...
0
votes
1answer
21 views

If $\{ \phi_n \} $ is an orthonormal basis for $L^2(c,d)$, then $\{ \phi_n \circ f \}$ is an orthonormal basis for $L^2_{f'}(a,b)$

Suppose $f : [a,b] \to [c,d]$ and $f'(x) > 0$ for $ x \in [a,b]$. Show that if $\{ \phi_n \}$ is an orthonormal basis for $L^2 (c,d)$, then $ \{ \phi_n \circ f \} $ is an orthonormal basis for ...
9
votes
2answers
216 views

Fourier transform of $\Gamma \left (\frac{1}{2}-i \frac{p}{2 \pi} \right) /\sqrt{ \cosh(p/2)}$

Is it possible to compute the following Fourier transform analytically? $$\psi(x) = \frac{1}{\sqrt{4 \pi}}\int \Gamma \left (\frac{1}{2}-i \frac{p}{2 \pi} \right) \frac{e^{i p x}}{\sqrt{ \cosh(p/2)}} ...