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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Fourier transform theorem

I have an expression: $c(s) = \int_{-\infty}^{\infty} d\omega e^{-i*\omega*s}*F(\omega)$ $G(W)=\int_{0}^{\infty} ds e^{i*W*s}*c(s) $ Is $G(W) = F(W)$? What is the relation of G(W) and F(W)? ...
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Laplace Transform: Basis

I tend to think of the Fourier Transform (FT) as projecting a function onto a basis of cosines and sines. The Laplace Transform (LT) has a similar form to the FT, except it has been generalised. ...
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47 views

Berlekamp Massey and DFT

I was looking into the Berlekamp Massey algortihm, for LFSR, over GF(2) wondering if there was any DFT(alternately FFT), for the above scheme. Also, is there any generalization to Fn, ie, start ...
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Fourier transforms similar $\Rightarrow$ functions similar?

I am wondering if there is a theorem that states something like the following? If $$\big|\;\tilde f(\omega)-\tilde g(\omega)\,\big| < \varepsilon\qquad \forall\omega$$ then there exists a ...
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44 views

Showing that complex exponentials of the Fourier Series are an orthonormal basis

I am revisiting the Fourier transform and I found great lecture notes by Professor Osgood from Standford (pdf ~30MB). On page 30 and 31 he show that the complex exponentials form an orthonormal ...
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Is this function square-integrable? Able to be Fourier expanded?

I want to do a 3-dimensional Fourier series expansion on this function$$\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left[(a+\sin (y)+\cos (z))^2+(b+\cos (x)+\sin (z))^2+(c+\sin ...
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Development of Hilbert transform relationship

In the development of Hilbert transform relationships, Prof. Oppenheim has chosen \begin{equation} ...
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53 views

Questions about the Fourier series

$$f(x)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty} (a_n \cos{(\frac{2 n \pi x}{L})}+b_n \sin{(\frac{2 n \pi x}{L})}) \ \ \ \ \ (*)$$ The symbol $\sim$ has the following meaning: We know that the right ...
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Rolling $n$ times with an $m$-sided dice. Closed, finite formula for the distribution of the sum? [duplicate]

My current idea is the following: practically we want to get the distribution of the sum of $n$-times of a discrete uniform distribution between $1,...,m$ . It is practically the discrete convolution ...
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32 views

counterexample of Riemann-Lebesgue lemma for non-Borel functions

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a Borel measurable function. Then $$ \lim_{\lambda\to\infty}\int_{\mathbb{R}}f(x)e^{i\lambda x}d\mu(x)=0. $$ I obtain this result by showing that it is ...
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39 views

Fourier transform - epicycles.

I have $f(x) = \sin(x)$. I thought that when I do Fourier Transform and construct epicycles, than those epicycles will draw that $\sin(x)$ function (but this is probably not case with $\sin(x)$, cause ...
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48 views

Fourier transform of a potential

I need help computing the distributional inverse Fourier transform of the function $1/|x|^2$ in dimension two. The integral makes sense written as \begin{align} 1/2\pi \int_{\mathbb{R}^2} e^{ix\xi} ...
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80 views

Fourier Transformation: an Animated GIF

Here I found the animated GIF below. I don't get it! Would someone explain it please?
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40 views

Hidden subgroup problem for $\mathbb{Z} mod 2$

The definition of the Hidden Subgroup Problem (HSP) is as follows (according to a lecture series by Pranab Sen), Let $G$ be a group, $S$ a set and $f : G \to S$ a function. We are given an ...
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61 views

Question regarding Fourier coefficients

I would like to express the product $$ \left( \sum_{k \in \mathbb{Z}} a_k \sin(k t) \right) \left( \sum_{k \in \mathbb{Z}} b_k \cos(k t) \right) $$ as $$ \sum_{k \in \mathbb{Z}} c_k \sin(k t). $$ ...
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Computing the inverse Fourier transform of $\frac{1}{1+|\xi|^2}$ for $\xi \in \mathbb{R}^n$.

I'm trying to compute the integral $$ \int_{\large\mathbb{R}^n} \frac{ e^{\large ix \cdot \xi}}{1 + |\xi|^2} ~d^n\xi. $$ I know that for an integral like $$\int_{\large\mathbb{R}^n} \frac{ 1}{1 + ...
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FFT of simultaneous signal sequences

There is a signal that is received at n antennas in a time period t. So the received signal at each antenna is $$ s_i = x_1, ..., x_t ~~~~~~~ i = 1,...n $$ where x is a feature of the signal. The ...
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33 views

How to solve this heat equation with fourier method

Solve this via Fourier method: $$u_t-u_{xx}=0 \quad\quad 0< x<\pi, \quad t >0, $$ $$u(0,t)=u_x(\pi,t)=0, \quad\quad t \ge 0$$ $$u(x,0)=2\sin\left(\frac{3x}{2}\right) \, ...
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What is Fourier Space

I know a some basics stuff regarding Fourier Analysis (Fourier series and Fourier transforms), but I've seen the term "Fourier Space" come up and I'm having trouble finding a definition for what this ...
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an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...
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32 views

A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$ with Dirac delta function $\delta$. ...
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How to find the Fourier components

I've got a sinusodial signal: \begin{equation} \Delta I=\cos(\Delta \omega t + \varphi)). \end{equation} and I would like to rewrite it as the sum of a cosine and a sine signal(without a phaseterm): ...
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36 views

Show that the Fourier transform of a a distribution is $C^{\infty}$

I am trying to understand the solution to the following problem: Let $u \in \mathcal{D}'(\mathbb{R}^{n})$ such that $u(x) = c \log(|x|)$ when $|x|>1$, where $c \in \mathbb{C}$. Show that $u \in ...
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39 views

Fourier and $Z$ transform of a signal?

We have $$X(k)=4[u(k-2)-u(k)* d(k-3)]$$ I need to find the Fourier transform,$Z$ transform,as well as dhe magnitude and phase spectra. First of all I think that I need to convert the $u(k)$ and ...
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38 views

Convolution of ring Delta function

Assume $f(r)=\delta(r-R)$ where $\delta(\cdot)$ is a ring delta function. In other word, $f$ is a circular delta function on a circle with radius $R$. I want to do the convolution of $f$ with itself ...
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23 views

convolution of lorentzian with cosine

Hi: I'm reading a text "Fourier Transforms for Pedestrians" and it's a nice text but it skips steps that I sometimes don't understand. The current example that I don't follow is one where the ...
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35 views

Units of frequency for Fourier transforms

I'm trying to get to grips with the relationship between a signal $x(t)$, its Fourier transform $X(F)$ and the graph representation of $t$ plotted against $|X(F)|$. I've been using Matlab to perform ...
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36 views

How do we determine the duration of a fundamental frequency using the DFT (or FFT)?

I'm still in the process of learning the details of the DFT (and FFT) and I've just made a test .wav file in Audacity by joining 3 one-second sine waves together. .wav file 1 = 440 Hz, sample rate ...
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Mean fourier coefficients of a $2\pi$-periodic function are just the usual Fourier coefficients.

Let $f$ be a continuous $2\pi$-periodic function on $\mathbb{R}$. I'm trying to show that \begin{align}\tag{1} \lim_{T \rightarrow \infty} \frac{1}{2T}\int_{-T}^{T} f(x)e^{-ix\xi}dx = ...
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Fourier transform causing problems in proof for Parsevals formula

$f^{*}$ is the complex conjugate and $\tilde{f}$ is the fourier transform of Ff$. If $g(-\xi)=f(\xi)^{*}$ how does this imply $\tilde{g}(k)=\tilde{f}(k)^{*}$. This result just does not seem true by ...
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Plotting the frequency spectrum of a signal

I've found this algorithm here on Mathematica.SE to plot the frequencies of a signal using Fourier. It works beautifully, but I'm having some trouble understanding ...
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Show that the Hilbert transform is a pseudo-differential operator of order 0

I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform $Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$ is a ...
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64 views

MATLAB: Plotting the inverse Fourier transform of a rectangular pulse.

I'm trying to repeat the results in the image below without using the rectangularPulse(a,b,z) function: Here is my unsuccessful attempt: Here is the code I used to create the first image: ...
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How did Fourier arrive at the following regarding his series and coefficients?

I am reading Karen Saxe's "Beginning Functional Analysis." Perhaps it is poor exposition on her part, but she states: ...Fourier begins with an arbitrary function $f$ on the interval from $-\pi$ ...
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54 views

Find distribution solving a differential equation

I think I have solved the following differential equation, but I am not sure of all steps are justified. Exercise: Find all distributions $u \in \mathcal{D}'(\mathbb{R})$ such that $x(u' -u) = ...
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In search of periodic solutions of a system of ODEs by means of Fourier series

Consider the following non-linear system of ODEs : \begin{cases} x' = y \\ y' = x^2-\lambda x. \end{cases} In search of a solution such that $y(0) = y(2 \pi) = 0$, I am being told to seek $x$ and $y$ ...
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If the Fourier transform of a probability measure goes to zero at infinity, can the measure have a point mass?

Let $\mu$ be a probability measure on $\mathbb{R}$. Is the following implication true? $$ \widehat{\mu}(y) \rightarrow 0 \text{ as } |y| \rightarrow \infty \quad \Rightarrow \quad \mu(\{x\})=0 \quad ...
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Fourier series with respect to orthonormal sequence

Let $H$ be the space of piecewise continuous $2 \pi$-periodic functions on the real line. For $f$ and $g$ in $H$, consider the inner product $<f,g>=\frac{1}{2\pi}\int_{- \pi}^{\pi}f(x)\overline ...
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Fourier transform time and frequency question?

So there is an example in my book where g(k) is converted to G(f) and its written $$g(k)\Longleftrightarrow G(f)$$ So: $$a^ku(k)\Longleftrightarrow \frac{1}{1-ae^{-j2\pi f}}$$ My question is, how ...
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56 views

Find limit of a sequence of distributions

I am trying to solve the following exercise: Determine the limit in $\mathcal{D}'(\mathbb{R})$ of $\lim_{t\rightarrow \infty} t^{2}xe^{itx}$, $u_{t} = t^{2}xe^{itx}$. I have tried evaluating the ...
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What wave does a “complex frequency” correspond to in the Fourier Transform?

The Fourier Transform takes a function $f$, you get another a function $g$. $g$ takes a complex frequency, and returns a sort of relative amplitude of a wave function in $f$. My question is how do you ...
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Prove $\sum_{n= -\infty}^{\infty} \frac{1}{(t+n)^2} = \frac{\pi^2 }{\sin^2(\pi t)}$ using f(x)=1-|x| and Poisson summation formula

I'd like to prove $$\sum_{n= -\infty}^{\infty} \frac{1}{(t+n)^2} = \frac{\pi^2 }{\sin^2(\pi t)}$$ by using the Poisson summation formula. There is a way to do it by firstly taking the Fourier ...
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Convolution of a sum of shifted delta functions

In the lecture notes for Fourier Transforms and it's Applications on page 212 by Bracewell he talks about representing a signal as a sum of distributions evenly spaced out by a distance p. ...
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12 views

Representing a real sampled signal with N samples as a complex sampled signal with N/2 samples

I am studying the discrete Fourier transform, and in its most basic definition it is an invertible linear transformation on the complex numbers. From Wikipedia: The sequence of $N$ complex numbers ...
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37 views

cant extract odd function with FFT

i cant correctly extract spectrum from data points of odd function (e.g. $\cos\left(\frac23\pi x\right)$, $16$-points vector $[1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1]$), instead of one function I get a bunch ...
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Formula for trace of particular operators

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R})$. View the Fourier transform as a unitary operator $\mathcal{F} \in B(\mathcal{H})$. For each function $f \in C_0(\mathbb{R})$, let $T(f) \in ...
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Meaning of multiplication by $\sin$ in $\omega$-domain

Multiplying some signal, a function of time, $m(t)$ by a cosine $\cos{\omega' t}$ causes a shift in frequency of $m(t)$, by $\pm\omega'$. But what about multiplication by a sine wave, such ...
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Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
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21 views

Inverse Fourier Transform

I need help solving the following Fourier transform question. Given, $$ X_s(f) = \frac{1}{\Delta T} \sum_{n = -\infty}^{\infty} X\left(f - \frac{n}{\Delta T} \right) $$ $$ H(f) = \begin{cases} 1 ...
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38 views

Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...