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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Given its spectrum, what can we know about a function?

Say we are given a well-behaved function $f(t)$ (either discrete or continuous) and are able to compute its spectrum $F(k)$ using the (discrete) Fourier transform. Then say we lose $f(t)$ and know ...
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Fourier expansion and transform - what about the phase of the waves that i am adding?

Say we have a wave on the surface of the water and we want to describe it as a sum of other waves. So we use Fourier expansion to add waves of different wavelengths. For simplicity, say we have to ...
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1answer
35 views

Fourier series of half of $\sin(\pi x)$

So my question is: Find the Fourier series (using integrals) for the half wave rectified sine function: $$f(x)= \begin{cases}0&-1<x<0\\ \sin(\pi x)& 0<x<1\end{cases}$$ ...
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56 views
+100

solving 2nd order pde with dirac delta

I want to find the functional form of the Green function G(x,t) for a parabolic differential equation: $$ ...
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1answer
14 views

Growth of Fourier coefficients of piecewise linear function

Suppose $f$ is a periodic continuous piecewise linear function. What can be said about the growth (or decay, rather) of the Fourier coefficients $\hat f(n)$ as $n\to\infty$, other than the ...
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1answer
24 views

'Converse' of Riemann-Lebesgue Lemma

So for a periodic function $f$ (of period $1$, say), I know the Riemann-Lebesgue Lemma which states that if $f$ is $L^1$ then the Fourier coefficients $F(n)$ go to zero as $n$ goes to infinity. And as ...
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Why the difference between definitions of the discrete/continuous Fourier transforms?

I should preface this question with the fact that I'm not familiar with the meaning/utility of the Fourier transform. Perhaps more accurately: I may have learned them, but have since forgotten; in any ...
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143 views

Exercise 22, Chapter 5 of Stein and Shakarchi's Fourier Analysis

I am working through Stein and Shakarchi's Fourier Analysis and am stuck on Exercise 22 of Chapter 5, which I quote below. Preliminary notation: $\mathcal{S}$ is the Schwartz space of functions on ...
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1answer
48 views

Solve one dimensional wave equation using fourier transform

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
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18 views

Von Neumann stability analysis of non-linear systems

The von-neumann stability analysis is based on the time and space discretisation schemes, what if the schemes are non-linear and too complicated to analyse. Is there a way to look at the matrices of ...
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1answer
43 views

Where does the $2 \pi$ come from in the Fourier Transform Equation?

So I was working through the Fourier transform equations that arise. I was wondering where the radical outside the integral originated from? $\hat{f}(k) = \sqrt{\frac{|b|}{(2 \pi)^{1-a}}} ...
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19 views

FFT multiplication

I'm currently implementing a specific polynomial multiplication algorithm for a project. The current goal is to implement chapter 2 of Daniel Bernstein's paper ...
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18 views

Positive-definiteness of a specific function

Is the following function positive-definite $$\varphi(t)=\max\left\{{1-\frac{n-\left|2|t|-n\right|}{2(m+1)},0}\right\}$$ where $0<m<n$ and $0\le |t|<n$. I'm aware of the Bochner's theorem ...
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1answer
18 views

Ambiguity in the Fourier transform of $f(x)=\cos(ax)$

I am slightly confused about two contradictory answers I am getting with regard to the Fourier transform of the function $f(x)=\cos(ax)$. The first method I used was \begin{align} ...
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1answer
33 views

Fourier transform of 2D function in terms of 1D function

$$f\left(x,y\right)=g\left(\frac{c_2 \cdot \left(x-y\right)}{c_1}+y\right)$$ $$\mathcal{F}\left(X,Y\right)=\frac{c_1}{c_2} \cdot \delta\left(Y-\left(\frac{c_1}{c_2}-1\right) \cdot X\right) \cdot ...
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1answer
18 views

Fourier transform of finite aperiodic signals

I know the fourier transform of most of the signals,but how about the fourier transform of aperiodic finite signals?
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30 views

Bochner-style theorem for SO(3)

Bochner's Theorem essentially provides necessary/sufficient conditions for when something is the Fourier transform of a nonnegative measure on a compact abelian group. I'm looking for a similar ...
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25 views

A discussion on fourier and laplace transforms and differential equations …?

i have read many of the answers and explanations about the similarities and differences between laplace and fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of ...
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134 views

What happens to Fourier Transform of function when the function's time scale is changed?

When a function $f(t)=exp(-|t|)$ for example undergoes Fourier Transformation, it gives $F(w)=\frac{-2}{1+w^2}$ But what happens to the result if the time scale is scaled and shifted, so that $t ...
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1answer
30 views

Norm triangle inequality for convolutions proof

I'm trying to prove that $$\|f*g\|_{L_1}\le{\|f\|_{L_1}\|g\|_{L_1}}$$ with respect to a Haar measure over a group G. Using Fubini's theorem, I'm up to ...
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29 views

Zero Gaussian curvature and restriction estimates

Let $$ \left(\int_M|\hat{f}|^qd\mu(\xi)\right)^{1/q}\leq c||f||_{L^p(\mathbb{R}^n)} $$ be a restriction estimate for a hypersurface $M\subset\mathbb{R}^n,~1<p<\infty$ and $\mu$ a ...
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56 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): $$\pi^{-s/2} \Gamma(s/2) \zeta (s)=\frac{1}{2} \int_{0}^{\infty}t^{\frac{s}{2}-1}(v(t)-1)dt$$ ...
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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 ...
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Complex Fourier integral of cosht [on hold]

Hi please help me with finding complex exponential Fourier integral for $f(t) = \begin{cases}cosht & |t|<p \\ 0 & |t| > p \end{cases}$
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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) = ...
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15 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 ?
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1answer
40 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 $\mathbb{R}^n$ $$ ...
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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 $T = 3$? ...
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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. ...
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1answer
94 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 ...
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491 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 ...
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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 ...
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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 ...
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210 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 ...
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1answer
291 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 ...
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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 ...
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83 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 ...
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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 ...
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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 ...
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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 ...
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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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Reducing or avoiding the Gibbs phenomenon.

What is your favourite method which would help reduce the Gibbs phenomenon in Fourier Series and Fourier Transforms. This could mean pre-processing or post-processing or altering the transform. With ...
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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?
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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 ...
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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 ...
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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}) = ...
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31 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 ...
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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 ...
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1answer
57 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 ...
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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 ( $\sigma\ll\mu$)?