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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How does 2D spatial Fourier (kx-ky) transform result responds to rotation of the original?

I have a 2D function $f:R\times R \rightarrow R$ that represents periodical axis-aligned spatial bumps at specific spatial periods (frequencies), like $f=\sin^2(2\pi \nu_1 x) \sin^2(2 \pi \nu_2 y)$. I ...
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5answers
9k views

Difference between Fourier transform and Wavelets

While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency whereas ...
3
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0answers
281 views

Is there a way to Fourier transform Cos[Sin[x]]

In my physics problem, I encountered a solution has the form like Cos[Sin[t]], and I need to do the Fourier transform to this solution. Is there a way to do the Fourier transform analytically to ...
4
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1answer
406 views

Fourier Sine Transform

There is a question from my book which I find hard. Here it goes: Consider $$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}-v_0\frac{\partial u}{\partial x} ...
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2answers
602 views

Interchange of sums and integral

Suppose I have a function $x:\mathbb{R}\rightarrow\mathbb{R}$ such that is square-integrable: $$\int_{-\infty}^\infty|x(t)|^2dt<\infty$$ Suppose also that $x(t)$ contains no higher frequencies ...
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1answer
124 views

The Fourier transform of a product of $f$ with $t^n$

There is a proof in my book I don't quite follow. We are supposed to prove that the Fourier transform of a product of $f$ with $t^n$ is given by $$\mathcal{F}[t^n f(t)] (\lambda) = i^n \frac{d^n}{d ...
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1answer
2k views

Fourier transform in cylindrical coordinates

I must implement Fourier transform in cylindrical co-ordinates. Matlab offer fft function. How can I use this function ?
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2k views

Why fourier transformation use complex number?

I know that the Fourier transform is as follows:$$\hat{f}(\xi)= \int_{-\infty}^{\infty}\exp(-\mathrm ix\xi)f(x)\mathrm{d}x$$ but I couldent understand why should use complex number $i$ in the ...
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346 views

Determining if something is a characteristic function

Suppose $X$ is a continuous random variable with pdf $f_X(x)$. We can compute its characteristic function as $\varphi_X(t)=\mathbb{E}[e^{itX}].$ Question: Given a function, say $\psi(t)$, how does ...
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1answer
216 views

Help proving Calderón reproducing formula (simple version)

Let $\phi$ be a real compactly supported smooth function on $\mathbb R$ with total integral zero. Define $\phi_t=\frac{1}{t} \phi(\frac{x}{t})$. I also suspect that they must be even, but the notes I ...
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2answers
109 views

How to express this convolution by the sum of integrals

If $$f\left(x\right)=\begin{cases} f_{1}\left(x\right), & x\in[0,1]\\ f_{2}\left(x\right), & x\in[1,\sqrt{5}]\\ 0, & \mbox{elsewhere} \end{cases}$$ what does the piecewise-defined function ...
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477 views

Can the differentiating and squaring process in the cochlea explain a reported dichotic stimulation experiment?

On this math.stackexchange on url What is Octave Equivalence? in an answer on the related ( octave equivalence ) question is stated: Mathematically, this signifies that the mammalian cochlea ...
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1answer
505 views

Fourier transform of compactly supported function on $\mathbb{R^n}$.

I am attempting to extend this question, which says that for a nonzero continuous $f$ with compact support on the real line, the Fourier transform cannot decay exponentially. Suppose $f \in ...
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1answer
477 views

On Vanishing Riemann Sums and Odd Functions

Let $ f: [-1,1] \to \mathbb{R} $ be a continuous function. Suppose that the $ n $-th midpoint Riemann sum of $ f $ vanishes for all $ n \in \mathbb{N} $. In other words, $$ \forall n \in ...
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1answer
103 views

Need numerical approximation for Fourier{max(0,f(x,y))} given Fourier{f(x,y)}

Given $\mathscr{F}\{f(x,y)\}$ is there a way to numerically approximate $\mathscr{F}\{max(0,f(x,y))\}$ ? I am not necessarily looking for a closed formula. Even some iterative method would be ...
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3answers
11k views

How to calculate the Fourier transform of a Gaussian function.

I would like to work out the Fourier transform for the Gaussian function: $f(x)=\exp(-n^2(x-m)^2)$. It seems likely that I will need to use differentiation and the shift rule at some point, but I ...
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1answer
36 views

When to use other transforms?

maple code int(g*f, x=-infinity..infinity) when $g$ is $\large exp^{i*t*x}$, Fourier transform between density function and characteristic function If $g$ are $x^t$, $|x^{t}|$, $t^{x}$, what do they ...
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2answers
2k views

Heaviside step function fourier transform and principal values

I found the following answer on SE: Fourier transform of unit step? However, it is still not clear to me and maybe somebody could explain it clearer. Problem I have the following in my notes of ...
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1answer
151 views

Obtaining Impulse Response from Graph

I want to know how to solve those types of problems.. is it by inspection ? Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the ...
5
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1answer
579 views

Fourier transform of $ \frac{\sinh(kx)}{\sinh(x)}$ [duplicate]

Possible Duplicate: Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$ Im trying to compute the integral of $$I = \int_{-\infty}^\infty ...
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2answers
3k views

Dirichlet Conditions and Fourier Analysis.

I read in my text book that the Dirichlet conditions are sufficient conditions for a real-valued, periodic function $f(x)$ to be equal to the sum of its Fourier series at each point where $f$ is ...
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2answers
178 views

Finding if the equation is even or odd

I am learning fourier transform and I came across this question in which author right away says the given equation is "even". How does this equation become "even"? $$x[n]=\begin{cases}A & -M\le ...
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1answer
194 views

Fourier Transform of an Operator

I need to calculate the fourier transform of an Operator. meaning I need to calculate the transform of the Operator's corresponding convolution kernel. so the question is: 1.given a 2d fourier ...
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1answer
383 views

Fourier transform as a Gelfand transform

One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?
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152 views

Is Fourier transform defined on $L^p(\mathbb{R})$ only for $p \in [1, 2]$?

Is Fourier transform defined on $L^p(\mathbb{R})$ only for $p \in [1, 2]$? From Lieb and Loss's Analysis, they extend the definition of Fourier transform from $L^1(\mathbb{R})$ to $L^p(\mathbb{R}), ...
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123 views

DFT shift theorem generalizations?

The DFT shift theorem implies that any circular shift in the input space is equivalent to a phase change in the frequency domain, while the absolute values are preserved. $$ ...
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211 views

Uniqueness of Haar Measures

Haar measure on a LCA (locally compact abelian) group $G$ is said to be unique ... up to a scaling factor. This is not as elegant as it might be expected because this requires a choice of a unit ...
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0answers
115 views

2-D DFT of a matrix PxP and 1-D DFT of a vector of size P^2?

What is the difference between the following two things: make a 2-D Discrete Fourier Transform of a certain matrix A[p,p], first reshape this matrix into a 1-D vector a[p^2,1], and compute the 1-D ...
3
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1answer
808 views

Normalizing scilab's FFT

I want to normalize the FFT used in scilab in a way so that the absolute values of the coefficients equal to the amplitudes of the time domain signal with that frequency. Example: I want an input ...
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0answers
195 views

convolution of L1 function with a harmonic oscillation

I have to show that the convolution of a function $f \in L^1(\mathbf{R})$ with the harmonic oscillation $\phi_\omega (t) = \exp(2 \pi i t \omega)$ is equal to the Fourier Transform of $f$, ...
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1answer
271 views

Can fourier transform of a function with not empty support be zero on whole range?

We consider $f : [a, b] \subset \mathbb{R} \rightarrow D \subseteq \mathbb{C}$ and support may consist of only one point.
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372 views

Fourier Transform of f(t)=|t|

This is a question on taking the Fourier transform of a function. How does one find the fourier transform of $f$, where $f(t) = |t|$?
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2answers
172 views

Fourier transform terms explained

I know there are lots of tutorial on different webpages.. but I am not an engineering student... those look quite complex to me.. Could anybody explain what is "Fourier transform" in the very simple ...
2
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1answer
104 views

Properties of Fourier transform of distributions

For distributions the scaling property, $f(ax) = \frac{1}{|a|} \mathcal{F(\frac{u}{a})}$, of the Fourier transform is no longer true. Is there a source that lists which properties of the Fourier ...
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213 views

Fourier transform of $\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$? Not Gaussian like with Fermi-Dirac statistics?

This equation $\bar n_i=\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$ is Fermi-Dirac statistics where variables are defined here. The classical equation i.e. the Maxwell Boltzman equation is Gaussian ...
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2answers
565 views

Combining bins of FFT ouput

I was trying to combine output of a $2n$ point Real FFT to generate custom FFT bins. For example the FFT generates components at equally spaced frequencies $f_0,f_1,f_2 ...f_{n-1}$ $f_0$ = ...
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2answers
601 views

Zeros of Fourier transform of a function in $C[-1,1]$

I am trying to prove the following: Let $g \in C[-1,1]$. Then the function $$G(z) = \int_{-1}^1 e^{itz}g(t)dt$$ has infinitely many zeros. I know that $G(z)$ is entire and $\lim_{x \to \pm ...
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2answers
801 views

Why is the absolute value needed with the scaling property of fourier tranforms?

I understand how to prove the scaling property of Fourier Transforms, except the use of the absolute value: If I transform $f(at)$ then I get $F\{f(at)\}(w) = \int f(at) e^{-jwt} dt$ where I can ...
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1answer
37 views

Quantization Uniform

Let the output of image sensor take values between 0 to 10. If the samples are quantized uniformly to 256 levels, show that transition and reconstruction levels are $$t_k=\dfrac{10(k-1)}{256},\, ...
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1answer
147 views

General questions on Cayley graphs

In Graph Theory mainly in Cayley graphs there are four general questions " according to Audery Terras" : 'Suppose A is the adjacency operator of a connected regular (undirected) graph $X$ of degree ...
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2answers
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Obtaining the $\frac{1}{2\pi}$ factor in the Fourier transform

This MathWorld page gives this definition of a Fourier transform: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x}dx.$$ But, I wish to speak in terms of linear frequency $\nu$ and time $t$ ...
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1answer
250 views

Sobolev inequality

If $f\in H^2(\mathbb R^2)$, I want to show that $||f||_{L^\infty}\le c||f||_{H^2}$ $||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$ For 1, I use $||f||_{L^\infty}\le \sup_{x\in \mathbb ...
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4answers
3k views

Fourier transform vs Fourier series

Why are Fourier series considered to be the subset of Fourier transform? It should have been other way around. Because a non periodic pulse is a subset of periodic pulse with period infinite. So ...
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196 views

Autocorrelation derivation using fourier transform

I am stuck with basic understanding of the Auto-correlation derivation of a simple signal and I would be pleased if you could help me out with that. Lets have a signal $x(t)=\cos(2\pi{f_{0}}{t})$. ...
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1answer
222 views

Sobolev space exercise

I need to show $|f|_{L^\infty}\leq c|f|_{H^2} = c(\int_{\mathbb R^n} (1+|\xi|^2)^2|\hat f(\xi)|^2 d\xi )^{1/2}$, assume $f\in H^2(\mathbb R^2)$ I think I can trasnfer $f\ = \int \hat f(\xi)e^{2\pi i ...
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1answer
150 views

What is this Hilbert space?

The space is $H^s(\mathbb R^d)$. If $f$ is in this space, it means $\int_\mathbb {R^n} (1+|\xi|^2)^s|\hat f(\xi)|^2d\xi < \infty$ where $\hat f$ is the fourier transform of $f$: $\hat ...
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0answers
112 views

Sampling Theorem Poisson Formula

Theorem If the Fourier transform $\hat{f}(w)$ of a signal function $f(x)$ is zero for all frequencies ouside the interval $-w_c\leq w \leq w_c$, then $f(x)$ can be uniquely determined from its sampled ...
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1answer
247 views

Orthogonal complete set of functions

Every square-integrable function on an interval can be written as a linear combination of e^inx (Fourier series). Are there any other orthogonal and complete set of functions for square integrable ...
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1answer
116 views

Is this a correct way of thinking of Fourier transforms

I am working on my understanding of various transforms and I have been thinking about the Fourier transform, what i "does" to the function it is applied to. The way I see it: The function $f$ that ...
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607 views

Zero-padding data for FFT

If I take a discrete Fourier transform of $\{ c_1, c_2, \ldots, c_n\}$ where $n$ is prime, I am rather limited in the FFT algorithms available to me and their performance. Additionally, having ...