Stack Exchange Network

Stack Exchange network consists of 175 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
0answers
9 views

Application of the operator $\exp\left( \alpha \dfrac{\partial^2}{\partial q^2}\right)$

I need to apply the operator $$\exp\left( \alpha \dfrac{\partial^2}{\partial q^2}\right) \tag{1} \label{1}$$ To the function $$M(x) N(y +C_{1}p)\mathcal{F}[f(q)](p) \tag{2} \label{2}$$ where $M(x)$ ...
1
vote
0answers
22 views

Sketching the Fourier Transform of a function

I am having trouble understanding how a Fourier transform can be sketched given an initial pulse function. The question asks to sketch a Fourier transform of a pulse, $𝑠(𝑡) = 1 + \cos(𝑡^2)$: It ...
0
votes
0answers
17 views

Estimating Fourier spectrum from a graph - exam tomorrow…

so basically i was doing a past paper question and cam across this.. PART B. Question which basically asks to plot an approximate magnitude of S vs frequency this is from looking at the graph period ...
-1
votes
0answers
28 views

Use Dirichlet Integral and Fourier Transform to evaluate Integrals [on hold]

Use the Dirichlet integral $\displaystyle\int_{0}^{\infty} \frac{\sin\left(x\right)}{x}\,\mathrm{d}x = \frac{\pi}{2}$ and the Fourier Transform $\displaystyle\hat{\mathrm{f}}\left(k\right) = \int_\...
0
votes
1answer
36 views

There is no function in $L^1$ whose Fourier transform is 1/log(x)

I need to prove that there is no odd function on $L^1$ whose Fourier transform is a continuous function $g:\mathbb{R}\to\mathbb{R}$ such that $g(\xi)=1/\log(\xi)$ for $\xi\geq 2$. I am suggested to ...
0
votes
0answers
21 views

Fourier Transform for heat equation on a half-plane

I'm trying to solve heat equation on a half-plane using fourier transform and I don't understand why it doesn't work (yes, I know we can use method of images or sine-transform, but I want to ...
0
votes
0answers
17 views

Issue w/ extracting coefficients from generating function using IDTFT

This q will make use of these 3 DTFT pairs... $$ \require{extpfeil}\Newextarrow{\xleftrightarrow}{15,15}{0x2194} \begin{array}{rcl} \alpha x_1[n] + \beta x_2[n] & \xleftrightarrow{\mathscr{F}} &...
3
votes
1answer
250 views

Fourier Transform of Airy Equation

I am trying to find $Y(k)$ of the equation $y''(x)-xy(x)=0$ and hence show that $$y(x)=\sqrt{\frac{2}{\pi}}\int_0^{\infty}\cos\left(\frac{k^3}{3}+kx\right) \ dk,$$ given $Y(0)=1$. Here, we use the ...
2
votes
2answers
35 views

How do I periodically extend a function?

I am given that $f(x) = -1-x$ on $[-1,1)$ and I'm asked to find the Fourier series for the $2$ period extension of $f(x)$. I get how to find a Fourier series for the given function on the given ...
1
vote
1answer
24 views

Question about the FFT version of the gradient of a function.

We know that for a sufficiently smooth function $f:\mathbb{R}^{3}\to\mathbb{R}$, its Fourier Transform $\hat{f}(\mathbf{k}) \colon= \mathcal{F}\{f\}$ should satisfy (using integration by parts): $$\...
0
votes
2answers
37 views

Computation of a 2D Fourier transform

Is there an easy to compute the Fourier transform of $\frac{1}{1+x_1^2+x_2^2}$ in two variables ? And more generally, the Fourier transform of $\frac{1}{1+x_1^2+...+x_N^2}$, where $N$ denotes the ...
0
votes
0answers
11 views

Fourier Transform on $L^1$ is not surjective.

Let $f\in L^1(\mathbb{R})$ and suppose that $\hat f$ is an odd function. Prove that there exists $C> 0$ s.t. for each $a>1$ we have $$ \left | \int_1 ^a \frac{\hat f(x)}{x} dx \right| \le C$$ ...
8
votes
0answers
113 views

Triple sum $\sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} \sum\limits_{c=1}^{\infty} \frac{\cos a \cos b \cos c}{a^2 + b^2 + c^2}$

We have poor water heating system in our countryside house (currently it takes 4 hours to warm up the water), and my father has decided to improve it; he bought a water tank and placed it up in the ...
0
votes
2answers
25 views

how can i solve this PDE (IVP) by using Fourier transform?

$$ u_t=\alpha^2u_{xx}\\ u(x,0)=e^{-x^2} $$ where t>0 i tried to solve it, but not certain and i got hint $$ \int_{-\infty}^{\infty} e^{-\xi^2+i\xi x}d\xi=\sqrt\pi e^{-x^2/4} $$ thanks for your ...
2
votes
1answer
50 views

Substitute $g(x)=Ae^{-\beta x^2}$ into $\theta(x,t)=\frac{1}{2\sqrt{\pi\alpha t}}\int_{-\infty}^{\infty} g(\eta)\exp(-(x-\eta)^2/4\alpha t) \ d\eta$

I am trying to show that by directly substituting $g(x)=Ae^{-\beta x^2}$ into $$\theta(x,t)=\frac{1}{2\sqrt{\pi\alpha t}}\int_{-\infty}^{\infty} g(\eta)\exp(-(x-\eta)^2/4\alpha t) \ d\eta,$$ we obtain ...
3
votes
2answers
34 views

Does Fourier imply Laplace?

Can we find a function $f(t)$ for which $$\int_{-\infty}^{+\infty}f(t)e^{-j\omega t}dt,$$ converges but $$\int_{-\infty}^{+\infty}f(t)e^{-st}dt,$$ does not ? Here, $j^2=-1$, $\omega$ is a real number ...
0
votes
2answers
26 views

Why is $\sum_{n=0}^{N-1}e^{-2i\pi(k+k_0)n/N}=N\delta(k-N+k_0)$?

EDIT: $\delta$ is the Dirac delta function and in the context it is defined as $\delta(0)=1$ and $0$ for all $n\neq 0$. I am having trouble concluding that $\sum_{n=0}^{N-1}e^{-2i\pi(k+k_0)n/N}=N\...
0
votes
0answers
13 views

Proving DTFT pair as special case of another.

Consider these 2 basic discrete-time Fourier transform (DTFT) pairs... $$ \require{extpfeil}\Newextarrow{\xleftrightarrow}{15,15}{0x2194} \begin{array}{rcl} u[n] & \xleftrightarrow{\mathscr{F}} &...
0
votes
0answers
12 views

Fourier shift theorem: What happens when the shift is very large (e.g., multiplying a signal by a shifting bandpass filter)?

I'm trying to apply the Fourier transform shift theorem and the convolution theorem to a problem. I thought I was using them logically, but I have a case where my logic has to break. I have a given ...
0
votes
0answers
16 views

Fourier transform of a product of two functions

How can I compute the Fourier transform of a function of this form $f(u)\vert u-v \vert^{a}??$ with v is a variable that appears in my computations and lives in $\mathbb{R}$. Any idea how to ...
0
votes
0answers
10 views

Fourier transformation of a discrete function; conclusion of having a finite sum

For a function $g$ defined at discrete points $x_n = n a$ with $n \in \{0, \ldots, N\}$ with periodicity $g(x_0) = g(x_N)$ the discrete Fourier transformation reads $g(x_n) = \frac{1}{Na} \sum_{q \in ...
5
votes
0answers
161 views
+100

Book recommendation for learning image processing as an application of Fourier analysis

I have searched around this website for some references of applications of Fourier transform in image processing, but did not find any satisfactory ones. I have a major in maths. In years I always ...
0
votes
0answers
33 views

Confused on Fourier Transform. What is amplitude spectrum and phase?

I can't understand how can I get the second formula by using the Euler formula on the first equation. How can I get ride of the summation symbol?
1
vote
0answers
16 views

Proving a that $C_c(TM)$ injects into $C_0(T^*M)$ through Fourier Transform

Let $M$ be a smooth manifold. Equip $TM$ with an Riemannian structure. We let $f \in C_c(TM)$ and define a homorphism into $C_0(T^*M)$ by $$ (x,w) \in T^*M, \hat{f}(x,w) = (2\pi)^{-\frac{n}{2}} \...
0
votes
0answers
36 views

Compute this integral over $\mathbb C$?

How can I compute this integral: $$I=\int_{\mathbb C} e^{-|z|^2} \, e^{i\left<z,w\right>} \, e^{a z + b \overline{z}} \, dz,$$ where $w\in \mathbb C$ and $a,b \in \mathbb R$ and where $\left&...
3
votes
0answers
70 views

Definitions of Sobolev Spaces - are they the same?

I have read two definitions of Sobolev spaces. Definition 1: We let $\lambda$ denote $\lambda^s(\xi)=(1+|\xi|^2)^\frac{s}{2}$ for $s \in \Bbb R$, $\xi \in \Bbb R^n$. We say that $u \in H^s$, if $u ...
0
votes
0answers
18 views

Fourier transform of magnitude of a vector

How would you do the Fourier transform of the magnitude of $v$ vector(i.e. $\sqrt{x^2 + y^2 + z^2}$ where $x,y,z$ are the components of the vector $v$)? Is it similar to the one dimensional case? I ...
0
votes
1answer
35 views

The Spectral Sets of $ \mathbb{Z}_{2^k}$(Related to the discrete version of the Fuglede's conjecture)

A subset $S\subseteq\mathbb{Z}_N$ is called a spectral if $L^2(S)$ has an orthogonal basis of exponentials (indexed by $\Lambda$). This is equivalent to the following two conditions to hold: Any ...
0
votes
0answers
9 views

Samples of the Fourier transform of a continuous-time signal

Assume that I manually compute the Fourier transform, $X_a(f)$, of a continuous-time aperiodic signal, $x_a(t)$, and then take equally-spaced samples of this Fourier transform, $X(k) \equiv X_a(k F_0)$...
0
votes
1answer
33 views

Fourier transform of fractional function [closed]

How can I calculate the Fourier transform of $ \ f(x)= \ e^{j2π(\frac{Dx^2+Bx+A}{Tx+C})} $ that A, B, C, D, T are constant.
1
vote
1answer
18 views

Fourier series of translated square

I can't seem to find the correct Fourier series coefficients ($s_n$) of the following periodic function. I know how to get the Fourier series of the same one that is not vertically translated and has ...
2
votes
0answers
33 views

Is the image of the Fourier Transform $\mathscr{F}: M(\mathbb{T}) \to BC(\mathbb{Z})$ dense in $BC(\mathbb{Z})$?

Let $M(\mathbb{T})$ be the space of (finite) complex measures on the circle ($[0,1]$ for calculation). Let $BC(\mathbb{Z}) = l^\infty (\mathbb{Z})$ be the bounded continuous functions on the integers ...
2
votes
0answers
23 views

Proof that orthogonal conjugation is self-inverse

I am trying to prove that given a cyclic gropu $G$ and a subgroup $H$ then $(H^\bot)^\bot = H$, where $H^\bot = \{ \alpha \in G | \chi_\alpha(x) = 1 \forall x \in H\}$ and $\chi_\alpha$ are fourier ...
1
vote
1answer
26 views

Convolution example

Let $f(x)$ be an even function on $(-\infty , \infty)$ and let $g(x) = \sin(ax)$, $a >0$. Show that $$(f \ast g)(x) = \int_{-\infty}^{\infty} f(t)\sin(a(x-t))\,dt= \sin(ax)\hat{f}(a).$$ I ...
2
votes
0answers
20 views

What do we mean by spectral multipliers?

I am reading a research paper in the context of geometric deep learning. In that paper, the following equation is defined $$f_l^{\text{out}} = \xi \left( \sum_{l'=1}^p \Phi_k \hat{G}_{l, l'} \Phi_k^T ...
1
vote
1answer
87 views

1D wave equation with Boundary Conditions: Fourier Transform solution

I am considering the 1D wave equation with $c=1$ for the sake of simplicity: $$u_{tt}-u_{xx}=0,\quad \forall x\in\mathbb R,\; \forall t\in\mathbb R\tag{1}\label{eq:1}$$ with the following boundary ...
0
votes
1answer
43 views

Integrating trignometric finctions with exponetial arguments

I want to integrate a trigonometric function where the argument contains an exponential function. $$f(t) = \cos\left(\omega_{0} \left(1 - e^{-t/\tau} \right) t \right)$$ where $\omega_{0}$, $\tau$, ...
1
vote
2answers
55 views

$ \mathscr{F}\Big\{e^{-(a-jb) t^2}\Big\}$

By using the definition of continuous-time-Fourier-transform: $$ \mathscr{F}\Big\{x(t)\Big\} \triangleq X(ω) = \int\limits_{-\infty}^{\infty}x(t)\, e^{-j 2 \pi ω t} \,\mathrm{d}t$$ and solving the ...
0
votes
0answers
19 views

Dirichlet conditions for the existence of the Fourier transform

Assuming a periodic function, Dirichlet conditions are sufficient (not necessary) conditions for Fourier series. 1) As they are defined for Fourier series, how Dirichlet conditions can also be ...
3
votes
1answer
89 views

Find function $f$ such that $(p*f)(x) = xf(x)$

Question: The function $p(x)$ is defined as $p(x) = e^{-x}$ for $x>0$ and $p(x) = 0$ for $x<0$. Find the Fourier Transform of $p(x)$ and use the convolution theorem to find $f(x)$ such that:...
0
votes
2answers
50 views

$L^2$ and Sobolev space

In Raymond's book on Pseudodifferential Operator page 18, he says , where $S'$ is the tempered distributions, we define sobolev space of exponent $s$ as $u \in S'$ with $\lambda^s \hat{u} \in L^2$. ...
0
votes
1answer
18 views

$L^2$ and Schwartz Space

It is stated, Introduction to the theory of Pseudodiffernetial Operators, by Raymond, pg 9, Theorem 16, if $S$ is the Schwartz space If $\varphi \mapsto U(\varphi)$ is a semilinear form on $ S$ ...
1
vote
0answers
14 views

Uniqueness of solutions when input is discontinuous at $t_0$.

I have recently been reading Signal Processing and Linear Systems by B. P. Lathi, but I also have some general knowledge of differential equations from reading parts of Elementary Differential ...
0
votes
1answer
30 views

Fourier transform of $\exp(-z^k)$: How can one quatify its decay?

Consider the Fourier transform of $\exp(-z^k)$ where $k$ is a positive integer. As the function is analytic, I expect it to have exponential decay at infinity. Is there some known theorem giving a ...
0
votes
0answers
12 views

Magnitude spectrum of a sampled time continuous signal

I have a signal $x(t)=\cos(2 \pi 200 t)+2\cos(2 \pi 400 t)+\cos(2 \pi 600 t)$. With sample frequency $F_s=1000 Hz$. How do I draw the magnitude spectrum of the sampled signal $x(n)$? I've tried ...
0
votes
0answers
18 views

STFT calculation with Gaussian Window

$$ f(t)=\exp(jat^2) \,\,\, and \,\,\, g(t)\,\,is\,\, a\,\, Gaussian\,\, Window:$$ $$ g(t)= \left (πσ^2\right)^{\frac{-1}{4}}\exp\left (\frac{-t^2}{2σ^2} \right ) , \,\,\,\,\,\,\left \|g(t) \right \|...
0
votes
0answers
18 views

How to solve a Semi-infinite Fourier Transform of the Laplace Equation with Dirac Delta boundaryCondition

I am given:enter image description here I know that I will need to use the Semi-infinite Sine transform to solve this problem, however I am unsure how to go about solving with a Dirac Delta function ...
1
vote
2answers
86 views

Inverse Fourier Transform of a half-integer Bessel function

Is there an analytical solution for the following inverse Fourier transform? $$f(x)=\frac{1}{\sqrt{2\pi}}\,\mathrm{j}^n \int_{-\infty}^\infty\frac{1}{\sqrt{k}\,(k^4-\lambda^4)}\mathrm{J}_{n+\frac{1}{...
0
votes
0answers
25 views

Discrete Fourier transform terms derivation

According to the paper Lecture 7 - Discrete Fourier Transform we can approximate a Fourier transform $$F(\omega ) = \int_{ - \infty }^\infty {f(t){e^{ - j\omega t}}}$$ by the series $$F(\omega ) = \...
3
votes
1answer
53 views

6D Fourier transform of Coulomb potential

I just wanted to check my result. Let's define the Fourier transform as (the integral over whole real line): $$g(k)=\frac{1}{\sqrt{2 \pi}} \int e^{-i k x} f(x) dx$$ We have the following function: ...