0
votes
1answer
27 views

Discrete Fourier Transform Interpretation

Using Mathematica I took the Discrete Fourier Transform (DFT) of a vector whose entries are volumes of a particular stock. The power spectrum is plotted below: There are two questions that I have ...
1
vote
0answers
33 views

Using the Modulation property of the Fourier Transform

I'm working on a problem: Let $X(w)$ be the Fourier transform of $x(t)$. Find the transform of $y(t)=x(5t+3)\sin(2t)$ in terms of X(w). I am table to take the Fourier transform of $x(5t+3)$ and ...
0
votes
1answer
64 views

abs(x)cos(x) in Fourier space

I am working on some problems concerning Fourier Transform and I am facing something I don't understand. I am trying to understand what is the representation of the function f(x)=abs(x)cos(x) in the ...
0
votes
1answer
60 views

Fourier Transform of rational function

So I have this function: $$f(t)=\frac{1}{(1-it)^{n+1}}$$ And I have the Fourier Transform defined as $$\hat{f}(\lambda)=\frac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(t)e^{-\lambda.it}dt $$ Now my ...
2
votes
0answers
37 views

Invariant functions under integral transforms

We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ...
0
votes
1answer
82 views

Fourier transformation: Determining the axis

I need some help with the Fourier transformation of my data. My original data is a Distance VS Time: upon doing a Fourier Transform, I get the following: I understand that normally after a ...
2
votes
1answer
64 views

If $f \in S(\mathbb R)$, can we say $\widehat{|f|} \in L^{1}(\mathbb R)$?

Let $f\in L^{1} (\mathbb R) := \{f:\mathbb R \rightarrow \mathbb C \ \text {measurable functions} : \int_{\mathbb R} | f(x)| dx < \infty \}$ and the Fourier transform of $f$, $\hat{f} (y) : = \int ...
0
votes
1answer
40 views

inverse transform of $Z(\omega) =\frac{a}{\alpha-i\omega}$

I am stuck at calculating the inverse transorm of $Z(\omega) =\frac{a}{\alpha-i\omega}$. Can someone help me please? thanks
1
vote
0answers
41 views

The Fourier Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
4
votes
2answers
122 views

Fourier, Laplace, … and other Integral-transformations

I know Laplace, Fourier and Mellin-Transformation. Is there a general theory of transformations? My main interest is about classification of transformations satisfying specified properties like ...
1
vote
1answer
67 views

Calculating Fourier Transform of $1/|t|^n$

I have found the Fourier Transform of $x(t)=|t|^{n}$ and i can't calculate the Fourier Transform of $x(t)=|t|^{-n}$. Any suggestions?
1
vote
1answer
360 views

Discrete Time Fourier Transform example: $x = [1 \; 2 \; 3 \; 4]^T \; \rightarrow \; X=?$

How do I find the Discrete Fourier Transform of the sequence below? $$ x = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$$ Show all steps.
1
vote
0answers
56 views

Help in implementing of peak function in Fourier transform

I have a function Peak function I know how to implement it in time range just need to caclulate $r$. first I initial $x$, y with a range and meshgrid them, after it calculate $r$ ...
1
vote
1answer
92 views

Fourier Transform - Time Shift

Could someone help me understand how a simultaneous time shift on two separate functions is possible? I am having trouble linking a property to this solution. Given the function: $$x(t) = ...
2
votes
0answers
48 views

Is there an intuitive understanding of what a walsh coefficient is?

I am working with Walsh coefficients. I know the intuitive understanding is almost that that they are the degree of connectivity, but it is there a better way of thinking about it? What is the ...
2
votes
2answers
150 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 ...
2
votes
1answer
167 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 ...
5
votes
2answers
555 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 ...
0
votes
2answers
255 views

Fourier Transformation

This expression: $x(t)=[e^{-3t+5}] u(t-1)$. I am trying to take the Fourier transformation of the above expression. I know that for $x(t)=[e^{-at}] u(t) \leftrightarrow \frac1{i\omega+a}$. But, ...
2
votes
0answers
70 views

bound on Hilbert transform

Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
2
votes
0answers
239 views

Understanding Discrete Cosine Transformation

I'm currently working on some software and a key component is 2D DCT. But my question is more general, as I'm trying to understand the DCT in general, let's say from engineers point of view. For ...
0
votes
0answers
102 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 ...
11
votes
2answers
1k views

Is a Fourier transform a change of basis, or is it a linear transformation?

I've frequently heard that a Fourier transform is "just a change of basis". However, I'm not sure whether that's correct, in terms of the terminology of "change of basis" versus "transformation" in ...