0
votes
0answers
48 views

Why does the Fourier transform include the base of the natural logarithm, the square root of -1 and $\pi$?

The formula itself, as a vector of summations of products of the original coefficients with some weight, itself a function the original and transformed coefficient indices, is not a hard pattern to ...
0
votes
0answers
14 views

Solution of definite integral of product of bessel function and exponential

I have an integral $I=\int_{\theta} \int_r J_m(k_1r)e^{-j[P_x r \cos(\theta)+P_y r \sin(\theta)]} r dr d\theta$ $0\leq\theta\leq2\pi; r<\infty$ is there any method to solve this?
0
votes
0answers
10 views

Spectral norm of a Hadamard product

Let $F$ be the $n\times n$ DFT matrix, i.e., $F_{i,j} = \exp\left(-\tfrac{2\pi(i-1)(j-1)}{n}\imath\right)$, for $1\le i,j\le n$ and $\imath =\sqrt{-1}$. Futhermore, let $\Vert \cdot\Vert$ and $\circ$ ...
3
votes
2answers
36 views

Is there any commonality between the use of Parseval's Identity in two different contexts?

In Fourier analysis, Parseval's Identity relates to "the summability of the Fourier series as a function." In inner product space analysis, the "identity" works as a "Pythagorean theorem" relating ...
0
votes
1answer
13 views

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 ...
1
vote
3answers
42 views

Understanding dot product of continuous functions

I'm reading about Fourier analysis and in my book the author speaks about dot product for continuous functions $f, g\in L^2(a,b)$(the set of functions which are square-integrable on the interval ...
1
vote
1answer
24 views

Proving that orthonormal set is an orthonormal basis

If I know that the set of functions $\{\phi_n\}_1^\infty$ forms an orthonormal basis on $L^2(a,b)$ and the set $\{\psi_n\}_1^\infty$ is an orthonormal set on $L^2(\frac{a-d}{c}, \frac{b-d}{c})$, with ...
1
vote
1answer
21 views

Proving that a set $\{\psi_n(x)\}_1^\infty = \{\sqrt{c}\;\phi_n(cx+d)\}_1^\infty$ is an orthonormal basis

I have the following problem I need to solve: Suppose $\{\phi_n\}_1^\infty$ is an orthonormal basis in $L^2(a,b)$ (set of square-integrable functions on $[a,b]$). Suppose $c>0$ and $d\in ...
1
vote
0answers
20 views

Let $f(x):= \sum^{\infty}_{-\infty} d_n e^{inx}$ for $x \in \mathbb R$. Find the Fourier series for $f$ and show uniform convergence against $f$.

Let $\sum^{\infty}_{-\infty} |d_n| < \infty$ and define $f(x):= \sum^{\infty}_{-\infty} d_n e^{inx}$ for $x \in \mathbb R$. Find the Fourier series for $f$ and show it converge uniformly on ...
0
votes
1answer
47 views

Multiply a circulant matrix by a vector with FFT.

I am asked to write a Matlab program to find the coefficients of the resulting polynomial which is the product of two other polynomials. However, I need someone to clarify the underlying concepts for ...
0
votes
0answers
20 views

Significance of Mutual Coherence

I am reading about compressive sensing and I am not able to understand the physical significance of mutual coherence. For tow matrices $\Phi$ and $\Psi$, mutual coherence is defined as $\mu(\Phi, ...
0
votes
0answers
27 views

8 X 8 FFT process

I'm having trouble figuring out how to compute F8 * c using the FFT algorithm. I understand that you factor F8 into ...
3
votes
2answers
76 views

Why are almost all prime sized circulant matrices non-singular?

Consider an $n$ by $n$ circulant matrix whose values are either $1$ or $0$. Now let $n$ be a prime. For any such $n$ there are exactly two circulant matrices that are singular (over $\mathbb{R}$). The ...
0
votes
0answers
43 views

Row sums of submatrix of the discrete Fourier transform

Let $p$ be a prime. Consider the $ p-1 \times \frac{p-1}{2}$ matrix $A$ given by $A_{m,n} = \cos(\frac{2\pi i mn}{p})$ Essentially $A$ is a the real part of the first $\frac{p-1}{2}$ columns (without ...
3
votes
1answer
36 views

Compare mixed derivatives to laplacian

Suppose $u,f$ periodic and smooth in $Q=[0,1]^n$ such that $\Delta u=f$. Show that for each $i,j$, $$\int_Q \left| \frac{\partial^2 u}{\partial x_i \, \partial x_j} \right|^2 \leq C \int_Q |f|^2.$$ ...
1
vote
0answers
77 views

Q: Bases and Frames using Fourier Series

Define $w: \Bbb R \rightarrow \Bbb C$ by \begin{equation} w(t) =\begin{cases} 1/\sqrt{2\pi} & t \in [0, 2\pi)\\ 0 & \text{otherwise}. \end{cases} \end{equation} and for $n \in \Bbb ...
1
vote
0answers
21 views

Is it possible to calculate a single frequency bin in $O(\log N)$ time (considering the $N\log N$ performance of FFT algorithms)?

Fast Fourier transform (FFT) algorithms are able to calculate the discrete Fourier transform (DFT) in only $O(N\log N)$ asymptotical time. Since there is roughly $N\log N$ operations for computing $N$ ...
0
votes
0answers
48 views

splitting up summations of product

I want to check the validity of splitting up summations of products. I am using the DFT matrix and am trying to get a simplified expression of it . In essence, I am trying to prove the following lemma ...
2
votes
5answers
148 views

Will spectral analysis help me understand digital signal processing better?

I am learning Fourier transforms, Z transforms etc. in Digital Signal Processing and I can work easily with integrals. However, I don't understand how a Fourier transform converts time domain signals ...
0
votes
0answers
50 views

The influence of eigendecomposition on the periodicity of a (rank 2) Hermitian matrix (of functions).

Let $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ be a Hermitian matrix (of Hermitian functions) with entries \begin{equation} r_{ij}(u,v) = 1 + Ae^{-2\pi i \phi_{ij}(ul_0 + vm_0)}; ...
1
vote
0answers
34 views

Fourier transform of a sequence by a matrix

Let $n$ be a positive integer and $H$ the Hilbert space $\ell^2(\mathbb Z^n,\mathbb C^n)$. For $u\in H$, denote by $\mathcal{F}(u)$ the Fourier transform of $u$, defined by $\displaystyle ...
1
vote
0answers
60 views

What sequence has this Discrete Fourier Transform?

Suppose $$ x[n]= \begin{cases} x_i &, i \in P\\ 0 &, i \notin P \end{cases} $$ where $P \subset \{0,1, \cdots,N-1 \}$ and $|P|=K$ and $x_i \geq 0$. Suppose these equalities hold : $$ ...
1
vote
1answer
99 views

FFT of a matrix and its square.

I am doing something computationally intensive that requires that I compute the fast fourier transform of a matrix, let's say $A$, and also compute the FFT of its square, $A^2$. I am wondering if ...
14
votes
4answers
522 views

How to interpret the adjoint?

Let $V \neq \{\mathbf{0}\}$ be a inner product space, and let $f:V \to V$ be a linear transformation on $V$. I understand the definition1 of the adjoint of $f$ (denoted by $f^*$), but I can't say I ...
5
votes
0answers
102 views

Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
1
vote
0answers
46 views

Conditions for matrix operator to preserve complex symmetry on DFT vector?

Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. ...
1
vote
0answers
110 views

DFT in complex form and Trigonometric Polynomial Interpolation: why different dimensions of basis vectors set?

I'm trying to figure out, how coefficients of Discrete Fourier Transform in complex form are converted into coefficients of Trigonometric Polynomials Interpolation: Say, I have a function vector with ...
2
votes
1answer
198 views

Confused as to how to prove the basis of dft is orthonormal

I have been stuck for hours trying to prove that the basis of discrete fourier transform is orthonormal can anyone point me in the direction of how to do so
1
vote
0answers
111 views

Matrix of Discrete fourier transform $F^4$ is identity

I already showed that Discrete fourier transform matrix is unitarian matrix. Now I would like to show that $F^4$ is identity. On wikipedia is written: "This can be seen from the inverse properties ...
1
vote
1answer
145 views

Sifting Property of Convolution

This is going to be a dumb question, but I can't figure it out, so here goes $ f(t)\quad \bigotimes \quad \delta \quad (t\quad -\quad { t }_{ o }) $ = $\int { f(\tau )\delta (t\quad -\quad { t }_{ ...
2
votes
0answers
110 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. $$ ...
3
votes
0answers
168 views

linear algebra approach to discrete cosine transform

I understand that the discrete Fourier transform simply changes basis to the discrete Fourier basis, which is an orthonormal basis of eigenvectors for any shift-invariant linear operator on $\mathbb ...
1
vote
1answer
164 views

Finding the eigenvalues of the sum of circulant and diagonal matrices - What am I doing wrong?

Saw this question about the eigenvalues of the sum of circulant and diagonal matrices on MO and, since I recall my prof mentioned circulant matrices and Robert Gray's book, I thought I'd give it a ...
1
vote
2answers
257 views

Understanding Matrix Formula with Scant Knowledge of Linear Algebra

$n$ is a power of $2$. $M =\pmatrix{ 1& x_0 & x_0^2 & \dots &x_0^{n-1}\\\ 1& x_1 & x_1^2 & \dots &x_1^{n-1}\\&& \vdots\\1& x_{n-1} & x_{n-1}^{2} ...
2
votes
1answer
684 views

Fourier transform over a diagonal matrix

Let $F$ be a $100 \times 100$ DFT matrix, and $U$ be a diagonal matrix with its diagonal entries being all positive, denoted by $U=\mathrm{diag}(u_1, u_2,\cdots, u_{100})$. My question is: Under ...
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 ...
4
votes
0answers
358 views

Fourier matrix - multiplicity of eigenvalues?

This question is Miscellaneous Exercise M.10 in Chapter 8 (Bilinear Forms) of Artin's Algebra. (The sentences in italics are due to me.) The row and column indices in the $n \times n$ Fourier ...
1
vote
0answers
59 views

Slowly varying vectors and coefficients of a sine transform

Let $u_k$ be the vector in $\mathbb{R}^n$ whose $i$'th entry is $\sin(\pi ki/n)$. The vectors $u_1,\ldots, u_n$ are orthogonal and correspondingly every vector in $\mathbb{R}^n$ can be decomposed as a ...
7
votes
3answers
620 views

Decomposing a discrete signal into a sum of rectangle functions

Hello math@stackexchange community ! I have a simple question that seems to have a non trivial answer. Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar ...
1
vote
1answer
3k views

Fourier basis functions

What are fourier basis functions? And how do I prove that fourier basis functions are orthonormal?
8
votes
1answer
796 views

How do I compute the eigenfunctions of the Fourier Transform?

In Andy's answer to the question "What are fixed points of the Fourier Transform" on Math Overflow, he shows that the Fourier Transform has eigenvalues $\{+1, +i, -1, -i \}$ and that the projections ...