3
votes
1answer
82 views

Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
0
votes
2answers
57 views

$\int_0^\pi\sin(2t)e^{-in2t}dt$ complex number integral for integer values of n

$$\int_0^\pi\sin(2t)e^{-in2t} \, dt$$ wolfram alpha say the answer is $$\frac{1-e^{-2 i n π}}{2-2 n^2}$$ although using the integral trig identity $$\int ...
1
vote
2answers
54 views

Finding complex Fourier coefficients

This is probably an easy question, but I'm a little bit stuck, so any help will be appreciated. PROBLEM Find the complex Fourier coefficients of: $$f(t) = \sin(2\pi t)$$ and $$f(t) = |\sin(2\pi ...
0
votes
0answers
7 views

How can we derive $\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)]$?

When I was reading digital communication theory, I couldn't derive following equation $$\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)] $$ ...
0
votes
0answers
19 views

Find Complex Fourier coefficients for $f(x) = \sum^5_{m=1} (\frac{-1}{2})^m \cos(2^mx)$

This is a question from a book that I'm trying to solve and I don't know how. Find Complex Fourier coefficients for $f(x) = \sum^5_{m=1} (\frac{-1}{2})^m \cos(2^mx)$ Can you please give me some ...
0
votes
0answers
16 views

Hyprecomplex datapoint

I was working on 2D dataset 512$\times$512. To be more precise,its the datapoints are collected in dimensions t1 and t2. All the data points are complex in the form a+b$i$. To get the frequency ...
1
vote
1answer
40 views

Complex Conjugate (FourierSeries)

Let $f$ be a piecewise continuous complex function on the interval $[\pi,\pi]$ and \begin{equation} f(x) \sim \sum_{n=-\infty}^{\infty}c_{n}e^{inx} \tag{*} \end{equation} be its complex Fourier ...
0
votes
1answer
116 views

Is the matrix Wn from the DFT a Hermitian operator?

A homework question asks me whether or not the matrix $W_N$ from the matrix representation of the Direct Fourier Transform is a Hermitian operator. From what I understand an Hermitian operator does ...
0
votes
2answers
56 views

Confused between multiple representations of Fourier Series' formula

I have never used the formula for Fourier Series with the representation that the instructor of the above video is using that involves $k$ and $\omega$. Instead, I use $n$ and $\pi$. Now, suppose ...
1
vote
1answer
211 views

What is the odd Fourier extension of $\sin x \cos(2x)$?

odd half range extension of $$f(x) = \sin x \cos(2x)\text{ with limits $0$ to $\pi$}$$
2
votes
5answers
297 views

Fourier Analysis

I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function? Can someone provide me simple information ...
1
vote
0answers
55 views

Proving that two representations of a Fourier series are the same

I have to show that $$\sum_{n=0}^\infty A_n\cos\left({xn\frac{2\pi}{T}-\theta_n}\right) \equiv \sum_{n=-\infty}^\infty c_n \mathrm{e}^{\left({ixn\frac{2\pi}{T}}\right)}$$ I have tried two ...
1
vote
1answer
322 views

Real part of an integral with complex argument

This is a paper about Fourier cosine series approximation to option pricing problem. The coefficient $A_k$ is defined as $$A_k=\frac{2}{b-a}\int_a^bf(x)\cos\left(k\pi\frac{x-a}{b-a}\right)dx$$ Then ...
5
votes
1answer
210 views

For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero

Consider an $n$-sided convex polygon $P$ that contains the origin in the complex plane. Let the $j$-th vertex be denoted $z_j = r_j e^{i\theta_j}$ ($0 \leq \theta_j < 2 \pi$) for $j= 1 \dots n$. ...
4
votes
2answers
446 views

Using complex exponentials as solution of ODE

I'm having trouble wrapping my head around the following issue. My book solves a problem without using complex exponential solution like $C_1 e^{it}$ and using either $A \cos(t) + B \sin(t)$ or $A ...