Tagged Questions

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 ...
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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 $$\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)]$$ ...
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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 ...
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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 ...
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Complex Conjugate (FourierSeries)

Let $f$ be a piecewise continuous complex function on the interval $[\pi,\pi]$ and $$f(x) \sim \sum_{n=-\infty}^{\infty}c_{n}e^{inx} \tag{*}$$ be its complex Fourier ...
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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 ...
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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 ...
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What is the odd Fourier extension of $\sin x \cos(2x)$?

odd half range extension of $$f(x) = \sin x \cos(2x)\text{ with limits 0 to \pi}$$
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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 ...
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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 ...
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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 ...
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 $j= 1 \dots n$. ...
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 ...