Fourier Series; Harmonic Analysis

Exercise 1 : Symmetric rectangular pulse

Obtain the Fourier series of the 2\pi-periodic function

$$ f(x)=\begin{cases} 0, & -\pi\le x< -\tfrac{\pi}{2},\\ 1, & -\tfrac{\pi}{2}\le x< \tfrac{\pi}{2},\\ 0, & \tfrac{\pi}{2}\le x\le \pi, \end{cases} $$

State whether the periodic extension is even or odd, compute the nonzero Fourier coefficients, and write the series in sine/cosine form.

Exercise 2 : Odd square wave

For the 2\pi-periodic function

$$ f(x)=\begin{cases} 1, & -\pi\le x<0,\\ -1, & 0\le x\le\pi, \end{cases} $$

Compute the Fourier series, indicate whether only sine terms appear, and derive the explicit formula for the coefficients.

Exercise 3 : Full-wave rectified sine

Find the Fourier series of

$$ f(x)=|\sin x|=\begin{cases} -\sin x, & -\piDetermine whether the function is even or odd, compute the cosine-series coefficients, and write the series in closed form.

Exercise 4 : Pulse with period 4/pi

Obtain the Fourier series for the function with period $4\pi$:

$$ f(x)=\begin{cases} 0, & -2\pi\le x< -\pi,\\ 1, & -\pi\le x< \pi,\\ 0, & \pi\le x<2\pi. \end{cases} $$

Give the Fourier coefficients and express the series.

Exercise 5 : General rectangular pulse

Let $f$ be the $T$-periodic rectangular pulse of width $t_0$ and unit height:

$$ f(t)=\begin{cases} 0, & -\tfrac{T}{2}\le t< -\tfrac{t_0}{2},\\ 1, & -\tfrac{t_0}{2}\le t\le \tfrac{t_0}{2},\\ 0, & \tfrac{t_0}{2}Derive the Fourier series (both real and complex forms) and simplify the coefficients to closed form (sinc-type expressions).

Exercise 6 : Triangular wave (continuous, even)

Consider the even, 2\pi-periodic triangular wave defined on $[-\pi,\pi]$ by

$$ f(x)=1-\frac{|x|}{\pi},\qquad -\pi\le x\le\pi. $$

Compute the Fourier cosine series and show that coefficients decay as $1/n^2$.

Exercise 7 : Sawtooth wave (odd)

For the 2\pi-periodic sawtooth function

$$ f(x)=\frac{x}{\pi},\qquad -\piobtain the Fourier sine series (odd extension) and derive the coefficient formula; discuss convergence at discontinuities.

Exercise 8 : Half-wave rectified sine

Define the half-wave rectified sine with period $2\pi$ by

$$ f(x)=\begin{cases} \sin x, & 0Find the Fourier series (both sine and cosine terms may appear) and compute the coefficients explicitly.

Exercise 9 : Even and odd extensions

Given a function $g(x)$ defined on $[0,\pi]$, explain how to form the even and odd $2\pi$-periodic extensions. Then, for $g(x)=x$ on $[0,\pi]$, compute both the Fourier cosine series (even extension) and the Fourier sine series (odd extension).

Exercise 10 : Parseval identity application

Use Parseval’s identity to compute the sum of squared Fourier coefficients. As a concrete task, compute the Fourier series of $|\sin x|$ and use Parseval’s identity to evaluate a related infinite sum (state which sum you evaluate and show the steps).

Exercise 11 : Gibbs phenomenon

For the 2\pi-periodic square wave defined by

$$ f(x)=\begin{cases} 1, & -\pi\le x<0,\\ -1, & 0\le x\le\pi, \end{cases} $$

compute the $N$-term partial sums of the Fourier series near a jump and numerically or analytically show the overshoot (Gibbs phenomenon). Quantify the limiting overshoot as $N\to\infty$.

Exercise 12 : Complex Fourier coefficients for pulses

Consider the $T$-periodic rectangular pulse of width $t_0$ and unit height:

$$ f(t)=\begin{cases} 0, & -\tfrac{T}{2}\le t< -\tfrac{t_0}{2},\\ 1, & -\tfrac{t_0}{2}\le t\le \tfrac{t_0}{2},\\ 0, & \tfrac{t_0}{2}Derive its complex Fourier coefficients $c_n$. Show equivalence between the complex form and the real sine/cosine form.