Homework 6

a) For some $N \in \mathbb{N}$, let $x \in l_2\Big( (-N, -(N-1), \cdots, N-1, N); \mathbb{C} \Big)$ with

$x(n) = 1_{\{|n| \le N_1\}}$

Find the Fourier series expansion of $x$. Study the case with $N_1 = N$, and the case with $N_1 = 0$.

b) For some $T \in \mathbb{R}_+$, let $x \in L_2([-\frac{T}{2}, \frac{T}{2}]; \mathbb{C})$ with

$x(t) = 1_{\{|t| \le T_1\}}$

Find the Fourier series expansion of $x$. Study the cases with (i) $T_1 = \frac{T}{2}$, and (ii) the case where, with $T_1 = \frac{1}{2n}$, we have

$x_n(t) = n1_{\{|t| \le \frac{1}{2n}\}},$

as $n \to \infty$.

In $l_2\Big( \{0, 1, \ldots, N-1\}; \mathbb{C} \Big)$, we observed that the Fourier series

$\frac{1}{\sqrt{N}} e^{i2\pi \frac{k}{N}n}, \quad k \in \{0, 1, \ldots, N-1\},$

provides a complete orthonormal sequence, hence, provides a basis. Let $x \in l_2\Big( \{0, 1, \ldots, N-1\}; \mathbb{C} \Big)$ be given as:

$x(n) = 2 + 3\cos\left(\frac{4\pi}{N}n\right) + \cos\left(\frac{6\pi}{N}n\right) + \cos\left(\frac{6\pi}{N}n + \frac{\pi}{2}\right),$

for all $n$.

What is the Fourier series expansion of $x$?

In class, we observed that in $L_2([-\frac{P}{2}, \frac{P}{2}]; \mathbb{C})$, the sequence of signals

$\left\{ \frac{1}{\sqrt{P}} e^{i2\pi \frac{k}{P}t}, k \in \mathbb{Z} \right\}$

provides a complete orthonormal sequence.

Let $f \in L_2([-\frac{P}{2}, \frac{P}{2}]; \mathbb{C})$ such that $f(-\frac{P}{2}) = f(\frac{P}{2})$. Suppose $f$ is differentiable.

Find the Fourier expansion of $g(t) = f'(t)$ for $t \in [-\frac{P}{2}, \frac{P}{2}]$ in terms of the Fourier series coefficients of $f$.

In this assignment, you are asked to generate (with the help of Matlab) the Fourier series expansion of the following function: $x \in l_2(\{-N, -(N-1), \cdots -1, 0, 1, \ldots, N-1, N\})$ where $x(n) = 1$ for $|n| \le N_1$ and $0$ elsewhere. That is

$x(n) = 1_{\{|n| \le N_1\}}, \quad -N \le n \le N$

with $1_{\{.\}}$ denoting the indicator function.

You are asked to obtain a representation of the form:

$\tilde{x}(n) = \frac{1}{\sqrt{2N+1}} \sum_{k=-N}^{N} a_k e^{ik(2\pi/(2N+1))n}$

where

$a_k = \frac{1}{\sqrt{2N+1}} \sum_{n=-N}^{N} x(n) e^{-ik(2\pi/(2N+1))n}.$

By defining a FourSeries function, obtain the expansion coefficients with Matlab. Your function should take the signal as its input and generate the Fourier series coefficients as the output.

Now, let $(N = 2, N_1 = 1)$; $(N = 20, N_1 = 5)$; and $(N = 20, N_1 = 20)$ and obtain $\tilde{x}(n)$ above for $n \in \mathbb{Z}, n \in [-25, 25]$. Plot the Fourier series coefficients for $x$.

For $N = 20, N_1 = 1$, plot $\tilde{x}(n)$ for $n \in \mathbb{Z}, n \in [-25, 25]$. What is the relationship between $x(n)$ and $\tilde{x}(n)$?

a) Let $x \in L_2([0,2]; \mathbb{C})$ be given by:

$x(t) = 1_{\{t \le 1\}}t + 1_{\{1 \le t \le 2\}}(1 - t)$

Let $\hat{x}(\frac{k}{2})$ denote the Fourier series coefficient corresponding to $\{\frac{1}{\sqrt{2}}e^{i2\pi \frac{k}{2}t}, t \in [0,2]\}$.

With Matlab, generate the plot of the signal

$x_N(t) = \sum_{k=-N}^{N} \hat{x}\left(\frac{k}{2}\right) \frac{1}{\sqrt{2}} e^{i2\pi \frac{k}{2}t}, \quad t \in [0,2]$

for $N = 5, 10$ and $15$. Here $\hat{x}(\frac{k}{P})$ are the Fourier Series expansion coefficients.

Observe that, the signal looks more and more like the original signal as $N$ gets larger.

b) Prove that $\lim_{N \to \infty} \int (x(t) - x_N(t))^2 dt = 0$.

Hint: Use the properties of Hilbert spaces and the fact that $\{\frac{1}{\sqrt{P}}e^{i2\pi \frac{k}{P}}\}$ forms a complete orthonormal sequence. We had proved a related theorem in class in the context of Hilbert spaces. You could invoke this result directly in your argument.

c) Does for a general $x \in L_2([0,2]; \mathbb{C})$,

$\sup_{t \in [0,2]} |x(t) - x_N(t)| \to 0,$

as $N \to \infty$? Explain your argument.

Summarize the Fourier transformations DDFT (Discrete-to-Discrete), CDFT (Continuous-to-Discrete), DCFT (Discrete-to-Continuous), and CCFT (Continuous-to-Continuous) by explicitly stating the input and the output spaces.

Follow your notes and study the theorem that CCFT is a continuous map from $\mathcal{S}$ to $\mathcal{S}$.

Show that CCFT is a unitary transformation from $L_2(\mathbb{R}; \mathbb{C})$ to itself.

Hint: You may use the construction we discussed in class where the $\mathcal{F}_{CC}$ is first defined on $\mathcal{S}$ and then extended to $L_2(\mathbb{R}; \mathbb{C})$.

Describe the Fast Fourier Transform (FFT) Algorithm and explain why the computational complexity is reduced when compared with the usual Fourier Transform computations.