Solved ECE113: DSP Homework 3

Description

Problem 1: Problem 6.4 in R1 (i.e., Proakis 4th Edition)
(Note: To obtain the Fourier Transform of x(n)=x0(nT)x(n)=x0​(nT) in this problem, recall the “differentiation in the frequency domain” property of FT, i.e., nx(n)↔jdX(f)2πdfnx(n)↔j2πdfdX(f)​)

Problem 2:
Consider the two CW tones given by:

$$a(t)=\cos (4000\pi t)$$$$b(t)=\cos (200\pi t)$$

These two tones are mixed (i.e., multiplied) and then sampled as shown in the following figure. What would be the minimum sampling rate, FsFs​, measured in Hz, that would result in a sequence $x(n)$ without any aliasing errors (i.e., no spectral replication overlap)?

Problem 3:
Consider an information-bearing signal, $r(t)$, with the following frequency spectrum:

R(F)↓2−100100F(KHz)R(F)↓2−100100F(KHz)​

The signal $r(t)$ is then modulated onto a carrier with frequency 1.0 MHz:

xa(t)=r(t)cos⁡(2π⋅106t)xa​(t)=r(t)cos(2π⋅106t)

(a) Sketch the frequency spectrum of xa(t)xa​(t) (i.e., Xa(F)Xa​(F)). Please show the values for all the important points on both axes.
(b) Now, assume xa(t)xa​(t) is sampled as shown below. What would be the minimum bandpass sampling rate, FsFs​, in KHz, in order to avoid any aliasing error?

xa(t)→ideal A/Dx(n),Fsxa​(t)ideal A/D​x(n),Fs​

(c) Now, assume Fs=800Fs​=800 KHz. Sketch the frequency spectrum, $X(\omega )$, for $-2\pi \le \omega \le 2\pi$. Please show the values for all the important points on both axes. Also please show values on the frequency axis in both KHz and rad/sample scales.

Problem 4:
(a) Consider a band-pass signal with the following frequency spectrum. What would be the minimum center frequency FcFc​, in terms of the signal bandwidth $B$, that would enable us to do bandpass sampling while avoiding any aliasing?

$$|X(F)|$$

(b) If a person wants to be classified as a soprano in classical opera, she must be able to sing notes in the frequency range of 247 Hz to 1175 Hz. Is bandpass sampling of full audio spectrum of a singing soprano possible? If yes, what would be the minimum FsFs​ sampling rate allowable for bandpass sampling? If no, why not?

Problem 5: Problem 11.1 in R1

Problem 6:
Given the input sequence xi(n)xi​(n), as shown in (a) in the following figure, whose magnitude spectrum is shown in (b) below, draw a rough sketch of the magnitude spectrum ∣Xc(F)∣∣Xc​(F)∣ of the system’s complex output sequence: xc(m)=xi(m)+jxo(m)xc​(m)=xi​(m)+jxo​(m). The frequency magnitude responses of the complex bandpass filter hBP(k)hBP​(k), and the real-valued highpass filter hHP(k)hHP​(k) are shown in (c) and (d) below.

MATLAB Exercises:

Part 1

Using the matrix-vector multiplication approach discussed in this chapter, write a MATLAB function to compute the DTFT of a finite-duration sequence. The format of the function should be:

function [X] = dtft(x, n, w)
% Computes Discrete-time Fourier Transform
% [X] = dtft(x, n, w)
% X = DTFT values computed at w frequencies
% x = finite duration sequence over n
% n = sample position vector
% w = frequency location vector

Note: Your function will be a single line performing a matrix-vector multiplication. It is fairly straightforward, and described in the following excerpt:

X(ejω)≜F[x(n)]=∑n=−∞∞x(n)e−jωnX(ejω)≜F[x(n)]=n=−∞∑∞​x(n)e−jωn

If $x(n)$ is of finite duration, then MATLAB can be used to compute X(ejω)X(ejω) numerically at any frequency $\omega$. The approach is to implement the sum directly. If, in addition, we evaluate X(ejω)X(ejω) at equispaced frequencies between $[0,\pi ]$, then it can be implemented as a matrix-vector multiplication operation.

Let the sequence $x(n)$ have $N$ samples between n1≤n≤nNn1​≤n≤nN​, and evaluate X(ejω)X(ejω) at

ωk≜πMk,k=0,1,…,Mωk​≜Mπ​k,k=0,1,…,M

Then:

X(ejωk)=∑ℓ=1Ne−j(π/M)knℓx(nℓ),k=0,1,…,MX(ejωk​)=ℓ=1∑N​e−j(π/M)knℓ​x(nℓ​),k=0,1,…,M

In matrix form:

$$\mathbf{X}=\mathbf{Wx}$$

where

W≜{e−j(π/M)knℓ}W≜{e−j(π/M)knℓ​}

In MATLAB, using row vectors for sequences and indices:

XT=xT[exp⁡(−jπMnTk)]XT=xT[exp(−jMπ​nTk)]

Implementation example:

k = [0:M]; n = [n1:n2];
X = x * (exp(-j*pi/M) .^ (n' * k));

Part 2

The following finite-duration sequences are called windows and are very useful in DSP:

• Rectangular:

  RM(n)={1,0≤n<M0,otherwiseRM​(n)={1,0,​0≤n<Motherwise​

• Triangular:

  TM(n)=[1−∣M−1−2n∣M−1]RM(n)TM​(n)=[1−M−1∣M−1−2n∣​]RM​(n)

• Hanning:

  CM(n)=0.5[1−cos⁡(2πnM−1)]RM(n)CM​(n)=0.5[1−cos(M−12πn​)]RM​(n)

• Hamming:

  HM(n)=[0.54−0.46cos⁡(2πnM−1)]RM(n)HM​(n)=[0.54−0.46cos(M−12πn​)]RM​(n)

For each of these windows, determine their DTFTs for $M=10,25,50,101$. Scale transform values so that the maximum value is equal to 1. Plot the magnitude of the normalized DTFT over $-\pi \le \omega \le \pi$. Study these plots and comment on their behavior as a function of $M$.

Note: Use the dtft function you wrote for Part 1. For the frequency vector input, you can use:

w = linspace(-pi, pi, 501);