Solved ECE113: DSP Homework 2

Description

Problem 1: Problem 2.24 in R1 (i.e., Proakis 4th Edition)

Problem 2: Problem 2.32 in R1

Problem 3: Problem 2.35 in R1

Problem 4: Problem 2.57 in R1

Problem 5: Problem 5.5 in R1

Problem 6: Problem 5.24 in R1

Problem 7: Problem 9.5 in R1 (In this problem, “transposed structure” refers to transposed Direct Form II).

Problem 8: Problem 9.9 in R1 (Find Direct Form I, Direct Form II, and Cascade realization for Part b only. Parallel realization optional).

Problem 9:
Consider a discrete-time sinewave sequence defined by

$$x(n)=\sin (\pi n/4)$$

which was obtained by sampling a CW tone x(t)=sin⁡(2πF0t)x(t)=sin(2πF0​t) with the frequency F0F0​ Hz. If the sampling rate was Fs=160Fs​=160 Hz, what are the possible positive frequency values for F0F0​, measured in Hz, that would result in the sequence $x(n)$?

MATLAB Exercises:

P2.19

A linear and time-invariant system is described by the difference equation

$$y(n)-0.5y(n-1)+0.25y(n-2)=x(n)+2x(n-1)+x(n-3)$$

1. Using the filter function, compute and plot the impulse response of the system over $0\le n\le 100$.

2. Determine the stability of the system from this impulse response.

3. If the input to this system is $x(n)=[5+3\cos (0.2\pi n)+4\sin (0.6\pi n)]u(n)$, determine the response $y(n)$ over $0\le n\le 200$ using the filter function.

P3.16

For a linear, shift-invariant system described by the difference equation

y(n)=∑m=0Mbmx(n−m)−∑ℓ=1Naℓy(n−ℓ)y(n)=m=0∑M​bm​x(n−m)−ℓ=1∑N​aℓ​y(n−ℓ)

the frequency-response function is given by

H(ejω)=∑m=0Mbme−jωm1+∑ℓ=1Naℓe−jωℓH(ejω)=1+∑ℓ=1N​aℓ​e−jωℓ∑m=0M​bm​e−jωm​

Write a MATLAB function freqresp to implement this relation. The format of this function should be:

function [H] = freqresp(b, a, w)
% Frequency response function from difference equation
% [H] = freqresp(b, a, w)
% H = frequency response array evaluated at w frequencies
% b = numerator coefficient array
% a = denominator coefficient array (a(1) = 1)
% w = frequency location array