Spring 2000 EE 6360 1
Homework # 5
Analysis of Discrete-time Signals
Due: March 9, 2000
A. RESPONSE OF DISCRETE-TIME SYSTEMS
1. Write a MATLAB program to determine the response of the following system:
y n x n ay n() . () ( )=+−45 1
where a=0.5, and the input signal x(n) is the sinewave, xn n() sin( . )=3202
π
. Assume
zero initial conditions, y(-1)=0. Plot y(n), for n=1, 2, …, 200.
2. Repeat (1) for a=0.9, a=1.2, a=-0.5. What happens when a=1.2? Why?
3. Write a MATLAB program to determine the response of the following system:
yn xn xn xn().().()()=+−+−45 23 2 4 4
where the input signal x(n) is the sinewave, xn n() sin( . )=3202
π
.
4. Square root algorithm
Most computers and calculators compute the square root of a positive number A
using the following recursive algorithm:
yn yn xn
yn
() ( ) ()
()
=−+
−
1
211
If we use as the input x(n) to this system (algorithm) a step function of amplitude A,
then y(n) will converge after several iterations to the square root of A.
Write a MATLAB program that implements the above algorithm to compute the
square root of: 16, 4, 5 and 3. How many iterations does it take to converge to the true
value assuming y(-1)=0.5? Is the algorithm sensitive to the initial conditions y(-1)?
B. IMPULSE RESPONSE
1. Write a MATLAB program to compute the impulse response of the following
systems:
(a) yn xn yn() . () . ( )=+−45 08 1 n=0,1,…,100
Plot the impulse response h(n) using stem(.). Theoretically, what is the
expression for h(n)?
(b) yn xn yn yn xn() () . ( ) . ( ) ( )=+ −− −+−05 1 05 4 3
(c ) yn xn xn xn().().()()=+−+−45 23 2 4 4
Plot the impulse response h(n), for n=0,1,…,100.
2. Of the impulse responses computed in (1), which impulse responses are infinite in
duration and which are finite? Which systems are FIR and which systems are IIR?
