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58 Chapter 2 DISCRETE-TIME SIGNALS AND SYSTEMS
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 ≤ n ≤ 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(0.2πn)+4sin(0.6πn)] u(n), determine the
response y(n) over0≤ n ≤ 200 using the filter function.
P2.20 A “simple” digital differentiator is given by
y(n) =x(n) − x(n − 1)
which computes a backward first-order difference of the input sequence. Implement this
differentiator on the following sequences, and plot the results. Comment on the
appropriateness of this simple differentiator.
1. x(n) =5[u(n) − u(n − 20)]: a rectangular pulse
2. x(n) =n [u(n) − u(n − 10)] + (20 − n)[u(n − 10) − u(n − 20)]: a triangular pulse
( ) πn
3. x(n) =sin [u(n) − u(n − 100)]: a sinusoidal pulse
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