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Problems 53
IIR filter If the impulse response of an LTI system is of infinite duration,
then the system is called an infinite-duration impulse response (or
IIR) filter. The following part of the difference equation (2.21):
N∑
a k y(n − k) =x(n) (2.26)
k=0
describes a recursive filter in which the output y(n) isrecursively computed
from its previously computed values and is called an autoregressive
(AR) filter. The impulse response of such filter is of infinite duration and
hence it represents an IIR filter. The general equation (2.21) also describes
an IIR filter. It has two parts: an AR part and an MA part. Such an IIR
filter is called an autoregressive moving average, oranARMA, filter. In
MATLAB, IIR filters are described by the difference equation coefficients
{b m } and {a k } and are implemented by the filter(b,a,x) function.
2.5 PROBLEMS
P2.1 Generate the following sequences using the basic MATLAB signal functions and the basic
MATLAB signal operations discussed in this chapter. Plot signal samples using the stem
function.
1. x 1(n) =3δ(n +2)+2δ(n) − δ(n − 3) + 5δ(n − 7), −5 ≤ n ≤ 15.
2. x 2(n) = ∑ 5
k=−5 e−|k| δ(n − 2k), −10 ≤ n ≤ 10.
3. x 3(n) =10u(n) − 5u(n − 5) − 10u(n − 10)+5u(n − 15).
4. x 4(n) =e 0.1n [u(n + 20) − u(n − 10)].
5. x 5(n) =5[cos(0.49πn)+cos(0.51πn)], −200 ≤ n ≤ 200. Comment on the waveform
shape.
6. x 6(n) =2sin(0.01πn) cos(0.5πn), −200 ≤ n ≤ 200. Comment on the waveform shape.
7. x 7(n) =e −0.05n sin(0.1πn + π/3), 0 ≤ n ≤ 100. Comment on the waveform shape.
8. x 8(n) =e 0.01n sin(0.1πn), 0 ≤ n ≤ 100. Comment on the waveform shape.
P2.2 Generate the following random sequences and obtain their histogram using the hist
function with 100 bins. Use the bar function to plot each histogram.
1. x 1(n) isarandom sequence whose samples are independent and uniformly distributed
over [0, 2] interval. Generate 100,000 samples.
2. x 2(n) isaGaussian random sequence whose samples are independent with mean 10 and
variance 10. Generate 10,000 samples.
3. x 3(n) =x 1(n)+x 1(n − 1) where x 1(n) isthe random sequence given in part 1 above.
Comment on the shape of this histogram and explain the shape.
4. x 4(n) = ∑ 4
y k=1 k(n) where each random sequence y k (n) isindependent of others with
samples uniformly distributed over [−0.5, 0.5]. Comment on the shape of this histogram.
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