Miscellaneous Exercises - Digital Signal Processing: An Introduction with MATLAB and Applications - page 294

z −1

x ( n )

Fig. B.2 Problem Q50

y ( n )

Q57: Implement the following digital system in cascade:

H ð z Þ¼ ð 2z 5 Þð z þ 1 Þ
z 3 þ 5z 2 þ 6z Þ : ð

Q58: Why do we normally choose complex poles as conjugates in digital

systems?
Q59: How can we equalize the distorting effect of a communication channel?
Q60: What are the advantages of FIR over IIR filters?
Q61: Explain Gibbs phenomenon and how it can be dealt with in digital systems.
Q62: Design an ideal FIR HPF with cutoff frequency = 20 Hz, knowing that the
sampling frequency is f s = 100 Hz. (Hint: use Tables to find h(n) for LPF,
then convert to HPF).
Q63: Plot the pole diagram of the sinusoidal digital oscillator. How can we
deduce the frequency of oscillation from these poles?
Q64: Simulate the following DSP systems using MATLAB (find the output time
signal for arbitrary causal input x(n) and the frequency response of the
system):
(a) The system shown in Fig. B.3 (b) The system shown in Fig. B.2 .
Q65: Design a digital DC blocker with f s = 20 kHz that attenuates the frequency
f = 20 Hz by -0.1 dB.
Q66: A signal x(t) = sin(t) is corrupted by AWGN n(t) with SNR = 0 dB.
Simulate the noisy signal s(t) = x(t)+n(t) using MATLAB and find the
spectra of all signals. Design a Butterworth analog LPF to reduce noise and
find the output of this filter.
Q67: A random signal n(t) is received and saved as a data file with f s = 1 kHz
and length N = 1000 samples. Write a MATLAB code to find the pdf of
this signal.
Q68: An analog signal has a time duration of 3 ms. How long would be the
frequency range of this signal?
Answer to Q4 (Hilbert Transformer):
1. We have y ð t Þ¼ x ð xt p = 2 Þ¼ x x t p = x

h i . Hence, there is a time-delay of

x , where the new time axis is t n ¼ t p = x . Therefore, the system has memory.
2. It is causal since there is no time-advance.
3. Consider only positive frequencies, hence the phase shift is -p/2 [see Tutorial
33]. Consider sinusoids in complex exponential form.
Let z ð t Þ¼ ae j ð x 1 t þ c Þ þ be j ð x 2 t þ d Þ , where a, b, c and d are constants. We have:

p = 2