Miscellaneous Exercises - Digital Signal Processing: An Introduction with MATLAB and Applications - page 300

Miscellaneous DSP Exercises: D

Q1: Prove that c M ð n Þ¼ M P M 1

k ¼ 0 e j2pkn = M .
Q2: Figure B.6 shows the spectrum of a band-pass signal. Show that down-
sampling this signal by M = 3 produces an alias-free spectrum.
Q3: Write a MATLAB code to change the sampling rate from 48 to 44.1 kHz as
explained in Example (2), Chap. 5 .
Q4: Consider the sinusoidal signal x(n) = cos (0.1 pn). Determine the frequency
spectrum of the 3-fold down-sampled version.
Q5: Prove the decimation and interpolation identities shown in Fig. 5.13.
Q6: Given the following specifications of the sampling rate conversion DSP
system shown in Fig. 5.11: L =2,M = 3 and the input audio signal is
sampled at 6 kHz, whereas the output signal sampling rate should be 9 kHz.
Determine the filter length and cutoff frequencies for the required filter
(window design method could be used).
Q7: The signal x(n) was sampled at a frequency f s = 10 kHz. Consider the
following two cases: (a) Resample x(n) at a new sampling frequency f s1 =22
kHz. (b) The signal signal x(n) is to be resampled to a new sampling
frequency f s2 = 8 kHz.
Q8: Consider the transfer function of a FIR filter: H(z) = 0.25 + 0.5 z -1 + 0.75
z -2 + z -3 + 1.25 z -4 + 1.5 z -5 . Use polyphase decomposition technique to
implement a factor of M = 3 decimator.
Q9: A single stage decimator structure is used to reduce the sampling rate of a
signal from 12000 to 500 Hz. The specifications of the single-stage
decimator low-pass FIR filter H(z) are: pass-band edge = 225 Hz, stop-band
edge = 250 Hz, pas-band ripple = 0.004, and stop-band attenuation = 0.001.
Assume H(z) as an equiripple linear-phase FIR filter.

X(e j ω )

0

− π π 3 π

Fig. B.6 Problem 2