Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 420

9.13 A CT band-limited signal x ( t ) is sampled at its Nyquist rate f s and trans-
mitted over a band-limited channel modeled with the transfer function

14 π f s ≤ω≤ 8 π f s
0

H ch ( ω ) =

otherwise .

Let the signal received at the end of the channel be x ch ( t ). Determine the
reconstruction system that recovers the CT signal x ( t ) from x ch ( t ).

9.14 If the quantization noise needs to be limited to p % of the peak-to-peak
value of the input signal, show that the number of bits in each PCM word
must satisfy the following inequality:

50
p

n ≥ 3 . 32 log 10 .

9.15 A voice signal with a bandwidth of 4 kHz and an amplitude range of
20 mV is converted to digital data using a PCM system.
(a) Determine the maximum sampling interval T s that can be used to
sample the voice signal.
(b) If the PCM system has an accuracy of 5% during the quantization
step, determine the length of the codewords in bits.
(c) Determine the data rate in bps (bits/s) of the resulting PCM sequence.

9.16 A baseband signal with a bandwidth of 100 kHz and an amplitude range
of 1 V is to be transmitted through a channel which is constrained to
a maximum transmission speed of 2 Mbps. Your task is to design a uni-
form quantizer that introduces minimum quantization error. Determine
the maximum number of levels L in the uniform quantizer. What is the
maximum distortion introduced by the uniform quantizer? Assume the
Nyquist rate for sampling.

9.17 Consider the input-output relationship of an ideal sampling system given
by

∞ ∞

x s ( t ) = x ( t ) δ ( t − kT s ) = x ( kT s ) δ ( t − kT s ) .

k =−∞ k =−∞

Determine if the ideal sampling system is (i) linear, (ii) time-invariant,
(iii) memoryless, (iv) causal, (v) stable, and (vi) invertible.

9.18 Consider the input-output relationship of a DT quantizer with L decision
levels, given by
y [ k ] = Q x [ k ] = 1

2 [ d m + d m + 1 ]

for d m ≤ x [ k ] < d m + 1 and

0 ≤ m < L .

Determine if the DT quantizer is (i) linear, (ii) time-invariant, (iii) mem-
oryless, (iv) causal, (v) stable, and (vi) invertible.