Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 418

9.6 Consider the following CT signal:

0 . 25(3 − t )0 ≤ t ≤ 3
0

x ( t ) =

otherwise .

(a) Calculate the CTFT X ( ω ). Determine the bandwidth of the signal and
the ideal Nyquist sampling rate.
(b) If the bandwidth is infinite, approximate the bandwidth as β Hz, such
that

X ( ω ) < 0 . 01 max X ( ω ) for ω > 2 πβ

and recalculate a practical Nyquist sampling rate.
(c) Discretize x ( t ) using a sampling interval of T s = 1 s. Plot the resulting
DT sequence x [ k ] corresponding to the duration − 5 ≤ t ≤ 5.
(d) Quantize the signal x [ k ] obtained in (c) with the uniform quantizer
derived in Example 9.3. Plot the quantization error with respect to k .
What is the maximum value of the quantization error?
(e) Repeat (d) using a uniform quantizer with L = 16 reconstruction levels
defined within the dynamic range [ − 1 , 1]. Plot the quantization error
with respect to k . What is the maximum value of the quantization error?
Compare the plot with your answer obtained in (d).

9.7 Show that the CTFS representation of the rectangular pulse train r ( t )as
defined in Eq. (9.17) is given by Eq. (9.19).

9.8 The spectrum of a CT signal x ( t ) satisfies the following conditions:

X ( ω ) = 0 for ω <ω 1 or ω >ω 2 with ω 2 >ω 1 > 0 .

In other words, the CTFT X ( ω )of x ( t ) is non-zero only within the range
of frequencies ω 1 ≤ω≤ω 2 . Such a signal is referred to as a bandpass
signal.
(a) Show that the bandpass signal x ( t ) can be sampled with an ideal
impulse train at a rate less than the Nyquist rate of 2( ω 2 / 2 π )
samples/s and can be perfectly reconstructed with a bandpass filter
with the following transfer function:

p ω ℓ ≤ω≤ω u

H bp ( ω ) =

0 elsewhere .

(b) Determine the minimum sampling rate for which perfect reconstruction
is possible.
(c) Compute the values of parameters p , ω ℓ , and ω u used to specify the
transfer function of the bandpass filter.

9.9 An alternative to the bandpass sampling procedure introduced in
Problem 9.8 is the system illustrated in Fig. P9.9. For a real-valued
bandpass signal x ( t ) with the spectrum shown in Fig. P9.9(a), the