Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 416

ideal impulse train sampling, the spectra of the two sampled signals are identical,
except that the shifted replicas in the spectrum of the pulse-train are attenuated
by a sinc function. Reconstruction of the original signal in rectangular pulse-
train sampling is also achieved by lowpass filtering the sampled signal. A second
practical implementation of sampling uses a zero-order hold circuit to sample
the CT signal; this is covered in Section 9.2.2.
To encode a CT signal into a digital waveform, Section 9.3 introduces the
process of quantization, in which the values of the samples are approximated to a
finite set of levels. This involves replacing the exact sample value with the closest
level defined by the L -level quantizer. In uniform quantization, the quantization
levels are distributed uniformly between the maximum and minimum ranges
of the input sequence. A uniform quantizer results in high quantization error
in most practical applications, where the distribution of the sample values is
skewed towards low amplitudes. In such cases, most of the quantization levels
in the uniform quantizer are rarely used. A non-uniform quantizer reduces the
overall quantization error by providing finer quantization at frequently occurring
lower amplitudes and coarser quantization at less frequent higher amplitudes.
Sampling is used in a number of important applications. Section 9.4 intro-
duces the compact disc (CD) and illustrates how sampling and quantization are
used to convert an analog music signal into binary data, which can be stored on
a CD. Since digital signals are less sensitive to distortion and interference than
analog signals, the audio CD provides excellent audio quality that surpasses
most analog storage mechanisms.

Problems

9.1 For the following CT signals, calculate the maximum sampling period T s
that produces no aliasing:
(a) x 1 ( t ) = 5 sinc(200 t );
(b) x 2 ( t ) = 5 sinc(200 t ) + 8 sin(100 π t );
(c) x 3 ( t ) = 5 sinc(200 t ) sin(100 π t );
(d) x 4 ( t ) = 5 sinc(200 t ) ∗ sin(100 π t ), where ∗ denotes the CT convolu-
tion operation.

9.2 A famous theorem known as the uncertainty principle states that a baseband
signal cannot be time-limited. By calculating the inverse CTFT of the
following baseband signals, show that the uncertainty principle is indeed
satisfied by the following signals (assume that ω 0 and W are real, positive
constants):
(a) X 1 ( ω ) = rect

ω
2 W

e − j2 ω ;

1 ω≤ W

(b) X 2 ( ω ) =

0 elsewhere;