Sampling and quantization - Continuous and Discrete Time Signals and Systems - page 394

Fig. 9.1. Processing CT signals
using DT systems.

x [ k ] y [ k ]

DT
system

x ( t ) y ( t )

sampling reconst.

flexible and can be reprogrammed such that the same hardware can be used in a
variety of different applications. In addition, the characteristics of CT systems
tend to vary with changes in the operating conditions and with age. The DT
systems have no such problems as the digital hardware used to implement these
systems does not drift with age or with changes in the operating conditions and,
therefore, can be self-calibrated easily. Digital signals, obtained by quantizing
DT sequences, are less sensitive to noise and interference than analog signals
and are widely used in communication systems. Finally, the data available from
the DT systems can be stored in a digital server so that the performance of the
system can be monitored over a long period of time. In summary, the advan-
tages of the DT system outweigh their limitations in most applications. Until
the late 1980s, most signal processing applications were implemented with CT
systems constructed with analog components such as resistors, capacitors, and
operational amplifiers. With the recent availability of cheap digital hardware,
it is a common practice now to perform signal processing in the DT domain
based on the hybrid setup shown in Fig. 9.1.
Although, a CT-DT hybrid setup similar to Fig. 9.1 is advantageous in many
applications, care should be taken during the design stage. For example, during
the sampling process some loss of information is generally inevitable. Conse-
quently, if the system is not designed properly, the performance of a CT-DT
hybrid setup may degrade significantly as compared with a CT setup. In this
chapter, we focus on the analysis of the sampling process and the converse
step of reconstructing a CT signal from its DT version. In addition, we also
analyze the process of quantization for converting an analog signal to a digi-
tal signal. Both time-domain and frequency-domain analyses are used where
appropriate.
The organization of Chapter 9 is as follows. Section 9.1 introduces the
impulse-train sampling process and derives a necessary condition, referred to as
the sampling theorem, under which a CT signal can be perfectly reconstructed
from its sampled DT version. We observe that violating the sampling theorem
leads to distortion or aliasing in the frequency domain. Section 9.2 introduces
the practical implementations for impulse-train sampling. These implementa-
tions are referred to as pulse-train sampling and zero-order hold.
In Section 9.3, we introduce another discretization process called quantiza-
tion, which, in conjunction with sampling, converts a CT signal into a digital
signal. In Section 9.4, we present an application of sampling and quantization