Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 64

signal is known as a PCM signal. In PCM, each sample is represented by one of 2 M
codewords, each of length M bits. If f s is the sampling rate, the data rate would be
Mf s bps. The binary encoding may correspond to the natural binary code (NBC),
which is a straightforward decimal-to-binary mapping of the index of the voltage
level (between 0 and 2 M - 1). Alternatively the encoding may use one of the Gray
Codes, which, because it reduces bit transmission errors, is particularly relevant
for long range transmission.
Practical multibit analog-to-digital converter (ADC) tend to perform all three
phases mentioned above within the one device: sampling, quantization, and binary
encoding. The accuracy (resolution) of multibit ADCs is strongly dependent on the
number of bits M, and increasing the number of bits typically improves accuracy.
2.1.1 Digital Systems
Digital systems are hardware or software systems that process digital signals. Most
digital systems are built up using binary or ''on-off'' logic, and so the operation of
digital processors can be described using binary arithmetic.
In contrast to the resistors, capacitors and inductors which make up analog
systems, the common building blocks for DSP systems are shift registers, adders
and multipliers. These building blocks may be realized in either hardware or
software, and if implemented in hardware, usually incorporate flip-flops, logic
gates and synchronously controlled clocks. Like analog systems, discrete-time and
digital systems can be analyzed in either the time or frequency domains. Both
forms of analysis are considered later in this topic.
2.2 Ideal Sampling and Reconstruction
Sampling is the process of selecting (and possibly storing) the values of a con-
tinuous-time signal x(t) at specific time instants which can be indexed by the ring
of integers Z = {…, -2, -1, 0, 1, 2, 3,… }. As long as the sampling arrangement
has been well designed, the discrete-time signal, denoted by x(n), reliably repre-
sents the original signal, at least under certain assumptions. In other words, with
appropriate design of the sampling strategy, it is possible to process analog signals
using digital systems without any loss of information.
2.2.1 Ideal Uniform Sampling
Suppose that an analog signal x(t) is sampled uniformly at a rate of f s = 1/T s , One
can represent this kind of sampling mathematically as multiplication of x(t) by the