Signals and Systems - Electrical Engineering

68 C H A P T E R 1: Continuous-Time Signals

As a result of the sampling, we obtain the discrete-time signal

v(nTs)=v(t)|t=nTs 0 ≤n≤N− 1 (1.2)

whereTs=(tf−tb)/Nand we have taken samples at timestb+Tsn. Clearly, this discretization of the

time variable reduces the number of values to enter into the computer, but the amplitudes of these

samples still can take possibly innumerable values. Now, to represent each of thev(nTs)values with a

certain number of bits, we also discretize the amplitude of the samples. To do so, the dynamic range

(the difference between the maximum and the minimum amplitude) of the analog signal is equally

divided into a certain number of levels. A sample value falling within one of these levels is allocated

a unique binary code. For instance, if we want each sample to be represented by 8 bits we have 2^8 or

256 possible levels. These operations are calledquantization and coding. The resulting signal is digital,

where each sample is represented as a binary number.

Given that many of the signals we encounter in practical applications are analog, if it is desirable

to process such signals with a computer, the above procedure is commonly done. The device that

converts an analog signal into a digital signal is called an analog-to-digital converter (ADC) and

it is characterized by the number of samples it takes per second (sampling rate 1/Ts) and by the

number of bits that it allocates to each sample. To convert a digital signal into an analog signal a

digital-to-analog converter (DAC) is used. Such a device inverts the ADC process: binary values are

converted into pulses with amplitudes approximating those of the original samples, which are then

smoothed out resulting in an analog signal. We will discuss in Chapter 7 how the sampling, binary

representation, and reconstruction of an analog signal is done.

Figure 1.1 shows how the discretization of an analog signal in time and amplitude can be understood,

while Figure 1.2 illustrates the sampling and quantization of a segment of speech.

Acontinuous-timesignal can be thought of as a real-(or complex-) valued function of time:

x(.):R→R(C)

t x(t) (1.3)

FIGURE 1.1

Discretization in time and amplitude of an analog

signal. The parameters are the sampling period

Tsand the quantization level 1. In time, samples

are taken at uniform times{nTs}, and in amplitude

the range of amplitudes is divided into a finite

number of levels so that each sample value is

approximated by them.

x(t)

Level

2 Ts

Ts

3 Ts

x(nTs)

4 Ts

t

− Δ

− Δ/ 2

Δ/ 2

Δ