Signals and Systems - Electrical Engineering

420 CHAPTER 7: Sampling Theory

the bridge between analog and discrete signals and systems and were the starting point for digital

signal processing as a technical area.

n Practical aspects of sampling—The device that samples, quantizes, and codes an analog signal is

called ananalog-to-digital converter(ADC), while the device that converts digital signals into ana-

log signals is called adigital-to-analog converter(DAC). These devices are far from ideal and thus

some practical aspects of sampling and reconstruction need to be considered. Besides the pos-

sibility of losing information by choosing too large of a sampling period, the ADC also loses

information in the quantization process. The quantization error is, however, made less signifi-

cant by increasing the number of bits used to represent each sample. The DAC interpolates and

smooths out the digital signal, converting it back into an analog signal. These two devices are

essential in the processing of continuous-time signals with computers.

7.2 UNIFORM SAMPLING

The first step in converting a continuous-time signalx(t)into a digital signal is to discretize the time

variable—that is, to consider samples ofx(t)at uniform timest=nTs, or

x(nTs)=x(t)|t=nTs n integer (7.1)

whereTsis the sampling period. The sampling process can be thought of as a modulation process, in

particular connected with pulse amplitude modulation (PAM), a basic approach in digital communi-

cations. A pulse amplitude modulated signal consists of a sequence of narrow pulses with amplitudes

the values of the continuous-time signal within the pulse. Assuming that the width of the pulses is

much narrower than the sampling periodTspermits a simpler analysis based on impulse sampling.

7.2.1 Pulse Amplitude Modulation

A PAM system can be visualized as a switch that closes everyTsseconds for 1 seconds, and remains

open otherwise. The PAM signal is thus the multiplication of the continuous-time signalx(t)by a

periodic signalp(t)consisting of pulses of width 1 , amplitude 1/1, and periodTs. Thus,xPAM(t)

consists of narrow pulses with the amplitudes of the signal within the pulse width. For a small pulse

width 1 , the PAM signal is approximately a train of pulses with amplitudesx(mTs)—that is,

xPAM(t)=x(t)p(t)≈

1

1

∑

m

x(mTs)[u(t−mTs)−u(t−mTs−1)] (7.2)

Now, as a periodic signal we representp(t)by its Fourier series

p(t)=

∑

k

Pkejk^0 t  0 =

2 π

Ts

wherePkare the Fourier series coefficients. Thus, the PAM signal can be expressed as

xPAM(t)=

∑

k

Pkx(t)ejk^0 t