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14.1 SIGNALS AND SPECTRAL ANALYSIS 629

used, the demodulator may process the received waveform and decide on whether the transmitted
bitisa0or1.Thesource decoderaccepts the output sequence from thechannel decoder, and
from the knowledge of the source encoding method used, attempts to reconstruct the original
signal from the source. Errors due to noise, interference, and practical system imperfections
do occur. The digital-to-analog (D/A) converter reconstructs an analog message that is a close
approximation to the original message. The difference, or some function of the difference, between
the original signal and the reconstructed signal is a measure of thedistortionintroduced by the
digital communication system.
The remainder of this chapter deals with basic methods for analyzing and processing analog
signals. A large number of building blocks in a communication system can be modeled by linear
time-invariant (LTI) systems. LTI systems provide good and accurate models for a large class of
communication channels. Some basic components of transmitters and receivers (such as filters,
amplifiers, and equalizers) are LTI systems.

Periodic Signals and Fourier Series

In the study of analog systems, predicting the response of circuits to a general time-varying voltage
or current waveformx(t) is a difficult task. However, ifx(t) can be expressed as asum of sinusoids,
then the principle of superposition can be invoked on linear systems and the frequency response
of the circuit can be utilized to expedite calculations. Expressing a signal in terms of sinusoidal
components is known asspectral analysis. Let us begin here by considering the Fourier-series
expansion of periodic signals, which has been introduced in Section 3.1.
Aperiodic signalhas the property that it repeats itself in time, and hence, it is sufficient to
specify the signal in the basic time interval called the period. A periodic signalx(t) satisfies the
property

x(t+kT )=x(t) (14.1.1)

for allt, all integersk, and some positive real numberT,called the period of the signal. For
discrete-time periodic signals, it follows that

x(n+kN)=x(n) (14.1.2)

for all integersn, all integersk, and a positive integerN, called the period. A signal that does not
satisfy the condition of periodicity is known as nonperiodic.

EXAMPLE 14.1.1

Consider the following signals, sketch each one of them and comment on the periodic nature:

(a)x(t)=Acos( 2 πf 0 t+θ), whereA,f 0 , andθare the amplitude, frequency, and phase of

the signal.

(b)x(t)=ej(^2 πf^0 t+θ), A>0.

(c) Unit step signalu− 1 (t) defined byu− 1 (t)=

{ 1 ,t> 0

(^1) / 2 ,t= 0
0 ,t< 0
.
(d) Discrete-time signalx[n]=Acos( 2 πf 0 n+θ), wherenis an integer.