Signals and Systems - Electrical Engineering

126 C H A P T E R 2: Continuous-Time Systems

n In many cases time invariance can be determined by identifying—if possible—the input and the output,

and letting the rest represent the parameters of the system. If these parameters change with time, the system

is time varying. For instance, if the input x(t)and the output y(t)of a system are related by the equation

y(t)=f(t)x(t)

the parameter of the system is the function f(t), and if it is not constant, the system is time varying. Thus,

the system y(t)=tx(t)is time varying as can be easily verified. Likewise, the AM modulation system given

by y(t)=cos( 0 t)x(t)is time varying as the function f(t)=cos(ot).

AM Communication Systems

Amplitude modulation (AM) communication systems arose from the need to send an acoustic sig-

nal, the “message,” over the airwaves using a reasonably sized antenna to radiate it. The size of the

antenna depends inversely on the frequencies present in the message, and voice and music have rel-

atively low frequencies. A voice signal typically has frequencies in the range of 100 Hz to about 5

KHz (the frequencies needed to make a telephone conversation intelligible), while music typically

displays frequencies up to about 22 KHz. The transmission of such signals with a practical antenna

is impossible. To make the transmission possible,modulationwas introduced—that is, multiplying

the messagem(t)by a periodic signal such as a cosine cos( 0 t), the carrier, with a frequency 0

much larger than those in the acoustic signal. Amplitude modulation provided the larger frequencies

needed to reduce the size of the antenna. Thus,y(t)=m(t)cos( 0 t)is the signal to be transmitted,

and we will see later that the effect of this multiplication is to change the frequency content of the

input. Such a system is clearly linear, but time-varying. Indeed, if the input ism(t−τ)the output

would bem(t−τ)cos( 0 t), which is noty(t−τ)=cos( 0 (t−τ))m(t−τ), as a time-invariant sys-

tem would give. Figure 2.3 illustrates the AM transmitter and receiver. In Chapter 6, we will discuss

AM and other modulation systems and will illustrate them with MATLAB simulations.

In comparison with the AM system, a frequency modulation (FM) system is represented by the

following input–output equation, wherem(t)is the input message andz(t)the output:

z(t)=cos(ct+

∫t

−∞

m(τ)dτ)

FIGURE 2.3

AM modulation: transmitter and receiver.

m(t)

cos (Ω 0 t)

y(t)

×

Transmitter

Receiver

m ̃ (t)