Signals and Systems - Electrical Engineering

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

n A property that is not obvious and that requires the frequency representation ofS(t)is that the

integral

∫∞

−∞

|S(t)|^2 dt= 1 (1.39)

Recall that we showed this in Chapter 0 using numeric and symbolic MATLAB.

The sinc signal will appear in several places in the rest of the book.

1.4.4 Basic Signal Operations—Time Scaling, Frequency Shifting, and

Windowing

Given a signalx(t), and real valuesα6= 0 or 1 , andφ > 0 :

n x(αt)is said to be contracted if|α|> 1 , and ifα < 0 it is also reflected.

n x(αt)is said to be expanded if|α|< 1 , and ifα < 0 it is also reflected.

n x(t)ejφtis said to be shifted in frequency byφradians.

n For a window signalw(t),x(t)w(t)displaysx(t)within the support ofw(t).

To illustrate the time scaling, consider a signalx(t)with a finite supportt 0 ≤t≤t 1. Assume that

α >1, thenx(αt)is defined int 0 ≤αt≤t 1 ort 0 /α≤t≤t 1 /α, a smaller support than the original

one. For instance, forα=2,t 0 =2, andt 1 =4, then the support ofx( 2 t)is 1≤t≤2, while the

support ofx(t)is 2≤t≤4. Ifα=−2, thenx(− 2 t)is not only contracted but also reflected. Similarly,

x(0.5t)would have a support of 2t 0 ≤t≤ 2 t 1 , which is larger than the original support.

Multiplication by an exponential shifts the frequency of the original signal. To illustrate this consider

the case of an exponentialx(t)=ej^0 tof frequency 0. If we multiplyx(t)by an exponentialejφt, then

x(t)ejφt=ej(^0 +φ)t=cos(( 0 +φ)t)+jsin(( 0 +φ)t)

so that the frequency of the new exponential is greater than 0 ifφ >0 or smaller ifφ <0. So we have

shifted the frequency ofx(t). If we have a sum of exponentials (they do not need to be harmonically

related as in the Fourier series we will consider later),

x(t)=

∑

k

Akejkt

then

x(t)ejφt=

∑

k

Akej(k+φ)t

so that each of the frequencies of the signalx(t)has been shifted. This shifting of the frequency is

significant in the development of amplitude modulation, and as such this frequency shift process

is calledmodulation—that is, the signalx(t)modulates the exponential andx(t)ejφtis the modulated

signal.