Signals and Systems - Electrical Engineering

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

FIGURE 1.3

Diagrams of basic signal operations: (a) adder,

(b) constant multiplier, (c) delay, and (d) time

windowing or modulation.

(a)

x(t) z(t)=x(t)+y(t)

y(t)

+ +

(d)

×

x(t)

w(t)

x(t) w(t)

(b)

α

x(t) α^ x(t)

(c)

x(t)

Delay τ

x(t−τ)

Given the simplicity of the first two operations we will only discuss the others. In this section we

consider time shifting and reflection (a special case of the time scaling) and leave the rest for a later

section.

In Figure 1.3 we show the diagrams used for the implementation of the addition of two signals, the

multiplication of a signal by a constant, the delay of a signal, and the time windowing or modulation

of a signal. These will be used in the block diagrams for systems in the next chapters.

It is important to understand that advancing or reflecting cannot be implemented inreal time—that

is as the signal is being processed. Delays can be implemented in real time. Advancing and reflection

require that the signal be saved or recorded. Thus, an acoustic signal recorded on magnetic tape can

be delayed or advanced with respect to an initial time, or played back, faster or slower, but it can only

be delayed if we have the signal coming from a live microphone.

We will see later in this chapter that shifting in frequency results in the process ofsignal modulation,

which is of great significance in communications. Scaling of the time variable results in a contracted

and expanded version of the original signal and causes changes in the frequency content of the signal.

n For a positive valueτ, a signalx(t−τ)is the original signalx(t)shifted right or delayedτseconds,

as illustrated in Figure 1.4(b). That the original signal has been shifted to the right can be verified

by finding that thex( 0 )value of the original signal appears in the delayed signal att=τ(which

results from makingt−τ=0).

n Likewise, a signalx(t+τ)is the original signalx(t)shifted left or advanced byτseconds as illus-

trated in Figure 1.4(c). The original signal is now shifted to the left—that is, the valuex( 0 )of the

original signal occurs now earlier (i.e., it has been advanced) at timet=−τ.

n Reflectionconsists in negating the time variable. Thus, the reflection ofx(t)isx(−t). This operation

can be visualized as flipping the signal about the origin. See Figure 1.4(d).

Given an analog signalx(t)andτ > 0 we have that with respect tox(t):

(a) x(t−τ)isdelayedorshifted rightτseconds.

(b) x(t+τ)isadvancedorshifted leftτseconds.