Signals and Systems - Electrical Engineering

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

FIGURE 1.15

Multiplication of a signalf(t)by an impulse signal

δ(t−t 0 ).

×=

tttt 0 t 0

f(t) δ(t−t 0 ) f(t 0 )δ(t−t 0 )

(f(t 0 ))

ttt

=+

2 Δ

x(0)

+ · · ·

xΔ(t)

x(t)

Δ Δ Δ

↔

x(Δ)

FIGURE1.16

Generic representation ofx(t)as an infinite sum of pulses of heightx(k1)and width 1 when 1 → 0 , so that the

sum becomes an integral of weighted impulse signals.

Bythe sifting property of the impulse functionδ(t), any signalx(t)can be represented by the followinggeneric

representation:

x(t)=

∫∞

−∞

x(τ)δ(t−τ)dτ (1.41)

Figure1.16 shows a generic representation. Equation (1.41) basically indicates that any signal can

be viewed as a stacking of pulsesx(k1)p 1 (t−k1), which in the limit as 1 →0 become impulses

x(τ)δ(t−τ).

Equation (1.41) provides a generic representation of a signal in terms of basic signals, in this case

impulse signals. As we will see in the next chapter, once we determine the response of a system to an

impulse we will use the generic representation to find the response of the system to any signal.

1.5 What Have We Accomplished? Where Do We Go from Here?....................

We have taken another step in our long journey. In this chapter we discussed the main classification of

signals and have started the study of deterministic, continuous-time signals. We have also discussed

important characteristics of signals such as periodicity, energy, power, evenness, and oddness, and

learned basic signal operations that will be useful as we will see in the next chapters. Interestingly,