Signals and Systems - Electrical Engineering

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

1.12. Reflection and time shifting

Do the reflection and the time-shifting operations commute? That is, do the two block diagrams in

Figure 1.17 provide identical signals (i.e., isy(t)equal toz(t))? To provide an answer to this consider the

signalx(t)shown in Figure 1.17. Reflectx(t)to getv(t)=x(−t)and then shift it to gety(t)=v(t− 2 ). Then

consider delayingx(t)to getw(t)=x(t− 2 ), and reflecting it to getz(t)=w(−t). Perform each of these

operations onx(t)to gety(t)andz(t); plot them and compare these plots. What is your conclusion? Explain

FIGURE 1.17

Problem 1.12.

t

1

1

Reflection

Reflection

Delay by 2

Delay by 2

x(t)

x(t)

x(t) v(t)

w(t) z(t)

y(t)

1.13. Contraction and expansion of signals

Letx(t)be the analog signal considered in Problem 1.12 (see Figure 1.17). In this problem we would like to

consider expanded and compressed versions of that signal.

(a) Plotx( 2 t)and determine if it is a compressed or expanded version ofx(t).

(b)Plotx(t/ 2 )and determine if it is a compressed or expanded version ofx(t).

(c)Supposex(t)is an acoustic signal—let’s say it is a music signal recorded in a magnetic tape. What

would be a possible application of the expanding and compression operations? Explain.

1.14. Even and odd decomposition and power

Consider the analog signalx(t)in Figure 1.18.

FIGURE 1.18

Problem 1.14.

1

0 1

t

x(t)

(a) Plot the even–odd decomposition ofx(t)(i.e., find and plot the evenxe(t)and the oddxo(t)components

ofx(t)).