Signals and Systems - Electrical Engineering

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

nExample 1.1

Characterize the sinusoidal signal

x(t)=

√

2 cos(πt/ 2 +π/ 4 ) −∞<t<∞

Solution

The signalx(t)is

n Deterministic, as the value of the signal can be obtained for any possible value oft.

n Analog, as there is a continuous variation of the time variabletfrom−∞to∞, and of the

amplitude of the signal between−

√

2 to

√

2.

n Of infinite support, as the signal does not become zero outside any finite interval.

The amplitude of the sinusoid is

√

2, its frequency is=π/2 (rad/sec), and its phase isπ/4 rad

(notice thatthas radians as units so that it can be added to the phase). Because of the infinite

support, this signal cannot exist in practice, but we will see that sinusoids are extremely important

in the representation and processing of signals. n

nExample 1.2

A complex signaly(t)is defined as

y(t)=( 1 +j)ejπt/^20 ≤t≤ 10

and zero otherwise. Expressy(t)in terms of the signalx(t)from Example 1.1. Characterizey(t).

Solution

Since 1+j=

√

2 ejπ/^4 , then using Euler’s identity:

y(t)=

√

2 ej(πt/^2 +π/^4 )=

√

2 [cos(πt/ 2 +π/ 4 )+jsin(πt/ 2 +π/ 4 )] 0 ≤t≤ 10

Thus, the real and imaginary parts of this signal are

Re[y(t)]=

√

2 cos(πt/ 2 +π/ 4 )

Im[y(t)]=

√

2 sin(πt/ 2 +π/ 4 )

for 0≤t≤10 and zero otherwise. The signaly(t)can be written as

y(t)=x(t)+jx(t− 1 ) 0 ≤t≤ 10

and zero otherwise. Notice that

x(t− 1 )=

√

2 cos(π(t− 1 )/ 2 +π/ 4 )=

√

2 cos(πt/ 2 −π/ 2 +π/ 4 )=

√

2 sin(πt/ 2 +π/ 4 )

The signaly(t)is

n Analog of finite support—that is, the signal is zero outside the interval 0≤t≤10.