Signals and Systems - Electrical Engineering

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

n If A is real, but a=j 0 , then we have

x(t)=Aej^0 t

=Acos( 0 t)+jAsin( 0 t)

where the real part of x(t)isRe[x(t)]=Acos( 0 t)and the imaginary part of x(t)isIm[x(t)]=

Asin( 0 t), and j=

√

− 1.

n If both A and a are complex, x(t)is a complex signal and we need to consider separately its real and

imaginary parts. For instance, the real part function is

g(t)=Re[x(t)]

=|A|ertcos( 0 t+θ)

The envelope of g(t)can be found by considering that

− 1 ≤cos( 0 t+θ)≤ 1

and that when multiplied by|A|ert> 0 , we have

−|A|ert≤|A|ertcos( 0 t+θ)≤|A|ert

so that

−|A|ert≤g(t)≤|A|ert

Whenever r< 0 the g(t)signal is a damped sinusoid, and when r> 0 then g(t)grows, as illustrated in

Figure 1.5.

n According to the above, several signals can be obtained from the complex exponential.

FIGURE 1.5

Analog exponentials:

(a) decaying exponential,

(b) growing exponential, and

(c–d) modulated exponential

(c) decaying and (d) growing.

− 2 0 2

1

2

3

4

t

− 2 0 2

1

2

3

4

t

−e

0.5

t

0.5e

t

(a) (b)

− 2 0 2

− 4

− 2

0

2

4

t

−e

0.5

t cos(2

πt

)

− 2 0 2

− 4

− 2

0

2

4

t

0.5e

t cos(2

πt

)

(c) (d)