Signals and Systems - Electrical Engineering

466 C H A P T E R 8: Discrete-Time Signals and Systems

Discrete-Time Complex Exponential

Given complex numbersA=|A|ejθandα=|α|ejω^0 , adiscrete-time complex exponentialis a signal of

the form

x[n]=Aαn

=|A||α|nej(ω^0 n+θ)

=|A||α|n[cos(ω 0 n+θ)+jsin(ω 0 n+θ)] (8.16)

whereω 0 is a discrete frequency in radians.

Remarks

n The discrete-time complex exponential looks different from the continuous-time complex exponential. This

can be explained by sampling the continuous-time complex exponential

x(t)=Ae(−a+j^0 )t

(for simplicity we let A be real) using as sampling period Ts. The sampled signal is

x[n]=x(nTs)=Ae(−anTs+j^0 nTs)=A(e−aTs)nej(^0 Ts)n

=Aαnejω^0 n

where we letα=e−aTsandω 0 = 0 Ts.

n Just as with the continuous-time complex exponential, we obtain different signals depending on the chosen

parameters A andα. For instance, the real part of x[n]in Equation (8.16) is a real signal

g[n]=Re[x[n]]=|A||α|ncos(ω 0 n+θ)

where when|α|< 1 it is a damped sinusoid, and when|α|> 1 it is a growing sinusoid (see Figure 8.3).

Ifα= 1 then the above signal is a sinusoid.

n It is important to realize that forα > 0 the real exponential

x[n]=(−α)n=(− 1 )nαn=αncos(πn)

nExample 8.12

Given the analog signal

x(t)=e−atcos( 0 t)u(t)

determine the values ofa>0, 0 , andTsthat permit us to obtain a discrete-time signal

y[n]=αncos(ω 0 n) n≥ 0