Signals and Systems - Electrical Engineering

8.2 Discrete-Time Signals 459

nExample 8.7

A “causal” sinusoid, obtained from a signal generator after it is switched on, is

x(t)=

{

2 cos( 0 t−π/ 4 ) t≥ 0

0 otherwise

The signalx(t)is sampled using a sampling period ofTs=0.1 sec to obtain a discrete-time signal

x[n]=x(t)|t=0.1n=2 cos(0.1 0 n−π/ 4 ) n≥ 0

and zero otherwise. Determine if this discrete-time signal has finite energy and finite power and

compare these characteristics with those of the continuous-time signalx(t)when 0 =πand when

 0 =3.2 rad/sec (an upper approximation ofπ).

Solution

The continuous-time signalx(t)has infinite energy, and so does the discrete-time signalx[n], for

both values of 0. Indeed, its energy is

εx=

∑∞

n=−∞

x[n]^2 =

∑∞

n= 0

4 cos^2 (0.1 0 n−π/ 4 )→∞

Although the continuous-time and the discrete-time signals have infinite energy, they have finite

power. That the continuous-time signal has finite power can be shown as indicated in Chapter 1.

For the discrete-time signalx[n], we have for the two frequencies:

1. For 0 =π,x 1 [n]=2 cos(πn/ 10 −π/ 4 )=2 cos( 2 πn/ 20 −π/ 4 )forn≥0 and zero other-
   wise. Thus,x[n] repeats everyN 0 =20 samples forn≥0, and its power is

Px= lim

N→∞

1

2 N+ 1

∑N

n=−N

|x 1 [n]|^2 = lim

N→∞

1

2 N+ 1

∑N

n= 0

|x 1 [n]|^2

= lim

N→∞

1

2 N+ 1

N



^1

N 0

N∑ 0 − 1

n= 0

|x 1 [n]|^2





︸ ︷︷ ︸

power of period,n≥ 0

=

1

2 N 0

N∑ 0 − 1

n= 0

|x 1 [n]|^2 <∞

where we used the causality of the signal(x 1 [n]=0 forn< 0 ), and consideredNperiods of

x 1 [n] forn≥0, and for each computed its power to get the final result. Thus, for 0 =π

the discrete-time signalx 1 [n] has finite power and can be computed using a period forn≥0.