1.3 Continuous-Time Signals 79
whereNandMare positive integers not divisible by each other so thatMT 1 =NT 0 becomes
the period ofw(t). That is,
w(t+MT 1 )=x(t+MT 1 )+u(t+MT 1 )=x(t+NT 0 )+u(t+MT 1 )=x(t)+u(t)
n
nExample 1.10
Letx(t)=ej^2 tandy(t)=ejπt, and consider their sumz(t)=x(t)+y(t), and their productw(t)=
x(t)y(t). Determine ifz(t)andw(t)are periodic, and if so, find their periods. Isp(t)=( 1 +x(t))( 1 +
y(t))periodic?
Solution
According to Euler’s identity,
x(t)=cos( 2 t)+jsin( 2 t)
y(t)=cos(πt)+jsin(πt)
indicatingx(t)is periodic of periodT 0 =π(the frequency ofx(t)is 0 = 2 = 2 π/T 0 ) andy(t)is
periodic of periodT 1 =2 (the frequency ofy(t)is 1 =π= 2 π/T 1 ).
Forz(t)to be periodic requires thatT 1 /T 0 be a rational number, which is not the case asT 1 /T 0 =
2 /π. Soz(t)is not periodic.
The product isw(t)=x(t)y(t)=ej(^2 +π)t=cos( 2 t)+jsin( 2 t)where 2 = 2 +π= 2 π/T 2 so
thatT 2 = 2 π/( 2 +π), sow(t)is periodic of periodT 2.
The terms 1+x(t)and 1+y(t)are periodic of periodT 0 =πandT 1 =2, and from the case of the
product above, one would hope this product be periodic. But sincep(t)= 1 +x(t)+y(t)+x(t)y(t)
andx(t)+y(t)is not periodic, thenp(t)is not periodic. n
n Analog sinusoids of frequency 0 > 0 are periodic of periodT 0 = 2 π/ 0. If 0 = 0 , the period is not
well defined.
n The sum of two periodic signalsx(t)andy(t), of periodsT 1 andT 2 , is periodic if the ratio of the periods
T 1 /T 2 is a rational numberN/M, withNandMbeing nondivisible. The period of the sum isMT 1 =NT 2.
n The product of two sinusoids is periodic. The product of two periodic signals is not necessarily periodic.
1.3.4 Finite-Energy and Finite Power Signals
Another possible classification of signals is based on their energy and power. The concepts of energy
and power introduced in circuit theory can be extended to any signal. Recall that for a resistor of unit
resistance itsinstantaneous poweris given by
p(t)=v(t)i(t)=i^2 (t)=v^2 (t)
