Page 38 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS

Hence,

Similarly,

and

in view of Eq. (1.76).

1.9. Show that the complex exponential signal

( t ) = ,j@d
is periodic and that its fundamental period is 27r/00.
By Eq. (1.7), x(t) will be periodic if

ei@dt + TI = eiwd
Since
eiw~(r + T) = eiqreiq,T
we must have
eimoT = 1 (1.78)

If w, = 0, then x(t) = 1, which is periodic for any value of T. If o0 # 0, Eq. (1.78) holds if
27T
ooT=m2r or T=m- m = positive integer
a0
Thus, the fundamental period To, the smallest positive T, of x(t) is given by 2r/oo.

1.10. Show that the sinusoidal signal
x(t) = cos(w,t + 8)

is periodic and that its fundamental period is 27r/wo.
The sinusoidal signal x(l) will be periodic if
cos[o,(t + T) + 81 = ws(oot + 8)
We note that
cos[w,(t + T) + 81 = cos[oot + 8 + woT] = cos(oot + 8)