Page 24 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS 13

Unlike the continuous-time unit impulse function S(f), S[n] is defined without mathe-
matical complication or difficulty. From definitions (1.45) and (1.46) it is readily seen that

which are the discrete-time counterparts of Eqs. (1.25) and (1.26), respectively. From
definitions (1.43) to (1.46), 6[n] and u[n] are related by
(1.49)

(1 SO)

which are the discrete-time counterparts of Eqs. (1.30) and (1.31), respectively.
Using definition (1.46), any sequence x[n] can be expressed as

which corresponds to Eq. (1.27) in the continuous-time signal case.

C. Complex Exponential Sequences:
The complex exponential sequence is of the form
x[n] =e~n~"

Again, using Euler's formula, x[n] can be expressed as

x [n] = eJnnn = cos Ron + j sin Ron (1.53)
Thus x[n] is a complex sequence whose real part is cos Ron and imaginary part is sin Ron.

In order for ejn@ to be periodic with period N (> O), Ro must satisfy the following
condition (Prob. 1.1 1):
no m
= - m = positive integer
2r N
Thus the sequence eJnon is not periodic for any value of R,. It is periodic only if R,/~IT is
a rational number. Note that this property is quite different from the property that the
continuous-time signal eJwo' is periodic for any value of o,. Thus, if R, satisfies the
periodicity condition in Eq. (1.54), !& f 0, and N and m have no factors in common, then
the fundamental period of the sequence x[n] in Eq. (1.52) is No given by

Another very important distinction between the discrete-time and continuous-time
complex exponentials is that the signals el"o' are all distinct for distinct values of w, but
that this is not the case for the signals ejRon.