Page 301 - Schaum's Outline of Theory and Problems of Signals and Systems
P. 301
Chapter 6
Fourier Analysis of Discrete-Time
Signals and Systems
6.1 INTRODUCTION
In this chapter we present the Fourier analysis in the context of discrete-time signals
(sequences) and systems. The Fourier analysis plays the same fundamental role in discrete
time as in continuous time. As we will see, there are many similarities between the
techniques of discrete-time Fourier analysis and their continuous-time counterparts, but
there are also some important differences.
6.2 DISCRETE FOURIER SERIES
A. Periodic Sequences:
In Chap. 1 we defined a discrete-time signal (or sequence) x[n] to be periodic if there
is a positive integer N for which
x[n + N] =x[n] all n (6.1)
The fundamental period No of x[n] is the smallest positive integer N for which Eq. (6.1) is
satisfied.
As we saw in Sec. 1.4, the complex exponential sequence
where no = 27r/Nu, is a periodic sequence with fundamental period Nu. As we discussed
in Sec. 1.4C, one very important distinction between the discrete-time and the continuous-
time complex exponential is that the signals el"^' are distinct for distinct values of wO, but
the sequences eiR~~", which differ in frequency by a multiple of 2rr, are identical. That is,
Let
and more generally,
rn = integer
*k[.I = *k+o~N,,[~l
Thus, the sequences qk[n] are distinct only over a range of No successive values of k.
