Page 311 - Schaum's Outline of Theory and Problems of Signals and Systems

298 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6

where @ denotes the periodic convolution defined by [Eq. (2.70)]

The multiplication property (6.59) is the dual property of Eq. (6.58).

N. Additional Properties:

If x[n] is real, let

where x,[n] and xo[n] are the even and odd components of x[n], respectively. Let

x [n] t, X(n) = A(R) + jB(R) = I X(R)leJe(n) (6.61)

Then

Equation (6.62) is the necessary and sufficient condition for x[n] to be real. From
Eqs. (6.62) and (6.61) we have

A( -R) =A(R) B(-R) = -B(R) (6.64a)

Ix(-fl)I= Ix(R)I +a) = -9(~) (6.646)

From Eqs. (6.63~)~ (6.636), and (6.64~) we see that if x[n] is real and even, then X(R) is
real and even, while if x[n] is real and odd, X(R) is imaginary and odd.

0. Parseval's Relations:

Equation (6.66 ) is known as Parseual's identity (or Parseual's theorem) for the discrete-time
Fourier transform.
Table 6-1 contains a summary of the properties of the Fourier transform presented in
this section. Some common sequences and their Fourier transforms are given in Table 6-2.