Page 306 - Schaum's Outline of Theory and Problems of Signals and Systems

CXV'. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS

B. Fourier Transform Pair:
The function X(R) defined by Eq. (6.23) is called the Fourier transform of x[n], and
Eq. (6.26) defines the inverse Fourier transform of X(R). Symbolically they are denoted by
m
X(R) = F{x[n]) = x x[n] ePJRn (6.27)
n= -m

and we say that x[n] and X(R) form a Fourier transform pair denoted by

44 ++X(fl) (6.29)
Equations (6.27) and (6.28) are the discrete-time counterparts of Eqs. (5.31) and (5.32).

C. Fourier Spectra:
The Fourier transform X(R) of x[n] is, in general, complex and can be expressed as

As in continuous time, the Fourier transform X(R) of a nonperiodic sequence x[n] is the
frequency-domain specification of x[n] and is referred to as the spectrum (or Fourier
spectrum) of x[n]. The quantity IX(R)I is called the magnitude spectrum of x[n], and #d R )
is called the phase spectrum of x[n]. Furthermore, if x[n] is real, the amplitude spectrum
IX(R)I is an even function and the phase spectrum 4((n) is an odd function of R.

D. Convergence of X(R):
Just as in the case of continuous time, the sufficient condition for the convergence of
X(R) is that x[n] is absolutely summable, that is,
m
C Ix[n]kw (6.31)
n= -oo

E. Connection between the Fourier Transform and the z-Transform:
Equation (6.27) defines the Fourier transform of x[n] as
D5
X(R) = z x[n] e-jnn
n= -m
The z-transform of x[n], as defined in Eq. (4.3), is given by
m
X(Z) = z x[n]z-"
n- -m
Comparing Eqs. (6.32) and (6.331, we see that if the ROC of X(z) contains the unit circle,
then the Fourier transform X(R) of x[n] equals X(z) evaluated on the unit circle, that is,
~(a) (6.34)
= ~(z)l,=,,~~
Note that since the summation in Eq. (6.33) is denoted by X(z), then the summation
in Eq. (6.32) may be denoted as X(ejn). Thus, in the remainder of this book, both X(R)