Page 314 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS 301

where Y(R), X(R), and H(R) are the Fourier transforms of y[n], x[n], and h[n],
respectively. From Eq. ( 6.68) we have

Relationships represented by Eqs. (6.67) and (6.68) are depicted in Fig. 6-3. Let

As in the continuous-time case, the function H(R) is called the frequency response of the
system, I H(R)l the magnitude response of the system, and BH(R) the phase response of the
system.

6InI hlnl
hlnl -
L
xfnl 40) y[n]=x[n] * h[n]
I t

X(W Y(n)=X(R)H(n)
Fig. 6-3 Relationships between inputs and outputs in an LTI discrete-time system.

Consider the complex exponential sequence

Then, setting z = ejRo in Eq. (4.1), we obtain

which indicates that the complex exponential sequence ejRnn is an eigenfunction of the
LTI system with corresponding eigenvalue H(Ro), as previously observed in Chap. 2
(Sec. 2.8). Furthermore, by the linearity property (6.42), if the input x[n] is periodic with
the discrete Fourier series

then the corresponding output y[n] is also periodic with the discrete Fourier series

If x[n] is not periodic, then from Eqs. (6.68) and (6.28) the corresponding output y[n] can
be expressed as