Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 74

Fig. 2.8 Discrete-time
domain representation of a
digital system

Digital System

y ( n ),
Output
Signal

x ( n ),
Input
Signal

h ( n )

y ( n ) = h ( n ) * x ( n )

y ð n Þ¼ x ð n Þ h ð n Þ¼ X

1

x ð k Þ h ð n k Þ ð 2 : 6 Þ

k ¼1

Note For causal digital systems h(n) = 0 for n \ 0.
A graphical illustration of a digital system is shown in Fig. ( 2.8 ).

Eigenfunctions of LTI Digital Systems

If the input signal to an LTI digital system is x ð n Þ¼ e jxnT s ¼ e jnX ; then the output
is given by:

y ð n Þ¼ x ð n Þ h ð n Þ¼ h ð n Þ x ð n Þ¼ X

1

h ð k Þ e j ð n k Þ X ¼ e jnX X

1

h ð k Þ e jkX :

k ¼1 k ¼1

If one defines H ð e jX Þ¼ P k ¼1 h ð k Þ e jkX ; then for an input of e jnX one obtains the
output y ð n Þ¼ e jnX H ð e jX Þ: Hence, e jnX is an eigenfunction, and the associated
eigenvalue is H ð e jX Þ (Compare with the analogous result obtained for continuous-
time systems in Sect. 1.2.3.4 ).

Analyzing the Stability of Digital Systems in the Time Domain

For a digital system to be BIBO stable, the output y(n) should be bounded when
the input x(n) is bounded, i.e.,

j y ð n Þj \ 1 when j x ð n Þj \A(which is a finite constant) ;8 n :

Using the inequality |a ? b| B |a| ? |b|, it follows that

j X

1

x ð k Þ h ð n k Þj X

1

j x ð k Þjj h ð n k Þj:

k ¼1 k ¼1

If |x(k)| \ AVk, then

j y ð n Þj¼j X

x ð k Þ h ð n k Þj A X

j h ð n k Þj A X

1 1 1

j h ð m Þj;

k ¼1 k ¼1 m ¼1

where m = n - k. Hence, a digital system is BIBO-stable if P k ¼1 j h ð k Þj \ 1:
(Compare with the condition for stability of analog systems found in Sect. 1.2.3.5 ).