PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

460 Practical MATLAB® Applications for Engineers

b. Time- or shift invariance

Let

f(n) → g(n)

Then

f(n − k) → g(n − k) for any arbitrary integer k (1, 2, 3, ...)

c. Causality

If

f(ni) = 0, over the range n < ni

Then

g(n) = 0 , over the range n < ni

This property states, in general terms, that the ith output sample g(ni) depends

only on the input samples over the range n ≤ ni, output g(n) of past and present

samples, and never on future samples.

d. Stability

There are many accepted defi nitions of stability for discrete-time systems. In

this chapter, the concept of stability is defi ned simply in the following way: A dis-

c r e t e s y s t e m i s u n s t a bl e i f a n d o n ly i f t h e r e e x i s t s a fi nite-bounded input sequence

f(n) that causes the output sequence g(n) to blow out or become infi nite.

If a given system is not unstable then it is stable. A stable system is a system

where a bounded (fi nite) input f(n) produces a bounded (fi nite) output g(n). This

defi nition of stability is often referred to as BIBO.

The energy associated with a discrete-time sequence, either an input or output,

must be satisfi ed by the following relations:

Energy offn fn

n

()
()

2


 





+

∑

Energy ofgn gn

n

()
()

 

 2

∑ 

e. Passive

A discrete-time system is passive if the output energy of [g(n)] does not exit the

input energy of [f(n)].

R.5.3 The discrete operators or system elements used to transform an input sequence f(n)
into an output sequence g(n) are illustrated graphically and defi ned in Figure 5.3.

R.5.4 This chapter deals mainly with SISO systems. The techniques developed for SISO
systems can easily be extended to multiple input–output systems by making use of
the superposition principle (assuming that the system considered is linear).

R.5.5 The discussions in this chapter are restricted mainly to discrete systems that are
linear, time invariant, and in most cases causal.