Wiley signals and systems e book TLFe BO 401

386

16. Stability arid Feedback Systems

To deteririirie the stability from the criteria in(16.3), wc investigate whether the
impulse response can be absohttely integrated:
1

Ie-‘’’E(L)l

tit =

bW

e-ncdt =

c1

for a > 0
(16.9)

cm otherwise

arid obtain the result: the system is stable for a > 0.
For m = 0, the impulse response takes the form of the step function h ( f )= ~ ( f ) .
We recognise the impulse respoiise of ail integrator froiri Chapter 8.4.4.1, and with
(16.9) we can say the following about the sgsteni stability: an integrator is not
BIBO-stable. This is not surprising when we recdl the response of an integrator
t o a step function ~ ( t )The
. integral over this bounded input signal grows with
incrcasing time ovcr all limits arid thus violatcs thc condition for BIBO-stabilit

Example 16.2

When we clealt with the sampling theorem in Chapter 11.3.2, we 1isc.d a system
with a rectangular frequency respoiise and a sint function impulse response (11.35)
as an interpolation filter for sampling. This is also called an d e a l low-pass filter.
We now want to invesligate whether such a systeni caii be realised.
Considering the impulse response in Figure 11.11, it can be immediately seen
that the impulse response of the ideal low-ptws is a bilateral signal. The ideal
low-pass is thrrrfore not c;iusal.
As the frequency iesponse of a stable system cannot ha~7eany tlisront iiiiiitie4,
we suspect that the ideal low-pass is also unstable. In ordrr t o confirm khis, the
unit of tiinc in (11 3 5 ) is chosen so that T = 7r for simplification, and the impulsc
response is
h ( t ) = si(t) .
(16.10)
W e iiow have to deterinine whether the integral

has a. finite vdue or grows hq7ond limit. We do this b.y estiinatiiig the area
under l h ( f ) l with a series of triangles whose areas are all less than the area of the
individual sitlelobes in Figure 16.1. The baselines of the trianglcs each have width
7r arid heights decreasing by 1/t ‘l’lic infinite sum of these triangle arm i s a lower
bound for the area uritler the iriagnitude component ofthe itnpdse response ( h ( l ) l :