Wiley signals and systems e book TLFe BO 402

16.1. BIBO, Iiiipulse Response arid El'fequency Kc>sponst:Curve

387

The slim itself is a known harmonic series, arid does riot oo~iverge,which tihows
that the impulse response of the ideal low-pass cannot bc absolutely integrated.
The ided hW-J2aSS is therefore neithel c i ~ u s dnor stable.

Q

Af'ter this rxample we have to ask whether it is worth examining such idralised
systcms, when they arc innpossible to impleiiirnt,. The answer is typical from tlic
point of vicw of' systems theory:
o

0

The ideal low-pass filter with an entiiely real rectaiigular frqiency plot i s
a very simple cority)t. It allows many operations to be coiisideied in tire
simplest form, for example, limiting spectra, and reconstructing contiriirour
sigiials fruiii samplPs.
The ideal lotv-pass rmriot be prrciscly rediwd lmt can bc apprtmimateil if
miiie coritaehsions can be accepted. A n cxarnple for this coiild be:

- Noncritical sampliiig.
Oversampling allows the steepn of thc edges to be rcduced (see Figure 11.13).'rhe irripiilsc rqmrrse eii drcays more quickly and therefore
can hc absolutely integrated and stable behaviour has beeii achieved
-

Permitting a time-delay.
filtcr can be tipproximated

Ttic impulse resporise of' the ideal lowwell by i i causal system if it is shifted far eiiough to t h c righl.