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: Ie-‘’’E(L)l tit = bW e-ncdt = c1 for a > (16.9) cm otherwise arid obtain the result: the system is stable for a > 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 ) 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 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 :