Re: are these stabability conditions the same?

From: Randy Yates (yates_at_ieee.org)
Date: 12/28/04 

Date: Tue, 28 Dec 2004 02:01:46 GMT

Hi Kiki,

First of all allow me to encourage you in your pursuit of
DSP. You are asking some amazingly-good questions and it is
clear you are going to skyrocket in this field. Keep it up!

Let me give an attempt at answering below.

"kiki" <lunaliu3@yahoo.com> writes:

> Hi all,
>
> I understood the two stability conditions:
>
> One is BIBO criteria in Fourier analysis... it says that if the impulse
> response is not absolute integrable, i.e. if Integrate(|h(t)|, t from -inf
> to inf)=inf then the system is not BIBO stable...
>
> From this criteria, an ideal low pass filter is not BIBO stable, since its
> impulse response is "sinc" function, which can be proved to have
> Integrate(|sinc(t)|, t from -inf to inf)=inf...

It does seem odd, but that's exactly right. I think the thing to
realize is that, in practice, a bounded signal that would cause the
output to grow without bound would be quite pathological since it
would be of the form s(t)*a, where s(t) = signum(sinc(t)) and "a"
is a constant.

> Another criteria is the negative real part of the poles criteria in Laplace
> analysis, which says that for a system to be stable, its poles should have
> negative real parts...(on the left plane on the complex s-domain)...
>
> According to this criteria, the Laplace transform of the impulse function
> sinc(t) is:
>
> -----------------------------------------------
>>> syms t s
>>> laplace(sin(pi*t)/(pi*t))
>
> ans =
>
> 1/pi*atan(pi/s)
> -----------------------------------------------
>
> But it has no pole at all... so sinc(t) is a stable system according to this
> Laplace analysis.

Right. This analysis is based on a study of the Region Of Convergence (ROC)
for the Laplace transform. Essentially, this is a sufficient condition for
convergence but not a necessary one. For example, this analysis assumes
the transfer function is a rational function, i.e., a ratio of polynomials.
What if it's not?

> Isn't this strange? Did I miss anything?

I don't think so - in fact what you've noticed slips by most engineers. 

-- 
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