Re: are these stabability conditions the same?

From: Tim Wescott (tim_at_wescottnospamdesign.com)
Date: 12/28/04 

Date: Mon, 27 Dec 2004 18:07:47 -0800

kiki wrote:

> 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...

How interesting. I never thought of that -- yet it's true. I'll have
to remember not to implement any ideal low pass filters in the near future.
>
> 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.
>
> Isn't this strange? Did I miss anything?
>
> Can anybody throw me some lights? And talk something about the relationships
> of these various conditions for stabality systems?
>
> Thanks a lot
>

If you define a "pole" of a transfer function H(s) as being a root of
the denominator polynomial of H(s) then no, the Laplace transform of
sinc(t) has no poles. If, however, you define a "pole" as being a spot
on the complex plane where the value of H(s) goes to infinity then H(s)
= 1/pi * atan(pi/s) has many poles on the imaginary line, including a
pair at s = +/- 1j * pi -- i.e. a pole pair with a non-negative real part. 

-- 
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com