[Numpy-discussion] polynomial fromroots

Fabrice Silva silva@lma.cnrs-mrs...
Sun Oct 10 14:28:02 CDT 2010


Le dimanche 10 octobre 2010 à 14:19 -0400, josef.pktd@gmail.com a 
> Good, this confirms the differences in convention z, or 1/z (and why I
> never remember if the roots are supposed to be inside or outside the
> unit circle)

Some tricks:
* in z-transform, 1-sample delay (in time domain) correspond to z**(-1)
multiplication:
H(z)		h[k]	present (current instant)
H(z)*z**(-N)	h[k-N]  past (N samples backward)
H(z)*z**(+N)	h[k+N]  future (N samples forward)

* in linear filter theory (or perhaps generally speaking), you may focus
on causal filter, i.e. filter where current output depends on past
samples. You then build a difference equation as a polynomial in
z**(-1).

* consider the following transform between z and w(complex)
	 z=exp(+jwt) for some arbitrary t>0.
  z inside unit circle associates with an positive imaginary part of w.
  t->exp(jwt) is then a decaying (stable) function of time
  
  z outside the unit circle lead them to unstable function. 

That's why you should prefer roots of the roots of z**(-1) polynomial
inside unit circle.

-- 
Fabrice Silva



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