This chapter discusses pole-zero analysis of digital filters. Every digital filter can be specified by its poles and zeros (together with a gain factor). Poles and zeros give useful insights into a filter's response, and can be used as the basis for digital filter design. This chapter additionally presents the Durbin step-down recursion for checking filter stability by finding the reflection coefficients, including matlab code.
Going back to Eq.(6.5), we can write the general transfer function for the recursive LTI digital filter as
The term ``pole'' makes sense when one plots the magnitude of
as a
function of z. Since
is complex, it may be taken to lie in a plane
(the
plane). The magnitude of
is real and therefore can be
represented by distance above the
plane. The plot appears as an
infinitely thin surface spanning in all directions over the
plane. The zeros are the points where the surface dips down to touch
the
plane. At high altitude, the poles look like thin, well,
``poles'' that go straight up forever, getting thinner the higher they
go.
Notice that the
feedforward coefficients from the general
difference quation, Eq.(5.1), give rise to
zeros. Similarly,
the
feedback coefficients in Eq.(5.1) give rise to
poles.
Recall that we defined the filter order as the maximum of
and
in Eq.(6.5). Therefore, the filter order equals the
number of poles or zeros, whichever is greater.