Pole-Zero Analysis

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

which is the same as Eq. (6.5) except that we have factored out the leading coefficient in the numerator (assumed to be nonzero) and called it g. (Here .) In the same way that can be factored into , we can factor the numerator and denominator to obtain

Assume, for simplicity, that none of the factors cancel out. The (possibly complex) numbers are the

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*.

- Filter Order = Transfer Function Order
- Graphical Amplitude Response
- Graphical Phase Response
- Stability Revisited

- Bandwidth of One Pole
- Time Constant of One Pole
- Unstable Poles--Unit Circle Viewpoint

- Poles and Zeros of the Cepstrum
- Conversion to Minimum Phase
- Hilbert Transform Relations
- Pole-Zero Analysis Problems

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