Pole-Zero Analysis

This chapter discusses *pole-zero analysis* of digital filters.
Every digital filter can be specified by its poles and zeros (plus 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. The
Durbin step-down recursion for checking filter stability by finding
the reflection coefficients is presented, 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'' really makes sense when you plot 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 coeficients in Eq. (5.1) give rise to
poles. This illustrates the general fact that zeros are caused by
adding a finite number of input samples together and poles are caused
by feedback. Recall that the filter order is the maximum of and
. If in Eq. (6.5), it then follows that *the
filter order equals the number of poles or zeros, whichever is greater*.

Recall that the *order of a polynomial* is defined as the highest
power of the polynomial variable. For example, the order of the
polynomial
is 2. From Eq. (8.1), we see that is
the order of the transfer-function numerator polynomial in .
Similarly, is the order of the denominator polynomial in .
Therefore, *the filter order is given by the maximum of the
numerator and denominator polynomial orders*.

- Graphical Amplitude Response
- Graphical Phase Response
- Stability Revisited
- Computing Reflection Coefficients
- Step-Down Procedure
- Testing Filter Stability in Matlab
- Bandwidth of One Pole
- Time Constant of One Pole

- Poles and Zeros of the Cepstrum
- Pole-Zero Analysis Problems

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