The Biquad Allpass Section
The general biquad transfer function was given in Eq.(B.8) to be
![$\displaystyle H(z) = g \frac{1 + \beta_1 z^{-1}+ \beta_2 z^{-2}}{1 + a_1 z^{-1}+ a_2 z^{-2}} \isdef
\frac{B(z)}{A(z)}.
$](http://www.dsprelated.com/josimages_new/filters/img1441.png)
![$\displaystyle \left\vert H(e^{j\omega T})\right\vert = 1.
$](http://www.dsprelated.com/josimages_new/filters/img1442.png)
![$ A(z)$](http://www.dsprelated.com/josimages_new/filters/img693.png)
![$ B(z)$](http://www.dsprelated.com/josimages_new/filters/img712.png)
![$\displaystyle B(z) = z^{-2}A(z^{-1}) = a_2 + a_1z^{-1}+ z^{-2}.
$](http://www.dsprelated.com/josimages_new/filters/img1444.png)
![$ g=a_2$](http://www.dsprelated.com/josimages_new/filters/img1445.png)
![$ \beta_1 = a_1/a_2$](http://www.dsprelated.com/josimages_new/filters/img1446.png)
![$ \beta_2=1/a_2$](http://www.dsprelated.com/josimages_new/filters/img1447.png)
In terms of the poles and zeros of a filter
, an
allpass filter must have a zero at
for each pole at
.
That is if the denominator
satisfies
, then the
numerator polynomial
must satisfy
. (Show this in
the one-pole case.) Therefore, defining
takes care of
this property for all roots of
(all poles). However, since we
prefer that
be a polynomial in
, we define
, where
is the order of
(the number of poles).
is then the flip of
.
For further discussion and examples of allpass filters (including muli-input, multi-output allpass filters), see Appendix C. Analog allpass filters are defined and discussed in §E.8.
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Allpass Filter Design
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