The BiQuad Section
The term ``biquad'' is short for ``bi-quadratic'', and is a common name for a two-pole, two-zero digital filter. The transfer function of the biquad can be defined as
where
![$ g$](http://www.dsprelated.com/josimages_new/filters/img435.png)
![$ z^{-1}$](http://www.dsprelated.com/josimages_new/filters/img91.png)
![$ z$](http://www.dsprelated.com/josimages_new/filters/img45.png)
![$ z^{-1}$](http://www.dsprelated.com/josimages_new/filters/img91.png)
![$ z$](http://www.dsprelated.com/josimages_new/filters/img45.png)
As derived in §B.1.3, for real second-order polynomials having
complex roots, it is often convenient to express the polynomial
coefficients in terms of the radius and angle
of the
positive-frequency pole. For example, denoting the denominator
polynomial by
, we have
![$\displaystyle A(z) = \left(1 - Re^{j\theta}z^{-1}\right)\left(1 - Re^{-j\theta}z^{-1}\right)
= 1 - 2R\cos(\theta)z^{-1}+ R^2z^{-2}.
$](http://www.dsprelated.com/josimages_new/filters/img1429.png)
![$ \theta$](http://www.dsprelated.com/josimages_new/filters/img1155.png)
![$ \theta=2\pi f_c T$](http://www.dsprelated.com/josimages_new/filters/img1430.png)
![$ f_c$](http://www.dsprelated.com/josimages_new/filters/img86.png)
![$ R$](http://www.dsprelated.com/josimages_new/filters/img61.png)
As discussed on page , a common setting for the zeros when
making a resonator is to place one at
(dc) and the other at
(half the sampling rate), i.e.,
and
in
Eq.
(B.8) above
.
This zero placement normalizes the peak gain of the resonator if it is
swept using the
parameter.
Using the shift theorem for z transforms, the difference equation for the biquad can be written by inspection of the transfer function as
![\begin{eqnarray*}
v(n) &=& g\, x(n) \\
y(n) &=& v(n) + \beta_1 v(n-1) + \beta_2 v(n-2) \\
& & \qquad - a_1 y(n-1) - a_2 y(n-2) .
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img1434.png)
where denotes the input signal sample at time
, and
is the output signal. This is the form that is typically implemented
in software. It is essentially the direct-form I implementation. (To obtain the official
direct-form I structure, the overall gain
must be not be pulled
out separately, resulting in feedforward coefficients
instead. See Chapter 9 for more about
filter implementation forms.)
Next Section:
Biquad Software Implementations
Previous Section:
Complex Resonator