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

$$H(z) = g \frac{1 + \beta_1 z^{-1}+ \beta_2 z^{-2}}{1 + a_1 z^{-1}+ a_2 z^{-2}} \protect$$ (11.8)

where $g$ is called the gain of the biquad. Since both the numerator and denominator of this transfer function are quadratic polynomials in $z$, the transfer function is said to be ``bi-quadratic'' in $z$.

As derived in §10.1.3, for real second-order polynomials having complex roots, it is often convenient to express the polynomial coefficients in terms of the radius $R$ and angle $\theta$ of the positive-frequency pole. For example, denoting the denominator polynomial by $A(z)=1 + a_1 z^{-1}+ a_2 z^{-2}$, we have

$$A(z) = \left(1 - Re^{j\theta}z^{-1}\right)\left(1 - Re^{-j\theta}z^{-1}\right) = 1 - 2R\cos(\theta)z^{-1}+ R^2z^{-2}.$$

This representation is most often used for the denominator of the biquad, and we think of $\theta$ as the resonance frequency (in radians per sample-- $\theta=2\pi f_c T$, where $f_c$ is the resonance frequency in Hz), and $R$ determines the ``Q'' of the resonance (see §10.1.3). The numerator is less often represented in this way, but when it is, we may think of the zero-angle as the antiresonance frequency, and the zero-radius affects the depth and width of the antiresonance (or notch).

As discussed in §10.6.4, a common setting for the zeros when making a resonator is to place one at $z = 1$ (dc) and the other at $z = -1$ (half the sampling rate), i.e., $\beta_1=0$ and $\beta_2=-1$ in Eq. (10.8) above $\Rightarrow B(z) = 1-\beta_2z^{-2}= (1-z^{-1})(1+z^{-1})$. This zero placement normalizes the peak gain of the resonator if it is swept using the $a_1$ parameter.

Using the shift theorem for z transforms, the difference equation for the biquad can be written by inspection of the transfer function as

$$\begin{eqnarray*} v(n) &=& g x(n) \\ y(n) &=& v(n) + \beta_1 v(n-1) + \beta_2 v(n-2) \\ & & \qquad - a_1 y(n-1) - a_2 y(n-2) . \end{eqnarray*}$$

where $x(n)$ denotes the input signal sample at time $n$, and $y(n)$ 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 $g$ must be not be pulled out separately, resulting in feedforward coefficients $[g,g\beta_1,g\beta_2]$ instead. See Chapter 9 for more about filter implementation forms.)