The BiQuad Section

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). The BiQuad Section

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.

<< Previous page  TOC  INDEX  Next page >>


The BiQuad Section

The term “biquad” is short for “bi-quadratic”, and is a common name for a two-pole, two-zero digital filter. Thetransfer function of a biquad can be defined as

\

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”.

The parameters $b_1$ and $b_2$ are called the numerator coefficients, and they determine the two zeros of the biquad. Using the quadratic formula for finding the roots of a second-order polynomial, we find that the zeros are located at

\

The parameters $a_1$ and $a_2$ are called the denominator coefficients, and they determine the two poles of the biquad. Using the quadratic formula, the poles are found to be

\

The biquad forms a resonator when the poles are complex, i.e., when $(a_1/2)^2<a_2$.

If a complex pole is denoted by $p=r e^{j\, then the resonancefrequency $f_r$ (in Hz) is related to pole angle $\ and sampling rate $f_s=1/T$ by the relation $\, i.e., $f_r = f_s\.

The magnitude $r$ of a complex pole $p=r e^{j\determines the damping or bandwidth of the resonator. (The damping may be defined as the reciprocal of the bandwidth.) A common definition for this relationship is

\

where $r$ is the pole radius, $B$ is the bandwidth in Hertz (cycles per second), and $T$ is the sampling interval in seconds.

The denominator coefficients of a resonator may be expressed in terms of $r$ and $\ as

\


Thus, $a_2$ depends only on the damping and is independent of the resonance frequency, while $a_1$ is a function of both.

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.,$b_1=0$ and $b_2=-1$ $\. This placement of the zeros 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

\


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.

<< Previous page  TOC  INDEX  Next page >>

 

© 1998-2017 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy