The BiQuad Section

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Elementary Filter Sections

Two-Pole Partial Fraction Expansion

Biquad Software Implementations

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

Click for https://ccrma.stanford.edu/~jos/filters/Time_Domain_Filter_Representations.html

Click for https://ccrma.stanford.edu/~jos/filters/Direct_Form_I.html

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

The difference equation is a recipe for computing samples of the output signal of a digital filter based on samples of the input signal and the filter coefficients. In an Infinite-Impulse-Response (IIR) digital filter, there are typically both feedforward and feedback coefficients in the difference equation. — Click for https://ccrma.stanford.edu/~jos/filters/Difference_Equation_I.html

Click for https://ccrma.stanford.edu/~jos/mdft/Shift_Theorem.html

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

dc stands for direct current in electrical engineering and signal processing (as opposed to 'alternating current'). The mean (average value) of a signal may be called its dc component. Similarly, frequency zero is called the dc frequency, because a sinusoid with zero frequency is a constant signal. — Click for https://ccrma.stanford.edu/~jos/filters/Mathematical_Sine_Wave_Analysis.html

A resonator is a recursive filter that boosts signal amplitude at a particular frequency. The simplest resonator is made using a two-pole filter. The impulse response of the two-pole resonator is an exponentially decaying sinusoidal oscillation, where the zero-crossing rate is (twice) the resonance frequency. — Click for https://ccrma.stanford.edu/~jos/filters/Two_Pole.html

The quality factor or Q of a resonator may be thought of as the number of cycles of oscillation at the resonant frequency in the impulse response of the resonator before it substantially decays to zero. — Click for https://ccrma.stanford.edu/~jos/filters/Quality_Factor_Q.html

The transfer function is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

A digital filter can be visualized as a 'black box' that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. — Click for https://ccrma.stanford.edu/~jos/filters/

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Elementary Filter Sections

Two-Pole Partial Fraction Expansion

Biquad Software Implementations

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

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

(B.8)

where

can be called the overall gain of the biquad. Since both the numerator and denominator of this transfer function are quadratic polynomials in $z^{-1}$ (or

), the transfer function is said to be ``bi-quadratic'' in $z^{-1}$ (or

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 $\theta$ 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

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

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

is the resonance frequency in Hz), and

determines the ``Q'' of the resonance (see §B.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 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., $\beta_1=0$ and $\beta_2=-1$ in Eq.(B.8) above $\Rightarrow B(z) = 1-z^{-2}= (1-z^{-1})(1+z^{-1})$ . 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

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

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 $[g,g\beta_1,g\beta_2]$ instead. See Chapter 9 for more about filter implementation forms.)

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

The BiQuad Section

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition) Copyright © 2024-09-03 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University