In signal processing

Signal processing

Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

, a digital biquad filter is a second-order recursive

Recursive filter

In signal processing, a recursive filter is a type of filter which re-uses one or more of its outputs as an input. This feedback typically results in an unending impulse response , characterised by either exponentially growing, decaying, or sinusoidal signal output components.However, a recursive...

 linear filter

Linear filter

Linear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity.This results from systems composed solely of components classified as having a linear response....

, containing two poles and two zeros. "Biquad" is an abbreviation of "biquadratic", which refers to the fact that in the Z domain

Z-transform

In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....

, its transfer function

Transfer function

A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

 is the ratio of two quadratic function

Quadratic function

A quadratic function, in mathematics, is a polynomial function of the formf=ax^2+bx+c,\quad a \ne 0.The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis....

s:


High-order recursive filters can be highly sensitive to quantization

Quantization (signal processing)

Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The error introduced by...

 of their coefficients, and can easily become unstable

BIBO stability

In electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for linear signals and systems that take inputs. BIBO stands for Bounded-Input Bounded-Output...

. This is much less of a problem with first and second-order filters; therefore, higher-order filters are typically implemented as serially-cascaded biquad sections (and a first-order filter if necessary).

Direct Form 1

The most straightforward implementation is the Direct Form 1, which has the following difference equation:


Here the , and coefficients determine zeros, and , determine the position of the poles.

Flow graph of biquad filter in Direct Form 1:

Direct Form 2

The Direct Form 1 implementation requires four delay registers. An equivalent circuit is the Direct Form 2 implementation, which requires only two delay registers:
The Direct Form 2 implementation is called the canonical form, because it uses the minimal amount of delays, adders and multipliers, yielding in the same transfer function as the Direct Form 1 implementation. The difference equations for DF2 are:


where