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## Difference Equation

The *difference equation* is a formula for computing an output
sample at time based on past and present input samples and past
output samples in the time domain.^{6.1}We may write the general, causal, LTI difference equation as follows:

where

is the input

signal,

is the output signal, and the
constants

,

are called the

*coefficients*
As a specific example, the difference equation

specifies a

digital filtering operation, and the coefficient sets

and

fully characterize the

filter. In this
example, we have

.

When the coefficients are real numbers, as in the above example, the
filter is said to be
*real*. Otherwise, it may be
*complex*.

Notice that a filter of the form of Eq.(5.1) can use ``past''
output samples (such as ) in the calculation of the
``present'' output . This use of past output samples is called
*feedback*. Any filter having one or more
feedback paths () is called
*recursive*. (By
the way, the minus signs for the feedback in Eq.(5.1) will be
explained when we get to transfer functions in §6.1.)

More specifically, the coefficients are called the
*feedforward coefficients* and the coefficients are called
the *feedback coefficients*.

A filter is said to be *recursive* if and only if for
some . Recursive filters are also called
*infinite-impulse-response (IIR)* filters.
When there is no feedback (
), the filter is said
to be a *nonrecursive* or
*finite-impulse-response (FIR)* digital filter.

When used for discrete-time physical modeling, the difference equation
may be referred to as an *explicit finite difference
scheme*.^{6.2}

Showing that a recursive filter is LTI (Chapter 4) is easy by
considering its *impulse-response representation* (discussed in
§5.6). For example, the recursive filter

has impulse response
,
. It is now
straightforward to apply the analysis of the previous chapter to find
that time-invariance, superposition, and the scaling property hold.

**Previous:** Time Domain Digital Filter Representations**Next:** Signal Flow Graph

**About the Author: ** Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at

Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See

http://ccrma.stanford.edu/~jos/ for details.