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

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 (finite-order) 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.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University