## 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

*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

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