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.6.1We may write the general, causal, LTI difference equation as follows:
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.