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:
where is the input signal, is the output signal, and the constants , are called the coefficients
As a specific example, the difference equation
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.)
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.
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.
Signal Flow Graph
Linearity and Time-Invariance Problems