Elementary Digital Filter Theory

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Difference Equation
Definition. The difference equation for a general linear time-invariant (LTI)digital filter is given by
$$\$ $$\$ $$\$ (42)
$$\$ $$\$ (43)

where is the input signal, is the output signal, and the constants $$\$ , $$i=0,1,2,\$ , $$\$ , $$i=1,2,\$ are called difference equation coefficients, or more simply, filter coefficients. When the and coefficients are real numbers, then the filter is said to be real.
Definition. Equation Eq. (43) represents only causal LTI filters. A filter is said to be causal when the output does not depend on any ``future'' inputs. (In more colorful terms, a filter is causal if it does not ``laugh'' before it is ``tickled.'')
Definition. The maximum time span, in samples, used in creating each output sample is called the order of the filter. In Eq. (43), the order is the larger of ${n_b}$ and ${n_a}$ . Since ${n_b}$ and ${n_a}$ in Eq. (43) are assumed finite, Eq. (43) represents the class of finite order causal LTI filters.
In addition to difference equation coefficients, any LTI filter may be represented in the time domain by its response to a specific signal called the impulse.
Definition. The impulse is denoted as $$\$ and is defined by

Definition. The impulse response of a filter is the response of the filter to $$\$ and is most often denoted .
Definition. A filter is said to be stable if the impulse response approaches zero as goes to infinity.

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Copyright © 2000-09-19 by Julius O. Smith III, Adpated from Techniques for Digital FilterDesign and System Identification, with Application to the Violin, Julius O. Smith III, Ph.D. Dissertation, CCRMA, Department of Electrical Engineering, Stanford University, June 1983.. This is an original HTML version reproduced by permission.

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