**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Difference Equation

Definition.Thedifference equationfor a general, causal, linear time-invariant (LTI)digital filter is given by

where is the input signal, is the output signal, and the constants , , , are calleddifference equation coefficients, or more simply,filter coefficients. When the and coefficients are real numbers, then the filter is said to bereal.

Definition.Equation Eq. (B.8) represents onlycausalLTI filters. A filter is said to becausalwhen 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.Themaximum time span, in samples, used in creating each output sample is called the order of the filter. In Eq. (B.8), the order is the larger of and . Since and in Eq. (B.8) are assumed finite, Eq. (B.8) represents the class offinite ordercausal 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.Theimpulse signalis denoted as and is defined by

Definition.Theimpulse responseof a filter is the response of the filter to and is most often denoted .

Definition.A filter is said to bestableif the impulse response approaches zero as goes to infinity.