Introduction to Digital Filter Theory
In this section, linearity, time-invariance and four basic representations of digital filters are defined: thedifference equation coefficients, impulse response, transfer function, and frequency response. Next the concepts ofphase delay, group delay, poles and zeros, and filter stability are defined. This elementary material was taken from course notes for a class given at Stanford by the author in 1979.
Definition. A real filter is defined as any real-valued functional of a signal for each integer . We express the input-output relation of the filter by
where is the entire input signal, and is the output at time .