are fundamental to digital audio processing. While we only have time here for a short overview of the essential features of filters, students are encouraged to pursue more advanced courses and references in filter analysis and design.
The processing of signals is called filtering. When applied to discrete-time signals, this processing is called digital filtering.
Digital filters are defined by their impulse response, , or the filter output given a unit sample impulse input signal. A discrete-time unit impulse signal is defined by:
The filtering operation in the time domain is referred to as convolution, defined as
where is the filter impulse response and is the input signal to the filter.
Digital filters are often more intuitively understood in terms of their frequency response. That is, how is a sinusoidal signal of a given frequency affected by the filter.
One way to find the frequency response of a digital filter is by taking the DFT (or FFT) of the filter impulse response.
The frequency response of a filter consists of its magnitude and phase responses. The magnitude response indicates the ratio of a filtered sine wave's output amplitude to its input amplitude. The phase response describes the phase “offset” or time delay experienced by a sine wave passing through a filter.