Digital filters are generally classified into two groups: finite impulse response (FIR) filters and infinite impulse response (IIR) filters. FIR filter are built from forward delay elements, while IIR filters require negative feedback.
A general FIR filter can be defined by the simple following discrete convolution:
:<math> y[n] = \sum_{k=0}{N} h[k] . x[n-k] </math>
where h[n] is the impulse function of the filter and x[n] and y[n] are the input and output sequences, respectively. Note that h[n] has nonzero values only for 0 ≤ n ≤ N. x[n] can be regarded as the samples of a continuous-time signal x(t), which is samples at t = 0, T, 2T, ..., NT, where T is the sampling period.
The above discrete convolution equation can be transformed into the z-domain as follows:
:<math> \frac{Y(z)}{X(z)} = H(z) = \sum_{k=0}{N} h[k] z{-k} </math>
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