Algorithms and Parallel Computing

9.3 THE 1-D FIR DIGITAL FILTER ALGORITHM

We are now ready to illustrate how to use the z-transform to obtain systolic structures. We use 1-D FIR. The 1-D FIR digital filter algorithm can be expressed as the set of difference equations

(9.6)

where a(k) is the filter coefficient and N is the filter length, which is the number of filter coefficients. Such an algorithm is a set of computations that is performed on input variables to produce output variables. The variables we might encounter are of three types: input, output, and intermediate or input/output (I/O) variables. An input variable is one that has its instances appearing only on the right-hand side (RHS) of the equations of the algorithm. An output variable is one that has its instances appearing only on the left-hand side (LHS) of the algorithm. An intermediate variable is one that has its instances appearing on the right-hand side and left-hand side of the equations. In Eq. 9.6, the variable y is an output variable, and variables x and a are input variables.

We study this algorithm using the z-transform of each side of the above equation to obtain

(9.7)

where X and Y are the z-transform of the signals x(n) and y(n), respectively. We can think of Eq. 9.7 as a polynomial expression in the different powers of z⁻¹.