The equation for the output of an FIR filter represented in the direct form structure is given as

Given the system function of the FIR filter is

Consider a sequence of FIR filer with system function Hm(z)=Am(z), m=0,1,2…M-1

There are several structures for implementing an FIR system, beginning with the simplest structure, called the direct form. There are several other methods like cascade form realization, frequency sampling realization and lattice realization which are used for implementing and FIR system.

The system function H(z) which is characterized by the set of frequency samples is obtained as

In frequency sampling realization, the system function H(z) is characterized by the set of frequency samples {H(k+ α)} instead of {h(n)}. We view this FIR filter realization as a cascade of two filters. One is an all-zero or a comb filter and the other consists of parallel bank of single pole filters with resonant frequencies.

The structure of the direct form realization, resembles a tapped delay line or a transversal system.

We know that the difference equation of an FIR system is given by y(n)=\\(\\sum_{k=0}^{M-1}b_k x(n-k)\\).

The single stage lattice filter is as shown below.

We have to do 3 multiplications for every second order equation. So, we have to do 6 multiplications if we combine two second order equations and we have to perform 3 multiplications by directly calculating the fourth order equation. Thus the number of multiplications are reduced by a factor of 50%.