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.

When the FIR system has linear phase, the unit sample response of the system satisfies either the symmetry or asymmetry condition, h(n)=±h(M-1-n)

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%.

We know that Hm(z)=Am(z) and Am(z) is a polynomial whose equation is given as Am(z)=\\(1+\\sum_{k=1}^m α_m (k)z^{-k}\\), m≤1 and A0(z)=1

The single stage lattice filter is as shown below.

Lattice filters are used extensively in digital signal processing and in the implementation of adaptive filters.

When two single stage lattice filters are cascaded, then the output from the first filter is given by the equation

The difference equation y(n)=\\(\\sum_{k=0}^{M-1}b_k x(n-k)\\) describes the FIR system.

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

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