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

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

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

The equation of the output from the second stage lattice filter is given by

We get the output from the third stage lattice filter as

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

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

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.