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

The single stage lattice filter is as shown below.

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)

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

The direct form realization follows immediately from the non-recursive difference equation given by y(n)=\\(\\sum_{k=0}^{M-1}b_k x(n-k)\\).

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

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

The FIR structure shown in the above figure is intimately related with the topic of linear prediction. Thus the top filter structure shown in the above figure is called a prediction error filter.