The single stage lattice filter is as shown below.

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)\\).

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

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

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.

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

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

To derive the frequency sampling structure, we specify the desired frequency response at a set of equally spaced frequencies, namely ωk=\\(\\frac{2\\pi}{M}\\)(k+α), k=0,1…(M-1)/2 for M odd