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

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

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)\\).

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

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)\\).

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.

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 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%.