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 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 get the output from the third stage lattice filter 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.

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.

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

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

Given the system function of the FIR filter is

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