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

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.

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

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

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

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

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)