Realization of FIR filters:

The realization of FIR filters A causal FIR filter can be characterized by the transfer function H(z) can be given by
 

or, by corresponding difference equation shown below

 
Some use notation below with M coefficients in spite of M + 1

 

The output y(n) is a weighted sum present and past input values; it does not depend on past output values such as y(n-1), etc. The block diagram is drawn below. It is also known as tapped delay line or transversal filter.

 



It can be seen that this is the similar to the direct form I or II shown earlier for the IIR filter, except that the coefficients a₁ through a_N are zero and a₀ = 1; the delay elements are arranged in a horizontal line. As in previous diagrammatic manipulation multipliers can all feed into the rightmost adder and remaining adders are removed.

  Other simplifications are possible which are based on the symmetry of the coefficients {b_r}, as in FIR filter design.

Cascade realization of FIR filters
When the system function is factored in terms of quadratic expressions in z^-1 the simplest form occurs as follows:

 

By selecting the quadratic terms to correspond to complex conjugate pairs of zeros of H(z) allows a realization in the terms of real coefficients b_0i, b_1i and b_2i. Each quadratic equation could then be realized by using the direct form (or the alternative structures) which is shown below.

Parallel realization of FIR filters

These are based on interpolation formulas by Lagrange,

Newton, or Hermite methods. Generally, these realizations require more multiplications, additions and delays than others.

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