We have many realizations of digital filter. Some of these are now discussed. Direct Form Realization - An important class of linear time -invariant systems is characterized by the transfer function.

A system with input and output could be realized by the following constant coefficient difference equation

A realization of the filter using equation (9.31) is shown in figure (9.6)

Fig 9.6  Direct form I

The output is seen to be weighted sum of input and past inputs and past outputs. Another realization can be obtained by uniting as product of two transfer functions and , where contains only the denominator or poles and contains only the numerator or zeros as follows

where

Fig 9.7

The output of the filter is obtained by calculating the intermediate result obtained from operating on the input with filter and then operating on w[n] with filter.Thus we obtain

or

and

or

The realization is shown in figure 9.8

Fig  9.8 

Upon close examination of Fig 9.8, it can be seen that the two branches of delay elements can be combined as they both refer to delayed versions of and upon simplification, the direct form II canonical realization is obtained as shown in figure 9.9.

Fig 9.9 Direct form II

In this form the number of delay element is max (M,N). It can be shown that this is the minimum number of delay elements that are required to implement the digital filter. This does not mean that this is the best realization. Immunity to roundoff and quantization are very important considerations.

An important special case that is used as building block occurs when. Thus is ratio of two qualities in , called biquadratic section, and is given by

The alternative form is found to be useful for amplitude scaling for improving performance file filter operation. This form is shown in figure 9.10.

Fig 9.10

Cascade Realizations: In the cascade realization is broken into productof transfer functions each a rational expression in as follows

Fig 9.11

could be broken up in many ways; however the most common method is to use biquadratic sections. Thus

by letting and equal to zero we get bilinear section. Even among the biquadratic sections we have many choices as how we pair poles and zeros. Also the order of the sections can be different

Example:

Using only real coefficients can be decompressed as

Divides both numerator and denominator by and factoring 8 as , one possible rearrangement for is

This can be realized as shown is figure 9.12

Fig 9.12

The transfer function H ( z ) could be written as a sum of transfer functions as follows:

One parallel form results when are all selected to be of the following form for

If , we will have a section of FIR filter, obtained by performing long division. Once denominator polynomial has degree more than the numerator polynomial we perform the partial fraction expansion. The resulting structure is shown in figure 9.13.

Fig 9.13

Using MATLAB program or otherwise we get

using direst form realization for individual section we get the structure shown in figure 9.14.

Fig 9.14

Apart from these there exist a number of other realizations like lattice form, state variable realization etc.