Realization of the digital filters:
Given H(z), the system function, or h(n), the impulse response, the difference equation can be obtained. The difference equation could be implemented by computer program, special purpose digital circuitry, or the special programmable integrated circuit. This direct evaluation of the difference equation is not the only possible realization of the digital filter. Alternative realizations of digital filter are possible by breaking up direct realization in some form.
The Direct Form realization of IIR filters is an important class of linear shift invariant systems can be characterized by following rational system function (where X(z) is the input, Y(z) output and we have taken a0 = 1 in comparison with the previous representation):
By cross multiplying and by taking inverse z-transform we get the difference equation
To construct a filter structure we shall require 3 types of block diagram elements: a delay element, a multiplier and an adder, shown below below:
We can construct realization of the filter called as Direct Form I by begning with y(n) and generating all the delayed versions y(n-1), y(n-2), ..., y(n-N); similarly beginning with x(n) and generating all the delayed versions x(n-1), x(n-2), ..., x(n-M). We multiply above terms by the respective coefficients and add them up. This is illustrated below.
This is the Nth order system N being the order of difference equation. There is no restriction as whether M should be greater than or less than or equal to N. The total number of delay elements = (N+M). It is not in canonic form as it uses more than the minimum
possible number of the delay elements. It is called as "Direct Form" as the multipliers are the actual filter coefficients {a1, a2, ..., aN, b0 , b1, b2, ..., bM}.
The difference equation of the realization continues to be
and will be referred to as Direct Form I difference equation. The entire number of multiplications can be counted and is seen to be (N+M+1). We can count and see that there are (N+M) additions. Ultimately, to calculate the value y(n) we are required to store N past values of y(.),
and M past values of x(.), which means that, a total of (N+M) storage locations (storage for present value of x(.) is not counted).
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