An IIR filter having numerator order M and denominator order N is to be realized using direct form II structure. How much total number of multiplications, additions and memory locations are required respectively ?

Question

Question

This question was previously asked in

ESE Electronics Prelims 2021 Official Paper

Answer 1

IIR Systems are represented in four different ways

1) Direct Form Structures Form I and Form II

2) Cascade Form Structure

3) Parallel Form Structure

4) Lattice and Lattice-Ladder structure.

Direct Form Structures II:

F1 Shubham Ravi 23.12.21 D1

Direct form realization of H(z) can be obtained by cascading the realization of H₁(z) which is all pole system and H₂(z) which is all zero system.
Two delay elements of all pole and all zero system can be merged into single delay element.
Direct Form II structure has reduced memory requirement of max of {M, N}, as compared to Direct form I structure requirement of M+N+1. Hence it is called canonic form.
The direct form II requires same number of multiplications (M+N+1) and additions (M+N) as that of direct form I.

Additional Information

Direct Form I Structure:

F1 Shubham Ravi 23.12.21 D2

Direct form I realization of H(z) can be obtained by cascading the realization of H₁(z) which is all zero system first and then H₂(z) which is all pole system.
There are M+N-1 unit delay blocks. One unit delay block requires one memory location. Hence direct form structure requires M+N-1 memory locations.
Direct Form I realization requires M+N+1 number of multiplications and M+N number of additions and M+N+1 number of memory locations.