Chapter: Digital Signal Processing - IIR Filter Design

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Structures For IIR Systems

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.

STRUCTURES FOR IIR SYSTEMS

 

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.


1.      

DIRECT FORM STRUCTURE FOR IIR SYSTEMS

 

IIR systems can be described by a generalized equations as


Overall IIR system can be realized as cascade of two function H1(z) and H2(z). Here H1(z) represents zeros of H(z) and H2(z) represents all poles of H(z).



FIG - DIRECT FORM I REALIZATION OF IIR SYSTEM

 

1.     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.

 

2.     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.

 

3.     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.

 

DIRECT FORM - II

 

1. 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.

 

2. Two delay elements of all pole and all zero system can be merged into single delay element.

 

3. Direct Form II structure has reduced memory requirement compared to Direct form I structure. Hence it is called canonic form.

 

4. The direct form II requires same number of multiplications(M+N+1) and additions (M+N) as that of direct form I.



CASCADE FORM STRUCTURE FOR IIR SYSTEMS

 In cascade form, stages are cascaded (connected) in series. The output of one system is input to another. Thus total K number of stages are cascaded. The total system function 'H' is given by

 

H= H1(z) . H2(z)……………………. Hk(z)                   (1)

H= Y1(z)/X1(z). Y2(z)/X2(z). ……………Yk(z)/Xk(z)         (2)


Each H1(z), H2(z)… etc is a second order section and it is realized by the direct form as shown in below figure.

System function for IIR systems


Thus Direct form of second order IIR system is shown as



PARALLEL FORM STRUCTURE FOR IIR SYSTEMS

 

System function for IIR systems is given as


The above system function can be expanded in partial fraction as follows

H(z) = C + H1(z) + H2(z)…………………….+ Hk(z)                     (3)

Where C is constant and Hk(z) is given as





IIR FILTER DESIGN

 IMPULSE INVARIANCE

 

BILINEAR TRANSFORMATION

 

BUTTERWORTH APPROXIMATION

 

IIR FILTER DESIGN - IMPULSE INVARIANCE METHOD

 


Impulse Invariance Method is simplest method used for designing IIR Filters. Important Features of this Method are

 

1. In impulse variance method, Analog filters are converted into digital filter just by replacing unit sample response of the digital filter by the sampled version of impulse response of analog filter. Sampled signal is obtained by putting t=nT hence

h(n) = ha(nT)                                                       n=0,1,2. ………….

where h(n) is the unit sample response of digital filter and T is sampling interval.

 

2. But the main disadvantage of this method is that it does not correspond to simple algebraic mapping of S plane to the Z plane. Thus the mapping from analog frequency to digital frequency is many to one. The segments

(2k-1)∏/T ≤ Ω ≤ (2k+1) ∏/T of j Ω axis are all mapped on the unit circle ∏≤ω≤∏. This takes place because of sampling.

 

3. Frequency aliasing is second disadvantage in this method. Because of frequency aliasing, the frequency response of the resulting digital filter will not be identical to the original analog frequency response.

 

4. Because of these factors, its application is limited to design low frequency filters like LPF or a limited class of band pass filters.

 

RELATIONSHIP BETWEEN Z PLANE AND S PLANE

Z is represented as re in polar form and relationship between Z plane and S plane is given as Z=eST where s= σ + j Ω.


Here we have three condition

1.     If σ = 0 then r=1

2.     If σ < 0 then 0 < r < 1

3.     If σ > 0 then r> 1

 

Thus

1.     Left side of s-plane is mapped inside the unit circle.

2.     Right side of s-plane is mapped outside the unit circle.

3.     jΩ axis is in s-plane is mapped on the unit circle.



 


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