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Chapter 11

Design of Recursive (IIR) Filters11.1 Introduction

11.2 Realizability Constraints11.3 Invariant Impulse-Response Method

11.4 Modified Invariant Impulse-Response Method11.5 Matched-zTransformation Method

Copyright c 2005- Andreas Antoniou

Victoria, BC, CanadaEmail: [email protected]

December 4, 2008

Frame # 1 Slide # 1 A. Antoniou Digital Signal Processing Secs. 11.1 to 11.5
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Introduction

Approximation methods for the design of recursive (IIR) filters

differ quite significantly from those used for the design ofnonrecursive filters.

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Introduction



The basic reason is that in recursive filters the transferfunction is a ratio of polynomials ofzwhereas in nonrecursivefilters it is a polynomial of negative powers ofz.

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Introduction



The basic reason is that in recursive filters the transferfunction is a ratio of polynomials ofzwhereas in nonrecursivefilters it is a polynomial of negative powers ofz.

In recursive filters, the approximation problem is usually solvedthrough indirect ordirect methods.

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Introduction Contd

In indirect methods, a continuous-time transfer function that

satisfies certain specifications is first obtained by using one ofthe standard analog-filter approximations.

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Introduction Contd



The continuous-time transfer function obtained is thenconverted into a discrete-time transfer function.

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Introduction Contd




In direct methods, the design problem is formulated as anoptimization problem which is then solved using standardoptimization methods.

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Introduction Contd




In direct methods, the design problem is formulated as anoptimization problem which is then solved using standardoptimization methods.

This presentation will deal with some indirect methods for thedesign of recursive filters.

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Introduction Contd

Several indirect approximation methods are available asfollows:

Invariant impulse-response method

Modified invariant impulse-response method

Matched-z transformation method

Bilinear transformation method

The most important among them are the invariant impulse-response and bilinear transformation methods.


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Introduction Contd







The first three are closely interrelated and will form the

subject matter of this presentation.

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Introduction Contd







The first three are closely interrelated and will form the

subject matter of this presentation.The bilinear transformation method will be discussed in thenext presentation.

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Realizability Constraints

Before we discuss the available approximation methods for recursivefilters, it is important to mention the constraints that are imposed

on the transfer function, which are as follows:

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It must be a rational function ofzwith real coefficients.

This follows from the fact that digital-filters comprise unitdelays, adders, and multipliers with real multiplier constants.


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Its poles must lie within the unit circle of the zplane to ensurethat the filter is stable.

The degree of the numerator polynomial must be equal to orless than the degree of the denominator polynomial to ensurethat the filter is causal.


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Its poles must lie within the unit circle of the zplane to ensurethat the filter is stable.

The degree of the numerator polynomial must be equal to orless than the degree of the denominator polynomial to ensurethat the filter is causal.

These constraints will ensure that the transfer function is realizablein the form of a stable digital-filter network and are, therefore, saidto be the realizability constraints.


I i I l R M h d
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Invariant Impulse-Response Method

Given an analog filter, a corresponding digital filter can beobtained by constructing an impulse-modulated filter F

Aas

shown in the figure where Sis an ideal impulse modulator andFA is an analog filter characterized by a continuous-timetransfer function HA(s).

FA

S

FA

^


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Invariant Impulse-Response Method Contd

On the basis of the Poison summation formula (see Chap. 6),the impulse-modulated filter can be represented by acontinuous-time transfer function HA(s) or, equivalently, by adiscrete-time transfer function HD(z) as follows:

HA(j) =HD(ejT) =

hA(0+)

2

+ 1

T

k=

HA(j+jks)

where

hA(t) =L1HA(s), hA(0+) = lims

[sHA(s)],

andHD(z) = ZhA(nT)


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Therefore, given an analog filter, a corresponding digital filtercan be obtained as follows:


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Deduce the impulse response of the analog filter as

hA(t) = L1HA(s)


Invariant Impulse Response Method C d
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hA(t) = L1HA(s)

Replace tby nT in hA(t) to obtain hA(nT).


Invariant Impulse Response Method C td
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hA(t) = L1HA(s)


Obtain the ztransform ofhA(nT), i.e.,

HD(z) = ZhA(nT)


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hA(t) = L1HA(s)


Obtain the ztransform ofhA(nT), i.e.,

HD(z) = ZhA(nT) The method is referred to as the invariant-impulse response

method because the impulse response of the digital filter isexactly equal to the impulse response of the analog filter att=nT forn= 0, 1, . . . ,.


Invariant Impulse Response Method Contd
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Invariant Impulse-Response Method Cont d

Obviously, we have a way of generating a discrete-timetransfer function from a given continuous-time transferfunction and, therefore, we can design a digital filter startingwith an analog filter.



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However, the following important questions will immediatelyarise:


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1. How is the frequency response of the derived digital filterrelated to that of the analog filter?


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Invariant Impulse Response Method Cont d




2. Would the digital filter obtained be stable if the analog filter isstable?


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2. Would the digital filter obtained be stable if the analog filter isstable?

3. Would the discrete-time transfer function obtained have realcoefficients?


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Let us examine the issue of the frequency response.


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As was mentioned earlier, the Poisson summation formula gives thefrequency response of the derived digital filter as

HA(j) =HD(ejT) =

h(0+)

2 +

1

T

k=

HA(j+jks)


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p p


As was mentioned earlier, the Poisson summation formula gives the

frequency response of the derived digital filter as

HA(j) =HD(ejT) =

h(0+)

2 +

1

T

k=

HA(j+jks)

If HA(j) 0 for || s2

then

k=,k=0

HA(j+jks)

0 for

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DSP-Ch11-S1-5

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