Digital signal processing By Er. Swapnil Kaware

A Seminar On,

“Design of Digital IIR filter”

Presented By,

Mr. Swapnil V. Kaware,svkaware@yaoo.co.in

svkaware@yahoo.co.in

Introduction

(i). Infinite Impulse Response (IIR) filters are the first

choice when:Speed is paramount.Phase non-linearity is acceptable.

(ii). IIR filters are computationally more efficient than

FIR filters as they require fewer coefficients due to

the fact that they use feedback or poles.

(ii). However feedback can result in the filter

becoming unstable if the coefficients deviate from

their true values.

Butterworth :- Maximally Flat Amplitude.

Chepyshev type I :- Equiriple in the passband.

Chepyshev type II :- Equiriple in the stopband.

Elliptic :- Equiripple in both the passband and stopband.

Filter Design Methods

Design ProcedureTo fully design and implement a filter five steps are required:

(1) Filter specification.

(2) Coefficient calculation.

(3) Structure selection.

(4) Simulation (optional).

(5) Implementation.

Filter Specification - Step 1

f(norm)fc : cut-off frequency

pass-band stop-band

pass-band stop-bandtransition band

pass-bandripple

stop-bandripple

fpb : pass-band frequency

fsb : stop-band frequency

f(norm)

fc : cut-off frequency

|H(f)|(dB)

|H(f)|(linear)

|H(f)|

IIR Filters Better magnitude response (sharper transition and/or lower stopband

attenuation than FIR with the same number of parameters: HW efficient)

Established filter types and design methods.

IIR filter design procedure:-

1) Set up digital filter specification,2) Determine the corresponding analog filter specification,

(frequency translation involved)3) Design the analog filter,4) Transform the analog filter to digital filter using various

transformation methods, Impulse invariant method Bilinear transformation

IIR Filters

Important parametersPassband ripple : Stopband attenuation : Discrimination factor :

Selectivity factor :

(-3dB) cutoff frequency :

spsvkaware@yahoo.co.in

IIR Filters

Frequency response Transfer function : Rational

Asymptotic attenuation at high frequency

Attenuation function: (rational or polynomial function)

If is monotone, so is If is oscillatory, exhibits ripple.

: Square magnitude frequency response

: reference frequency

IIR Filters

Frequency response For real rational transfer function

Stability requirement

must include all poles of

on the left half of the s plane and only those.

Analog filter typesButterworthChebyshevElliptic

Butterworth Filters

The Butterworth filter is a type of signal processing filter

designed to have as flat a frequency response as possible in the

passband. It is also referred to as a maximally flat magnitude

filter. It was first described in 1930 by the British engineer and

Physicist Stephen Butterworth in his paper entitled "On the Theory

of Filter Amplifiers"

Butterworth Filters

The frequency response of the Butterworth filter is maximally flat

(i.e. has no ripples) in the passband and rolls off towards zero in the stopband.

When viewed on a logarithmic Bode plot the response slopes off linearly towards negative infinity.

A first-order filter's response rolls off at −6 dB per octave (−20 dB

per decade) (all first-order lowpass filters have the same normalized frequency response).

A second-order filter decreases at −12 dB per octave, a third-order at −18 dB and so on.

Butterworth filters have a monotonically changing magnitude function with ω, unlike other filter types that have non-monotonic ripple in the passband and/or the stopband.

• Passband is designed to be maximally flat.• The magnitude-squared function is of the form

c The Cutoff frequency

N The order of the filter

12,1,0 ,2

Nkes N

poles LHP

Butterworth Lowpass Filters

Butterworth filters

Magnitude Squared Response

Properties of a LP Butterworth filterMagnitude response : monotonically decreasingMaximum gain : 0 at Asymptotic attenuation at high frequency : Maximally flat at DC (maximally flat filter)

12)1( p

2|)(| H

: -3 dB point

Butterworth filters

Transfer function

2N poles: : N poles are on the left side

of the complex plane All pole filter

Normalized transfer function : Nth-order LP Butterworth filter

Butterworth filters

LP Butterworth filter design procedure1. Set up filter spec :

2. Compute N, using

3. Choose using

4. Compute the poles , using

5. Compute , using

Chebyshev Filters

Chebyshev filters are analog or digital filters having a steeper roll-offand more passbandripple(type I) or stopband ripple (type II) than Butterworth filters.

Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband.

This type of filter is named after Pafnuty Chebyshv because its mathematical characteristics are derived from Chebyshev polynomials.

Because of the passband ripple inherent in Chebyshev filters, the ones that have a smoother response in the passband but a more irregular response in the stopband are preferred for some applications.

• Equiripple in the passband and monotonic in the stopband.• Or equiripple in the stopband and monotonic in the passband.

Type IType II

xcosNcosxV /V1

)]/([1

Chebyshev Filters

Chebyshev filters

Chebyshev polynomial of degree

Monotone only in one band Chebyshev Type I : equiripple in the passband Chebyshev Type II : equiripple in the stopband

Sharper than Butterworth due to the ripples ! Why ? Sharpest if equiripple in both bands, pass- and stop-bands. Phase response : Better for maximally flat or monotonic mag

response filters

Recursive formula:

4,3,2,1N

If N is even(odd), so is

Chebyshev-I : Chebyshev Filter of the first kind

Properties All-pole filter For

For Monotonically decreasing because asymptotic attenuation :

Chebyshev-I : Chebyshev Filter of the first kind

Poles of a Nth-order LP Chebyshev-I filter

Transfer function0a

(N=3) case

Chebyshev-II : Chebyshev Filter of the second kind

Inverse Chebyshev filter or Chebyshev-II

Properties Passband : monotonic Stopband : equi-ripple Contains both the poles and zeros

for all

: monotonically decreasing

Elliptic filter(i). An elliptic filter (also known as a Cauer filter, named after

Wilhelm Cauer) is a signal processing filter with equalized ripple (equiripple) behavior in both thepassband and the stopband.

(ii). The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition ingain between the passbandand the stopband, for the given

values of ripple (whether the ripple is equalized or not).

(iii). Alternatively, one may give up the ability to independently adjust the passband and stopband ripple, and instead design a filter which is maximally insensitive to componenAn elliptic filter (also known as a Cauer filter,

Elliptic Filters

Overview Equiripple in both the passband and the stop band Minimum possible order for a given spec : Sharpest (optimum)

Magnitude Squared Response: LP elliptic filter

: Jacobian elliptic function of degree N Even(odd) function of for even(odd) For , oscillates between -1 and +1

oscillates between 1 and for

oscillates between and for

Elliptic FiltersExample

Elliptic Filters

Equiripple both in stopband and in the passband

Jacobian Elliptic function)(1

Frequency transformation

Analog filter design1. Design a LPF (Butterworth, Chebyshev, elliptic)

2. Frequency transformation to obtain HPF, BPF, BRF

Definitions : rational function ( , ) Transfer function of a LP filter : Transformed filter

: rational function of Class and stability of the filter is preserved after transformation.

Design domain : Target domain :

Frequency transformation

LP to LP transformation

LP to HP transformation

LP to BP transformation

LP to BS transformation

Digital IIR filter design

Digital IIR filter design1. Digital filter spec -> analog filter spec

2. Design analog filter

3. Transformation : Analog filter to digital filter

Transformation Goal

Requirements for Real, causal, stable, rational The order of should not be greater than that of if possible. should be close to where •transform should be simple, convenient to implement and applicable

to all analog filter types and classes

Impulse Invariant Transformation Definition

Procedure

3. High-pass filter cannot be

transformed !! Filter orders are not changed

After transformation

Example)

Bilinear Transform Definition and Properties

(Approximation of continuous-time integration by discrete-time trapezoidal integration)

1) # of poles are preserved. => Preserve the filter order2) # of zeros increase from q to p if p > q (p-1 zeros at z=-1)

If => Preserve the stability

z-plane

1) Frequency warping : One-to-one mapping, 2) 3) Can be used for all filter types

z-plane

Bilinear Transform Prewarping

Prewarp the analog frequencies -> Bilinear transform -> desired digital frequencies

For convenience, set => Prewarping -> BLT gives the same result.

IIR filter Design procedure using BLT

1. Convert each specified band-edge frequency of the digital filter to a corresponding band-edge freq of an analog filter, using (A)

- Leave the ripple values unchanged.

2. Design

3. using BLT

Bilinear TransformationBilinear Transformation

Transformation is unaffected by scaling. Consider inverse transformation with scale factor equal to unity

and so

)1()1(

oo zjj

10 zo10 zo10 zo

Mapping of s-plane into the z-plane

For with unity scalar we have

)2/tan(11

)2/tan(

Mapping is highly nonlinear

Complete negative imaginary axis in the s-plane from to is mapped into the lower half of

the unit circle in the z-plane from to

Complete positive imaginary axis in the s-plane from to is mapped into the upper half of the unit circle in the z-plane from to

Nonlinear mapping introduces a distortion in the

frequency axis called frequency warping

Effect of warping shown below,

References

(1). J.G. Proakis and D.G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, Prentice Hall, 3rd Edition, 1996, ISBN 013373762- 4.

(2). S.S. Soliman and M.D. Srinath, Continuous and Discrete Signals and Systems, Prentice Hall, 1998, ISBN 013518473-8.

(3). A.V. Oppenheim and R.W. Schafer, Digital Signal Processing, Prentice Hall, 1975, ISBN 013214635-5.

(4). L.R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing, Prentice Hall, 1975, ISBN 013914101-4.

(5). E.O. Brigham, The Fast Fourier Transform and Its Applications, Prentice Hall, 1988, ISBN 013307505-2.

(6). M.H. Hayes, Digital Signal Processing , Schaum’s Outline Series, McGraw Hill, 1999, ISBN 0-07-027389-8

Thank You!!

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