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Filter design

Mohammad Fathi

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Outline

Filter types Analog filters IIR and FIR filters IIR Impulse invariant Bilinear Applications: speech equalization, ….

FIR Windowing Applications

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Filter

Any discrete-time system that modifies certain frequencies

Frequency-selective filters pass only certain frequencies

Filters Lowpass Highpass Bandpass Bandstop

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Lowpass & highpass filter

Low frequency components are passed through the filter while the high-frequency components are attenuated.

keeps high-frequency components and rejects low-frequency components.

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Lowpass & highpass filter

Passband the frequency range with the amplitude gain of the filter

response being approximately unity. Stopband

The frequency range over which the filter magnitude response is attenuated to eliminate the input signal whose frequency components are within that range.

Transision band the frequency range between the passband and stopband

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Lowpass & highpass filter

: passband cutoff frequency : stopband cutoff frequency : ripple (fluctuation) of the frequency response in

the passband. : ripple of the frequency response in the stopband.

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Bandpass filter

The bandpass filter attenuates both low- and high-frequency components while keeping the middle frequency components. Ω : lower stopband cutoff frequency Ω : lower passband cutoff frequency

Ω : higher passband cutoff frequency

Ω : higher stopband cutoff frequency

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Bandstop (notch) filter

rejects the middle-frequency components and accepts both the low- and the high-frequency components.

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Continuous-time filters

Butterworth Chebyshev Type I Type II

Ellipitic

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Butterworth Lowpass Filters

Magnitude response is maximally flat in the pass band. Magnitude response is monotonic in the passband and

stopband. The magnitude-squared function is of the form

N2

c

2c j/j1

1jH

N2

c

2c j/s1

1sH

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Butterworth Lowpass Filters

There are 2N poles equally spaced in angle on circle of radius .

Poles occurs in pairs,

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Chebyshev Filters

Type I Equiripplein the passband and monotonic in the stopband

| Ω | ripples between 1 and for 0 ΩΩ

1 and

decreases monotonically for ΩΩ

1

xcosNcosxV /V1

1jH 1N

c2N

22

c

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Chebyshev Filters

Type II Monotonic in the passband and equiripple in the stopband One approch to design type II filter is to first design type I

filter and then apply the above transformation.

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Elliptic filter

Equiripple in both passband and stopband

Jacobianellipitic function

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Digital filters

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Ideal filters are non causal and nonrealizable

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Discrete-time processing of continuous-time signals

We already studied the use of discrete-time systems to implement a continuous-time system If the input is band limited and the sampling frequency is high

to avoid aliasing, then the overall system behaves as a LTI continuous-time system.

If our specifications are given in continuous time we can obtain discrete-time specifications.

/ jeff

T

H e H j T

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Filter Specifications Specifications Passband

Stopband

Parameters

Specs in dB Ideal passband gain =20log(1) = 0 dB Max passband gain = 20log(1.01) = 0.086dB Max stopband gain = 20log(0.001) = -60 dB

200020 01.1jH99.0 eff

30002 001.0jHeff

1

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4

0.010.001

2 2000 2 2000 (10 ) 0.4

2 3000 2 3000 (10 ) 0.6p p

s s

Filter implementation

all kinds of digital filters are implemented using FIR or IIR systems. Infinite impulse response (IIR)

finite impulse response (FIR)

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IIR or FIR?

Compared with FIR filter, the IIR filter offers a much smaller filter size. Filter operation requires a fewer number of computations. Not linear phase

FIR Linear phase much higher filter order than IIR filters

the design methods often are iterative in nature requiring computer-aided techniques.

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Filter design

Filter Design Steps Specification

Problem or application specific

Approximation of specification with a discrete-time system Our focus is to go from spec to discrete-time system

Implementation Realization of discrete-time systems depends on target technology

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IIR filter design

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IIR

The techniques for the design of standard frequency selective discrete-time IIR filters are based on well-developed continuous-time filter design methods. Impulse invariant Bilinear transformation

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IIR filter design using impulse invariant

Sampling the impulse response of continuous-time filter. it preserves the shape of the impulse response.

If the continuous-time filter is bandlimited to

Continuous-time and discrete-time frequencies are related by the linear scaling = Td .

Because of the aliasing effect, the impulse-invariance method is only meaningful for bandlimited filters, like lowpass and bandpass filters.

2jd c d c

k d d

h n T h nT H e H j j kT T

0 / jc d c

d

H j T H e H jT

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Impulse Invariance of System Functions Consider the CT system function with partial fraction

expansion

Corresponding impulse response

Impulse response of discrete-time filter

Then, DT system function is

Pole s=sk in s-domain transform into pole at

N

1k k

kc ss

AsH

0t0

0teAthN

1k

tsk

c

k

N

1k

nTskd

N

1k

nTskddcd nueATnueATnThTnh dkdk

N

1k1Ts

kd

ze1ATzH

dk

dkTse

Not one-to-one mapping

Ω Ω

ΩΩ

= Ω

A strip of height 2π/Td is mapped into the entire z-plane.26

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Example Impulse invariance applied to Butterworth

Since sampling rate Td cancels out we can assume Td=1 Map spec to continuous time

Butterworth filter is monotonic so spec will be satisfied if

Determine N and c to satisfy these conditions

3.0 0.17783eH

2.00 1eH89125.0j

j

3.0 0.17783jH

2.00 1jH89125.0

0.17783 3.0jH and 89125.0 2.0jH cc

N2

c

2c j/j1

1jH

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Example Cont’d

Satisfy both constrains

Solve these equations to get

Poles of transfer function

2N2

c

2N2

c 17783.013.01 and

89125.012.01

70474.0 and 68858.5N c

0,1,...,11kfor ej1s 11k212/jcc

12/1k

Example Cont’d

For causality and stability, we select poles in the left half of s-plane.

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Example Cont’d

The transfer function

Mapping to z-domain

4945.0s3585.1s4945.0s9945.0s4945.0s364.0s12093.0sH 222

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1

21

1

21

1

z257.0z9972.01z6303.08557.1

z3699.0z0691.11z1455.11428.2

z6949.0z2971.11z4466.02871.0zH

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Freq. response

Bilinear transformation

To avoid the limitations of impulse-invariance transformation caused by the aliasing effect,

we need a one-to-one mapping from the s-plane to the z-plane.

The bilinear transformation is an invertible nonlinear mapping between the s-plane and the z-plane defined by

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Filter Design by Bilinear Transformation Avoid the aliasing problem of impulse invariance Map the entire jΩ-axis in the s-plane to one revolution of the unit-circle

in the z-plane. Nonlinear transformation Frequency response subject to warping

Bilinear transformation

Transformed system function

We can solve the transformation for z as

1

1

d z1z1

T2s

1

1

dc z1

z1T2HzH

2/Tj2/T1

2/Tj2/T1s2/T1s2/T1z

dd

dd

d

d

js

Maps the left-half s-plane into the inside of the unit-circle in z Causal Stable continuous-time filter map into the causal stable discrete-time

filter

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Bilinear Transformation

On the unit circle the transform becomes

To derive the relation between and

Which yields

j

d

d e2/Tj12/Tj1z

2tan

Tj2

2/cose22/sinje2

T2j

e1e1

T2s

d2/j

2/j

dj

j

d

2Tarctan2or

2tan

T2 d

d

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Bilinear Transformation

Design method

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Example Bilinear transform applied to Butterworth

Apply bilinear transformation to specifications

We can assume Td=1 and apply the specifications to

To get

3.0 0.17783eH

2.00 1eH89125.0j

j

23.0tan

T2 0.17783jH

22.0tan

T20 1jH89125.0

d

d

N2

c

2c /1

1jH

2N2

c

2N2

c 17783.0115.0tan21 and

89125.011.0tan21

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Example Cont’d Solve N and c

The resulting transfer function has the following poles

Resulting in

Applying the bilinear transform yields

6305.51.0tan15.0tanlog2

189125.0

1117783.0

1log

N

22

766.0c

0,1,...,11kfor ej1s 11k212/jcc

12/1k

5871.0s4802.1s5871.0s0836.1s5871.0s3996.0s20238.0sH 222c

21

2121

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z2155.0z9044.011

z3583.0z0106.11z7051.0z2686.11z10007378.0zH

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Example Cont’d

Design example

low pass filter Specifications

Butterworth filter N=14

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Design example

Chebyshev type I N=8

Chebyshev type II N=8

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Design example

Elliptic filter N=6

For a fixed specifications, the lowest order filter is obtained when the approximation error ripples equally between the extremes of the two bands.

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Frequency transformation of lowpass IIR filters

Design highpass, bandpass and bandstop filters first design a low pass filter. then using an algebraic transformation, derive the desired filter.

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FIR filter design

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FIR filter

Linear phase Window method Simplest way of designing FIR filters Start with ideal desired frequency response

Ideal impulse responses are noncausal and of infinite length The easiest way to obtain a causal FIR filter from ideal is to

truncate the ideal impulse response.

n

njd

jd enheH

deeH21nh njj

dd

else0

Mn0nhnh d

else0

Mn01nw where nwnhnh d

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Windowing in frequency domain Windowed frequency response

The windowed version is smeared version of desired response

If w[n]=1 for all n, then W(ej) is pulse train with 2 period

deWeH21eH jj

dj

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Properties of Windows

It is desired that is concentrated in a narrow band of frequency to have Less

smearing. is as short as possible in duration to minimize computation in

implementing of the filter.

These are conflicting requirements! Example: Rectangular window

0

1/2 sin 1 / 21

1 sin / 2

Mj j n

n

j Mj M

j

W e e

Me ee

else0

Mn01nw

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Rectangular window

By tapering the window smoothly to zero at each end, the height of the sidelobs can be diminished. This is achieved at the expense of a wider mainlob and a wider

transition at the discontinuity.

Narrowest main lob 4/(M+1) Sharpest transitions at

discontinuities in frequency

Large side lobs -13 dB Large oscillation around

discontinuities

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Bartlett (Triangular) Window Medium main lob 8/M

Side lobs -25 dB

Hamming window performs better

Simple equation

else0Mn2/MM/n222/Mn0M/n2

nw

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Hanning Window

Medium main lob 8/M

Side lobs -31 dB

Hamming window performs better

Same complexity as Hamming

else0

Mn0Mn2cos1

21

nw

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Hamming Window• Medium main lob

8/M

• Good side lobs -41 dB

• Simpler than Blackman

else0

Mn0Mn2cos46.054.0nw

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Blackman Window

• Large main lob 12/M

• Very good side lobs -57 dB

• Complex equation

else0

Mn0M

n4cos08.0Mn2cos5.042.0nw

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Some properties

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Non rectangular windows have wider main lobes and lower sidelobes.

The width of transition band which is controlled by the width of the mainlob can be reduced by increasing the order M of the filter.

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Generalized Linear Phase

It is desirable to obtain causal systems with linear phase.

All windows are symmetric about point M/2.

/2 /2

00

[ ] [ / 2] ( ) ( ) jw jw jwM j j j Me e e

w M n n Mw n

else

w n w n M W e W e e W e W e e

where is a real and even function of is linear phase. j jeW e W e

/2

if [ ] [ ] [ ] [ ] [ ] Linear phase

d d d

j j j Me

h M n h n h n h n w n

H e H e e

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Linear-Phase Lowpass filter Desired linear phase response

Corresponding impulse response

Using a symmetric window, then a linear phase system will result.

c

c2/Mj

jlp 0

eeH

2/Mn

2/Mnsinnh clp

nw

2/Mn2/Mnsinnh c

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Some notes

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Peak overshoot δ. Peak undershoot δ. Distance between the peak ripples is approximately

the mainlobe width.

Design procedure

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Determine the cut-off frequency: Determine From the table choose the window function that provides the

smallest stopband attenuation greater than A. For this window, determine the required value of M by

selecting the corresponding value of ∆w. If M is odd, we may increase it by one to have flexible filter. Determine the impulse response of the ideal lowpass filter by

Compute

( ) / 2c p sw w w= +

1020 log , .s pA w w wd= - D = -

[ ] [ ] [ ]dh n h n w n=

Example

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Specification

Hamming window M= 80 approximately, M=66 exactly

0.25 , 0.35 , 0.0032p sw wp p d= = =

0.3 , 0.1 , 50cw Ap w p = D = =

Typical application: speech noise reduction

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Typical application: speech noise reduction

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Impulse response of standard FIR filters

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Low pass

High pass

Bandpass

Bandstop

0Low pass: [ ]

sin( ) 0

1 0High pass: [ ]

sin( ) 0

0Band pass: [ ]

sin( ) sin( ) 0

1 0Band stop: [ ]

sin( ) sin

c

c

c

c

H L

H L

H L

H

nh n

n nn

nh n

n nn

nh n

n n nn n

nh n

nn

( ) 0Ln nn

Copyright (C) 2005 Güner66

Kaiser Window Filter Parameterized equation

forming a set of windows Parameter to change main-lob

width and side-lob area trade-off

I0(.) represents zeroth-order modified Bessel function of 1st

kind

else0

Mn0I

2/M2/Mn1I

nw0

2

0

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Determining Kaiser Window Parameters

Given filter specifications Kaiser developed empirical equations Given the peak approximation error or in dB as A=-20log10 and transition band width

The shape parameter should be

The filter order M is determined approximately by

21A050A2121A07886.021A5842.0

50A7.8A1102.04.0

ps

285.2

8AM

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Example: Kaiser Window Design of a Lowpass Filter

Specifications Window design methods assume Determine cut-off frequency Due to the symmetry we can choose it to be

Compute

And Kaiser window parameters

Then the impulse response is given as

001.0,01.0,6.0,4.0 21pp

001.021

5.0c

2.0ps 60log20A 10

653.5 37M

else0

Mn0653.5I

5.185.18n1653.5I

5.18n5.18n5.0sinnh

0

2

0

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Example Cont’d

Approximation Error

Example

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Specification 0.25 , 0.35 , 0.0032p sw wp p d= = =

0.3 , 0.1 , 50

4.5, 59 60cw A

M

p w pb

= D = = = =

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General Frequency Selective Filters

A general multiband impulse response can be written as

Window methods can be applied to multiband filters Example multiband frequency response Special cases of

Bandpass Highpass Bandstop

mbN

1k

k1kkmb 2/Mn

2/MnsinGGnh

Filter design tool

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References

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Discrete-Time Signal Processing, 2e by Oppenheim, Shafer

Lecture notes by Güner Arslan Dept. of Electrical and Computer Engineering, The University of Texas at Austin