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EECS 247 Lecture 2: Filters 2005 H.K. Pag e 1

EE247 - Lecture 2

Filters

Material covered today:

Nomenclature

Filter specifications

Quality factor

Frequency characteristics

Group delay

Filter types Butterworth

Chebyshev I

Chebyshev II

Elliptic

Bessel

Group delay comparison example


Filters

Filters Provide frequency selectivity and/or phase shaping

nV

FilteroutV

( )jH

( )jH

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0

x 10Frequency (Hz)

Lowpass Filter Frequency Characteristics

( )jH

( )jH

( )0H

Passband Ripple (Rpass)

TransitionBand

cPassband

Passband

Gain

s topfStopband

Frequency

StopbandRejection

f

( )jH

3dBf

dB3


Quality Factor (Q)

The term Quality Factor (Q) has different definitions:

Component quality factor (inductor & capacitor

Q)

Pole quality factor

Bandpass filter quality factor

Next 3 slides clarifies each

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Component Quality Factor (Q)

For any component with a transfer function:

Quality factor is defined as:

( )( ) ( )

( )( )

Energy StoredAverage Powe r Diss ipation

1H jR jX

XQ

R

=+

=


Inductor & Capacitor Quality Factor

Inductor Q :

Capacitor Q :

Rs LL L1 LY Q

Rs j L Rs

= =+

Rp

C

C C1Z Q CRp1 jRp

C

= =+

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Pole Quality Factor

x

j

P

xPo lex

Q2

=

s-Plane

EECS 247 Lecture 2: Filters 2005 H.K. Page 10

Bandpass Filter Quality Factor (Q)

-20

-15

-10

-5

0

0.1 1 10f1 fcenter f2

0

-3dB

f = f2 - f1

( )H jf

Frequency

Magnitude(dB)

Q= fcenter/f

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Consider a continuous time filter with s-domain transfer function G(s):

Let us apply a signal to the filter input composed of sum of two

sinewaves at slightly different frequencies (

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What is Group Delay?

Signal Magnitude and Phase Impairment

{ ]}[vOUT(t) = A1 G(j) sin t +()

+

{ ]}[+ A2 G[j(+)]sin (+) t +d()

d()

+()

-( )

If the second term in the phase of the 2nd sin wave is non-zero, then the

filters output at frequency + is time-shifted differently than thefilters output at frequency Phase distortion

If the second term is zero, then the filters output at frequency +

and the output at frequency are each delayed in time by -()/ PD -()/ is called the phase delay and has units of time


Phase distortion is avoided only if:

Clearly, if ()=k, k a constant, no phase distortion

This type of filter phase response is called linear phase

Phase shift varies linearly with frequency

GR -d()/d is called the group delay and also has unitsof time. For a linear phase filter GR PD =k

GR= PD implies linear phase

Note: Filters with()=k+c are also called linear phase filters, buttheyre not free of phase distortion

What is Group Delay?Signal Magnitude and Phase Impairment

d()d

()- = 0

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What is Group Delay?

Signal Magnitude and Phase Impairment

IfGR= PD No phase distortion

[ )](vOUT(t) = A1 G(j) sin t -GR +

[+ A2 G[j(+)] sin (+) )]( t -GR

If alsoG( j)=G[ j(+)] for all input frequencies withinthe signal-band, vOUT is a scaled, time-shifted replica of the

input, with no signal magnitude distortion :

In most cases neither of these conditions are realizable exactly


Phase delay is defined as:

PD -()/ [ time]

Group delay is defined as :

GR -d()/d [time]

If ()=k, k a constant, no phase distortion

For a linear phase filter GR PD =k

SummaryGroup Delay

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Filter Types

Lowpass Butterworth Filter

Maximally flat amplitude

within the filter passband

Moderate phase distortion

-60

-40

-20

0

Magnitude(dB)

1 2

-400

-200

Normalized Frequency

Phase(degrees)

5

3

1

N

ormalizedGroupDelay

0

0

Example: 5th Order Butterworth filter

N

0

d H( j )0

d

=

=


Lowpass Butterworth Filter

All poles

Poles located on the unit

circle with equal angless-plane

j

Example: 5th Order Butterworth Filter

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Filter Types

Chebyshev I Lowpass Filter

Chebyshev I filter

Ripple in the passband

Sharper transition band

compared to Butterworth

Poorer group delay

As more ripple is allowed in

the passband:


Poorer phase response

1 2

-40

-20

0


Magnitude(dB)

-400

-200

0

Phase(degrees)

0

Example: 5th Order Chebyshev filter

35

0 No

rmalizedGroupDelay


Chebyshev I Lowpass Filter Characteristics

All poles

Poles located on an ellipse

inside the unit circle

Allowing more ripple in the

passband:

_Narrower transition band

_Sharper cut-off

_Higher pole Q

_Poorer phase response

Example: 5th Order Chebyshev I Filter

s-planej

Chebyshev I LPF 3dB passband ripple

Chebyshev I LPF 0.1dB passband ripple

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Bode Diagram

Frequency [Hz]

Phase(deg)

Magnitude(dB)

0 0.5 1 1.5 2

-360

-270

-180

-90

0

-60

-40

-20

0

Filter Types

Chebyshev II Lowpass

Chebyshev II filter

Ripple in stopband

Sharper transition

band compared to

Butterworth

Passband phase

more linear

compared to

Chebyshev I

Example: 5th Order Chebyshev II filter


Filter Types

Chebyshev II Lowpass

Example:

5th Order

Chebyshev II Filter

s-plane

j

Both poles & zeros

No. of poles n

No. of zeros n-1

Poles located both inside

& outside of the unit circle

Zeros located onj axis

Ripple in the stopband

only

poles

zeros

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Filter Types

Elliptic Lowpass Filter

Elliptic fi lter

Ripple in passband

Ripple in the stopband


compared to Butterworth &

both Chebyshevs

Poorest phase response

Magnitude(dB)

Example: 5th Order Elliptic filter

-60

1 2Normalized Frequency

0-400

-200

0

Phase(degrees)

-40

-20

0


Filter Types

Elliptic Lowpass Filter

Example: 5th Order Elliptic Filter

s-plane

j

Pole

Zero

Both poles & zeros

No. of poles n

No. of zeros n-1

Zeros located onj axis

Sharp cut -off

_Narrower transition

band

_Pole Q higher

compared to the

previous filters

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Filter Types

Bessel Lowpass Filter

s-planej

Bessel

All poles

Maximally flat group

delay

Poor amplitude

attenuation

Poles outside unit circle

(s-plane)

Relatively low Q poles

Example: 5th Order Bessel filter

Pole


Magnitude Response of a Bessel Filter as aFunction of Filter Order (n)


Magnitude[dB]

0.1 100-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

n=1

2

3

4

5

76

Filte

rOrder

Increased

1 10

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Filter Types

Comparison of Various Type LPF Magnitude Response

-60

-40

-20

0


Magnitude(dB)

1 20

BesselButterworth

Chebyshev I

Chebyshev II

Elliptic

All 5th order filters with same corner freq.

Magnitude(dB)


Filter Types

Comparison of Various LPF Singularities

s-plane

j

Poles Bessel

Poles Butterworth

Poles Elliptic

Zeros Elliptic

Poles Chebyshev I 0.1dB

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Comparison of Various LPF Groupdelay

Bessel

Butterworth

Chebyshev I

0.5dB Passband Ripple

Ref: A. Zverev,Handbook of filter synthesis, Wiley, 1967.

1

12

1

28

1

1

10

5

4


Group Delay Comparison

Example

Lowpass filter with 100kHz corner frequency

Chebyshev I versus Bessel

Both filters 4th order- same -3dBpoint

Passband ripple of 1dBallowed for Chebyshev I

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Magnitude Response

Bode Magnitude Diagram

Frequency [Hz]

Magnitude(dB)

104

105

106

-70

-60

-50

-40

-30

-20

-10

0

4th Order Chebychev 14th Order Bessel


Phase Response

0 0.5 1 1.5 2

x 105

-350

-300

-250

-200

-150

-100

-50

0

Frequency [Hz]

Phase[degrees]

4th Order Chebychev 14th Order Bessel

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Group Delay

104

105

106

0

2

4

6

8

10

12

14

Frequency [Hz]

GroupDelay[s

]

4th Ord. Chebychev 14th Ord. Bessel


Normalized Group Delay

104

105

106

0

0.5

1

1.5

2

2.5

3

Frequency [Hz]

GroupDelay[normalized]


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Step Response

Time (sec)

Amplitude

0 0.5 1 1.5 2

x 10-5

0

0.2

0.4

0.6

0.8

1

1.2

1.44th Order Chebychev 14th Order Bessel


Intersymbol Interference (ISI)ISI Broadening of pulses resulting in interference between successive transmitted

pulses

Example: Simple RC filter

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Pulse Broadening

Bessel versus Chebyshev

1.1 1.2 1. 3 1 .4 1. 5 1 .6 1.7 1. 8 1. 9 2

x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5

1.1 1.2 1. 3 1 .4 1. 5 1 .6 1. 7 1 .8 1.9 2

x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5

8th order Bessel 4th order Chebyshev I

Chebyshev has more pulse broadening compared to Bessel More ISI

InputOutput


0 0.2 0.4 0.6 0.8 1 1.2 1.4

x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5

1111011111001010000100010111101110001001

1111011111001010000100010111101110001001 1111011111001010000100010111101110001001

Response to Random Data

Chebyshev versus Bessel

4th order Bessel 4th order Chebyshev I

Input Signal:

Symbol rate 1/130kHz

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Measure of Signal Degradation

Eye Diagram

Eye diagram is a useful graphical illustration for signal degradation

Consists of many overlaid traces of a signal using an oscilloscope

where the symbol timing serves as the scope trigger

It is a visual summary of all possible intersymbol interference

waveforms The vertical opening immunity to noise

Horizontal opening timing jitter


Measure of Signal Degradation

Eye Diagram

Random data with symbol rates: 1/50kHz

1/100kHz

1/130kHz

Frequency [Hz]

Magnitude(dB)

2 4 6 8 10 12 14

x 104

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

BesselChebychev 1

2 4 6 8 10 12 14

x 104

0

0.5

1

1.5

2

2.5

3

Frequency [Hz]

GroupDelay[normalized]


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1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

x 10-5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time

4th Order Chebychev

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

x 10-5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time

4th Order Bessel

0.6 0.8 1 1.2 1.4 1.6

x 10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Input Signal

Eye Diagram

Chebyshev versus Bessel

4th order Chebyshev I4th order Bessel

Input Signal

Random data

Symbol rate:1/ 130kHz


Eye Diagrams

2.5 3 3.5 4 4.5 5

x 10

-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

4th Order Chebychev

2.5 3 3.5 4 4.5 5

x 10

-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

4th Order Bessel

100%

Eye openin g

70%

Eye openin g

Random data maximum power symbol rate 1/50kHz

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Eye Diagrams

Filter with constant group delay More open eye Lower BER (bit-error-rate)

Random data maximum symbol rate 1/100kHz

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

x 10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

4th Order Bessel

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

x 10-5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

4th Order Chebychev

80%

Eye opening

40%

Eye openin g


Summary

Filter Types

Filters with high signal attenuation per pole _poor phase response

For a given signal attenuation requirement ofpreserving constant groupdelay Higher orderfilter In the case of passive filters _ higher component count

Case of integrated active filters _ higher chip area &power dissipation

In cases where filter is followed by ADC and DSP Possible to digitally correct for phase non-linearities

incurred by the analog circuitry by using phase equalizers

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RLC Filters

Bandpass filter:

Singularities: Complex conjugate poles + zeros and zero & infinity

o

so RC

2 2in oQ

o

oo

VV s s

1 LC

RQ RCL

=+ +

=

= =

oVR

CLnV


RLC Filters

Design a bandpass filter with:

Center frequency of 1kHz

Q of 20

Assume that the inductor has series R resulting in an

inductor Q of 40

What is the effect of finite inductor Q on the overall Q?

oVR

CLnV

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RLC Filters

Effect of Finite Component Q

ideal f i l t ind.f i l t

1 1 1Q QQ

= +

Q=20 (ideal L)

Q=13.3 (Qind.=40)

eComponent Q must be much higher compared

to desired filter Q


RLC Filters

Question:

Can RLC filters be integrated on-chip?

oVR

CLnV

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Monolithic Inductors

Feasible Quality Factor & Value

vRef: Radio Frequency Filters, Lawrence Larson; Mead workshop presentation 1999

c Feasible monolithic inductor in CMOS tech.

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Monolithic Filters

Desirable to integrate filters with critical frequencies

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