EE247 - Lecture 2 FiltersEECS 247 Lecture 2: Filters © 2007 H.K. Page 9 Bandpass Filter Quality Factor (Q) 0.1 1 10f 1 fcenter f2 0-3dB Δf = f2 -f1 Hjf() Frequency Magnitude [dB]

EECS 247 Lecture 2: Filters © 2007 H.K. Page 1

EE247 - Lecture 2Filters

• From last lecture:– Dynamic range of analog circuits

• Filters: – Nomenclature– Specifications

• Quality factor• Frequency characteristics• Group delay

– Filter types• Butterworth• Chebyshev I & II• Elliptic• Bessel

– Group delay comparison example– Biquads


NomenclatureFilter Types

( )ωjH( )ωjH

Lowpass Highpass Bandpass Band-reject(Notch)

ω ω ω

Provide frequency selectivity

( )ωjH( )ωjH

ω ω

All-pass

( )ωjH

Phase shaping or equalization


Filter Specifications• Frequency characteristics (lowpass filter):

– Passband ripple (Rpass)– Cutoff frequency or -3dB frequency – Stopband rejection– Passband gain

• Phase characteristics:– Group delay

• SNR (Dynamic range)• SNDR (Signal to Noise+Distortion ratio)• Linearity measures: IM3 (intermodulation distortion), HD3

(harmonic distortion), IIP3 or OIP3 (Input-referred or output-referred third order intercept point)

• Power/pole & Area/pole


0

x 10Frequency (Hz)

Lowpass Filter Frequency Characteristics

( )ωjH

( )ωjH

( )0H

Passband Ripple (Rpass)

Transition Band

cfPassband

Passband Gain

stopf Stopband Frequency

Stopband Rejection

f

( )H j [ dB]ω3dBf−

dB3


Quality Factor (Q)

• The term quality factor (Q) has different definitions in different contexts:– Component quality factor (inductor &

capacitor Q)– Pole quality factor– Bandpass filter quality factor

• Next 3 slides clarifies each


Component Quality Factor (Q)

• For any component with a transfer function:

• Quality factor is defined as:

( ) ( ) ( )

( )( )

Energy Stored per uni t t imeAverage Power Dissipat ion

1H j R jX

XQ R

ω ω ω

ωω →

= +

=


Inductor & Capacitor Quality Factor

• Inductor Q :

• Capacitor Q :

Rs LL L1 LY QRs j L Rsω

ω= =+

Rp

CC C1Z Q CRp1 jRp C

ωω

= =+


Pole Quality Factor

xσ

xω

ωj

σ

PωpPole

xQ

2ωσ=

s-Plane


Bandpass Filter Quality Factor (Q)

0.1 1 10f1 fcenter f2

0

-3dB

Δf = f2 - f1

( )H jf

Frequency

Mag

nitu

de [d

B]

Q= fcenter /Δf


• 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 twosinewaves at slightly different frequencies (Δω<<ω):

• The filter output is:

What is Group Delay?

vIN(t) = A1sin(ωt) + A2sin[(ω+Δω) t]

G(jω) ≡ ⏐G(jω)⏐ejθ(ω)

vOUT(t) = A1 ⏐G(jω)⏐ sin[ωt+θ(ω)] +

A2 ⏐G[ j(ω+Δω)]⏐ sin[(ω+Δω)t+ θ(ω+Δω)]


What is Group Delay?

{ ]}[vOUT(t) = A1 ⏐G(jω)⏐ sin ω t + θ(ω)

ω +

{ ]}[+ A2 ⏐G[ j(ω+Δω)]⏐ sin (ω+Δω) t + θ(ω+Δω)ω+Δω

θ(ω+Δω)ω+Δω ≅ θ(ω)+ dθ(ω)

dω Δω[ ][ 1ω )( ]1 - Δω

ω

dθ(ω)dω

θ(ω)ω +

θ(ω)ω-( ) Δω

ω≅

Δωω <<1Since then Δω

ω 0[ ]2


What is Group Delay?Signal Magnitude and Phase Impairment

{ ]}[vOUT(t) = A1 ⏐G(jω)⏐ sin ω t + θ(ω)

ω +

{ ]}[+ A2 ⏐G[ j(ω+Δω)]⏐sin (ω+Δω) t + dθ(ω)dω

θ(ω)ω +

θ(ω)ω-( )Δω

ω

• If the second term in the phase of the 2nd sin wave is non-zero, then the filter’s output at frequency ω+Δω is time-shifted differently than the filter’s output at frequency ω

“Phase distortion”• If the second term is zero, then the filter’s output at frequency ω+Δω

and the output at frequency ω are each delayed in time by -θ(ω)/ω• τPD ≡ -θ(ω)/ω is called the “phase delay” and has units of time


• Phase distortion is avoided only if:

• Clearly, if θ(ω)=kω, k a constant, no phase distortion• This type of filter phase response is called “linear phase”

Phase shift varies linearly with frequency• τGR ≡ -dθ(ω)/dω is called the “group delay” and also has units

of time. For a linear phase filter τGR ≡ τPD =k τGR= τPD implies linear phase

• Note: Filters with θ(ω)=kω+c are also called linear phase filters, but they’re not free of phase distortion


dθ(ω)dω

θ(ω)ω- = 0



• If τGR= τPD No phase distortion

[ )](vOUT(t) = A1 ⏐G(jω)⏐ sin ω t - τGR +

[+ A2 ⏐G[ j(ω+Δω)]⏐ sin (ω+Δω) )]( t - τGR

• If also⏐G( jω)⏐=⏐G[ j(ω+Δω)]⏐ for all input frequencies within the 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 exactly realizable


• Phase delay is defined as:τPD ≡ -θ(ω)/ω [ time]

• Group delay is defined as :τGR ≡ -dθ(ω)/dω [time]

• If θ(ω)=kω, k a constant, no phase distortion

• For a linear phase filter τGR ≡ τPD =k

SummaryGroup Delay


Filter Types Lowpass Butterworth Filter

• Maximally flat amplitude within the filter passband

• Moderate phase distortion

-60

-40

-20

0

Mag

nitu

de (d

B)

1 2

-400

-200

Normalized Frequency

Phas

e (d

egre

es)

5

3

1

Nor

mal

ized

Gro

up D

elay0

0

Example: 5th Order Butterworth filter

N

0

d H( j )0

dω

ωω

=

=


Lowpass Butterworth Filter

• All poles• Poles located on the unit

circle with equal angless-plane

jω

σ

Example: 5th Order Butterworth Filter


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:• Sharper transition band• Poorer phase response

1 2

-40

-20

0


Mag

nitu

de (d

B)

-400

-200

0

Phas

e (d

egre

es)

0

Example: 5th Order Chebyshev filter

35

0 Nor

mal

ized

Gro

up D

elay


Chebyshev I Lowpass Filter Characteristics

• All poles• Poles located on an ellipse

inside the unit circle• Allowing more ripple in the

passband:Narrower transition bandSharper cut-offHigher pole QPoorer phase response

Example: 5th Order Chebyshev I Filter

s-planejω

σ

Chebyshev I LPF 3dB passband rippleChebyshev I LPF 0.1dB passband ripple



Phas

e (d

eg)

Mag

nitu

de (d

B)

0 0.5 1 1.5 2-360

-270

-180

-90

0

-60

-40

-20

0

Filter Types Chebyshev II Lowpass

• Chebyshev II filter– No ripple in passband

– Nulls or notches 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

jω

σ

• Both poles & zeros– No. of poles n– No. of finite zeros n-1

• Poles located both inside & outside of the unit circle

• Complex conjugate zeros located on jω axis

• Nulls in stopband

poleszeros


Filter Types Elliptic Lowpass Filter

• Elliptic filter

– Ripple in passband

– Nulls in the stopband

– Sharper transition band compared to Butterworth & both Chebyshevs

– Poorest phase response

Mag

nitu

de (d

B)

Example: 5th Order Elliptic filter

-60

1 2Normalized Frequency

0-400

-200

0

Phas

e (d

egre

es)

-40

-20

0


Filter Types Elliptic Lowpass Filter

Example: 5th Order Elliptic Filter

s-plane

jω

σ

PoleZero

• Both poles & zeros– No. of poles: n– No. of finite zeros: n-1

• Zeros located on jω axis

• Sharp cut-offNarrower transition

bandPole Q higher

compared to the previous filters


Filter TypesBessel Lowpass Filter

s-planejω

σ

• Bessel–All poles

–Maximally flat group delay

–Poor out-of-band attenuation

–Poles outside unit circle

–Relatively low Q poles

Example: 5th Order Bessel filter

Pole


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


Mag

nitu

de [d

B]

0.1 100-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

n=1

2

34

5

76

Filter O

rder In

crease

d

1 10


Filter Types Comparison of Various Type LPF Magnitude Response

-60

-40

-20

0


Mag

nitu

de (d

B)

1 20

Bessel ButterworthChebyshev IChebyshev IIElliptic

All 5th order filters with same corner freq.

Mag

nitu

de (d

B)


Filter Types Comparison of Various LPF Singularities

s-plane

jω

σ

Poles BesselPoles ButterworthPoles EllipticZeros EllipticPoles Chebyshev I 0.1dB


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 -3dB point

– Passband ripple of 1dB allowed for Chebyshev I


Magnitude Response4th Order Chebyshev I versus Bessel

Frequency [Hz]

Mag

nitu

de (d

B)

104 105 106

-60

-40

-20

0

4th Order Chebychev 14th Order Bessel


Phase Response4th Order Chebyshev I versus Bessel

0 50 100 150 200-350

-300

-250

-200

-150

-100

-50

0

Frequency [kHz]

Phas

e [d

egre

es]

4th order Chebyshev 1

4th order Bessel


Group Delay4th Order Chebyshev I versus Bessel

10 100 10000

2

4

6

8

10

12

14

Frequency [kHz]

Gro

up D

elay

[use

c]



4th order Bessel


Normalized Group Delay4th Order Chebyshev I versus Bessel

10 100 10000

0.5

1

1.5

2

2.5

3

Frequency [kHz]

Gro

up D

elay

[nor

mal

ized

]


4th order Bessel


Step Response4th Order Chebyshev I versus Bessel

Time (usec)

Am

plitu

de

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4


4th order Bessel


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

pulsesExample: Simple RC filter


Pulse ImpairmentBessel versus Chebyshev

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x 10- 4

-1.5

-1

-0.5

0

0.5

1

1.5

8th order Bessel 4th order Chebyshev I

Note that in the case of the Chebyshev filter not only the pulse has broadened but it also has a long tail

More ISI compared to Bessel

InputOutput

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x 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 1.4x 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 1.4x 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 1.4x 10-4

-1.5

-1

-0.5

0

0.5

1

1.5

1111011111001010000100010111101110001001

1111011111001010000100010111101110001001 1111011111001010000100010111101110001001

Response to Psuedo-Random DataChebyshev versus Bessel

4th order Bessel 4th order Chebyshev I

Input Signal: Symbol rate 1/130kHz


SummaryFilter Types

– Filters with high signal attenuation per pole poor phase response

– For a given signal attenuation, requirement of preserving constant groupdelay Higher order filter• In the case of passive filters higher component count• For integrated active filters higher chip area &

power dissipation

– In cases where filter is followed by ADC and DSP• Possible to digitally correct for phase impairments incurred

by the analog circuitry by using digital phase equalizers


RLC Filters

•Bandpass filter:

Singularities: Pair of complex conjugate poles Zeros @ f=0 & f=inf.

o

so RC

2 2in oQ

o

o o

VV s s

1 LCRQ RC L

ω ω

ωω ω

=+ +

=

= =

oVR

CLinV

jω

σ

s-Plane


RLC Filters

• Design a bandpass filter with:

Center frequency of 1kHzQuality factor of 20

• First assume the inductor is ideal• Next consider the case where the inductor has series R

resulting in a finite inductor Q of 40• What is the effect of finite inductor Q on the overall Q?

oVR

CLinV


RLC FiltersEffect of Finite Component Q

idealf i l t ind.f i l t

1 1 1Q QQ

= + Qfilt.=20 (ideal L)

Qfilt. =13.3 (QL.=40)

Need to have component Q much higher compared to desired filter Q


RLC Filters

Question:Can RLC filters be integrated on-chip?

oVR

CLinV


Monolithic InductorsFeasible Quality Factor & Value

Ref: “Radio Frequency Filters”, Lawrence Larson; Mead workshop presentation 1999

Feasible monolithic inductor in CMOS tech. <10nH with Q <7


Monolithic LC Filters• Monolithic inductor in CMOS tech.

– L<10nH with Q<7

• Max. capacitor size (based on realistic chip area)– C< 20pF

LC filters in the monolithic form feasible: - Frequency >350MHz - Only low quality factor filters

Learn more in EE242


Integrated Filters• Implementation of RLC filters in CMOS technologies requires

on-chip inductors– Integrated L<10nH with Q<10 – Combined with max. cap. 20pF

LC filters in the monolithic form feasible: freq>350MHz

• Analog/Digital interface circuitry require fully integrated filters with critical frequencies << 350MHz

• Hence:

Need to build active filters built without inductor


Filters2nd Order Transfer Functions (Biquads)

• Biquadratic (2nd order) transfer function:

PQjH

jH

jH

P=

=

=

=

∞→

=

ωω

ω

ω

ω

ω

ω

)(

0)(

1)(0

2

2P P P

2 22

2P PP

P 2P

P

1P 2

1H( s )s s

1Q

1H( j )

1Q

Biquad poles @: s 1 1 4Q2Q

Note: for Q poles are real , complex otherwise

ω ω

ωω ωω ω

ω

=

+ +

=⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

⎛ ⎞= − ± −⎜ ⎟⎝ ⎠

≤


Biquad Complex Poles

Distance from origin in s-plane:

( )2

22

2 1412

P

PP

P QQ

d

ω

ω

=

−+⎟⎟⎠

⎞⎜⎜⎝

⎛=

12

2

Complex conjugate poles:

s 1 4 12

P

PP

P

Q

j QQ

ω

> →

⎛ ⎞= − ± −⎜ ⎟⎝ ⎠

poles

ωj

σ

d

S-plane


s-Plane

poles

ωj

σ

Pω radius =

P2Q- part real Pω=

PQ21arccos

2s 1 4 12

PP

Pj Q

Qω ⎛ ⎞= − ± −⎜ ⎟

⎝ ⎠


Implementation of Biquads• Passive RC: only real poles can’t implement complex conjugate

poles

• Terminated LC– Low power, since it is passive– Only fundamental noise sources load and source resistance– As previously analyzed, not feasible in the monolithic form for

f <250MHz

• Active Biquads– Many topologies can be found in filter textbooks! – Widely used topologies:

• Single-opamp biquad: Sallen-Key• Multi-opamp biquad: Tow-Thomas• Integrator based biquads


Active Biquad Sallen-Key Low-Pass Filter

• Single gain element• Can be implemented both in discrete & monolithic form• “Parasitic sensitive”• Versions for LPF, HPF, BP, …

Advantage: Only one opamp used Disadvantage: Sensitive to parasitic – all pole no zeros

outV

C1

inV

R2R1

C2

G

2

2

1 1 2 2

1 1 2 1 2 2

( )1

1

1 1 1

P P P

P

PP

GH ss sQ

R C R C

Q GR C R C R C

ω ω

ω

ω

=+ +

=

= −+ +


Addition of Imaginary Axis Zeros

• Sharpen transition band• Can “notch out” interference• High-pass filter (HPF)• Band-reject filter

Note: Always represent transfer functions as a product of a gain term,poles, and zeros (pairs if complex). Then all coefficients have a physical meaning, and readily identifiable units.

2

Z2

P P P

2P

Z

s1

H( s ) Ks s

1Q

H( j ) Kω

ω

ω ω

ωωω→∞

⎛ ⎞+ ⎜ ⎟⎝ ⎠=

⎛ ⎞+ + ⎜ ⎟

⎝ ⎠

⎛ ⎞= ⎜ ⎟

⎝ ⎠


Imaginary Zeros• Zeros substantially sharpen transition band• At the expense of reduced stop-band attenuation at

high frequencyPZ

P

P

ffQ

kHzf

32100

===

104 105 106 107-50

-40

-30

-20

-10

0

10

Frequency [Hz]

Mag

nitu

de [

dB]

With zerosNo zeros

Real Axis

Imag

Axi

s

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2x 106

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

6

Pole-Zero Map


Moving the Zeros

PZ

P

P

ffQ

kHzf

===

2100

104 105 106 107-50

-40

-30

-20

-10

0

10

20

Frequency [Hz]

Mag

nitu

de[d

B]

Pole-Zero Map

Real AxisIm

ag A

xis

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

x105

x105


Tow-Thomas Active Biquad

Ref: P. E. Fleischer and J. Tow, “Design Formulas for biquad active filters using three operational amplifiers,” Proc. IEEE, vol. 61, pp. 662-3, May 1973.

• Parasitic insensitive• Multiple outputs


Frequency Response

( ) ( )

( ) ( )01

21001020

01

3

012

012

22

012

0021122

1

1asas

babasabbakV

Vasasbsbsb

VV

asasbabsbabk

VV

in

o

in

o

in

o

++−+−−=

++++=

++−+−−=

• Vo2 implements a general biquad section with arbitrary poles and zeros

• Vo1 and Vo3 realize the same poles but are limited to at most one finite zero


Component Values

8

72

173

2821

111

21732

80

6

82

74

81

6

8

111

21753

80

1

1

RRk

CRRCRRk

CRa

CCRRRRa

RRb

RRRR

RR

CRb

CCRRRRb

=

=

=

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

=

827

2

86

20

015

112124

10213

20

12

111

111

11

1

RkRbRR

Cbak

R

CbbakR

CakkR

CakR

CaR

=

=

=

−=

=

=

=

821 and , ,,, given RCCkba iii

thatfollowsit

11

21732

8

CRQCCRRR

R

PP

P

ω

ω

=

=

Documents

EE247 - Lecture 2 FiltersEECS 247 Lecture 2: Filters © 2007 H.K. Page 9 Bandpass Filter Quality Factor (Q) 0.1 1 10f 1 fcenter f2 0-3dB Δf = f2 -f1 Hjf() Frequency Magnitude [dB]