EE247 - Lecture 2 Filters - University of California, Berkeleyinst.eecs.berkeley.edu/~ee247/fa09/files07/lectures/L2_2... · 2009. 9. 1. · EECS 247 Lecture 2: Filters © 2009 H.K

EECS 247 Lecture 2: Filters © 2009 H.K. Page 1

EE247 - Lecture 2Filters

• Filters: – Nomenclature– Specifications

• Quality factor• Frequency characteristics• Group delay

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

– Group delay comparison example– Biquads


NomenclatureFilter Types wrt Frequency Range Selectivity

( )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:– 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)

• Area/pole & Power/pole


0

x 10Frequency (Hz)

Filter Frequency CharacteristicsExample: Lowpass

( )ωjH

( )ωjH

( )0H

Passband Ripple (Rpass)

Transition Band

cfPassband

Passband Gain

stopf Stopband Frequency

Stopband Rejection

f

( )H j [ dB]ω3dBf−

dB3


Filters


• Frequency characteristics• Quality factor• Group delay




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 →

= +

=


Component Quality Factor (Q) Inductor & Capacitor Quality Factor

• Inductor Q :

• Capacitor Q :

RsLs s

L L1 LY QR j L Rω

ω= =+

Rp

CpC C

p

1Z Q CR1 jR Cω

ω= =

+


Pole Quality Factor

xσ

xω

ωj

σ

Pω

pP o l e

xQ

2ωσ

=

s-Plane• Typically filter singularities include pairs of complex conjugate poles.

• Quality factor of complex conjugate poles are defined as:


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


Filters






• 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

• τPD ≡ -θ(ω)/ω is called the “phase delay” and has units of time

• If the second delay 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

• 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”


• 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


Filters






Filter Types wrt Frequency ResponseLowpass 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 Ph

ase

(deg

rees

)

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

• Number of poles equal to filter order

• Poles located on the unit circle with equal angles

s-plane

jω

σ

Example: 5th Order Butterworth Filter

pole


Filter Types Chebyshev I Lowpass Filter

• Chebyshev I filter– Ripple in the passband– Sharper transition band

compared to Butterworth (for the same number of poles)

– Poorer group delay compared to Butterworth

– More ripple in passband poorer phase response

1 2

-40

-20

0

Normalized Frequency

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

• Poles & finite zeros– No. of poles n

(n −> filter order)– No. of finite zeros: n-1

• Poles located both inside & outside of the unit circle

• Complex conjugate zeros located on jω axis

• Zeros create nulls in stopband pole

zero


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

• Poles & finite 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 filter types Pole

Zero


Filter TypesBessel Lowpass Filter

s-planejω

σ

• Bessel

–All poles

–Poles outside unit circle

–Relatively low Q poles

–Maximally flat group delay

–Poor out-of-band attenuation

Example: 5th Order Bessel filter

Pole


Magnitude Response Behavioras a Function of Filter Order

Example: Bessel Filter


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

n Filter order


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


Filters






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


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

4th 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 for Chebyshev 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

– Filter types 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 & thus reducing the required filter order


Filters






RLC Filters

• Bandpass filter (2nd order):

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 FiltersExample

• Design a bandpass filter with:

Center frequency of 1kHzFilter quality 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 filter

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 Spiral Inductors

Top View


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

Typically, on-chip inductors built as spiral structures out of metal/s layers

QL= (ω L/R)

QL measured at frequencies of operation ( >1GHz)


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 (Learn more in EE242 & RF circuit courses)

• 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 <350MHz

• 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 finite 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 frequenciesPZ

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

=

=


Higher-Order Filters in the Integrated Form

• One way of building higher-order filters (n>2) is via cascade of 2nd

order biquads & 1st order , e.g. Sallen-Key,or Tow-Thomas

2nd orderFilter ……

Nx 2nd order sections Filter order: n=2N

1 2 Ν

Cascade of 1st and 2nd order filters:☺ Easy to implement

Highly sensitive to component mismatch -good for low Q filters only

For high Q applications good alternative: Integrator-based ladder filters

2nd orderFilter

1st or 2nd orderFilter


Integrator Based Filters• Main building block for this category of filters

Integrator• By using signal flowgraph techniques

Conventional RLC filter topologies can be converted to integrator based type filters

• Next lecture:– Introduction to signal flowgraph techniques– 1st order integrator based filter– 2nd order integrator based filter– High order and high Q filters

Documents

EE247 - Lecture 2 Filters - University of California, Berkeleyinst.eecs.berkeley.edu/~ee247/fa09/files07/lectures/L2_2... · 2009. 9. 1. · EECS 247 Lecture 2: Filters © 2009 H.K