introduction to filters - university of california,...

EECS 247 Lecture 2: Introduction to Filters © 2002 B. Boser 1A/DDSP

Introduction to Filters

• Filtering = frequency-selective signal processing– It’s the most common type of signal processing– Examples:

• Extract desired signal from many (radio)• Separating signal and noise• Amplifier bandwidth limitations

• Where to start– Perfectionist: ideal (low-pass) filter– Engineer: continuous time, first-order low-pass filter


First-Order RC Filter (LPF1)

Steady-state frequency response:

kHzRC

ssVsV

sH

o

o

in

out

10021

with

1

1)()(

)(

×==

+==

πω

ω


Poles and Zeros

s-plane (pzmap):

∞→−=

+=

zp

ssH

o

o

:Zero :Pole

1

1)(

ω

ω

jω

σp=-ωo


Magnitude Response

-5

0

5

x 105

-5

0

5

x 105

0

0.5

1

1.5

2

2.5

3

Frequency [Hz]

Magnitude Response (s-plane)

Sigma [Hz]

Mag

nitu

de [l

inea

r]

L02_bode3_lpf1.m


Frequency ResponseAsymptotes:- 20 dB/dec rolloff- 90 degrees phase shift per 2 decades

Matlab code (L02_bode_lpf1.m):wo = 2*pi*100e3;s = tf('s');h = 1 / (1+s/wo);bodehz(h, logspace(1, 10, 100));

Note: bodehz is same as bode, but frequency axis is in Hz, rather than rad/s.

-120

-100

-80

-60

-40

-20

0Bode Diagram

Frequency [Hz]P

ha

se (

de

g)

Ma

gn

itu

de

(d

B)

101

102

103

104

105

106

107

108

109

1010

-90

-60

-30

0

0)(

1)(0

==

==

∞→

=

ω

ω

ω

ω

jsH

jsH


Parasitics

Can we really get 100dB attenuation at 10GHz?– Probably not– Parasitics limit the performance of analog

components– E.g.

• Shunt capacitance• Feed-through capacitance• Finite inductor, capacitor Q


LPF2

( )P

P

CCsRsRC

sH++

+=

11

)( ( )

P

P

RCz

RCCCRp

1 :Zero

11 :Pole

−=

−≈+

−=


Frequency Response

dB

CC

CCC

jH

jH

P

P

P

6010

)(

1)(

3

0

−==

≈

+=

=

−

∞→

=

ω

ω

ω

ω

LPF2

Frequency [Hz]

Pha

se (

deg)

Mag

nitu

de (

dB)

-80

-70

-60

-50

-40

-30

-20

-10

0

102

103

104

105

106

107

108

109

1010

-90

-45

0

• Why not just make C larger?• Beware of other parasitics not included in this model …


Continuous Time Analog

• Analog passive components aren’t ideal– Extra real poles/zeroes result from parasitics– Parasitic effects begin to appear “50dB beyond” desired

component characteristics– Common sense helps you anticipate them

• Digital filters do not suffer from these effects


Second-Order LPF• Improved attenuation (compared to 1st order)• Complex poles (rather than multiple real ones)

– Why?– Visualize 3D s-plane plot!

• Biquadratic (2nd order) transfer function:

2

2

1

1)(

PPP

sQs

sH

ωω++

=

PQjH

jH

jH

P=

=

=

=

∞→

=

ωω

ω

ω

ω

ω

ω

)(

0)(

1)(0


Biquad Poles

( )

otherwisecomplex real, are poles for

4112

s at poles has

1

1)(

21

2

2

2

≤

−±−=

++=

P

PP

P

PPP

Q

QQ

sQs

sH

ω

ωω


Complex Poles

( )1412

s 2

21

−±−=

>

PP

P

P

QjQ

Q

ω

Distance from origin in s-plane:

( )2

2

2

2 1412

P

PP

P QQ

d

ω

ω

=

−+

=


s-Plane

poles

ωj

σ

Pω radius =

P2Q- part real Pω

=


LPF3

101002

=×=

P

P

QkHzπω

-2

-1

0

1

2

x 105

-2

-1

0

1

2

x 105

0

0.5

1

1.5

2

2.5

3

Frequency [Hz]

Magnitude Response (s-plane)

Sigma [Hz]

Mag

nitu

de [l

inea

r]


Bode Diagram

Frequency (rad/sec)

Ph

ase

(d

eg

)M

ag

nit

ud

e (

dB

)

-200

-150

-100

-50

0

50

102

103

104

105

106

107

108

109

1010

-180

-135

-90

-45

0

Frequency Response

-40 dB/dec

-180o


Varying Q … Magnitude

Gain at ωp:

20 log Q [dB]

104

105

106

-50

-40

-30

-20

-10

0

10

20

30

40LPF3 Magnitude Response

Frequency [Hz]

Mag

nitu

de

[dB

]

Q = 0.5Q = 10.0Q = 100.0


Phase

Slope at ωp :

-45 Q deg/decade

104

105

106

-180

-160

-140

-120

-100

-80

-60

-40

-20

0LPF3 Phase Response

Frequency [Hz]

Pha

se [

degr

ees]

Q = 0.5Q = 10.0Q = 100.0


Implementation of Biquads

• Passive RC: only real poles

• Terminated LC– “lowest power” (well … it’s passive!)– No noise (except load and source)

• Active Biquad– Filter texts give you dozens of topologies.

Who needs or wants that many choices?– Single-opamp biquad: Sallen-Key– Two-opamp biquad: Tow-Thomas


Sallen-Key LPF

• Single gain element• “Parasitic sensitive”• Versions for LPF, HPF, BP, …

Ref: K. L. Su, “Analog Filters,” Chapman & Hall, 1996, pp. 215.

221211

2211

2

2

111

1

1)(

CRG

CRCR

Q

CRCR

sQsG

sH

PP

P

PPP

−++

=

=

++=

ω

ω

ωω


Component Sizing Choice 1

4 unknowns: R1, R2, C1, C2

2 knowns: ωP, QP

à problem is underdetermined

Choice 1: minimum component spread

9.21

3

6.11

1

121

21

=−=

Ω===

==

P

P

QG

kC

RR

nFCC

ω


SK Magnitude Response 1

10% increase of R1more than doubles QP!

à The circuit is very sensitiveto component variations.

104

105

106

-50

-40

-30

-20

-10

0

10

20

30

40Sallen-Key Choice 1 Magnitude Response

Frequency [Hz]

Mag

nitu

de

[dB

]

nominal R1R1 10% large


Component Sizing Choice 2

Choice 2: minimum sensitivity

pFRQ

C

nFR

QC

kRRG

PP

P

P

402

1

162

21

12

11

21

==

==

Ω===

ω

ω4004 2

2

1 == PQCC

Note also:

Huge element spread àThis topology is suitable only for low-Q filter implementations.


SK Magnitude Response 2

10% increase of R1 has only small effect on response!

à The circuit is NOT very sensitive to component variations.

104

105

106

-50

-40

-30

-20

-10

0

10

20

30Sallen-Key Choice 2 Magnitude Response

Frequency [Hz]

Mag

nitu

de

[dB

]

nominal R1R1 10% large


Sensitivity

• Implementation and component sizing have huge impact on sensitivity

• High-sensitivity circuits are problems in practice

• No theory for finding a low-sensitivity architecture

• Ladder filters are usually low sensitivity

• Use proven circuits & check!0 2 Choice

%955.9

5.95.0 1 Choice

Example

with

Definition

1

1

1

1

1

1

1

=

=∆

≈∆

=−=

∆=

∆

=

∆=

∆

P

P

P

QR

P

P

PQR

QR

P

P

yx

yx

S

RR

QQ

QS

RR

SQQ

dxdy

yx

S

xx

Syy

Common sense: Sensitivity is a first order approximation, correct only for infinitesimally small errors


Summary

• Frequency Response– Poles and zeros are like tent poles and pegs– Frequency response is evaluated on jω axis– Poles and zeros close to jω axis dominate resonse

• Practical Implementation Constraints– Components are not ideal– Avoid solutions requiring large element spread– Beware of high-sensitivity architectures

introduction to filters - university of california,...

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