EE 508 Lecture 16

Filter TransformationsLowpass

to Bandpass

Lowpass

to HighpassLowpass

to Band-reject

Standard LP to BP TransformationFrequency and s-domain Mappings -

Denormalized

(subscript variable in LP approximation for notational convenience)

X

2 2M

s

s +ωs•BW

↓

2 2M

X

s +ωss•BW

→2 2

MX

ω - ωωω•BW

→

( )2 2X X Ms •BW ± BW s - 4ω

s2•

←

( )2 2X X Mω •BW ± BW ω 4ω

ω2• +

←

Exercise: Resolve the dimensional consistency in the last equation

Review from Last Time

Standard LP to BP TransformationFrequency and s-domain Mappings -

Denormalized

(subscript variable in LP approximation for notational convenience)

All three approaches give same approximation

X sNs

N

sW

↓ ↓2N

N N

s +1 s •BW

TLPN(sx)

TBP(s)

X

2 2M

s

s +ωs•BW

↓

Ns

N

sW

↓

s↓2

N

s +1 s•BW

Which is most practical to use? Often none of them !

Review from Last Time

Standard LP to BP TransformationPole Mappings

Can show that the upper hp pole maps to one upper hp pole and one lower hp pole as shown. Corresponding mapping of the lower hp pole is also shown

Re

Im

Re

Im

{ }0BPH LBPHω ,Q

{ }0LP LPω ,Q

{ }0BPL LBPLω ,Q

Review from Last Time

Standard LP to BP TransformationPole Mappings

( )2

X N N Xp •BW ± BW p - 4p

2•

←

Re

Im

Re

Im

multipliity 6

Note doubling of poles, addition of zeros, and likely Q enhancement

Review from Last Time

LP to BP TransformationPole Q of BP Approximations

Define: 0LPM

BW ωω

δ⎛ ⎞

= ⎜ ⎟⎝ ⎠

ω

( )LPN

T jω

ω

( )BP

T jω

1

1

ωM

BW

ωL ωH

H LBW = ω - ω

M H Lω ω ω=

{ }0BPH BPHω ,Q

{ }0LP LPω ,Q

{ }0BPL BPLω ,Q

It can be shown that

2

2 2 2

4 4 41 12LP

BPL BPH 22LP

QQ QQδ δ δ

⎛ ⎞= = + + + −⎜ ⎟⎝ ⎠

For δ

small, LPBP

2QQδ

It can be shown that 2

42

M BP BP0BP

LP LP

ω Q QωQ Q

δ δ⎡ ⎤⎛ ⎞⎢ ⎥= ± −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Note for

δ

small, QBP

can get very large

Review from Last Time

Standard LP to BP Transformation

–

Standard LP to BP transform is a variable mapping transform–

Maps jω

axis to jω

axis–

Maps LP poles to BP poles–

Preserves basic shape but warps frequency axis–

Doubles order–

Pole Q of resultant band-pass functions can be very large for narrow pass-band

–

Sequencing of frequency scaling and transformation does not affect final function

2

N

s +1ss•BW

→

Review from Last Time

Filter Transformations

Lowpass

to Bandpass

(LP to BP)

Lowpass

to Highpass

(LP to HP)

Lowpass

to Band-reject (LP to BR)

•

Approach will be to take advantage of the results obtained for the standard LP approximations

•

Will focus on flat passband

and zero-gain stop-band transformations

LP to BS Transformation

Strategy: As was done for the LP to BP approximations, will use

a variable mapping strategy that maps the imaginary axis in the s-plane to the imaginary axis in the s-plane so the basic shape is preserved.

( )LPNT s ( )BST s( )s f s→

( ) ( )( )BS LPNT s T f s=

( )

T

T

m

Ti=0n

Ti=0

a sf s =

b s

i

i

i

i

∑

∑

LP to BS Transformation

( )T jω ( )BS

T jω

( )T jω ( )BS

T jω

N BN ANBW =ω -ω

1AN BNω ω =

Standard LP to BS TransformationMapping Strategy:

map s=0 to s=±

j∞map s=0 to s= j0map s=j1 to s=jωAmap s=j1 to s=-jωBmap s=-j1 to s=jωBmap s=-j1 to s=-jωA

FN

(s) should

map ω=0 to ω

= ±∞map ω=0 to ω

= 0map ω=1 to ω

= ωAmap ω=1 to ω= -ωBmap ω= –1 to ω= ωBmap ω= –1 to ω= -ωA

Variable Mapping Strategy to Preserve Shape of LP function:

( )T jω ( )BS

T jω

ω

1

1

ω

1

( )T jω

( )BS

T jω

-1

ω0-ω0 1-1

Standard LP to BS Transformation

map ω=0 to ω

= ±∞map ω=0 to ω

= 0map ω=1 to ω

= ωAmap ω=1 to ω= -ωBmap ω= –1 to ω= ωBmap ω= –1 to ω= -ωA

Standard LP to BS Transformation

Mapping Strategy: consider variable mapping transform

FN

(s) should

Consider variable mapping

( ) ( )1N

2s BWLPN BSN

s

T ( ) =TN sF s s •

=+

1N

2

s BWss•

→+

map s=0 to s=±

j∞map s=0 to s= j0map s=j1 to s=jωAmap s=j1 to s=-jωBmap s=-j1 to s=jωBmap s=-j1 to s=-jωA

map ω=0 to ω

= ±∞map ω=0 to ω

= 0map ω=1 to ω

= ωAmap ω=1 to ω= -ωBmap ω= –1 to ω= ωBmap ω= –1 to ω= -ωA

Comparison of Transforms

1N

2

s BWss•

→+

2

N

s +1ss•BW

→

LP to BP

LP to BS( )T jω ( )

BST jω

( )LPN

T jω ( )BP

T jω

Standard LP to BS TransformationFrequency and s-domain Mappings

(subscript variable in LP approximation for notational convenience)

1

X

N2

s

s BWs

↓•+

1N

X 2

s BWss•

→+

NX 2

ω BWω1-ω•

→

12

2

N N

X X

BW BW1s ± - 4s 2 s

⎛ ⎞← ⎜ ⎟

⎝ ⎠

1 1 42 2

N N

X X

BW BWωω ω

⎛ ⎞−← ± +⎜ ⎟

⎝ ⎠

Standard LP to BS TransformationUn-normalized Frequency and s-domain Mappings

(subscript variable in LP approximation for notational convenience)

X

2 2M

s

s BWs ω

↓•+

X 2 2M

s BWss ω•

→+

X 2 2M

ω BWωω -ω•

→

12

2

2M

X X

BWBW 1s ± - 4ωs 2 s

⎛ ⎞← ⎜ ⎟

⎝ ⎠

1 1 42 2

2M

X X

BW BWω ωω ω

⎛ ⎞−← ± +⎜ ⎟

⎝ ⎠

Standard LP to BS TransformationPole Mappings

Can show that the upper hp pole maps to one upper hp pole and one lower hp pole as shown. Corresponding mapping of the lower hp pole is also shown

{ }0BPH LBPHω ,Q

{ }0LP LPω ,Q

{ }0BPL LBPLω ,Q

LP to BS TransformationPole Q of BS Approximations

Define:M 0LP

BWω ω

γ⎛ ⎞

= ⎜ ⎟⎝ ⎠

BN ANBW = ω - ω

M AN BNω ω ω=

It can be shown that

2

2 2 2

4 4 41 12LP

BSL BSH 2LP

QQ QQγ γ γ

⎛ ⎞= = + + + −⎜ ⎟

⎝ ⎠

For γ

small, LPBS

2QQγ

It can be shown that 2

42

BS BSM0BS

LP LP

Q QωωQ Q

γ γ⎡ ⎤⎛ ⎞⎢ ⎥= ± −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Note for

γ

small, QBS

can get very large

{ }0BPH LBPHω ,Q

{ }0LP LPω ,Q

{ }0BPL LBPLω ,Q

( )T jω ( )BP

T jω

Standard LP to BS TransformationPole Mappings

( )2

N N

X X

BW BW± - 4p pp

2←

Note doubling of poles, addition of zeros, and likely Q enhancement

Re

Im

Re

Im

multipliity 6

multipliity 6

Standard LP to BS Transformation

• Standard LP to BS transformation is a variable mapping transform

• Maps jω

axis to jω

axis in the s-plane

• Preserves basic shape of an approximation but warps frequency

axis

• Order of BS approximation is double that of the LP Approximation

• Pole Q and ω0 expressions are identical to those of the LP to BP transformation

• Pole Q of BS approximation can get very large for narrow BW

• Other variable transforms exist but the standard is by far the

most popular

X 2 2M

s BWss ω•

→+

Filter Transformations

Lowpass

to Bandpass

(LP to BP)

Lowpass

to Highpass

(LP to HP)

Lowpass

to Band-reject (LP to BR)

•

Approach will be to take advantage of the results obtained for the standard LP approximations

•

Will focus on flat passband

and zero-gain stop-band transformations

LP to HP Transformation

Strategy: As was done for the LP to BP approximations, will use

a variable mapping strategy that maps the imaginary axis in the s-plane to the imaginary axis in the s-plane so the basic shape is preserved.

( )LPNT s ( )HPT s( )s f s→

( ) ( )( )HP LPNT s T f s=

( )

T

T

m

Ti=0n

Ti=0

a sf s =

b s

i

i

i

i

∑

∑

( )T jω ( )HP

T jω

LP to HP Transformation

( )T jω ( )HP

T jω

Standard LP to HP TransformationMapping Strategy:

map s=0 to s=±

j∞map s=j1 to s=-j1map s= –j1 to s=j1

FN

(s) should

map ω=0 to ω=∞map ω=1 to ω=-1map ω= –1 to ω=1

Variable Mapping Strategy to Preserve Shape of LP function:

( )LP

T jω ( )HP

T jω

Standard LP to HP Transformation

Mapping Strategy: consider variable mapping transform

FN

(s) should

Consider variable mapping

( ) ( ) 1LPN LPNs

T ( ) =Ts

F s s=

1ss

→

map s=0 to s=±

j∞map s=j1 to s=-j1map s= –j1 to s=j1

map ω=0 to ω=∞map ω=1 to ω=-1map ω= –1 to ω=1

Comparison of Transforms

1N

2

s BWss•

→+

2

N

s +1ss•BW

→

LP to BP

LP to BS( )T jω ( )

BST jω

( )LPN

T jω ( )BP

T jω

LP to HP

1ss

→

( )LP

T jω ( )HP

T jω

LP to HP Transformation

( )T jω

( )HP

T jω

(Normalized Transform)

Standard LP to HP TransformationFrequency and s-domain Mappings

(subscript variable in LP approximation for notational convenience)

Xs

1s

↓

X

1ss

→

X

-1ωω

→

X

1ss

←

X

-1ωω

←

Standard LP to HP TransformationPole Mappings

Xs

1s

↓

X

1pp

→

X

1pp

←

Claim: With a variable mapping transform, the variable mapping naturallydefines the mapping of the poles of the transformed function

Standard LP to HP TransformationPole Mappings

Xs

1s

↓

X

1pp

←

If pX

=α+jβ

2 2

1 α-jβp= =α+jβ α +β

and pX

=α-jβ

2 2

1 α+jβp= =α+jβ α +β

Standard LP to HP TransformationPole Mappings

Xs

1s

↓

X

1pp

←

If pX

=α+jβ

2 2

1 α-jβp= =α+jβ α +β

and pX

=α-jβ

2 2

1 α+jβp= =α+jβ α +β

Highpass

poles are scaled in magnitudebut make identical angles with imaginary axis

HP pole Q is same as LP pole Q

Order is preserved

Standard LP to HP Transformation

0ωss

→

(Un-normalized variable mapping transform)

( )T jω

( )HP

T jω

Filter Design/Synthesis Considerations

There are many different filter architectures that can realize a given transfer function

Considerable effort has been focused over the years on “inventing”

these architectures and on determining which is

best suited for a given application

Filter Design/Synthesis ConsiderationsMost designs today use one of the following three basic architectures

( ) 1 2 mT s T T T= ii

I1(s)

Integrator

I2(s)

Integrator

I3(s)

Integrator

I4(s)

Integrator

Ik(s)

IntegratorVIN

VOUTIk-1(s)

Integrator

a2a1

+

Cascaded Biquads

Leapfrog

Multiple-loop Feedback (less popular)

Filter Design/Synthesis Considerations

( )2BPT jω

H2

H

Review: Second-order bandpass

transfer function

( ) H

0

2BP2 20

0

ωsQT sωs + s + ωQ

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

0B A

ωBW = ω -ω = Q

PEAK 0ω = ω

( )T s

Filter Design/Synthesis Considerations

There are many different filter architectures that can realize a

given transfer function

Will first consider second-order Bandpass

filter structures

( )T s

( ) H

0

2 200

ωsQT sωs + s + ωQ

⎛ ⎞⎜ ⎟⎝ ⎠=⎛ ⎞⎜ ⎟⎝ ⎠

0B A

ωBW = ω -ω = Q

PEAK 0ω = ω