ee 508 lecture 2 - iowa state universityclass.ece.iastate.edu/ee508/lectures/ee 508 lect 16 fall...

EE 508

Lecture 16

Filter Transformations

Lowpass to Bandpass

Lowpass to Highpass

Lowpass to Band-reject

Flat Passband/Stopband Filters

ω

T j

ω

T j

ω

T j

ω

T j

Lowpass Bandpass

Highpass Bandreject

Review from Last Time

Standard LP to BP Transformation

Verification of mapping Strategy:2

N

s +1s

s•BW

The mapping is termed the standard LP to BP transformation

2

N

s +1s

s•BW

map s=0 to s= j1

map s=j1 to s=jωBN

map s= –j1 to s= jωAN

map ω=0 to ω=1

map ω=1 to ω=ωBN

map ω= –1 to ω=ωAN

s-domain ω-domain

TLPN(f(s))

Image of Im axis:2

N

s +1jω =

s•BW

2 2

N N N Njω•BW ± BW jω - 4 ω•BW ± BW ω + 4

s j2 2

solving for s, obtain

this has no real part so the imaginary axis maps to the imaginary axis

Can readily show this mapping maps PB to PB and SB to SB



ω

1

1

ω

1

1

ωAN ωBN

LP

T j

BP

T j

BWN

-1

-1- ωAN - ωBN

BWN

TLPN(s)

TBPN(s)

2

N

s

s +1

s•BW


Standard LP to BP TransformationPole Mappings

2

X N N Xp •BW ± BW p - 4

p2

Re

Im

Re

Im

0BPH LBPHω ,Q

0LP LPω ,Q

0BPL LBPLω ,Q

Image of the cc pole pair is the two pairs of poles



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



2

X N N Xp •BW ± BW p - 4

p2

Re

Im

Re

Im

multipliity 6

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


LP to BP Transformation

TLPN(sx)

TBPN(s)

X

2

s

f (s)

Claim: Other variable mapping transforms exist that satisfy the

imaginary axis mapping properties needed to obtain the LP to BP

transformation but are seldom, if ever, discussed. The Standard

LP to BP transform Is by far the most popular and most authors

treat it as if it is unique.


Pole Q of BP Approximations

ω

LPN

T j

ω

BP

T j

1

1

ωM

BW

ωL ωH

H LBW = ω - ω

M H Lω ω ω

Consider a pole in the LP approximation characterized by {ω0LP,QLP}

It can be shown that the corresponding BP poles have the same Q (i.e. both bp poles lie on a common radial line)

Re

Im

Re

Im

0BPH LBPHω ,Q

0LP LPω ,Q

0BPL LBPLω ,Q

LP to BP TransformationPole Q of BP Approximations

Define: 0LP

M

BWω

ω

ω

LPN

T j

ω

BP

T j

1

1

ωM

BW

ωL ωH

H LBW = ω - ω

M H Lω ω ω

Re

Im

Re

Im

0BPH BPHω ,Q

0LP LPω ,Q

0BPL BPLω ,Q

It can be shown that

2

2 2 2

4 4 41 1

2

LPBPL BPH 2

2LP

QQ Q

Q

For d small, LPBP

2QQ

It can be shown that 2

42

M BP BP0BP

LP LP

ω Q Qω

Q Q

Note for d small, QBP can get very large

LP to BP TransformationPole Q of BP Approximations

2

2 2 2

4 4 41 1

2

LPBPL BPH 2

2LP

QQ Q

Q

0LP

M

BWω

ω


Pole locations vs QLP and

0LP

M

BWω

ω


Classical BP Approximations

ω

LPN

T j

ω

BP

T j

1

1

ωM

BW

ωL ωH

Butterworth

Chebyschev

Elliptic

Bessel

Obtained by the LP to BP transformation of the corresponding LP approximations


– 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 +1s

s•BW

Example 1: Obtain an approximation that meets the following specifications

ω

AM

AR

AS

ωMωA ωB ωBHωAL

2 2 2 2

M AL BH M

AL BH

ω -ω ω - ω

ω •BW ω •BW

B ABW=ω -ω

M B Aω = ω ω

Assume that ωAL, ωBH and ωM satsify

Standard LP to BP TransformationFrequency and s-domain Mappings - Denormalized

(subscript variable in LP approximation for notational convenience)

TLPN(sx)

TBP(s)

X

2 2

M

s

s +ω

s•BW

2 2

M

X

s +ωs

s•BW

2 2

M

X

ω - ωω

ω•BW

2 2

X X Ms •BW ± BW s - 4ω

s2

2 2

X X Mω •BW ± BW ω 4ω

ω2

Exercise: Resolve the dimensional consistency in the last equation

Recall from last lecture


2 2 2 2

M AL BH M

AL BH

ω -ω ω - ω

ω •BW ω •BW

B ABW=ω -ω

M B Aω = ω ω

R

2M

A1=

A1+ε

2

M

R

A-1

A

R

RN

M

AA =

A

2 2

M AL

S

AL

ω -ω ω

ω •BW

S

SN

M

AA =

A

1 ω

ω

AM

AR

AS

ASN

ωS

ωMωA ωB ωBHωAL

1ARN

(actually –ωA and –ωAL that map to 1 and ωS respectively but show ωA and ωAL for convenience)


2 2 2 2

M AL BH M

AL BH

ω -ω ω - ω

ω •BW ω •BW

B ABW=ω -ω

M B Aω = ω ω

ωMωA ωB ωBHωAL ω

AM

AR

ASH

ASL

In this example,


B ABW=ω -ω

M B Aω = ω ω

ωMωA ωB ωBHωAL ω

ω ω

ASN

1ARN

ASN

1ARN

1 ωS1 1 ωS2

AM

AR

ASH

ASL

R

RN

M

AA =

A

R

2M

A1=

A1+ε

2

M

R

A-1

A

2 2

M AL

S1

AL

ω - ω ω

ω •BW

2 2

BH M

S2

BH

ω - ωω

ω •BW

SH SL

SN

M M

A AA =min ,

A A

SN S1 S2ω min ω ,ω


B ABW=ω -ω

M B Aω = ω ω

R

RN

M

AA =

A

R

2M

A1=

A1+ε

2

M

R

A-1

A

2 2

M AL

S1

AL

ω -ω ω

ω •BW

2 2

BH M

S2

BH

ω - ωω

ω •BW

SH SL

SN

M M

A AA =min ,

A A


ω ω

ASN

1ARN

ASN

1ARN

1 ωS1 1 ωS2

ω

ASN

1ARN

1 ωSN



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.

XIN XOUT

LPNT sXIN XOUT

BST s s f s

BS LPNT s T f s

T

T

m

Ti

=0

n

Ti

=0

a s

f s =

b s

i

i

i

i

LP to BS Transformation

1

1

ω1

1

ω

T j BS

T j

Normalized

BWN

ωBNωAN

1

1

ω1

1

ω

T j BS

T j

Normalized

-1-1

BWNBWN

ωBN-ωBN ωAN-ωAN

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= j0

map s=j1 to s=jωA

map s=j1 to s=-jωB

map s=-j1 to s=jωB

map s=-j1 to s=-jωA

FN(s) should

map ω=0 to ω = ±∞

map ω=0 to ω = 0

map ω=1 to ω = ωA

map ω=1 to ω= -ωB

map ω= –1 to ω= ωB

map ω= –1 to ω= -ωA

Variable Mapping Strategy to Preserve Shape of LP function:

1

1

ω1

1

ω

T j BS

T j

Normalized

-1-1

BWNBWN

ωBN-ωBN ωAN-ωAN

ω

1

1

ω

1

T j

BS

T j

-1

ω = ∞ω = - ∞

ω0-ω0 1-1

Standard LP to BS Transformation


map ω=0 to ω = 0






Mapping Strategy: consider variable mapping transform

TLPN(s) TBSN(s)FN(s)

FN(s) should

Consider variable mapping

1

N2

s BWLPN BSN

s

T ( ) =TN s

F s s

1N

2

s BWs

s


map s=0 to s= j0

map s=j1 to s=jωA

map s=j1 to s=-jωB

map s=-j1 to s=jωB

map s=-j1 to s=-jωA


map ω=0 to ω = 0





Comparison of Transforms

1N

2

s BWs

s

2

N

s +1s

s•BW

LP to BP

LP to BS

1

1

ω1

1

ω

T j BS

T j

BWN

ωBNωAN

ω

LPN

T j

ω

BP

T j

1

1

ωM

BW

ωL ωH

Standard LP to BS TransformationFrequency and s-domain Mappings


TLPN(sx)

TBSN(s)

1

X

N

2

s

s BW

s

1N

X 2

s BWs

s

N

X 2

ω BWω

1-ω

1

2

2

N N

X X

BW BW1s ± - 4

s 2 s

1 14

2 2N N

X X

BW BWω

ω ω

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


TLPN(sx)

TBS(s)

X

2 2

M

s

s BW

s ω

X 2 2

M

s BWs

s ω

X 2 2

M

ω BWω

ω -ω

1

2

2

2

M

X X

BWBW 1s ± - 4ω

s 2 s

1 14

2 2

2

M

X X

BW BWω ω

ω 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

Re

Im

Re

Im

0BPH LBPHω ,Q

0LPN LPω ,Q

0BPL LBPLω ,Q

LP to BS TransformationPole Q of BS Approximations

Define:M 0LPN

BW

ω ω

BN ANBW = ω - ω

M AN BNω ω ω

It can be shown that

2

2 2 2

4 4 41 1

2

LPBSL BSH 2

LP

QQ Q

Q

For γ small, LPBS

2QQ

It can be shown that 2

42

BS BSM0BS

LP LP

Q Qωω

Q Q

Note for γ small, QBS can get very large

Re

Im

Re

Im

0BSH LBSHω ,Q

0LPN LPω ,Q

0BSL LBSLω ,Q

1

1

ω1

1

ω

T j BP

T j

BWN

ωBNωAN

Standard LP to BS TransformationPole Mappings

2

N N

X X

BW BW± - 4

p 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 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 2

M

s BWs

s ω


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.

XIN XOUT

LPNT sXIN XOUT

HPT s s f s

HP LPNT s T f s

T

T

m

Ti

=0

n

Ti

=0

a s

f s =

b s

i

i

i

i

1

1

ω1

1

ω

T j HP

T j

Normalized


1

1

ω1

1

ω

T j HP

T j

Normalized

-1-1

Standard LP to HP Transformation

Mapping Strategy:


map s=j1 to s=-j1

map s= –j1 to s=j1

FN(s) should

map ω=0 to ω=∞

map ω=1 to ω=-1

map ω= –1 to ω=1

Variable Mapping Strategy to Preserve Shape of LP function:

1

1

ω1

1

ω

LP

T j HP

T j

Normalized

-1-1


Mapping Strategy: consider variable mapping transform

TLPN(s) THPN(s)FN(s)

FN(s) should

Consider variable mapping

1LPN LPNs

T ( ) =Ts

F s s

1s

s


map s=j1 to s=-j1

map s= –j1 to s=j1

map ω=0 to ω=∞

map ω=1 to ω=-1

map ω= –1 to ω=1

Comparison of Transforms

1N

2

s BWs

s

2

N

s +1s

s•BW

LP to BP

LP to BS

1

1

ω1

1

ω

T j BS

T j

BWN

ωBNωAN

ω

LPN

T j

ω

BP

T j

1

1

ωM

BW

ωL ωH

LP to HP

1s

s

1

1

ω1

1

ω

LP

T j HP

T j


ω

1

1

ω

1

1

T j

HP

T j

-1

-1

ω = ∞ω = - ∞

(Normalized Transform)

Standard LP to HP TransformationFrequency and s-domain Mappings


TLPN(sx)

THPN(s)

Xs

1

s

X

1s

s

X

-1ω

ω

X

1s

s

X

-1ω

ω

Standard LP to HP TransformationPole Mappings

TLPN(sx)

THPN(s)

Xs

1

s

X

1p

p

X

1p

p

Claim: With a variable mapping transform, the variable mapping naturally

defines the mapping of the poles of the transformed function


TLPN(sx)

THPN(s)

Xs

1

s

X

1p

p

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


TLPN(sx)

THPN(s)

Xs

1

s

X

1p

p

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 magnitude

but make identical angles with imaginary

axis

HP pole Q is same as LP pole Q

Order is preserved


0ωs

s

(Un-normalized variable mapping transform)

ω

1

1

ω

1

T j

HP

T j

-1

ω = ∞ω = - ∞

ω0-ω0

Filter Design

Process

Establish

Specifications

- possibly TD(s) or HD(z)

- magnitude and phase

characteristics or restrictions

- time domain requirements

Approximation

- obtain acceptable transfer

functions TA(s) or HA(z)

- possibly acceptable realizable

time-domain responses

Synthesis

- build circuit or implement algorithm

that has response close to TA(s) or

HA(z)

- actually realize TR(s) or HR(z)

Filter

End of Lecture 16

ee 508 lecture 2 - iowa state universityclass.ece.iastate.edu/ee508/lectures/ee 508 lect 16 fall...

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