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EE 508 Lecture 15

Filter TransformationsLowpass

to Bandpass

Lowpass

to HighpassLowpass

to Band-reject

Thompson and Bessel ApproximationsUse of Bessel Filters: XIN(s) XOUT(s)( )T s

( ) -shT s = e

( )T jω = 1 ( )T jω = -hω∠ = hGτ

A filter with a constant group delay and unity magnitude introduces a constant delay

Bessel filters are filters that are used to approximate a constant delay

Bessel filters are attractive for introducing constant delays in

digital systems

Some authors refer to Bessel filters as “Delay Filters”

An ideal delay filter would -

introduce a time-domain shift of a step input by the group delay-

introduce a time-domain shift each spectral component by the group delay-

introduce a time-domain shift of a square wave by the group delay

It is challenging to build filters with a constant delay

Review from Last Time

Filter Transformations

If the imaginary axis in the s-plane is mapped to the imaginary axis in the s-plane with a variable mapping function, the basic shape of the function T(s) will be preserved in the function F(T(s)) but the frequency axis may be warped and/or folded in the magnitude domain

Claim:

Preserving basic shape, in this context, constitutes maintaining

features in the magnitude response of F(T(s)) that are in T(s) including, but not limited to, the peak amplitude, number of ripples, peaks of ripples,

( )T s ( )MT s( )s f s→

( ) ( )( )MT s T f s=


LP to BP Filter Transformations

( )LPNT s ( )BPT s( )s f s→

( ) ( )( )BP LPNT s T f s=

ω

( )LP

T jω

ω

( )BP

T jω

Lowpass Bandpass

( )

T

T

m

Ti=0n

Ti=0

a sf s =

b s

i

i

i

i

∑

∑

Will consider rational fraction mappings


Standard LP to BP TransformationMapping Strategy:

Consider variable mapping

2

N

s +1ss•BW

→

map s=0 to s= j1map s=j1 to s=jωBNmap s= –j1 to s= jωAN

map ω=0 to ω=1map ω=1 to ω=ωBNmap ω= –1 to ω=ωAN

s-domain ω-domain

( )2

T2 T1 T0

T1 T0

a s a s+af s = b s+b+

With this mapping, there are 5 D.O.F and 3 mathematical constraints and the additional constraint that the Im

axis maps to the Im

axis

Will now show that the following mapping will meet these constraints

( )2

N

s +1f ss•BW

= or equivalently

This is the lowest-order mapping that will meet these constrains

TLPN

(f(s))


Standard LP to BP Transformation

( )LP

T jω

( )BP

T jω

2

N

s

s +1s•BW

↓


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


Denormalized


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 !


Denormalized


BPN N

X s↓2N

N N

s +1 s •BW

Often most practical to synthesize directly from the TBPN

and then do the frequency scaling of components at the circuit level rather than

at the approximation level

Standard LP to BP TransformationFrequency and s-domain Mappings


2

XN

s +f 1ss•BW

→ ( )1 2

Nf

X N Xs •BW ± BW s - 4s

2

− •←

solving for s

Poles and Zeros of the BP approximations

Since this relationship maps the complex plane to the complex plane, it also maps the poles and zeros of the LP approximation to the poles and zeros of the BP approximation

( ) 0LPN xT p =

( ) ( )( )BP LPNT s T f s=

( )( ) 0LPNT f p =

( ) ( )( ) 0BP LPNT p T f p= =

Standard LP to BP TransformationPole Mappings

X

2

N

s

s +1s•BW

↓

2

XN

p +1pp•BW

→

( )2

X N N Xp •BW ± BW p - 4p

2•

←

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

Exercise: Resolve the dimensional consistency in the last equation


( )2


2•

←

{ }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


2•

←

Re

Im

Re

Im

multipliity 6

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

LP to BP Transformation

X

2

s

f (s)↓

Claim: Other variable mapping transforms exist that satisfy theimaginary axis mapping properties needed to obtain the LP to BPtransformation 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.

LP to BP TransformationPole Q of BP Approximations ( )

LPNT jω ( )

BPT jω

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

Re

Im

Re

Im

{ }0BPH LBPHω ,Q

{ }0LP LPω ,Q

{ }0BPL LBPLω ,Q

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

LP to BP TransformationPole Q of BP Approximations

2

2 2 2

4 4 41 12LP

BPL BPH 22LP

QQ QQδ δ δ

⎛ ⎞= = + + + −⎜ ⎟⎝ ⎠

0LPM

BW ωω

δ⎛ ⎞

= ⎜ ⎟⎝ ⎠

LP to BP TransformationPole locations vs

QLP

and

0LPM

BW ωω

δ⎛ ⎞

= ⎜ ⎟⎝ ⎠

δ

LP to BP Transformation

Classical BP Approximations

ω

( )LPN

T jω

ω

( )BP

T jω

1

1

ωM

BW

ωL ωH

ButterworthChebyschevEllipticBessel

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

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

→

Example 1: Obtain an approximation that meets the following specifications

M

R

S

MA B BHAL

2 2 2 2M AL BH M

AL BH

ω -ω ω - ωω •BW ω •BW

=

B ABW=ω -ω

M B Aω = ω ω•

Assume that ωAL

, ωBH

and ωM

satsify


2 2 2 2M AL BH M

AL BH


=B ABW=ω -ω

M B Aω = ω ω•

R2

M

A1 =A1+ε

2

M

R

A -1A

ε⎛ ⎞

= ⎜ ⎟⎝ ⎠

RRN

M

AA =A

2 2M AL

SAL

ω -ω ωω •BW

=

SSN

M

AA =A

(actually is is

–ωA

and –ωAL

that map to 1 and ωS

respectively but show ωA

and ωAL

for convenience)


2 2 2 2M AL BH M

AL BH


≠

B ABW=ω -ω

M B Aω = ω ω•

In this example,


B ABW=ω -ω

M B Aω = ω ω•

RRN

M

AA =A

R2

M

A1 =A1+ε

2

M

R

A -1A

ε⎛ ⎞

= ⎜ ⎟⎝ ⎠

2 2M AL

S1AL

ω - ω ωω •BW

=2 2BH M

S2BH

ω - ωωω •BW

=

SH SLSN

M M

A AA =min ,A A

⎧ ⎫⎨ ⎬⎩ ⎭

{ }SN S1 S2ω min ω ,ω=


B ABW=ω -ω

M B Aω = ω ω•

RRN

M

AA =A

R

2M

A1 =A1+ε

2

M

R

A -1A

ε⎛ ⎞

= ⎜ ⎟⎝ ⎠

2 2M AL

S1AL

ω -ω ωω •BW

=

2 2BH M

S2BH

ω - ωωω •BW

=

SH SLSN

M M

A AA =min ,A A

⎧ ⎫⎨ ⎬⎩ ⎭



ee 508 lecture 2 - class.ece.iastate.educlass.ece.iastate.edu/ee508/lectures/ee 508 lect 15 fall...

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