Normalized Lowpass Filters

“All-pole” lowpass filters, such as Butterworth and Chebyshev filters, have transfer functions of the form:

012

21

1 ...)(

asasasas

CsH

NN

NN

N

Where N is the filter order.

Filter functions are tabulated in “normalized” form

Normalized Lowpass Filters

Normalized form means the tabulated functions are for filter prototypes with:

)sec 1( Hz 2

1

gain) passband(unity 1)0(

1-

ccf

H

1cIf we can writec

so 1c

cc

Unity gain means that in the transfer function

012

21

1 ...)(

asasasas

CsH

NN

NN

N

Normalized Lowpass Filters

0aC

An Nth order filter has N poles.

If N is odd, one pole is purely real (It’s imaginary part is zero, so it lies on the real axis.

Normalized Lowpass FiltersIf N is even, no pole lies on the real axis.

If a pole is not on the real axis (it’s imaginary part is not zero) then it’s complex conjugate is also a pole.

If N is even, the transfer function may be factored into

2

12

2

2

2

1

2

2

22

2

2

22

2

2

11

2

1

2

Re21

1

1

1

...1

1

N

k kk

k

N

k kk

k

NN

N

pps

p

psps

p

psps

p

psps

p

psps

psH

Normalized Lowpass FiltersIf N is even, the transfer function may be factored into

2

12

2

2

2

1

2

2

22

2

2

22

2

2

11

2

1

2

Re21

1

1

1

...1

1

N

k kk

k

N

k kk

k

NN

N

pps

p

psps

p

psps

p

psps

p

psps

psH

For Butterworth filters, = 0

Normalized Lowpass Filters

If N is odd, the transfer function may be factored into

2

12

2

0

0

21

1

2

0

0

22

2

2

22

2

2

11

2

1

0

0

Re2

...

N

k kk

k

N

k kk

k

NN

N

pps

p

ps

p

psps

p

ps

p

psps

p

psps

p

psps

p

ps

psH

Normalized Lowpass Filters

x

x

j

1

1

-1

-1

Second order

2

2

2

2

2

2

2

2

1

1

jp

jp

Normalized Lowpass Filters

x

x

j

1

1

-1

-1

Third order

2

3

2

1

2

3

2

1

1

1

1

0

jp

jp

p

x

Normalized Lowpass Filters

The pole locations are tabulated for Butterworth filters of other filter orders, and for Chebyshev filters of orders up to 8 and various ripple factors, in the textbook. Specialized filter references contain far more extensive tabulations for these and other filter types (Bessel, elliptic, etc.)

Lowpass to Lowpass Transformation

Denormalizing the normalized filter

We will denormalize a prototype lowpass filter (c = 1) by scaling it so it’s cutoff frequency is c. Take the normalized transfer function H(s), and replace s with

c

s

Lowpass to Lowpass Transformation

Denormalizing the normalized filter

For a second-order Butterworth, the normalized prototype is:

7071.7071.7071.7071.

1

1414.1

1)(

2 jsjssssH

If we’re designing a filter with 210 10 cc f

Lowpass to Lowpass Transformation

Denormalizing the normalized filter

071.7071.7071.7071.7

100

10014.14

100

110

414.1100

1

110

414.110

1

1414.1

1)(

22

22

jsjs

ssss

sssssH

cc

Lowpass to Lowpass Transformation

Denormalizing the normalized filter

This illustrates how we can denormalize a complex-conjugate pole pair, or second-order section.

Problems

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