Frequency Selective Filters Lecture

8/4/2019 Frequency Selective Filters Lecture

1/19

Lecture 2 - Frequency SelectiveFilters

2.1 Applications

Low-pass : to extract short-term average or to eliminate high-frequency fluctua-

tions (eg. noise filtering, demodulation, etc.)

High-pass : to follow small-amplitude high-frequency perturbations in presence

of much larger slowly-varying component (e.g. recording the electrocardio-

gram in the presence of a strong breathing signal)

Band-pass : to select a required modulated carrier frequency out of many (e.g.

radio)

Band-stop : to eliminate single-frequency (e.g. mains) interference (also known

as notch filtering)

2.2 Design of Analogue Filters

We will start with an analysis of analogue low-pass filters, since a low-pass filter

can be mathematically transformed into any other standard type.

Design of a filter may start from consideration of

The desired frequency response.

The desired phase response.

11


2/19

low-pass high-passband-pass

notch

G( )| |

Figure 2.1: Standard filters.

The majority of the time we will consider the first case. Consider some desired

response, in the general form of the (squared) magnitude of the transfer function,

i.e.

. This response is given as

where

denotes complex conjugation. If

represents a stable filter (its poles

are on the LHS of the s-plane) then

is unstable (as its poles will be on the

RHS).

The design procedure consists then of

Considering some desired response

as a polynomial in even powers

of

.

Designing the filter with the stable part of !

.

This means that, for any given filter response in the positive frequency domain, a

mirror image exists in the negative frequency domain.

2.2.1 Ideal low-pass filter

Any frequency-selective filter may be described either by its frequency response

(more common) or by its impulse response. The narrower the band of frequencies

transmitted by a filter, the more extendedin time is its impulse responsewaveform.

Indeed, if the support in the frequency domain is decreased by a factor of $ (i.e.

made narrower) then the required support in the time domain is increased by a

factor of $ (you should be able to prove this).

12


3/19

Consider an ideal low-pass filter with a brick wall amplitude cut-off and no

phase shift, as shown in Fig. 2.2.

| G( ) |

c

1

0

Figure 2.2: The ideal low-pass filter. Note the requirement of response in the

negative frequency domain.

Calculate the impulse response as the inverse Fourier transform of the fre-

quency response:

%

&

)

0 1

2 3

5

3

6 8 @ A B D E 8

)

0 1

2

B G

5

B G

) H

@ A B D E 8

)

0 1

6 &

@ A BG

D R @

5

A BG

D

hence,

%

&

8 U

1

W X Y

8 U &

8 U &

Figure 2.3 shows the impulse response for the filter (this is also referred to as the

filter kernel).

The output starts infinitely long before the impulse occurs i.e. the filter is

not realisable in real time.

13


4/19

5 4 3 2 1 0 1 2 3 4 50.5

0

0.5

1

1.5

2

t

g(t)

Figure 2.3: Impulse response (filter kernel) for the ILPF. The zero crossings occur

at integer multiples of1 b

8 U

.

A delay of timed

such that

%

&

8 U

1

W X Y

8 U & R

d

8 U & R

d

would ensure that mostof the response occurred after the input (for larged

). The

use of such a delay, however, introduces a phase lag proportional to frequency,

since$ f

% h

6 8 p

8

d. Even then, the filter is still not exactly realisable; instead

the design of analogue filters involves the choice of the most suitable approxima-

tion to the ideal frequency response.

2.3 Practical Low-Pass Filters

Assume that the low-pass filter transfer function

is a rational function in

.The types of filters to be considered in the next few pages are all-pole designs,

which means that

will be of the form:

)

$ q

q r

$ q

5 t

q

5 t

r v v v

$

t

r

$ x

y

6 8

)

$ q

6 8

qr

$ q

5 t

6 8

q

5 t

r v v v

$

t

6 8

r

$ x

The magnitude-squared response is 6 8

6 8

H

R 6 8

. The de-

nominator of 6 8

is hence a polynomial in even powers of8

. Hence the

14


5/19

task of approximating the ideal magnitude-squared characteristic is that of choos-

ing a suitable denominator polynomial in8

, i.e. selecting the function

in the

following expression:

6 8

)

)

r

h

B

BG

p

where8 U

nominal cut-off frequency and

rational function of

B

BG

.

The choice of

is determined by functions such that ) r

h

8

b

8 U

p

is close

to unity for8 8 U

and rises rapidly after that.

2.4 Butterworth Filters

h

8

8 U

p

h

8

8 U

p

q

8

8 U

q

i.e.

6 8

)

)

r

B

BG

q

where is the order of the filter. Figure 2.4 shows the response on linear (a) and

log (b) scales for various orders .

2.4.1 Butterworth filter notes

1.

t

for8

8 U

(i.e. magnitude response is 3dB down at cut-off fre-quency)

2. For large

:

in the region8 8 U

, 6 8

)

in the region8 8 U

, the steepness of

is a direct function of

.

3. Response is known as maximally flat, because

E

q

E 8

q

B

x

for

)

!

0

!

v v v

0

R

)

15


6/19

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1(a)

n=1

n=2

n=6

|G()|

102

101

100

101

100

(b)

log

log|G()|

Figure 2.4: Butterworth filter response on (a) linear and (b) log scales. On a

log-log scale the response, for8 8 U

falls off at approx -20db/decade.

Proof

Express 6 8

in a binomial expansion:

6 8

h

)

r

8

8 U

q

p

5

)

R

)

0

8

8 U

q

r jl

8

8 U

m

q

R n

) o

8

8 U

q

r v v v

It is then easy to show that the first0

R

) derivatives are all zero at the

origin.

2.4.2 Transfer function of Butterworth low-pass filter

6 8

6 8 R 6 8

Since 6 8

is derived from

using the substitution 6 8

, the reverse

operation can also be done, i.e.8 R 6

16


7/19

R

)

)

r

R 6

BG

q

or R

)

)

r

5

A

B G

q

Thus the poles of

)

)

r

5

A

B G

q

belong either to

or R

. The poles are given by:

R 6

8 U

q

R

)

@ A z

| }

t ~

!

!

)

!

0

!

v v v

!

0

R

)

ThusR 6

8 U

@ A z

| }

t ~

Since6

@

A

, then we have the final result:

8 U 6

1

0

r

0

r

)

1

0

i.e. the poles have the same modulus8 U

and equi-spaced arguments. For

example, for a fifth-order Butterworth low-pass filter (LPF),

n :1

0

)

l

1

0

r

0

r

)

1

0

r

)

l

!

n

!

!

)

0

o

!

v v v

i.e. the poles are at:

)

l

!

)

!

)

l

!

0

) o

!

0

n

0

!

in L.H. s-plane therefore stable

0

l l

!

j

0

!

j

o

!

j

o

!

j

0

in R.H.S. s-plane therefore unstable

17


8/19

1.5 1 0.5 0 0.5

0.5

0

0.5

Figure 2.5: Stable poles of 5th order Butterworth filter.

We want to design a stable filter. Since each unstable pole is R

)

a stable pole,

we can let the stable ones be in

, and the unstable ones in R

v

Therefore the poles of

are8 U @

A

t

x

! 8 U @

A

t

m m

! 8 U @

A

t

x

! 8 U @

A

t

! 8 U @

A

as shown in Figure 2.5. Hence,

)

)

R

)

)

r

B G

h

)

r

0

y

W

0

B G

r

B G

p

h

)

r

0

y

W

j

o

B G

r

B G

p

or, multiplying out:

)

)

r

j

v

0

j

o )

BG

r

n

v

0

j

o )

BG

r

n

v

0

j

o )

BG

r

j

v

0

j

o )

BG

m

r

BG

Note that the coefficients are palindromic (read the same in reverse order) this

is true for all Butterworth filters. Poles are always on same radii, at

q angularspacing, with half-angles at each end. If

is odd, one pole is real.

$

t

$

$

$

m

$

$

$ $

) )

v

0

)

v

)

) )

v

j

0

v

0

v

)

v

0

v

o )

j

)

j

v

)

0 0

v

o )

j

) )

v

n

j

v

0

j

o )

n

v

0

j

o )

n

v

0

j

o )

j

v

0

j

o ) )

v

o

j

v

l

o

j

v

o

)

v

)

) o

v

o

)

j

v

l

o

j

)

v

v

)

v

l

)

v

n

)

l

)

v

n

)

l

)

v

l

v

)

v

l

n

v

)

0

n

l

)

j

v

)

j

)

0

)

v

l

o

0 0

n

v

o

l l

0

)

v

l

o

0

)

j

v

)

j

)

n

v

)

0

n

l

)

v

18


9/19

Butterworth LPF coefficients for$

l

2.4.3 Design of Butterworth LPFs

The only design variable for Butterworth LPFs is the order of the filter

. If a

filter is to have an attenuation

at a frequency8

:

B

)

)

)

r

B

BG

q

i.e.

y

R

)

0

y

B

B G

or since usually

)

!

y

y

B

BG

Butterworth design example

Design a Butterworth LPF with at least 40 dB attenuation at a frequency of 1kHz

and 3dB attenuation atU

= 500Hz.

Answer

40dB

= 100;

8

0

1

and8 U

)

1

rads/sec

Therefore

y

t

x

)

y

t

x

0

0

v

j

)

o

v

o

Hence = 7 meets the specification

Check: Substitute

6

0

into the transfer function from the above table for

6

0

)

l

v

j

l

R

j

v

n

6

which gives

)

0

l

.

19


10/19

2.5 Equiripple Filters

You may have noticed that the Butterworth response is monotone i.e. it has no

ripple. If a certain amount of ripple is allowed in the pass-band and/or the stop-

band, filters which have a sharper cut-off than the Butterworth filter (for a given

order ) can be designed. There are three types ofequi-ripple filters:

Chebyshev Type I (equi-ripple in the pass-band)

Chevyshev Type II (also known as inverse Chebyshev equi-ripple in the

stop-band)

Elliptic (equi-ripple on both pass-band and stop-band)

2.5.1 Chebyshev polynomials

The

-order Chebyshev polynomiald q

can be expressed as:

d q

y

W

y

W

5 t

for

) ; and

d q

y

W

y

W

5 t

for

) .

These two expressions are equivalent in that the latter can be derived from

the former and so the result is a single Chebyshev polynomial which applies for

.Alternatively,d q

can be expressed as a polynomial in

, which can be

evaluated for any

:

d

t

y

W

y

W

5 t

d

y

W

0

where

y

W

5 t

i.e. d

0

y

W

R

)

0

R

)

20


11/19

Obviously, the same results are also valid with the

y

W function.

Generally,

d q

}

t

r

d q

5 t

y

W

r

)

r

y

W

R

)

0

y

W

y

W

0

d q

Thus:

d q

}

t

0

d q

R

d q

5 t

is a recurrence relationship which allows the computation ofd q

}

t

from the

two previous polynomials.

Using the recurrence relationship,

d

0

d

R

d

t

0

0

R

)

R

R

j

d

m

0

d

R

d

0

R

j

R

0

R

)

l

m

R

l

r

)

The graphs of Fig. 2.6 below show that, for large

,d q

diverges very

rapidly for

) ; for example,d

m

)

v

n

0

j

v

n . This is just the kind of behaviour

we need for a good filter function.

For a Chebyshev Type I filter (i.e. equi-ripple in the pass-band):

h

8

8 U

p

d

q

8

8 U

X

v

6 8

)

)

r

d

B

B G

Chebyshev type I notes

1. The parameter

)

sets the ripple amplitude in the ripple pass-

band which is defined as

8

8 U

. Since, for8

8 U !

d

q

)

! 6 8

will fluctuate between ) andt

t

}

.

e.g. with

v

n

l l

, the amplitude will vary between ) andt

t

, ie.

v

l

or 1dB ripple since0

y

t

x

v

l

R

)

v

.

2. At8

!

d

q

h

y

W

y

W

5 t

p

21


12/19

2 1 0 1 22

1

0

1

2

T1(x)

x2 1 0 1 2

2

0

2

4

6

8

T2(x)

x

2 1 0 1 230

20

10

0

10

20

30

T3(x)

x2 1 0 1 2

20

0

20

40

60

80

100

T4(x)

x

Figure 2.6: Chebyshev functions.

ie. d

q

if is odd and 1 if is even

3.8 U

, in this case, is notthej

E

cut-off frequency.

By definition,8

is given by:

)

)

r

d

q

B

B G

)

0

ie.

d

q

B

B G

) which means that8

8 U

, since

)

4. For8 8 U !

d

q

gets large and 6 8

decreases monotonically, as for the

Butterworth LPF, but much more rapidly.

22


13/19

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1(a)

|G()|

n=4

n=1

Figure 2.7: Response of Chebyshev type I filter. The lower limit of ripple in the

pass-band is )b

)

r

. Here

v

n

l l

.

2.5.2 Transfer function of Chebyshev Type I LPF

As before, put 6 8

6 8 R 6 8

, and let8 R 6

R

)

)

r

d

5

A

B G

The poles are given by d q

5

A

BG

6

5 t

. Of the0

roots, assign the stable ones

to

.

Example

j

t

d

R 6

8 U

R 6

8 U

R

j

R 6

8 U

6

8 U

r

6

H

j

8 U

6

5 t

6

0

i.e. 2 cubics

8 U

r

j

8 U

R

0

D

G

}

5

A

and

8 U

r

j

8 U

r

0

D

G

5

}

A

Select as stable poles

R

t

8 U !

R

t

m

6

t

t

8 U

Therefore

|

t

)

)

R

)

h

)

r

0

BG

p

h

)

r

t

BG

r

BG

p

(factorised form)

23


14/19

(Note

t

mfor complex pair it was

t

for Butterworth filter)

Hence

)

)

r

0

t

B G

r

0

B G

r

0

B G

(polynomial form)

In general, the poles lie on an ellipse (this is only meaningful for

n ; if

, an ellipse can be drawn through any setof poles).

A large

means a larger ripple, narrower ellipse and less damped complex

poles, ie. a more oscillatory impulse or step response buta steeper cut-off.

0.4 0.6 0.8 1 1.2

0.3

0.2

0.1

0

0.1

0.2

0.3

Figure 2.8: Stable poles of 5th order Chebyshev Type I filter.

2.5.3 Design of Chebyshev Type I LPFs

If a filter is to have an attenuation

at a frequency8

:

B

)

)

)

r

d

q

B

BG

ie.

R

)

d

q

8

8 U

y

W

y

W

5 t

8

8 U

24


15/19

i.e.

y

W

y

W

5 t

8

8 U

R

)

or

y

W

5 t

5 t

y

W

5 t

B

BG

Example

Design a Chebyshev Type I LPF to meet the following specifications:

Maximum ripple of 1dB in pass-band from 0 to 50Hz

3dB cut-off frequency at = 65 Hz

Attenuation of at least 40 dB for

250 Hz

Answer

1. As shown on page 21, 1dB ripple corresponds to a value of

v

n

l l

for

.

2. 3dB cut-off:8

)

j

1

rads/sec. Minimum

given by:

y

W

5 t

t

x

x

y

W

5 t

)

v

j

)

v

)

Therefore

meets the 3 dB cut-off requirement.

3. For 40dB attenuation,

)

. Minimum

given by:

5 t

t

x

5 t

~

v

n

l l

y

W

5 t

n

0

v

o

o

ie. meets the overall specification.

25


16/19

2.5.4 Chebyshev Type II low-pass filters

These exhibit equiripple behaviour in the stop-bandbut have the same behaviour

as a Butterworth filter (ie. maximally flat around8

in the pass-band. This be-

haviour cannot be achieved with an all-pole filter; Chebyshev Type II filters have

a transfer function

which includes zeroes on the imaginary axis as well aspoles in the left-half

-plane.

6 8

)

)

r

z

G

~

z

~

where8

= lowest frequency at which stop-band loss attains a specified value.

The magnitude-squared response for Chebyshev Type I and II filters in Fig.

2.9. For both these filters, the pass-band edge is at8

8 U

where

t

t

}

andthe stop-band edge is at

8

8

where

t

.

1/(1+ )2

1/A2

c r

|G( )|2

Type I

1/(1+ )2

1/A2

c r

|G( )|2

Type II

Figure 2.9: Response of Chebyshev type I, II filters.

Thus, for

n :

h

)

r

B

p

h

)

r

B

p

)

r

$

t

B G

r

$

B G

r

$

B G

r

$

m

B G

m

r

$

B G

with zeroes at

6 8

t

,

6 8

26


17/19

2.5.5 Design of Chebyshev Type II LPFs

Example

Design a Chebyshev Type II LPF to meet the following specifications:

3dB cut-off frequency at 50 Hz

Attenuation of at least 50dB for

100Hz.

Answer

1. For Chebyshev Type II LPFs, the pass-band behaviour is the same as that

of a Butterworth LPF; ie.8 U

8

. Therefore

B G

t

t

t

}

which

gives

)

2.

When8

8

!

)

)

)

r

z

G

~

z

t ~

Since d

q

)

) for all , the expression reduces to:

B

)

)

r

d

q

B

BG

Therefore, from section 2.5.3,

y

W

5 t

R

)

y

W

5 t

B

B G

)

!

in this case

For an attenuation of 50dB,

= 10

and thus:

y

W

5 t

)

R

)

y

W

5 t

0

v

l

ie. meets the overall specification.

27


18/19

2.5.6 Elliptic low-pass filters

Elliptic filters3 have a magnitude response which is equi-ripple in both the pass-

band and the stop-band. Elliptic filters are optimum in the sense that, for a given

order and for given ripple specifications, no other filter achieves a faster transition

between the pass-band and the stop-band ie. has a narrower transition band-width.

The magnitude-squared response of a low-pass elliptic filter is of the form:

6 8

)

)

r

q

B

B G

!

where

q

B

B G

!

is called a Chebyshev rational function and

is a parameter de-

scribing the ripple properties of

q

B

BG

!

v

Figure 2.10 shows the magnitude-squared response of a typical elliptic low-

pass filter. The frequencies8

represent the edges of the pass- and stop-bands

and it is noted that the cut-off frequency is given as8 U

8

8

. The latter, once

again, is not the 3dB point though.

2.5.7 Design of Elliptic Filters

In order to design an elliptic LPF with arbitrary attenuations in both pass-band

and stop-band, three of the four parameters:

filter order

in-band loss or ripple

out-of-band loss or attenuation

transition ratio

B

B

)

can be chosen and the fourth parameter is uniquely defined.

The theory behind the determination of the function

q

B

BG

!

involves an

understanding of Jacobian elliptic functions, which are beyond the scope of this

3also known as Cauer, or Darlington, filters

28


19/19

1/(1+ )2

1/A2

|G( )|2

p s

Figure 2.10: Elliptic filter response.

course. Hence the design of elliptic filters will not be considered in detail in this

lecture course. In any case, elliptic filters are usually designed with the help of

graphical procedures (see, for example, Rabiner & Gold, Theory and Application

of Digital Signal Processing) or computer programmes(see, for example, Daniels,

Approximation methods for the design of passive, active and digital filters).

You should remember, however, that for any given specification, the elliptic

filter will always be of lower order than any other.

29

Documents

Frequency Selective Filters Lecture