Frequency Selective Filters Lecture 4b2a3x
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Lecture 2 - Frequency Selective Filters 2.1 Applications Low- : to extract short-term average or to eliminate high-frequency fluctuations (eg. noise filtering, demodulation, etc.) High- : to follow small-amplitude high-frequency perturbations in presence of much larger slowly-varying component (e.g. recording the electrocardiogram in the presence of a strong breathing signal) Band- : 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- filters, since a low- 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
notch | G(ω ) |
ω low-
high-
band-
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, . This response is given as i.e.
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 of .
Deg the filter with the stable part of
as a polynomial in even powers
.
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- 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 extended in time is its impulse response waveform. Indeed, if the in the frequency domain is decreased by a factor of (i.e. made narrower) then the required in the time domain is increased by a factor of (you should be able to prove this).
12
Consider an ideal low- filter with a “brick wall” amplitude cut-off and no phase shift, as shown in Fig. 2.2.
| G(ω ) | 1
ωc
0
ω
Figure 2.2: The ideal low- filter. Note the requirement of response in the negative frequency domain. Calculate the impulse response as the inverse Fourier transform of the frequency response: */
"!
%$'& ()+*,.-'&
()+*,.-'& 10
#
!
* #
3()+*
/ ,546( #
)+*
/ ,
$2
/
hence, &87
&87 9:<;
&87=
Figure 2.3 shows the impulse response for the filter (this is also referred to as the filter kernel). The output starts infinitely long before the impulse occurs – i.e. the filter is not realisable in real time. 13
2 1.5
g(t)
1 0.5 0 −0.5 −5
−4
−3
−2
−1
0
1
2
3
4
5
t
Figure 2.3: Impulse response (filter kernel) for the ILPF. The zero crossings occur at integer multiples of . &87
?>
A delay of time
such that @
&87
.
&87.84
9:<; @ &A7A4
@
would ensure that most of the response occurred after the input (for large ). The use of such a delay, however, introduces a phase lag proportional to frequency, since . Even then, the filter is still not exactly realisable; instead the design of analogue filters involves the choice of the most suitable approximation to the ideal frequency response. @
CB
5D
E$'&1F
&
@
2.3 Practical Low- Filters
Assume that the low- 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:
CG
O2P
E$'&
GIH
2G
#KJ
G
#KJ
HMLLL
J
H
CN
%$'& 2G
G
H
E$'& CG
#KJ
G
#KJ
%$2& R
HMLQLL
$'&
H
J
%$2&
CN +4 $'&
The magnitude-squared response is nominator of is hence a polynomial in even powers of 0
%$2& R
14
&
. The de. Hence the
task of approximating the ideal magnitude-squared characteristic is that of choosing a suitable denominator polynomial in , i.e. selecting the function in the following expression: &
S
%$2& R T H
&A7
D
S
*
*
F
/
where nominal cut-off frequency and rational function of The choice of is determined by functions such that to unity for and rises rapidly after that. S
S &"WX&87
H
S
D
U&
>
*
* / &87V
F
. is close
2.4 Butterworth Filters & D S
&
F
&87
D
&
F
G
&87
G
&87
i.e.
E$'& Y H
*
* /
G
where is the order of the filter. Figure 2.4 shows the response on linear (a) and log (b) scales for various orders . Z
Z
2.4.1 Butterworth filter – notes 1.
[
\
J
for
&
&87
(i.e. magnitude response is 3dB down at cut-off fre-
quency) 2. For large : Z
in the region in the region
&]WX&87
&]^X&87
,
E$'& R
_
, the steepness of
is a direct function of . Z
3. Response is known as maximally flat, because G -2&
dc Ga` ` *'b ` `
for
N
15
Z
LLL
'e
4
(a) 1
|G(ω)|
0.8 n=1
0.6 0.4
n=2
0.2
n=6
0
0
0.2
0.4
0.6
0.8
1
1.2
ω
1.4
1.6
1.8
2
(b)
0
log |G(ω)|
10
−1
10
−2
−1
10
10
0
10
log ω
Figure 2.4: Butterworth filter response on (a) linear and (b) log scales. On a log-log scale the response, for falls off at approx -20db/decade. &"^f&A7
Proof Express
%$'&
in a binomial expansion: &
E$'&
D
H
&87
& G F
#hi g
4
& G
&A7
Hkjl
&87
& Vm G
4
n
&A7
p G
HMLQLL
Ro
It is then easy to show that the first origin.
4 Z
derivatives are all zero at the
2.4.2 Transfer function of Butterworth low- filter E$'&
E$'&
Since is derived from operation can also be done, i.e.
rq
&us
%$'& 4T$2&
using the substitution
4T$v
16
ts
$'&
, the reverse
q
4w
x
*
+4a
or
4T$ y H
G
/
H
) y #
G
*R/
Thus the poles of
belong either to
+4a
or
H
) y #
*/
G
. The poles are given by:
4T$v G
&87
4
(){z V|}
J~€
dc
LQLL
4
Z
Thus 4T$v
()z V|‚}
J~„iUƒ …
&87
Since
$
( )
ƒi
, then we have the final result:
&87v†{‡„ˆT$5‰
H
H ZTŠ
&87
i.e. the poles have the same modulus and equi-spaced arguments. For example, for a fifth-order Butterworth low- filter (LPF), :
n
Z
lŒ‹
s
H
Ž2c
H
Z
‹
l'‹
H
‹
‹
2c
‹
n
‚†‘“’
LLL
o
Z
i.e. the poles are at: c ”
l ‹
‹
2
l
c •Q–
‹
‹
Ro
n
‹
—
in L.H. s-plane therefore stable
”
l2l ‹
‹
j
c j
o • – Q
‹
‹
j
o
‹
j
—
in R.H.S. s-plane therefore unstable 17
0.5
0
−0.5
−1.5
−1
−0.5
0
0.5
Figure 2.5: Stable poles of 5–th order Butterworth filter. +4
We want to design a stable filter. Since each unstable pole is we can let the stable ones be in , and the unstable ones in
&87+(
Therefore the poles of are as shown in Figure 2.5. Hence,
4
y
œ
H
“ y
D
*R/
’
H
O
&87V(
) J NV™=š
)
J mVm š
&87+( )
Œ˜
a stable pole,
+4a
&87V(
J ™VN=š
L
‹
95ž
y
H
*/
y
F D
*R/
’
H
‹ O 9
&87V(
) J p š
y
o j
H
*R/
) V›V š
y
F
*R/
or, multiplying out:
H
y
L j
oŸ j
*
H /
L n
oŸ j
y
*
H
/
L n
+ y
o„ j
*
H
/
L j
m y
oŸ j
*
H
/
› y
* /
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 angular spacing, with “half-angles” at each end. If is odd, one pole is real. G
Z
J c'c2c2c
Z
L
L
c'c2c2c
L
j
L
o„
L
n o
j
L o
j lž
n
L
c
n
L
L L
o„
o c2
L
j
ž
j
›
p
¢¡
C™
Ro
L
ž
o
o
n
n L
L
o
L
oŸ j
L
l
c2c'c2c
L
j
ln
c2c2c2c L
oŸ j
L
l
L
n
ž L
j
L
oŸ j
ž c
L
n
l
m
c'c2c2c L
j
ž
j
L
L
oŸ
l j
c2c2c2c
c2c2c2c L
j
l
l2l
l L
cj
o
18
ž l
j
c2 L
L
o
L
c2c2c2c L
lž
L j
c
j
c2c2c2c L
ž
n
L
l
n
L
c2c2c'c
l
Butterworth LPF coefficients for
£
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 frequency : Z
&A¥
¤
_
*R¦
¤
¨§
i.e.
*R¦ H
O2©
* /
4
¤
Z
G
*
O2© §
¦ *R/
¤dª
O2© §
or since usually
Z¬«
¤ *R¦
§
O2©
* /
Butterworth design – example Design a Butterworth LPF with at least 40 dB attenuation at a frequency of 1kHz and 3dB attenuation at = 500Hz. Answer 7
40dB
s ¤
&8¥
= 100;
c2c2c
Therefore Hence Z
Check: Substitute
$
and §
§
&87
c2c2c
rads/sec
c'c
O2©
Z¬«
J N
c
O'©
J N
o c
L j
l
l ž
which gives
¤
l
o
into the transfer function from the above table for
ž
E$
L
= 7 meets the specification Z
L j
. 19
4
L j
$® n
2.5 Equiripple Filters You may have noticed that the Butterworth response is monotone i.e. it has no ripple. If a certain amount of ripple is allowed in the -band and/or the stopband, filters which have a sharper cut-off than the Butterworth filter (for a given order ) can be designed. There are three types of equi-ripple filters: Z
Chebyshev Type I (equi-ripple in the -band) Chevyshev Type II (also known as inverse Chebyshev – equi-ripple in the stop-band) Elliptic (equi-ripple on both -band and stop-band)
2.5.1 Chebyshev polynomials The -order Chebyshev polynomial Z
¯°
¯°
’
c ¯ £
£
’
9
O
¯°
#KJ 9
Z
; and
.¯°
’
O
@°G
for
O
@5G
for
can be expressed as:
@5G
’
9“± Z
¯°
#KJ O
9±
¯²^
. These two expressions are equivalent in that the latter can be derived from the former and so the result is a single Chebyshev polynomial which applies for . Alternatively, can be expressed as a polynomial in , which can be evaluated for any : c
¯
£
£M³
¯
.¯°
¯
@5G
.¯°
¯° @
’
O
i.e.
¯° @
’
.’ O
J @
Ϋ
9
’ O
9
20
¯
9
where
9
¯°
#KJ O
´
´
4
’
O
#KJ
¯
9
¯
4
’
Obviously, the same results are also valid with the Generally, .¯° @°G
}
.¯° H
J
’
O
#KJ
@5G
9
´ H
Z
’ H
O
4
9
O
function.
9“±
´
’
Z
’ ´
O
´
O
9
9
¯
.¯°
Z
@5G
Thus:
.¯° }
@°G
¯
.¯°A4
J
¯°
@°G
#KJ
@5G
¯°
is a recurrence relationship which allows the computation of two previous polynomials. Using the recurrence relationship, @
.¯° @
m
.¯°
¯
.¯°µ4
¯ @
¯°
¯
4
A4¶¯
¯8
¯
4
¯°A4
¯
@
¯8
¯
4
¯ j
l
4
¯
m
l 4
¯
j
from the
J
J @
¯°h4
.¯°
@
}
@5G
H
.¯°
The graphs of Fig. 2.6 below show that, for large , diverges very rapidly for ; for example, . This is just the kind of behaviour we need for a good filter function. For a Chebyshev Type I filter (i.e. equi-ripple in the -band): ¯hv^
m
@
L
Z
L
n
n
j
& &
S
† L
:
D
@°G
¸·
F
&87 @
E$'& R
&87
G
· H
@IZ
*
* /
Chebyshev type I – notes ·
c
·
W
W
sets the ripple amplitude in the ripple 1. The parameter band which is defined as . Since, for will fluctuate between and .
c
&
&87
£
'E$'&
£
\
·»¼c
L
c
&A7“ c
&
£
l2l
} º i J Ÿ
c
2. At
&
dc
@ G
¯°
c
O2©
§
D
’
9
¾
L
J N
O
l
’ Z
4
O 9
#KJ
21
c
{F
c
L
£
G
J
e.g. with , the amplitude will vary between and or 1dB ripple since . n
£¹@
\
J JV½ p
, ie.
c
l L
T (x)
T (x)
1
2
2
8 6
1
4 0 2 −1
0
−2 −2
−1
0
1
−2 −2
2
−1
0
x T (x) 3
100
20
80
10
60
0
40
−10
20
−20
0 −1
2
1
2
4
30
−30 −2
1
x T (x)
0
1
−20 −2
2
−1
0
x
x
Figure 2.6: Chebyshev functions. ie. 3.
&87
@
, in this case, is not the
By definition,
&
VÀÁ
Žc
¯°
if
G
-v¿
is odd and 1 if Z
is given by:
·
@ G
* ÂUÃÄ
*R/
&¼^d&87‚
is even
cut-off frequency.
j
ie.
Z
H
·
Å
@
* ÂUÃ=Ä
*R/ G
which means that
&
E$'& R
VÀÁ
^f&A7
, since
· W
4. For gets large and decreases monotonically, as for the Butterworth LPF, but much more rapidly. @
G
22
(a) 1
|G(ω)|
0.8 n=1
0.6 0.4 n=4
0.2 0
0
0.2
0.4
0.6
0.8
1
1.2
ω
1.4
1.6
1.8
2
Figure 2.7: Response of Chebyshev type I filter. The lower limit of ripple in the -band is . Here . >¢Æ
·
H
·¾Mc
L
c
l2l
n
2.5.2 Transfer function of Chebyshev Type I LPF
E$'&
As before, put
E$'& 4T$'& q
Z
·»
*
kÇ
· $
#KJ
) y #
*R/
. Of the
/
@IZ
roots, assign the stable ones Z
J
j„È
4T$v
4T$v @
) y #
@5G
· H
The poles are given by to . Example
4 $v
4w
&6s
, and let
4T$v
&87
&A7
&87
&87
A4
Mc
•Q– ‹=‹
, y8Ê Ë /
} b
Select as stable poles
—
ig Ì #
0
)
Íg Î {
4
Æ
+4
‹=‹
, yAÊ Ë /
x $
#KJ
dÇ $
Žc
H
•Q–
Ç J
&87
j
” É
H
&87
gEÏ g%Ð
J &A7
· $
&87 j
and
&A7 j
dÇ $
H
H
” É
$
j
&87
i.e. 2 cubics
4
J ›
m
b #
ig Ì Q
— }
)
Íg Î
Æ
gEÏ g%Ð
V&87
J p
Therefore
Ñ
| b
J
4 œ
y
D
H
*
23
F y
/
D
H
y J
*
/
H
y
* /
F
(factorised form)
(Note
Ò
J
J
for complex pair – it was
m
Hence
for Butterworth filter)
H
J
y
*R/
H
y
H
*R/
(polynomial form)
y
*R/
In general, the poles lie on an ellipse (this is only meaningful for ; if , an ellipse can be drawn through any set of poles). A large means a larger ripple, narrower ellipse and less damped complex poles, ie. a more oscillatory impulse or step response but a steeper cut-off. Z¼Ó
ZÔ£
·
0.3 0.2 0.1 0 −0.1 −0.2 −0.3 0.4
0.6
0.8
1
1.2
Figure 2.8: Stable poles of 5–th order Chebyshev Type I filter.
2.5.3 Design of Chebyshev Type I LPFs If a filter is to have an attenuation _
*
at a frequency ¤
¦
&8¥
:
¤
H
·
*
@
¦
* G
/
&8¥
ie. ¤
M·
4 @
G
&8¥
M·
&A7
24
’ O
9±
Z
’ O
#KJ 9“±
&87
n
&A¥ ’
i.e.
O
’
9“± Z
9±
’
or
¤ ·
&87
\
#KJ O
Õ i
9“± ’
Z
#KJ
º *¦
#KJ O
4
Æ
#KJ O
9“± *
/
Example Design a Chebyshev Type I LPF to meet the following specifications: Maximum ripple of 1dB in -band from 0 to 50Hz 3dB cut-off frequency at
= 65 Hz
Attenuation of at least 40 dB for
250 Hz
ÖÓ
Answer
1. As shown on page 21, 1dB ripple corresponds to a value of &
2. 3dB cut-off:
VÀÁ
c
rads/sec. Minimum
j ’
#KJ O
c2c
x
#KJ
J N
×àßÛá
Í
#KJ ~
¨ÜÝ
ie.
L
ž
c n
l2l ·
for .
given by: Z
j
. Minimum
¤
Þ
Z
L
9±
L
meets the 3 dB cut-off requirement.
×ÙØÛÚ
3. For 40dB attenuation,
J › N ½ V N ™V™
9± ’
Z
Therefore
#KJ O
c
’ O
#KJ 9“±
c L
n
c
Z
given by: l2l L
c o
o
n
meets the overall specification. 25
2.5.4 Chebyshev Type II low- filters These exhibit equiripple behaviour in the stop-band but have the same behaviour in the -band. This beas a Butterworth filter (ie. maximally flat around 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 as poles in the left-half -plane. dc
&
%$2& R T
H
where
& É
·
i z Ëã Ë / i Ëã … z Ë ~
…
â â
~
= 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 -band edge is at where and the stop-band edge is at where .
&
&
&
_
Õ
É
|G(ω )|
J
&87
|G(ω )|
2
1/(1+ ε2)
2
2
1/A
1/A ω
ωc ωr
ω
ωc ωr
Type I
Type II
Figure 2.9: Response of Chebyshev type I, II filters. Thus, for
: n
Z
D
with zeroes at
J i J }ټ
i
2
1/(1+ ε2)
[
dÇ
H
J
$'& J
y
*R/
,
Ç
H
H
y
F D
* y
*/
g H
$'&
26
H
y *R/
y
* H
F
i
m
y *R/
m H
›
y *R/
›
2.5.5 Design of Chebyshev Type II LPFs Example Design a Chebyshev Type II LPF to meet the following specifications: 3dB cut-off frequency at 50 Hz Attenuation of at least 50dB for
100Hz.
ÔÓ
Answer
1. For Chebyshev Type II LPFs, the -band behaviour is the same as that of a Butterworth LPF; ie. . Therefore which gives
&87
&
[
VÀÁ
*
·ä
J
/
J i J }ټ
2. &
When
&
_ T
É
¤
Since
@ G
H
·
â
i z Ë ã ~ Ë / i z J ~ … â …
for all , the expression reduces to: Z
_
*
ã
· H
*
@
ã
* G
/
Therefore, from section 2.5.3, ’ Z
O
#KJ 9“±
’ O
#KJ 9“±
Z
×àßæå
*
·¾
in this case
ã
*R/
’
ig
For an attenuation of 50dB,
ie.
4
¤
›
= 10 and thus:
O
¤
#KJ ’ O
c
4 ›
9“±
#KJ
ig
l
L
ž
9±
meets the overall specification. 27
2.5.6 Elliptic low- filters Elliptic filters3 have a magnitude response which is equi-ripple in both the band and the stop-band. Elliptic filters are optimum in the sense that, for a given order and for given ripple specifications, no other filter achieves a faster transition between the -band and the stop-band – ie. has a narrower transition bandwidth. The magnitude-squared response of a low- elliptic filter is of the form: E$'& R Y
ç
· H
ça
G
*
‚èT
*R/
èY
*
è
where is called a Chebyshev rational function and scribing the ripple properties of G
*
/
ç
G
èY
*
*
/
is a parameter de-
L
Figure 2.10 shows the magnitude-squared response of a typical elliptic low filter. The frequencies represent the edges of the - and stop-bands and it is noted that the cut-off frequency is given as . The latter, once again, is not the 3dB point though. &
œ
Ì
y
&87
&
Æ
&
y
œ
2.5.7 Design of Elliptic Filters In order to design an elliptic LPF with arbitrary attenuations in both -band and stop-band, three of the four parameters: filter order
Z ·
in-band loss or ripple
out-of-band loss or attenuation transition ratio
ç
*‘é *
+W Ê
¤
can be chosen and the fourth parameter is uniquely defined.
ç
èY
*
The theory behind the determination of the function involves an understanding of Jacobian elliptic functions, which are beyond the scope of this G
3
also known as Cauer, or Darlington, filters
28
*
/
|G(ω )|
2
1/(1+ ε2)
2
1/A
ωp ωs
ω
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 ive, active and digital filters). You should , however, that for any given specification, the elliptic filter will always be of lower order than any other.
29
2.2 Design of Analogue Filters We will start with an analysis of analogue low- filters, since a low- 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
notch | G(ω ) |
ω low-
high-
band-
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, . This response is given as i.e.
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 of .
Deg the filter with the stable part of
as a polynomial in even powers
.
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- 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 extended in time is its impulse response waveform. Indeed, if the in the frequency domain is decreased by a factor of (i.e. made narrower) then the required in the time domain is increased by a factor of (you should be able to prove this).
12
Consider an ideal low- filter with a “brick wall” amplitude cut-off and no phase shift, as shown in Fig. 2.2.
| G(ω ) | 1
ωc
0
ω
Figure 2.2: The ideal low- filter. Note the requirement of response in the negative frequency domain. Calculate the impulse response as the inverse Fourier transform of the frequency response: */
"!
%$'& ()+*,.-'&
()+*,.-'& 10
#
!
* #
3()+*
/ ,546( #
)+*
/ ,
$2
/
hence, &87
&87 9:<;
&87=
Figure 2.3 shows the impulse response for the filter (this is also referred to as the filter kernel). The output starts infinitely long before the impulse occurs – i.e. the filter is not realisable in real time. 13
2 1.5
g(t)
1 0.5 0 −0.5 −5
−4
−3
−2
−1
0
1
2
3
4
5
t
Figure 2.3: Impulse response (filter kernel) for the ILPF. The zero crossings occur at integer multiples of . &87
?>
A delay of time
such that @
&87
.
&87.84
9:<; @ &A7A4
@
would ensure that most of the response occurred after the input (for large ). The use of such a delay, however, introduces a phase lag proportional to frequency, since . Even then, the filter is still not exactly realisable; instead the design of analogue filters involves the choice of the most suitable approximation to the ideal frequency response. @
CB
5D
E$'&1F
&
@
2.3 Practical Low- Filters
Assume that the low- 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:
CG
O2P
E$'&
GIH
2G
#KJ
G
#KJ
HMLLL
J
H
CN
%$'& 2G
G
H
E$'& CG
#KJ
G
#KJ
%$2& R
HMLQLL
$'&
H
J
%$2&
CN +4 $'&
The magnitude-squared response is nominator of is hence a polynomial in even powers of 0
%$2& R
14
&
. The de. Hence the
task of approximating the ideal magnitude-squared characteristic is that of choosing a suitable denominator polynomial in , i.e. selecting the function in the following expression: &
S
%$2& R T H
&A7
D
S
*
*
F
/
where nominal cut-off frequency and rational function of The choice of is determined by functions such that to unity for and rises rapidly after that. S
S &"WX&87
H
S
D
U&
>
*
* / &87V
F
. is close
2.4 Butterworth Filters & D S
&
F
&87
D
&
F
G
&87
G
&87
i.e.
E$'& Y H
*
* /
G
where is the order of the filter. Figure 2.4 shows the response on linear (a) and log (b) scales for various orders . Z
Z
2.4.1 Butterworth filter – notes 1.
[
\
J
for
&
&87
(i.e. magnitude response is 3dB down at cut-off fre-
quency) 2. For large : Z
in the region in the region
&]WX&87
&]^X&87
,
E$'& R
_
, the steepness of
is a direct function of . Z
3. Response is known as maximally flat, because G -2&
dc Ga` ` *'b ` `
for
N
15
Z
LLL
'e
4
(a) 1
|G(ω)|
0.8 n=1
0.6 0.4
n=2
0.2
n=6
0
0
0.2
0.4
0.6
0.8
1
1.2
ω
1.4
1.6
1.8
2
(b)
0
log |G(ω)|
10
−1
10
−2
−1
10
10
0
10
log ω
Figure 2.4: Butterworth filter response on (a) linear and (b) log scales. On a log-log scale the response, for falls off at approx -20db/decade. &"^f&A7
Proof Express
%$'&
in a binomial expansion: &
E$'&
D
H
&87
& G F
#hi g
4
& G
&A7
Hkjl
&87
& Vm G
4
n
&A7
p G
HMLQLL
Ro
It is then easy to show that the first origin.
4 Z
derivatives are all zero at the
2.4.2 Transfer function of Butterworth low- filter E$'&
E$'&
Since is derived from operation can also be done, i.e.
rq
&us
%$'& 4T$2&
using the substitution
4T$v
16
ts
$'&
, the reverse
q
4w
x
*
+4a
or
4T$ y H
G
/
H
) y #
G
*R/
Thus the poles of
belong either to
+4a
or
H
) y #
*/
G
. The poles are given by:
4T$v G
&87
4
(){z V|}
J~€
dc
LQLL
4
Z
Thus 4T$v
()z V|‚}
J~„iUƒ …
&87
Since
$
( )
ƒi
, then we have the final result:
&87v†{‡„ˆT$5‰
H
H ZTŠ
&87
i.e. the poles have the same modulus and equi-spaced arguments. For example, for a fifth-order Butterworth low- filter (LPF), :
n
Z
lŒ‹
s
H
Ž2c
H
Z
‹
l'‹
H
‹
‹
2c
‹
n
‚†‘“’
LLL
o
Z
i.e. the poles are at: c ”
l ‹
‹
2
l
c •Q–
‹
‹
Ro
n
‹
—
in L.H. s-plane therefore stable
”
l2l ‹
‹
j
c j
o • – Q
‹
‹
j
o
‹
j
—
in R.H.S. s-plane therefore unstable 17
0.5
0
−0.5
−1.5
−1
−0.5
0
0.5
Figure 2.5: Stable poles of 5–th order Butterworth filter. +4
We want to design a stable filter. Since each unstable pole is we can let the stable ones be in , and the unstable ones in
&87+(
Therefore the poles of are as shown in Figure 2.5. Hence,
4
y
œ
H
“ y
D
*R/
’
H
O
&87V(
) J NV™=š
)
J mVm š
&87+( )
Œ˜
a stable pole,
+4a
&87V(
J ™VN=š
L
‹
95ž
y
H
*/
y
F D
*R/
’
H
‹ O 9
&87V(
) J p š
y
o j
H
*R/
) V›V š
y
F
*R/
or, multiplying out:
H
y
L j
oŸ j
*
H /
L n
oŸ j
y
*
H
/
L n
+ y
o„ j
*
H
/
L j
m y
oŸ j
*
H
/
› y
* /
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 angular spacing, with “half-angles” at each end. If is odd, one pole is real. G
Z
J c'c2c2c
Z
L
L
c'c2c2c
L
j
L
o„
L
n o
j
L o
j lž
n
L
c
n
L
L L
o„
o c2
L
j
ž
j
›
p
¢¡
C™
Ro
L
ž
o
o
n
n L
L
o
L
oŸ j
L
l
c2c'c2c
L
j
ln
c2c2c2c L
oŸ j
L
l
L
n
ž L
j
L
oŸ j
ž c
L
n
l
m
c'c2c2c L
j
ž
j
L
L
oŸ
l j
c2c2c2c
c2c2c2c L
j
l
l2l
l L
cj
o
18
ž l
j
c2 L
L
o
L
c2c2c2c L
lž
L j
c
j
c2c2c2c L
ž
n
L
l
n
L
c2c2c'c
l
Butterworth LPF coefficients for
£
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 frequency : Z
&A¥
¤
_
*R¦
¤
¨§
i.e.
*R¦ H
O2©
* /
4
¤
Z
G
*
O2© §
¦ *R/
¤dª
O2© §
or since usually
Z¬«
¤ *R¦
§
O2©
* /
Butterworth design – example Design a Butterworth LPF with at least 40 dB attenuation at a frequency of 1kHz and 3dB attenuation at = 500Hz. Answer 7
40dB
s ¤
&8¥
= 100;
c2c2c
Therefore Hence Z
Check: Substitute
$
and §
§
&87
c2c2c
rads/sec
c'c
O2©
Z¬«
J N
c
O'©
J N
o c
L j
l
l ž
which gives
¤
l
o
into the transfer function from the above table for
ž
E$
L
= 7 meets the specification Z
L j
. 19
4
L j
$® n
2.5 Equiripple Filters You may have noticed that the Butterworth response is monotone i.e. it has no ripple. If a certain amount of ripple is allowed in the -band and/or the stopband, filters which have a sharper cut-off than the Butterworth filter (for a given order ) can be designed. There are three types of equi-ripple filters: Z
Chebyshev Type I (equi-ripple in the -band) Chevyshev Type II (also known as inverse Chebyshev – equi-ripple in the stop-band) Elliptic (equi-ripple on both -band and stop-band)
2.5.1 Chebyshev polynomials The -order Chebyshev polynomial Z
¯°
¯°
’
c ¯ £
£
’
9
O
¯°
#KJ 9
Z
; and
.¯°
’
O
@°G
for
O
@5G
for
can be expressed as:
@5G
’
9“± Z
¯°
#KJ O
9±
¯²^
. These two expressions are equivalent in that the latter can be derived from the former and so the result is a single Chebyshev polynomial which applies for . Alternatively, can be expressed as a polynomial in , which can be evaluated for any : c
¯
£
£M³
¯
.¯°
¯
@5G
.¯°
¯° @
’
O
i.e.
¯° @
’
.’ O
J @
Ϋ
9
’ O
9
20
¯
9
where
9
¯°
#KJ O
´
´
4
’
O
#KJ
¯
9
¯
4
’
Obviously, the same results are also valid with the Generally, .¯° @°G
}
.¯° H
J
’
O
#KJ
@5G
9
´ H
Z
’ H
O
4
9
O
function.
9“±
´
’
Z
’ ´
O
´
O
9
9
¯
.¯°
Z
@5G
Thus:
.¯° }
@°G
¯
.¯°A4
J
¯°
@°G
#KJ
@5G
¯°
is a recurrence relationship which allows the computation of two previous polynomials. Using the recurrence relationship, @
.¯° @
m
.¯°
¯
.¯°µ4
¯ @
¯°
¯
4
A4¶¯
¯8
¯
4
¯°A4
¯
@
¯8
¯
4
¯ j
l
4
¯
m
l 4
¯
j
from the
J
J @
¯°h4
.¯°
@
}
@5G
H
.¯°
The graphs of Fig. 2.6 below show that, for large , diverges very rapidly for ; for example, . This is just the kind of behaviour we need for a good filter function. For a Chebyshev Type I filter (i.e. equi-ripple in the -band): ¯hv^
m
@
L
Z
L
n
n
j
& &
S
† L
:
D
@°G
¸·
F
&87 @
E$'& R
&87
G
· H
@IZ
*
* /
Chebyshev type I – notes ·
c
·
W
W
sets the ripple amplitude in the ripple 1. The parameter band which is defined as . Since, for will fluctuate between and .
c
&
&87
£
'E$'&
£
\
·»¼c
L
c
&A7“ c
&
£
l2l
} º i J Ÿ
c
2. At
&
dc
@ G
¯°
c
O2©
§
D
’
9
¾
L
J N
O
l
’ Z
4
O 9
#KJ
21
c
{F
c
L
£
G
J
e.g. with , the amplitude will vary between and or 1dB ripple since . n
£¹@
\
J JV½ p
, ie.
c
l L
T (x)
T (x)
1
2
2
8 6
1
4 0 2 −1
0
−2 −2
−1
0
1
−2 −2
2
−1
0
x T (x) 3
100
20
80
10
60
0
40
−10
20
−20
0 −1
2
1
2
4
30
−30 −2
1
x T (x)
0
1
−20 −2
2
−1
0
x
x
Figure 2.6: Chebyshev functions. ie. 3.
&87
@
, in this case, is not the
By definition,
&
VÀÁ
Žc
¯°
if
G
-v¿
is odd and 1 if Z
is given by:
·
@ G
* ÂUÃÄ
*R/
&¼^d&87‚
is even
cut-off frequency.
j
ie.
Z
H
·
Å
@
* ÂUÃ=Ä
*R/ G
which means that
&
E$'& R
VÀÁ
^f&A7
, since
· W
4. For gets large and decreases monotonically, as for the Butterworth LPF, but much more rapidly. @
G
22
(a) 1
|G(ω)|
0.8 n=1
0.6 0.4 n=4
0.2 0
0
0.2
0.4
0.6
0.8
1
1.2
ω
1.4
1.6
1.8
2
Figure 2.7: Response of Chebyshev type I filter. The lower limit of ripple in the -band is . Here . >¢Æ
·
H
·¾Mc
L
c
l2l
n
2.5.2 Transfer function of Chebyshev Type I LPF
E$'&
As before, put
E$'& 4T$'& q
Z
·»
*
kÇ
· $
#KJ
) y #
*R/
. Of the
/
@IZ
roots, assign the stable ones Z
J
j„È
4T$v
4T$v @
) y #
@5G
· H
The poles are given by to . Example
4 $v
4w
&6s
, and let
4T$v
&87
&A7
&87
&87
A4
Mc
•Q– ‹=‹
, y8Ê Ë /
} b
Select as stable poles
—
ig Ì #
0
)
Íg Î {
4
Æ
+4
‹=‹
, yAÊ Ë /
x $
#KJ
dÇ $
Žc
H
•Q–
Ç J
&87
j
” É
H
&87
gEÏ g%Ð
J &A7
· $
&87 j
and
&A7 j
dÇ $
H
H
” É
$
j
&87
i.e. 2 cubics
4
J ›
m
b #
ig Ì Q
— }
)
Íg Î
Æ
gEÏ g%Ð
V&87
J p
Therefore
Ñ
| b
J
4 œ
y
D
H
*
23
F y
/
D
H
y J
*
/
H
y
* /
F
(factorised form)
(Note
Ò
J
J
for complex pair – it was
m
Hence
for Butterworth filter)
H
J
y
*R/
H
y
H
*R/
(polynomial form)
y
*R/
In general, the poles lie on an ellipse (this is only meaningful for ; if , an ellipse can be drawn through any set of poles). A large means a larger ripple, narrower ellipse and less damped complex poles, ie. a more oscillatory impulse or step response but a steeper cut-off. Z¼Ó
ZÔ£
·
0.3 0.2 0.1 0 −0.1 −0.2 −0.3 0.4
0.6
0.8
1
1.2
Figure 2.8: Stable poles of 5–th order Chebyshev Type I filter.
2.5.3 Design of Chebyshev Type I LPFs If a filter is to have an attenuation _
*
at a frequency ¤
¦
&8¥
:
¤
H
·
*
@
¦
* G
/
&8¥
ie. ¤
M·
4 @
G
&8¥
M·
&A7
24
’ O
9±
Z
’ O
#KJ 9“±
&87
n
&A¥ ’
i.e.
O
’
9“± Z
9±
’
or
¤ ·
&87
\
#KJ O
Õ i
9“± ’
Z
#KJ
º *¦
#KJ O
4
Æ
#KJ O
9“± *
/
Example Design a Chebyshev Type I LPF to meet the following specifications: Maximum ripple of 1dB in -band from 0 to 50Hz 3dB cut-off frequency at
= 65 Hz
Attenuation of at least 40 dB for
250 Hz
ÖÓ
Answer
1. As shown on page 21, 1dB ripple corresponds to a value of &
2. 3dB cut-off:
VÀÁ
c
rads/sec. Minimum
j ’
#KJ O
c2c
x
#KJ
J N
×àßÛá
Í
#KJ ~
¨ÜÝ
ie.
L
ž
c n
l2l ·
for .
given by: Z
j
. Minimum
¤
Þ
Z
L
9±
L
meets the 3 dB cut-off requirement.
×ÙØÛÚ
3. For 40dB attenuation,
J › N ½ V N ™V™
9± ’
Z
Therefore
#KJ O
c
’ O
#KJ 9“±
c L
n
c
Z
given by: l2l L
c o
o
n
meets the overall specification. 25
2.5.4 Chebyshev Type II low- filters These exhibit equiripple behaviour in the stop-band but have the same behaviour in the -band. This beas a Butterworth filter (ie. maximally flat around 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 as poles in the left-half -plane. dc
&
%$2& R T
H
where
& É
·
i z Ëã Ë / i Ëã … z Ë ~
…
â â
~
= 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 -band edge is at where and the stop-band edge is at where .
&
&
&
_
Õ
É
|G(ω )|
J
&87
|G(ω )|
2
1/(1+ ε2)
2
2
1/A
1/A ω
ωc ωr
ω
ωc ωr
Type I
Type II
Figure 2.9: Response of Chebyshev type I, II filters. Thus, for
: n
Z
D
with zeroes at
J i J }ټ
i
2
1/(1+ ε2)
[
dÇ
H
J
$'& J
y
*R/
,
Ç
H
H
y
F D
* y
*/
g H
$'&
26
H
y *R/
y
* H
F
i
m
y *R/
m H
›
y *R/
›
2.5.5 Design of Chebyshev Type II LPFs Example Design a Chebyshev Type II LPF to meet the following specifications: 3dB cut-off frequency at 50 Hz Attenuation of at least 50dB for
100Hz.
ÔÓ
Answer
1. For Chebyshev Type II LPFs, the -band behaviour is the same as that of a Butterworth LPF; ie. . Therefore which gives
&87
&
[
VÀÁ
*
·ä
J
/
J i J }ټ
2. &
When
&
_ T
É
¤
Since
@ G
H
·
â
i z Ë ã ~ Ë / i z J ~ … â …
for all , the expression reduces to: Z
_
*
ã
· H
*
@
ã
* G
/
Therefore, from section 2.5.3, ’ Z
O
#KJ 9“±
’ O
#KJ 9“±
Z
×àßæå
*
·¾
in this case
ã
*R/
’
ig
For an attenuation of 50dB,
ie.
4
¤
›
= 10 and thus:
O
¤
#KJ ’ O
c
4 ›
9“±
#KJ
ig
l
L
ž
9±
meets the overall specification. 27
2.5.6 Elliptic low- filters Elliptic filters3 have a magnitude response which is equi-ripple in both the band and the stop-band. Elliptic filters are optimum in the sense that, for a given order and for given ripple specifications, no other filter achieves a faster transition between the -band and the stop-band – ie. has a narrower transition bandwidth. The magnitude-squared response of a low- elliptic filter is of the form: E$'& R Y
ç
· H
ça
G
*
‚èT
*R/
èY
*
è
where is called a Chebyshev rational function and scribing the ripple properties of G
*
/
ç
G
èY
*
*
/
is a parameter de-
L
Figure 2.10 shows the magnitude-squared response of a typical elliptic low filter. The frequencies represent the edges of the - and stop-bands and it is noted that the cut-off frequency is given as . The latter, once again, is not the 3dB point though. &
œ
Ì
y
&87
&
Æ
&
y
œ
2.5.7 Design of Elliptic Filters In order to design an elliptic LPF with arbitrary attenuations in both -band and stop-band, three of the four parameters: filter order
Z ·
in-band loss or ripple
out-of-band loss or attenuation transition ratio
ç
*‘é *
+W Ê
¤
can be chosen and the fourth parameter is uniquely defined.
ç
èY
*
The theory behind the determination of the function involves an understanding of Jacobian elliptic functions, which are beyond the scope of this G
3
also known as Cauer, or Darlington, filters
28
*
/
|G(ω )|
2
1/(1+ ε2)
2
1/A
ωp ωs
ω
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 ive, active and digital filters). You should , however, that for any given specification, the elliptic filter will always be of lower order than any other.
29
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