frequency selective filters lecture
TRANSCRIPT
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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.
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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).
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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.
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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
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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
)
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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
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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
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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
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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
.
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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
)
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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
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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.
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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)
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(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
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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.
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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
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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.
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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
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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