Comb Filters
? The simple filters discussed so far are
characterized either by a single passband
and/or a single stopband
? There are applications where filters with
multiple passbands and stopbands are
required
? The comb filter is an example of such filters
Comb Filters
? In its most general form,a comb filter has a
frequency response that is a periodic
function of w with a period 2p/L,where L is
a positive integer
? If H(z) is a filter with a single passband
and/or a single stopband,a comb filter can
be easily generated from it by replacing
each delay in its realization with L delays
resulting in a structure with a transfer
function given by )()( LzHzG ?
Comb Filters
? If exhibits a peak at,then
will exhibit L peaks at,
in the frequency range
? Likewise,if has a notch at,
then will have L notches at,
in the frequency range
? A comb filter can be generated from either
an FIR or an IIR prototype filter
|)(| wjeH
|)(| wjeH
|)(| wjeG
|)(| wjeGpw
ow
Lkp /w
Lko /w
10 ??? Lk
10 ??? Lk
p?w? 20
p?w? 20
Comb Filters
? For example,the comb filter generated from
the prototype lowpass FIR filter
has a transfer function
? has L notches
at w = (2k+1)p/L and L
peaks at w = 2p k/L,
)( 121 1 ?? z
?)( zH 0
)()()( LL zzHzG ???? 12100
10 ??? Lk,in the
frequency range
p?w? 20
|)(| 0 wjeG
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
w / p
M
a
gni
t
ude
C om b f i l t e r f r om l ow pa s s pr ot ot ype
Comb Filters
? For example,the comb filter generated from
the prototype highpass FIR filter
has a transfer function
? has L peaks
at w = (2k+1)p/L and L
notches at w = 2p k/L,
|)(| 1 wjeG
)( 121 1 ?? z
?)( zH 1
)()()( LL zzHzG ???? 12111
10 ??? Lk,in the
frequency range
p?w? 20
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
w / p
M
a
gni
t
ude
C om b f i l t e r f r om hi ghpa s s pr ot ot ype
Comb Filters
? Depending on applications,comb filters
with other types of periodic magnitude
responses can be easily generated by
appropriately choosing the prototype filter
? For example,the M-point moving average
filter
has been used as a prototype
)()( 11
1
?
?
?
??
zM
z MzH
Comb Filters
? This filter has a peak magnitude at w = 0,
and notches at,
? The corresponding comb filter has a transfer
function
whose magnitude has L peaks at,
and notches at
,
1?M M/2 ?p?w 11 ??? M?
)()( L
ML
zM
zzG
?
?
?
??
1
1
Lk /2 p?w
10 ??? Lk )( 1?ML LMk /2 p?w
)( 11 ??? MLk
Allpass Transfer Function
Definition
? An IIR transfer function A(z) with unity
magnitude response for all frequencies,i.e.,
is called an allpass transfer function
? An M-th order causal real-coefficient
allpass transfer function is of the form
w?w a l lf o r,1|)(| 2jeA
M
M
M
M
MM
MM
M zdzdzd
zzdzddzA
???
?
?
????
?
????
??????
1
1
1
1
1
1
1
1
1,..
...
)(
Allpass Transfer Function
? If we denote the denominator polynomial of
as,
then it follows that can be written as,
? Note from the above that if is a pole
of a real coefficient allpass transfer function,
then it has a zero at
)(zDM)(zAM
MMMMM zdzdzdzD ????? ????? 11111,..)(
)(zAM
)(
)()(
zD
zDz
M M M
MzA 1????
?? jrez
??? jr ez 1
Allpass Transfer Function
? The numerator of a real-coefficient allpass
transfer function is said to be the mirror-
image polynomial of the denominator,and
vice versa
? We shall use the notation to denote
the mirror-image polynomial of a degree-M
polynomial,i.e.,
)( zDM~
)( zDM
)()( zDzzD MMM ??~
Allpass Transfer Function
? The expression
implies that the poles and zeros of a real-
coefficient allpass function exhibit mirror-
image symmetry in the z-plane
)(
)()(
zD
zDz
M M M
MzA 1????
321
321
3 2.018.04.01
4.018.02.0)(
???
???
???
?????
zzz
zzzzA
-1 0 1 2 3
- 1.5
-1
- 0.5
0
0.5
1
1.5
R e a l P a r t
I
m
a
gi
na
r
y P
a
r
t
Allpass Transfer Function
? To show that we observe that
? Therefore
? Hence
)(
)(1
1)( ???
?
zD
zDz
M M M
MzA
)(
)(
)(
)(1
1
1)()(
?
????
zD
zDz
zD
zDz
MM M M
M
M
MMzAzA
1|)(| ?wjM eA
1)()(|)(| 12 ?? w??w jezMMjM zAzAeA
Allpass Transfer Function
? Now,the poles of a causal stable transfer
function must lie inside the unit circle in the
z-plane
? Hence,all zeros of a causal stable allpass
transfer function must lie outside the unit
circle in a mirror-image symmetry with its
poles situated inside the unit circle
Allpass Transfer Function
? Figure below shows the principal value of
the phase of the 3rd-order allpass function
? Note the discontinuity by the amount of 2p
in the phase q(w)
321
321
3 2.018.04.01
4.018.02.0)(
???
???
???
?????
zzz
zzzzA
0 0.2 0.4 0.6 0.8 1
-4
-2
0
2
4
w / p
P
ha
s
e
,de
gr
e
e
s
P r i nc i pa l va l ue of pha s e
Allpass Transfer Function
? If we unwrap the phase by removing the
discontinuity,we arrive at the unwrapped
phase function indicated below
? Note,The unwrapped phase function is a
continuous function of w
)(wqc
0 0.2 0.4 0.6 0.8 1
- 10
-8
-6
-4
-2
0
w / p
P
ha
s
e
,de
gr
e
e
s
U nw r a ppe d ph a s e
Allpass Transfer Function
? The unwrapped phase function of any
arbitrary causal stable allpass function is a
continuous function of w
Properties
? (1) A causal stable real-coefficient allpass
transfer function is a lossless bounded real
(LBR) function or,equivalently,a causal
stable allpass filter is a lossless structure
Allpass Transfer Function
? (2) The magnitude function of a stable
allpass function A(z) satisfies,
? (3) Let t(w) denote the group delay function
of an allpass filter A(z),i.e.,
??
?
?
?
??
??
??
1f o r1
1f o r1
1f o r1
z
z
z
zA
,
,
,
)(
)]([)( wq??wt w cdd
Allpass Transfer Function
? The unwrapped phase function of a
stable allpass function is a monotonically
decreasing function of w so that t(w) is
everywhere positive in the range 0 < w < p
? The group delay of an M-th order stable
real-coefficient allpass transfer function
satisfies,
)(wqc
p?w? wt
p
Md
0
)(
Allpass Transfer Function
A Simple Application
? A simple but often used application of an
allpass filter is as a delay equalizer
? Let G(z) be the transfer function of a digital
filter designed to meet a prescribed
magnitude response
? The nonlinear phase response of G(z) can be
corrected by cascading it with an allpass
filter A(z) so that the overall cascade has a
constant group delay in the band of interest
Allpass Transfer Function
? Since,we have
? Overall group delay is the given by the sum
of the group delays of G(z) and A(z)
1|)(| ?wjeA
|)(||)()(| www ? jjj eGeAeG
G(z) A(z)
Minimum-Phase and Maximum-
Phase Transfer Functions
? Consider the two 1st-order transfer functions,
? Both transfer functions have a pole inside the
unit circle at the same location and are
stable
? But the zero of is inside the unit circle
at,whereas,the zero of is at
situated in a mirror-image symmetry
11121 ???? ???? bazHzH azbzaz bz,,)(,)(
az ??
bz ??
bz 1??
)(zH1
)(zH 2
Minimum-Phase and Maximum-
Phase Transfer Functions
? Figure below shows the pole-zero plots of
the two transfer functions
)(
1 zH )(2 zH
Minimum-Phase and Maximum-
Phase Transfer Functions
? However,both transfer functions have an
identical magnitude function as
? The corresponding phase functions are
1122111 ?? ?? )()()()( zHzHzHzH
w? w?w? w?w ?? c o ss i n1c o ss i n11 t a nt a n)](a r g [ abjeH
w? w?w? w?w ?? c o ss i n1c o s1 s i n12 t a nt a n)](a r g [ abbjeH
Minimum-Phase and Maximum-
Phase Transfer Functions
? Figure below shows the unwrapped phase
responses of the two transfer functions for
a = 0.8 and b = 5.0?
0 0.2 0.4 0.6 0.8 1
-4
-3
-2
-1
0
1
2
w / p
P
ha
s
e
,de
gr
e
e
s
H
1
( z )
H
2
( z )
Minimum-Phase and Maximum-
Phase Transfer Functions
? From this figure it follows that has
an excess phase lag with respect to
? Generalizing the above result,we can show
that a causal stable transfer function with all
zeros outside the unit circle has an excess
phase compared to a causal transfer
function with identical magnitude but
having all zeros inside the unit circle
)(2 zH
)(1 zH
Minimum-Phase and Maximum-
Phase Transfer Functions
? A causal stable transfer function with all
zeros inside the unit circle is called a
minimum-phase transfer function
? A causal stable transfer function with all
zeros outside the unit circle is called a
maximum-phase transfer function
? Any nonminimum-phase transfer function
can be expressed as the product of a
minimum-phase transfer function and a
stable allpass transfer function
Complementary Transfer
Functions
? A set of digital transfer functions with
complementary characteristics often finds
useful applications in practice
? Four useful complementary relations are
described next along with some applications
Complementary Transfer
Functions
Delay-Complementary Transfer Functions
? A set of L transfer functions,,
,is defined to be delay-
complementary of each other if the sum of
their transfer functions is equal to some
integer multiple of unit delays,i.e.,
where is a nonnegative integer
)}({ zH i
10 ??? Li
0,)(
1
0
???? ?
?
?
? on
L
i
i zzH
on
Complementary Transfer
Functions
? A delay-complementary pair
can be readily designed if one of the pairs is
a known Type 1 FIR transfer function of
odd length
? Let be a Type 1 FIR transfer function
of length M = 2K+1
? Then its delay-complementary transfer
function is given by
)}(),({ 10 zHzH
)()( 01 zHzzH K ?? ?
)(0 zH
Complementary Transfer
Functions
? Let the magnitude response of be
equal to in the passband and less than
or equal to in the stopband where and
are very small numbers
? Now the frequency response of can be
expressed as
where is the amplitude response
)(0 zH
p??1
s? p?
s?
)(0 zH
)()( 00 w? w?w HeeH jKj~
)(0 wH~
Complementary Transfer
Functions
? Its delay-complementary transfer function
has a frequency response given by
? Now,in the passband,
and in the stopband,
? It follows from the above equation that in the
stopband,and in the
passband,
)(1 zH
)](1[)()( 011 w??w? w?w?w HeHeeH jKjKj~ ~
,1)(1 0 pp H ???w??? ~
ss H ??w??? )(0
~
pp H ??w??? )(1
~
ss H ???w??? 1)(1 1
~
Complementary Transfer
Functions
? As a result,has a complementary
magnitude response characteristic to that of
with a stopband exactly identical to
the passband of,and a passband that
is exactly identical to the stopband of
? Thus,if is a lowpass filter,will
be a highpass filter,and vice versa
)(1 zH
)(0 zH
)(0 zH
)(1 zH)(0 zH
)(0 zH
Complementary Transfer
Functions
? The frequency at which
the gain responses of both filters are 6 dB
below their maximum values
? The frequency is thus called the 6-dB
crossover frequency
ow
ow
5.0)()( 10 ?w?w oo HH ~ ~
Complementary Transfer
Functions
? Example - Consider the Type 1 bandstop
transfer function
? Its delay-complementary Type 1 bandpass
transfer function is given by
)45541()1()( 12108424264 1 ?????? ??????? zzzzzzzH BS
)45541()1( 121084242641 ?????? ??????? zzzzzz
)()( 10 zHzzH BSBP ?? ?
Complementary Transfer
Functions
? Plots of the magnitude responses of
and are shown below
)( zH BS
)( zH BP
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w / p
M
a
gni
t
ude
H
BS
( z ) H
BP
( z )
Complementary Transfer
Functions
Allpass Complementary Filters
? A set of M digital transfer functions,,
,is defined to be allpass-
complementary of each other,if the sum of
their transfer functions is equal to an allpass
function,i.e.,
)}({ zH i
10 ??? Mi
)()(
1
0
zAzH
M
i
i ??
?
?
Complementary Transfer
Functions
Power-Complementary Transfer Functions
? A set of M digital transfer functions,,
,is defined to be power-
complementary of each other,if the sum of
their square-magnitude responses is equal to
a constant K for all values of w,i.e.,
)}({ zH i
10 ??? Mi
w??
?
?
w a llf o r,)(
1
0
2 KeHM
i
j
i
Complementary Transfer
Functions
? By analytic continuation,the above
property is equal to
for real coefficient
? Usually,by scaling the transfer functions,
the power-complementary property is
defined for K = 1
)(zH i
w??
?
?
? a llf o r,)()(
1
0
1 KzHzH
M
i
ii
Complementary Transfer
Functions
? For a pair of power-complementary transfer
functions,and,the frequency
where,is
called the cross-over frequency
? At this frequency the gain responses of both
filters are 3-dB below their maximum
values
? As a result,is called the 3-dB cross-over
frequency
ow
ow
)(0 zH )(1 zH
5.0|)(||)(| 2120 ?? ww oo jj eHeH
Complementary Transfer
Functions
? Example - Consider the two transfer functions
and given by
where and are stable allpass
transfer functions
? Note that
? Hence,and are allpass
complementary
)(0 zH )(1 zH
)]()([)( 10210 zAzAzH ??
)(0 zA )(1 zA
)]()([)( 10211 zAzAzH ??
)()()( 010 zAzHzH ??
)(0 zH )(1 zH
Complementary Transfer
Functions
? It can be shown that and are
also power-complementary
? Moreover,and are bounded-
real transfer functions
)(0 zH )(1 zH
)(0 zH )(1 zH
Complementary Transfer
Functions
Doubly-Complementary Transfer Functions
? A set of M transfer functions satisfying both
the allpass complementary and the power-
complementary properties is known as a
doubly-complementary set
Complementary Transfer
Functions
? A pair of doubly-complementary IIR
transfer functions,and,with a
sum of allpass decomposition can be simply
realized as indicated below
)(0 zH )(1 zH
?
?)(1 zA
)(0 zA)(zX )(0 zY
)(1 zY1?
2/1
)( )(00 )( zX zYzH ? )( )(11 )( zX zYzH ?
Complementary Transfer
Functions
? Example - The first-order lowpass transfer
function
can be expressed as
where
??????? ?
?
??
???
1
1
1
1
2
1)(
z
z
LP zH
)]()([)( 1021
1
121 1
1 zAzAzH
z
z
LP ???
??
?
?? ?
??
????
?
?
1
1
1 1)( ?
?
??
????
z
zzA
,)( 10 ?zA
Complementary Transfer
Functions
? Its power-complementary highpass transfer
function is thus given by
? The above expression is precisely the first-
order highpass transfer function described
earlier
???????? ?
?
??
?????
1
1
1
1211021 )]()([)(
z
z
HP zAzAzH
??????? ?
?
??
???
1
1
1
1
2
1
z
z
Complementary Transfer
Functions
? Figure below demonstrates the allpass
complementary property and the power
complementary property of and )( zH LP
)( zH HP
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w / p
M
a
gni
t
ude
|H
HP
(e
j
w
)|
|H
LP
(e
j
w
)|
|H
LP
(e
j
w
) + H
HP
(e
j
w
)|
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w / p
M
a
gni
t
ude
|H
HP
(e
j
w
)|
2
|H
LP
(e
j
w
)|
2
|H
LP
(e
j
w
)|
2
+ | H
HP
(e
j
w
)|
2
Complementary Transfer
Functions
Power-Symmetric Filters
? A real-coefficient causal digital filter with a
transfer function H(z) is said to be a power-
symmetric filter if it satisfies the condition
where K > 0 is a constant
KzHzHzHzH ???? ?? )()()()( 11
Complementary Transfer
Functions
? It can be shown that the gain function G(w) of a
power-symmetric transfer function at w = p is
given by
? If we define,then it follows from
the definition of the power-symmetric filter that
H(z) and G(z) are power-complementary as
c o n s t a n ta)()()()( 11 ?? ?? zGzGzHzH
)()( zHzG ??
dBK 3lo g10 10 ?
Complementary Transfer
Functions
? If a power-symmetric filter has an FIR
transfer function H(z) of order N,then the
FIR digital filter with a transfer function
is called a conjugate quadratic filter of
H(z) and vice-versa
)()( 11 ??? zHzzG
Complementary Transfer
Functions
? It follows from the definition that G(z) is
also a power-symmetric causal filter
? It also can be seen that a pair of conjugate
quadratic filters H(z) and G(z) are also
power-complementary
Complementary Transfer
Functions
? Example - Let
? We form
? Thus,H(z) is a power-symmetric transfer function
)3621)(3621( 32321 zzzzzz ??????? ???
)()()()( 11 ?? ??? zHzHzHzH
321 3621)( ??? ???? zzzzH
)3621)(3621( 32321 zzzzzz ??????? ??? )345043( 313 ?? ????? zzzz
1 0 0)345043( 313 ??????? ?? zzzz
Algebraic Stability Test
? We have shown that the BIBO stability of a
causal rational transfer function requires
that all its poles be inside the unit circle
? For very high-order transfer functions,it is
very difficult to determine the pole
locations analytically
? Root locations can of course be determined
on a computer by some type of root finding
algorithms
Algebraic Stability Test
The Stability Triangle
? For a 2nd-order transfer function the
stability can be easily checked by
examining its denominator coefficients
Algebraic Stability Test
? Let
denote the denominator of the transfer
function
? In terms of its poles,D(z) can be expressed
as
? Comparing the last two equations we get
22111)( ?? ??? zdzdzD
2211211211 )(1)1)(1()( ???? ????????????? zzzzzD
212211 ),( ???????? dd
Algebraic Stability Test
? The poles are inside the unit circle if
? Now the coefficient is given by the
product of the poles
? Hence we must have
? It can be shown that the second coefficient
condition is given by
1||,1|| 21 ????
2d
1|| 2 ?d
21 1|| dd ??
Algebraic Stability Test
? The region in the ( )-plane where the
two coefficient condition are satisfied,
called the stability triangle,is shown below
21,dd
Stability region
Algebraic Stability Test
? Example - Consider the two 2nd-order
bandpass transfer functions designed
earlier,
21
2
3 7 6 3 817 3 4 3 4 2 401
11 8 8 1 90
??
?
??
???
zz
zzH
BP,..)('
21
2
7 2 6 5 4 2 5 305 3 3 5 3 101
11 3 6 7 30
??
?
??
??
zz
zzH
BP,..)("
Algebraic Stability Test
? In the case of,we observe that
? Since here,is unstable
? On the other hand,in the case of,we
observe that
? Here,and,and hence
is BIBO stable
)(' zH BP
3 7 6 3 8 1 9.1,7 3 4 3 4 2 4.0 21 ??? dd
7 2 6 5 4 2 5 2 8.0,5 3 3 5 3 0 9 8.0 21 ??? dd
)(' zH BP1|| 2 ?d
)(" zH BP
21 1|| dd ??1|| 2 ?d
)(" zH BP