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Copyright © 2001, S. K. Mitra1
Comb FiltersComb Filters
• The simple filters discussed so far arecharacterized either by a single passbandand/or a single stopband
• There are applications where filters withmultiple passbands and stopbands arerequired
• The comb filter is an example of suchfilters
Copyright © 2001, S. K. Mitra2
Comb FiltersComb Filters• In its most general form, a comb filter has a
frequency response that is a periodicfunction of ω with a period 2π/L, where L isa positive integer
• If H(z) is a filter with a single passbandand/or a single stopband, a comb filter canbe easily generated from it by replacingeach delay in its realization with L delaysresulting in a structure with a transferfunction given by )()( LzHzG =
Copyright © 2001, S. K. Mitra3
Comb FiltersComb Filters
• If exhibits a peak at , thenwill 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
|)(| ωjeH
|)(| ωjeH|)(| ωjeG
|)(| ωjeGpω
oω
Lkp /ω
Lko /ω
10 −≤≤ Lk
10 −≤≤ Lk
π<ω≤ 20
π<ω≤ 20
Copyright © 2001, S. K. Mitra4
Comb FiltersComb Filters• For example, the comb filter generated from
the prototype lowpass FIR filter has a transfer function
• has L notchesat ω = (2k+1)π/L and Lpeaks at ω = 2π k/L,
)( 121 1 −+ z
=)(zH0
)()()( LL zzHzG −+== 121
00
10 −≤≤ Lk , in thefrequency range
π<ω≤ 20
|)(| 0ωjeG
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
Comb filter from lowpass prototype
Copyright © 2001, S. K. Mitra5
Comb FiltersComb Filters• For example, the comb filter generated from
the prototype highpass FIR filter has a transfer function
• has L peaksat ω = (2k+1)π/L and Lnotches at ω = 2π k/L,
|)(| 1ωjeG
)( 121 1 −− z
=)(zH1
)()()( LL zzHzG −−== 121
11
10 −≤≤ Lk , in thefrequency range
π<ω≤ 200 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
Comb filter from highpass prototype
Copyright © 2001, S. K. Mitra6
Comb FiltersComb Filters
• Depending on applications, comb filterswith other types of periodic magnituderesponses can be easily generated byappropriately choosing the prototype filter
• For example, the M-point moving averagefilter
has been used as a prototype)(
)( 111
−
−
−−=
zMz M
zH
Copyright © 2001, S. K. Mitra7
Comb FiltersComb Filters• This filter has a peak magnitude at ω = 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 lπ=ω 11 −≤≤ Ml
)()( L
ML
zMzzG −
−
−−=1
1
Lk/2π=ω10 −≤≤ Lk )( 1−ML
LMk/2π=ω )( 11 −≤≤ MLk
Copyright © 2001, S. K. Mitra8
Allpass Allpass Transfer FunctionTransfer FunctionDefinition• 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
ω=ω allfor,1|)(| 2jeA
MM
MM
MMMM
M zdzdzdzzdzddzA −+−
−−
−+−−−
++++++++±= 1
11
1
11
11
1 ...
...)(
Copyright © 2001, S. K. Mitra9
AllpassAllpass Transfer Function Transfer Function• If we denote the denominator polynomials
of as :
then it follows that can be written as:
• Note from the above that if is apole of a real coefficient allpass transferfunction, then it has a zero at
)(zDM)(zAMM
MM
MM zdzdzdzD −+−−
− ++++= 11
111 ...)(
)(zAM
)()()(
zDzDz
MM
MM
zA1−−
±=φ= jrez
φ−= jr ez 1
Copyright © 2001, S. K. Mitra10
AllpassAllpass Transfer Function Transfer Function
• The numerator of a real-coefficient allpasstransfer function is said to be the mirror-image polynomial of the denominator, andvice versa
• We shall use the notation to denotethe mirror-image polynomial of a degree-Mpolynomial , i.e.,
)(zDM~
)(zDM
)()( zDzzD MM
M−=
~
Copyright © 2001, S. K. Mitra11
AllpassAllpass Transfer Function Transfer Function• The expression
implies that the poles and zeros of a real-coefficient allpass function exhibit mirror-image symmetry in the z-plane
)()()(
zDzDz
MM
MM
zA1−−
±=
321
3213 2.018.04.01
4.018.02.0)( −−−
−−−
−+++++−=
zzzzzzzA
-1 0 1 2 3
-1.5
-1
-0.5
0
0.5
1
1.5
Real Part
Imag
inar
y P
art
Copyright © 2001, S. K. Mitra12
AllpassAllpass Transfer Function Transfer Function
• To show that we observe that
• Therefore
• Hence
)()(1
1)( −±=−zD
zDzM
M
MM
zA
)()(
)()(1
1
1)()( −
−−=−
zDzDz
zDzDz
MMM
MM
M
MM
zAzA
1|)(| =ωjM eA
1)()(|)(| 12 == ω=−ω
jezMMj
M zAzAeA
Copyright © 2001, S. K. Mitra13
AllpassAllpass Transfer Function Transfer Function
• Now, the poles of a causal stable transferfunction must lie inside the unit circle in thez-plane
• Hence, all zeros of a causal stable allpasstransfer function must lie outside the unitcircle in a mirror-image symmetry with itspoles situated inside the unit circle
Copyright © 2001, S. K. Mitra14
AllpassAllpass Transfer Function Transfer Function• Figure below shows the principal value of
the phase of the 3rd-order allpass function
• Note the discontinuity by the amount of 2πin the phase θ(ω)
321
3213 2.018.04.01
4.018.02.0)( −−−
−−−
−+++++−=
zzzzzzzA
0 0.2 0.4 0.6 0.8 1-4
-2
0
2
4
ω/π
Pha
se, d
egre
es
Principal value of phase
Copyright © 2001, S. K. Mitra15
AllpassAllpass Transfer Function Transfer Function• If we unwrap the phase by removing the
discontinuity, we arrive at the unwrappedphase function indicated below
• Note: The unwrapped phase function is acontinuous function of ω
)(ωθc
0 0.2 0.4 0.6 0.8 1-10
-8
-6
-4
-2
0
ω/π
Pha
se, d
egre
es
Unwrapped phase
Copyright © 2001, S. K. Mitra16
AllpassAllpass Transfer Function Transfer Function
• The unwrapped phase function of anyarbitrary causal stable allpass function is acontinuous function of ω
Properties• (1) A causal stable real-coefficient allpass
transfer function is a lossless bounded real(LBR) function or, equivalently, a causalstable allpass filter is a lossless structure
Copyright © 2001, S. K. Mitra17
AllpassAllpass Transfer Function Transfer Function• (2) The magnitude function of a stable
allpass function A(z) satisfies:
• (3) Let τ(ω) denote the group delay functionof an allpass filter A(z), i.e.,
<>==><
1for11for11for1
zzz
zA,,,
)(
)]([)( ωθ−=ωτω cdd
Copyright © 2001, S. K. Mitra18
AllpassAllpass Transfer Function Transfer Function
• The unwrapped phase function of astable allpass function is a monotonicallydecreasing function of ω so that τ(ω) iseverywhere positive in the range 0 < ω < π
• The group delay of an M-th order stablereal-coefficient allpass transfer functionsatisfies:
)(ωθc
π=ω∫ ωτπ
Md0
)(
Copyright © 2001, S. K. Mitra19
AllpassAllpass Transfer Function Transfer FunctionA 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 prescribedmagnitude response
• The nonlinear phase response of G(z) can becorrected by cascading it with an allpassfilter A(z) so that the overall cascade has aconstant group delay in the band of interest
Copyright © 2001, S. K. Mitra20
AllpassAllpass Transfer Function Transfer Function
• Since , we have
• Overall group delay is the given by the sumof the group delays of G(z) and A(z)
1|)(| =ωjeA|)(||)()(| ωωω = jjj eGeAeG
G(z) A(z)
Copyright © 2001, S. K. Mitra21
Minimum-Phase and Maximum-Minimum-Phase and Maximum-Phase Transfer FunctionsPhase Transfer Functions
• Consider the two 1st-order transfer functions:
• Both transfer functions have a pole inside theunit circle at the same location and arestable
• But the zero of is inside the unit circleat , whereas, the zero of is at
situated in a mirror-image symmetry
11121 <<== +
+++ bazHzH az
bzazbz ,,)(,)(
az −=
bz −=
bz 1−=
)(zH1)(zH2
Copyright © 2001, S. K. Mitra22
Minimum-Phase and Maximum-Minimum-Phase and Maximum-Phase Transfer FunctionsPhase Transfer Functions
• Figure below shows the pole-zero plots ofthe two transfer functions
)(1 zH )(2 zH
Copyright © 2001, S. K. Mitra23
Minimum-Phase and Maximum-Minimum-Phase and Maximum-Phase Transfer FunctionsPhase Transfer Functions
• However, both transfer functions have anidentical magnitude function as
• The corresponding phase functions are11
221
11 == −− )()()()( zHzHzHzH
ω+ω−
ω+ω−ω −= cos
sin1cos
sin11 tantan)](arg[ ab
jeH
ω+ω−
ω+ω−ω −= cos
sin1cos1
sin12 tantan)](arg[ ab
bjeH
Copyright © 2001, S. K. Mitra24
Minimum-Phase and Maximum-Minimum-Phase and Maximum-Phase Transfer FunctionsPhase Transfer Functions
• Figure below shows the unwrapped phaseresponses of the two transfer functions fora = 0.8 and b = 5.0−
0 0.2 0.4 0.6 0.8 1-4
-3
-2
-1
0
1
2
ω/π
Pha
se, d
egre
es
H1(z)
H2(z)
Copyright © 2001, S. K. Mitra25
Minimum-Phase and Maximum-Minimum-Phase and Maximum-Phase Transfer FunctionsPhase Transfer Functions
• From this figure it follows that hasan excess phase lag with respect to
• Generalizing the above result, we can showthat a causal stable transfer function with allzeros outside the unit circle has an excessphase compared to a causal transferfunction with identical magnitude buthaving all zeros inside the unit circle
)(2 zH)(1 zH
Copyright © 2001, S. K. Mitra26
Minimum-Phase and Maximum-Minimum-Phase and Maximum-Phase Transfer FunctionsPhase Transfer Functions
• A causal stable transfer function with allzeros inside the unit circle is called aminimum-phase transfer function
• A causal stable transfer function with allzeros outside the unit circle is called amaximum-phase transfer function
• Any nonminimum-phase transfer functioncan be expressed as the product of aminimum-phase transfer function and astable allpass transfer function
Copyright © 2001, S. K. Mitra27
Complementary TransferComplementary TransferFunctionsFunctions
• A set of digital transfer functions withcomplementary characteristics often findsuseful applications in practice
• Four useful complementary relations aredescribed next along with some applications
Copyright © 2001, S. K. Mitra28
Complementary TransferComplementary TransferFunctionsFunctions
Delay-Complementary Transfer Functions• A set of L transfer functions, ,
, is defined to be delay-complementary of each other if the sum oftheir transfer functions is equal to someinteger multiple of unit delays, i.e.,
where is a nonnegative integer
)}({ zHi10 −≤≤ Li
0,)(1
0≠ββ= −
−
=∑ onL
ii zzH
on
Copyright © 2001, S. K. Mitra29
Complementary TransferComplementary TransferFunctionsFunctions
• A delay-complementary paircan be readily designed if one of the pairs isa known Type 1 FIR transfer function ofodd length
• Let be a Type 1 FIR transfer functionof length M = 2K+1
• Then its delay-complementary transferfunction is given by
)}(),({ 10 zHzH
)()( 01 zHzzH K −= −
)(0 zH
Copyright © 2001, S. K. Mitra30
Complementary TransferComplementary TransferFunctionsFunctions
• Let the magnitude response of beequal to in the passband and less thanor 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 zHpδ±1sδ pδ
sδ)(0 zH
)()( 00 ω= ω−ω HeeH jKj ~
)(0 ωH~
Copyright © 2001, S. K. Mitra31
Complementary TransferComplementary TransferFunctionsFunctions
• 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 inthe stopband, and in thepassband,
)(1 zH)](1[)()( 011 ω−=ω= ω−ω−ω HeHeeH jKjKj ~ ~
,1)(1 0 pp H δ+≤ω≤δ− ~
ss H δ≤ω≤δ− )(0~
pp H δ≤ω≤δ− )(1~
ss H δ+≤ω≤δ− 1)(1 1~
Copyright © 2001, S. K. Mitra32
Complementary TransferComplementary TransferFunctionsFunctions
• As a result, has a complementarymagnitude response characteristic to that of
with a stopband exactly identical tothe passband of , and a passband thatis exactly identical to the stopband of
• Thus, if is a lowpass filter, willbe a highpass filter, and vice versa
)(1 zH
)(0 zH
)(0 zH
)(1 zH)(0 zH
)(0 zH
Copyright © 2001, S. K. Mitra33
Complementary TransferComplementary TransferFunctionsFunctions
• The frequency at which
the gain responses of both filters are 6 dBbelow their maximum values
• The frequency is thus called the 6-dBcrossover frequency
oω
oω5.0)()( 10 =ω=ω oo HH ~~
Copyright © 2001, S. K. Mitra34
Complementary TransferComplementary TransferFunctionsFunctions
• Example - Consider the Type 1 bandstoptransfer function
• Its delay-complementary Type 1 bandpasstransfer function is given by
)45541()1()( 121084242641 −−−−−− +−++−+= zzzzzzzHBS
)45541()1( 121084242641 −−−−−− +++++−= zzzzzz
)()( 10 zHzzH BSBP −= −
Copyright © 2001, S. K. Mitra35
Complementary TransferComplementary TransferFunctionsFunctions
• Plots of the magnitude responses ofand are shown below
)(zHBS)(zHBP
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
HBS
(z) HBP
(z)
Copyright © 2001, S. K. Mitra36
Complementary TransferComplementary TransferFunctionsFunctions
Allpass Complementary Filters• A set of M digital transfer functions, ,
, is defined to be allpass-complementary of each other, if the sum oftheir transfer functions is equal to an allpassfunction, i.e.,
)}({ zHi10 −≤≤ Mi
)()(1
0zAzH
M
ii =∑
−
=
Copyright © 2001, S. K. Mitra37
Complementary TransferComplementary TransferFunctionsFunctions
Power-Complementary Transfer Functions• A set of M digital transfer functions, ,
, is defined to be power-complementary of each other, if the sum oftheir square-magnitude responses is equal toa constant K for all values of ω, i.e.,
)}({ zHi10 −≤≤ Mi
ω=∑−
=
ω allfor,)(1
0
2KeH
M
i
ji
Copyright © 2001, S. K. Mitra38
Complementary TransferComplementary TransferFunctionsFunctions
• By analytic continuation, the aboveproperty is equal to
for real coefficient• Usually, by scaling the transfer functions,
the power-complementary property isdefined for K = 1
)(zHi
ω=∑−
=
− allfor,)()(1
0
1 KzHzHM
iii
Copyright © 2001, S. K. Mitra39
Complementary TransferComplementary TransferFunctionsFunctions
• For a pair of power-complementary transferfunctions, and , the frequencywhere , iscalled the cross-over frequency
• At this frequency the gain responses of bothfilters are 3-dB below their maximumvalues
• As a result, is called the 3-dB cross-over frequency
oω
oω
)(0 zH )(1 zH5.0|)(||)(| 2
12
0 == ωω oo jj eHeH
Copyright © 2001, S. K. Mitra40
Complementary TransferComplementary TransferFunctionsFunctions
• Example - Consider the two transfer functions and given by
where and are stable allpasstransfer functions
• Note that• Hence, and are allpass
complementary
)(0 zH )(1 zH)]()([)( 102
10 zAzAzH +=
)(0 zA )(1 zA)]()([)( 102
11 zAzAzH −=
)()()( 010 zAzHzH =+)(0 zH )(1 zH
Copyright © 2001, S. K. Mitra41
Complementary TransferComplementary TransferFunctionsFunctions
• It can be shown that and arealso power-complementary
• Moreover, and are bounded-real transfer functions
)(0 zH )(1 zH
)(0 zH )(1 zH
Copyright © 2001, S. K. Mitra42
Complementary TransferComplementary TransferFunctionsFunctions
Doubly-Complementary Transfer Functions• A set of M transfer functions satisfying both
the allpass complementary and the power-complementary properties is known as adoubly-complementary set
Copyright © 2001, S. K. Mitra43
Complementary TransferComplementary TransferFunctionsFunctions
• A pair of doubly-complementary IIRtransfer functions, and , with asum of allpass decomposition can be simplyrealized as indicated below
)(0 zH )(1 zH
+
+)(1 zA
)(0 zA)(zX
)(0 zY
)(1 zY1−
2/1
)()(0
0 )( zXzYzH = )(
)(11 )( zX
zYzH =
Copyright © 2001, S. K. Mitra44
Complementary TransferComplementary TransferFunctionsFunctions
• Example - The first-order lowpass transferfunction
can be expressed as
where
= −
−
α−+α−
1
1
11
21)(
zz
LP zH
)]()([)( 1021
11
21
1
1zAzAzH
zz
LP +
= =
α−+α−+ −
−
1
1
1 1)( −
−
α−+α−=
zzzA,)( 10 =zA
Copyright © 2001, S. K. Mitra45
Complementary TransferComplementary TransferFunctionsFunctions
• Its power-complementary highpass transferfunction is thus given by
• The above expression is precisely the first-order highpass transfer function describedearlier
=− −
−
α−+α−−= 1
1
11
21
1021 )]()([)(
zz
HP zAzAzH
= −
−
α−−α+
1
1
11
21
zz
Copyright © 2001, S. K. Mitra46
Complementary TransferComplementary TransferFunctionsFunctions
• Figure below demonstrates the allpasscomplementary property and the powercomplementary property of and)(zHLP
)(zHHP
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
|HHP
(ejω)|
|HLP
(ejω)|
|HLP
(ejω) + HHP
(ejω)|
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de |HHP
(ejω)|2
|HLP
(ejω)|2
|HLP
(ejω)|2 + |HHP
(ejω)|2
Copyright © 2001, S. K. Mitra47
Complementary TransferComplementary TransferFunctionsFunctions
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 constantKzHzHzHzH =−−+ −− )()()()( 11
Copyright © 2001, S. K. Mitra48
Complementary TransferComplementary TransferFunctionsFunctions
• It can be shown that the gain function G(ω)of a power-symmetric transfer function at ω= π is given by
• If we define , then it followsfrom the definition of the power-symmetricfilter that H(z) and G(z) are power-complementary as
constanta)()()()( 11 =+ −− zGzGzHzH
)()( zHzG −=dBK 3log10 10 −
Copyright © 2001, S. K. Mitra49
Complementary TransferComplementary TransferFunctionsFunctions
Conjugate Quadratic Filter• If a power-symmetric filter has an FIR
transfer function H(z) of order N, then theFIR digital filter with a transfer function
is called a conjugate quadratic filter ofH(z) and vice-versa
)()( 11 −−= zHzzG
Copyright © 2001, S. K. Mitra50
Complementary TransferComplementary TransferFunctionsFunctions
• It follows from the definition that G(z) isalso a power-symmetric causal filter
• It also can be seen that a pair of conjugatequadratic filters H(z) and G(z) are alsopower-complementary
Copyright © 2001, S. K. Mitra51
Complementary TransferComplementary TransferFunctionsFunctions
• Example - Let• We form
• H(z) is a power-symmetric transferfunction
)3621)(3621( 32321 zzzzzz ++−++−= −−−)()()()( 11 −− −−+ zHzHzHzH
321 3621)( −−− ++−= zzzzH
)3621)(3621( 32321 zzzzzz −++−+++ −−−
)345043( 313 −− ++++= zzzz100)345043( 313 =−−+−−+ −− zzzz
Copyright © 2001, S. K. Mitra52
Digital Two-PairsDigital Two-Pairs
• The LTI discrete-time systems consideredso far are single-input, single-outputstructures characterized by a transferfunction
• Often, such a system can be efficientlyrealized by interconnecting two-input, two-output structures, more commonly calledtwo-pairs
Copyright © 2001, S. K. Mitra53
Digital Two-PairsDigital Two-Pairs• Figures below show two commonly used
block diagram representations of a two-pair
• Here and denote the two outputs, andand denote the two inputs, where the
dependencies on the variable z has beenomitted for simplicity
1Y 2Y1X 2X
1Y
2Y1X
2X1Y1X 2Y
2X
Copyright © 2001, S. K. Mitra54
Digital Two-PairsDigital Two-Pairs• The input-output relation of a digital two-
pair is given by
• In the above relation the matrix τ given by
is called the transfer matrix of the two-pair
=
21
22211211
21
XX
tttt
YY
=
22211211
ttttτ
Copyright © 2001, S. K. Mitra55
Digital Two-PairsDigital Two-Pairs
• It follows from the input-output relation thatthe transfer parameters can be found asfollows:
02
112
01
111
12 ====
XX XYt
XYt ,
02
222
01
221
12 ====
XX XYt
XYt ,
Copyright © 2001, S. K. Mitra56
Digital Two-PairsDigital Two-Pairs• An alternate characterization of the two-pair
is in terms of its chain parameters as
where the matrix Γ given by
is called the chain matrix of the two-pair
=
22
11
XY
DCBA
YX
= DC
BAΓ- -
Copyright © 2001, S. K. Mitra57
Digital Two-PairsDigital Two-Pairs• The relation between the transfer
parameters and the chain parameters aregiven by
ACt
At
ABCADt
ACt −==−== 22211211
1,,,
2122112112
2111
2122
211
tttttDt
tCttBtA −
==−== ,,,
Copyright © 2001, S. K. Mitra58
Two-Pair InterconnectionTwo-Pair InterconnectionSchemesSchemes
Cascade Connection - Γ-cascade
• Here
'1Y
'1X
"1Y
'2Y "
2Y"1X
"2X'
2X
''''
DCBA
- -
""""
DCBA
--
=
'2
'2
''''
'1
'1
XY
DCBA
YX
=
"2
"2
""""
"1
"1
XY
DCBA
YX
Copyright © 2001, S. K. Mitra59
Two-Pair InterconnectionTwo-Pair InterconnectionSchemesSchemes
• But from figure, and• Substituting the above relations in the first
equation on the previous slide andcombining the two equations we get
• Hence,
'2
"1 YX = '
2"
1 XY =
=
"2
"2
""""
''''
'1
'1
XY
DCBA
DCBA
YX
=
""""
''''
DCBA
DCBA
DCBA
Copyright © 2001, S. K. Mitra60
Two-Pair InterconnectionTwo-Pair InterconnectionSchemesSchemes
Cascade Connection - τ-cascade
• Here
'1Y'
1X "1Y
'2Y
"2Y
"1X
"2X
'2X
'22'21
'12'11tttt
- - --
"22"21
"12"11tttt
=
'2
'1'22'21
'12'11'2
'1
XX
tttt
YY
=
"2
"1
"22"21
"12"11"2
"1
XX
tttt
YY
Copyright © 2001, S. K. Mitra61
Two-Pair InterconnectionTwo-Pair InterconnectionSchemesSchemes
• But from figure, and• Substituting the above relations in the first
equation on the previous slide andcombining the two equations we get
• Hence,
'1
"1 YX = '
2"2 YX =
=
'2
'1
'22'21
'12'11"22"21
"12"11"2
"1
XX
tttt
tttt
YY
=
'22'21
'12'11"22"21
"12"1122211211
tttt
tttt
tttt
Copyright © 2001, S. K. Mitra62
Two-Pair InterconnectionTwo-Pair InterconnectionSchemesSchemes
Constrained Two-Pair
• It can be shown that1Y1X 2Y
2XG(z)
H(z)
)()()(
1
1zGBAzGDC
XYzH
⋅+⋅+
==
)(1)(
22
211211 zGt
zGttt−
+=
Copyright © 2001, S. K. Mitra63
Algebraic Stability TestAlgebraic Stability Test• We have shown that the BIBO stability of a
causal rational transfer function requiresthat all its poles be inside the unit circle
• For very high-order transfer functions, it isvery difficult to determine the polelocations analytically
• Root locations can of course be determinedon a computer by some type of root findingalgorithms
Copyright © 2001, S. K. Mitra64
Algebraic Stability TestAlgebraic Stability Test
• We now outline a simple algebraic test thatdoes not require the determination of polelocations
The Stability Triangle• For a 2nd-order transfer function the
stability can be easily checked byexamining its denominator coefficients
Copyright © 2001, S. K. Mitra65
Algebraic Stability TestAlgebraic Stability Test• Let
denote the denominator of the transferfunction
• In terms of its poles, D(z) can be expressedas
• Comparing the last two equations we get
22
111)( −− ++= zdzdzD
221
121
12
11 )(1)1)(1()( −−−− λλ+λ+λ−=λ−λ−= zzzzzD
212211 ),( λλ=λ+λ−= dd
Copyright © 2001, S. K. Mitra66
Algebraic Stability TestAlgebraic Stability Test• The poles are inside the unit circle if
• Now the coefficient is given by theproduct of the poles
• Hence we must have
• It can be shown that the second coefficientcondition is given by
1||,1|| 21 <λ<λ
2d
1|| 2 <d
21 1|| dd +<
Copyright © 2001, S. K. Mitra67
Algebraic Stability TestAlgebraic Stability Test
• The region in the ( )-plane where thetwo coefficient condition are satisfied,called the stability triangle, is shown below
21,dd
Stability region
Copyright © 2001, S. K. Mitra68
Algebraic Stability TestAlgebraic Stability Test
• Example - Consider the two 2nd-orderbandpass transfer functions designedearlier:
21
2
3763817343424011188190 −−
−
+−−−=
zzzzHBP
...)('
21
2
726542530533531011136730 −−
−
+−−=
zzzzHBP
...)("
Copyright © 2001, S. K. Mitra69
Algebraic Stability TestAlgebraic 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
)(' zHBP3763819.1,7343424.0 21 =−= dd
726542528.0,53353098.0 21 =−= dd
)(' zHBP1|| 2 >d)(" zHBP
21 1|| dd +<1|| 2 <d)(" zHBP
Copyright © 2001, S. K. Mitra70
Algebraic Stability TestAlgebraic Stability Test
A General Stability Test Procedure• Let denote the denominator of an
M-th order causal IIR transfer function H(z):
where we assume for simplicity• Define an M-th order allpass transfer
function:
)(zDM
∑ =−= M
ii
iM zdzD 0)(
10 =d
)()( 1
)( zDzMDz
M M
MzA
−−=
Copyright © 2001, S. K. Mitra71
Algebraic Stability TestAlgebraic Stability Test
• Or, equivalently
• If we express
then it follows that
MM
MM
MMMMM
zdzdzdzdzzdzdzdd
zMA −+−−
−−
−+−−−
−−
++++++++++
= 11
22
11
11
22
11
....1
....)(
∏ =−λ−= M
ii
iM zzD 1 )1()(
∏ = λ−= Mi i
MMd 1)1(
Copyright © 2001, S. K. Mitra72
Algebraic Stability TestAlgebraic Stability Test
• Now for stability we must have ,which implies the condition
• Define
• Then a necessary condition for stability of , and hence, the transfer function
H(z) is given by
1|| <λi1|| <Md
MMM dAk =∞= )(
)(zAM
12 <Mk
Copyright © 2001, S. K. Mitra73
Algebraic Stability TestAlgebraic Stability Test• Assume the above condition holds• We now form a new function
• Substituting the rational form of inthe above equation we get
)(zAM
−
−=
−
−=− )(1)(
)(1)()(1 zAd
dzAzzAk
kzAzzAMM
MM
MM
MMM
)1('1
)2('2
1'1
)1()2('1
1'2
'1
....1
....)(1 −−
−−−
−−
−−−−−−−
++++++++
=− MM
MM
MMMM
zdzdzdzzdzdd
zMA
Copyright © 2001, S. K. Mitra74
Algebraic Stability TestAlgebraic Stability Test
where
• Hence, is an allpass function oforder
• Now the poles of are given bythe roots of the equation
oλ )(1 zAM −
11,1 2
' −≤≤−
−= − Mi
dddddM
iMMii
MkoMA 1)( =λ
)(1 zAM −1−M
Copyright © 2001, S. K. Mitra75
Algebraic Stability TestAlgebraic Stability Test• By assumption• Hence• If is a stable allpass function, then
• Thus, if is a stable allpass function,then the condition holds only if
12 <Mk1|)(| >λoMA
)(zAM
<>==><
1for,11for,11for,1
)(zzz
zAM
)(zAM1|)(| >λoMA
1<λo
Copyright © 2001, S. K. Mitra76
Algebraic Stability TestAlgebraic Stability Test
• Or, in other words is a stableallpass function
• Thus, if is a stable allpass functionand , then is also a stableallpass function of one order lower
• We now prove the converse, i.e., ifis a stable allpass function and , then
is also a stable allpass function
)(1 zAM −
)(1 zAM −
)(zAM12 <Mk
)(1 zAM −
)(zAM
12 <Mk
Copyright © 2001, S. K. Mitra77
Algebraic Stability TestAlgebraic Stability Test• To this end, we express in terms of
arriving at
• If is a pole of , then
• By assumption holds
)(1)()(
11
11
zAzkzAzkzA
MM
MMM
−−
−−
++=
)(1 zAM −
)(zAM
)(zAM
MkoMo A 11
1 )( −=ζζ −−
12 <Mk
oζ
Copyright © 2001, S. K. Mitra78
Algebraic Stability TestAlgebraic Stability Test
• Therefore, i.e.,
• Assume is a stable allpass function• Then for• Now, if , then because of the above
condition• But the condition reduces
to if
1|)(| 11 >ζζ −
−oMo A
)(1 zAM −1|)(| 1 ≤− zAM 1|| ≥z
1|| ≥ζo
|||)(| 1 ooMA ζ>ζ−
1|)(| 1 ≤ζ− oMA|||)(| 1 ooMA ζ>ζ−
1|)(| 1 >ζ− oMA 1|| ≥ζo
Copyright © 2001, S. K. Mitra79
Algebraic Stability TestAlgebraic Stability Test
• Thus there is a contradiction• On the other hand, if then from
we have• The above condition does not violate the
condition
1|| <ζo1||for1|)(| 1 <>− zzAM
1|)(| 1 >ζ− oMA
|||)(| 1 ooMA ζ>ζ−
Copyright © 2001, S. K. Mitra80
Algebraic Stability TestAlgebraic Stability Test
• Thus, if and if is a stableallpass function, then is also a stableallpass function
• Summarizing, a necessary and sufficient setof conditions for the causal allpass function
to be stable is therefore:(1) , and(2) The allpass function is stable
)(1 zAM −)(zAM
12 <Mk
)(zAM
)(1 zAM −
12 <Mk
Copyright © 2001, S. K. Mitra81
Algebraic Stability TestAlgebraic Stability Test• Thus, once we have checked the condition
, we test next for the stability of thelower-order allpass function
• The process is then repeated, generating aset of coefficients:
and a set of allpass functions of decreasingorder:
12 <Mk)(1 zAM −
121 ,,...,, kkkk MM −
1)(),(),(.,..),(),( 0121 =− zAzAzAzAzA MM
Copyright © 2001, S. K. Mitra82
Algebraic Stability TestAlgebraic Stability Test• The allpass function is stable if and
only if for i• Example - Test the stability of
• From H(z) we generate a 4-th order allpassfunction
• Note:
12 <ik)(zAM
432
23
14
12
23
34
4
41
412
213
434
432
213
414
41
411
)(dzdzdzdzzdzdzdzd
zzzz
zzzzzA
++++++++
=++++
++++=
12341)( 234 ++++
=zzzz
zH
1)(41
444 <==∞= dAk
Copyright © 2001, S. K. Mitra83
Algebraic Stability TestAlgebraic Stability Test• Using
we determine the coefficients of thethird-order allpass function from thecoefficients of :
31,1 2
4
44' ≤≤−
−= − i
ddddd ii
i
}{ 'id
}{ id )(4 zA)(3 zA
151
522
15113
15112
523
151
'3
2'2
3'1
'1
2'2
3'3
31
11
)(+++
+++=
++++++
=zzz
zzz
zdzdzdzdzdzd
zA
Copyright © 2001, S. K. Mitra84
Algebraic Stability TestAlgebraic Stability Test• Note:• Following the above procedure, we derive
the next two lower-order allpass functions:
1)(151'
333 <==∞= dAk
22479
2241592
2241592
22479 1
)(2 ++
++=
zz
zzzA
10153
10153 1
)(1 +
+=
zz
zA
Copyright © 2001, S. K. Mitra85
Algebraic Stability TestAlgebraic Stability Test
• Note:
• Since all of the stability conditions aresatisfied, and hence H(z) are stable
• Note: It is not necessary to derivesince can be tested for stability usingthe coefficient conditions
1)(22479
22 <=∞= Ak
1)(10153
11 <=∞= Ak
)(4 zA)(3 zA
)(2 zA