Delay Complementary
Transfer Functions
Complementary Transfer
Functions
• A set of L transfer functions,
{H i ( z )} ,
0 ≤ i ≤ L − 1, is defined to be delaycomplementary of each other if the sum of
their transfer functions is equal to some
integer multiple of unit delays, i.e.,
• 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
L −1
∑ H i ( z ) = βz −no ,
β≠0
i =0
where no is a nonnegative integer
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Copyright © 2010, S. K. Mitra
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Delay Complementary
Transfer Functions
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• A delay-complementary pair {H 0 ( z ), H1( z )}
can be readily designed if one of the pairs is
a known Type 1 FIR transfer function of
odd length
• Let H 0 ( z ) be a Type 1 FIR transfer function
of length M = 2K+1
• Then its delay-complementary transfer
function is given by
H1( z ) = z − K − H 0 ( z )
Copyright © 2010, S. K. Mitra
Delay Complementary
Transfer Functions
• Let the magnitude response of H 0 ( z ) be
equal to 1 ± δ p in the passband and less than
or equal to δ s in the stopband where δ p and
δ s are very small numbers
• Now the frequency response of H 0 ( z ) can be
expressed as
(
H 0 (e jω ) = e − jKωH 0 (ω)
(
where H 0 (ω) is the amplitude response
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Delay Complementary
Transfer Functions
Copyright © 2010, S. K. Mitra
Copyright © 2010, S. K. Mitra
Delay Complementary
Transfer Functions
• Its delay-complementary transfer function
H1( z ) has a frequency response given by
(
(
H1 (e jω ) = e − jKωH1(ω) = e − jKω[1 − H 0 (ω)]
(
• Now, in the passband, 1 − δ p ≤ H0 (ω) ≤ 1 + δ p ,
(
and in the stopband, − δ s ≤ H 0 (ω) ≤ δ s
• It follows from the above
equation that in
(
the stopband, − δ p ≤ H1 (ω) ≤ δ p and in the
(
passband, 1 − δ s ≤ H1(ω) ≤ 1 + δ s
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Copyright © 2010, S. K. Mitra
• As a result, H1( z ) has a complementary
magnitude response characteristic to that of
H 0 ( z ) with a stopband exactly identical to
the passband of H 0 ( z ), and a passband that
is exactly identical to the stopband of H 0 ( z )
• Thus, if H 0 ( z ) is a lowpass filter, H1( z ) will
be a highpass filter, and vice versa
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Copyright © 2010, S. K. Mitra
1
Delay Complementary
Transfer Functions
Delay Complementary
Transfer Functions
• The frequency ωo at which
(
(
H 0 (ωo ) = H1(ωo ) = 0.5
the gain responses of both filters are 6 dB
below their maximum values
• The frequency ωo is thus called the 6-dB
crossover frequency
• Example - Consider the Type 1 bandstop
transfer function
H BS ( z ) =
Copyright © 2010, S. K. Mitra
H BP ( z ) = z −10 − H BS ( z )
1 (1 − z − 2 ) 4 (1 + 4 z − 2
64
+ 5 z − 4 + 5 z −8 + 4 z −10 + z −12 )
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Copyright © 2010, S. K. Mitra
Delay Complementary
Transfer Functions
Allpass Complementary
Transfer Functions
• Plots of the magnitude responses of H BS ( z )
and H BP (z ) are shown below
• A set of M digital transfer functions, {H i ( z )} ,
0 ≤ i ≤ M − 1, is defined to be allpasscomplementary of each other, if the sum of
their transfer functions is equal to an allpass
function, i.e.,
1
H (z)
H (z)
BS
BP
Magnitude
0.8
0.6
M −1
∑ H i ( z ) = A( z )
0.4
i =0
0.2
0
0
0.2
0.4
0.6
0.8
1
ω/π
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Copyright © 2010, S. K. Mitra
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Power-Complementary
Transfer Functions
M −1
∑
i =0
2
H i ( e jω ) = K ,
• By analytic continuation, the above
property is equal to
M −1
−1
∑ H i ( z ) H i ( z ) = K , for all ω
i =0
for real coefficient H i (z )
• Usually, by scaling the transfer functions,
the power-complementary property is
defined for K = 1
for all ω
Copyright © 2010, S. K. Mitra
Copyright © 2010, S. K. Mitra
Power-Complementary
Transfer Functions
• A set of M digital transfer functions, {H i ( z )} ,
0 ≤ i ≤ M − 1, is defined to be powercomplementary of each other, if the sum of
their square-magnitude responses is equal to
a constant K for all values of ω, i.e.,
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+ 5 z − 4 + 5 z −8 − 4 z −10 + z −12 )
• Its delay-complementary Type 1 bandpass
transfer function is given by
=
7
1 (1 + z − 2 ) 4 (1 − 4 z − 2
64
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Copyright © 2010, S. K. Mitra
2
Power-Complementary
Transfer Functions
Power-Complementary
Transfer Functions
• For a pair of power-complementary transfer
functions, H 0 ( z ) and H1( z ) , the frequency ωo
where | H 0 (e jωo )| 2 = | H1 (e jωo )| 2 = 0.5 , 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, ωo is called the 3-dB crossover frequency
• Example - Consider the two transfer functions
H 0 ( z ) and H1( z ) given by
H 0 ( z ) = 12 [A 0( z ) + A 1( z )]
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Copyright © 2010, S. K. Mitra
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Power-Complementary
Transfer Functions
Copyright © 2010, S. K. Mitra
• A set of M transfer functions satisfying both
the allpass complementary and the powercomplementary properties is known as a
doubly-complementary set
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• A pair of doubly-complementary IIR
transfer functions, H 0 ( z ) and H1( z ) , with a
sum of allpass decomposition can be simply
realized as indicated below
A 0( z )
+
Y0 ( z )
+
Y1( z )
X (z )
A1( z )
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H0 ( z) =
Y0 ( z )
X (z)
−1
H1 ( z ) =
Y1( z )
X (z)
Copyright © 2010, S. K. Mitra
Copyright © 2010, S. K. Mitra
Doubly-Complementary
Transfer Functions
Doubly-Complementary
Transfer Functions
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Copyright © 2010, S. K. Mitra
Doubly-Complementary
Transfer Functions
• It can be shown that H 0 ( z ) and H1( z ) are
also power-complementary
• Moreover, H 0 ( z ) and H1( z ) are boundedreal transfer functions
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H1( z ) = 12 [A 0( z ) − A 1( z )]
where A 0 ( z ) and A1 ( z ) are stable allpass
transfer functions
• Note that H 0 ( z ) + H1( z ) = A 0( z )
• Hence, H 0 ( z ) and H1( z ) are allpass
complementary
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• Example - The first-order lowpass transfer
function
−1
H LP ( z ) = 1− α ⎛⎜ 1+ z −1 ⎞⎟
2 ⎝ 1− α z ⎠
can be expressed as
−1
H LP ( z ) = 1 ⎛⎜ 1+ −α + z−1 ⎟⎞ = 1 [A 0( z ) + A 1( z )]
2 ⎝ 1−α z ⎠ 2
where
− α + z −1
A 0 ( z ) = 1 , A1( z ) =
1 − α z −1
Copyright © 2010, S. K. Mitra
3
Doubly-Complementary
Transfer Functions
Doubly-Complementary
Transfer Functions
• Figure below demonstrates the allpass
complementary property and the power
complementary property of H LP ( z ) and
H HP ( z )
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Copyright © 2010, S. K. Mitra
|H (ejω) + H
Magnitude
1
20
22
|H (e )|
LP
HP
(ejω)|2
0.8
|H
0.4
0.6
ω/π
0.8
1
(ejω)|2
0.4
|H (ejω)|2
0.2
0.2
HP
0.6
0
0
LP
0.2
0.4
0.6
0.8
1
ω/π
Copyright © 2010, S. K. Mitra
Copyright © 2010, S. K. Mitra
Conjugate Quadratic Filters
• If a power-symmetric filter has an FIR
transfer function H(z) of order N, then the
FIR digital filter with a transfer function
= (1 − 2 z −1 + 6 z − 2 + 3 z −3 )(1 − 2 z + 6 z 2 + 3 z 3 )
G ( z ) = z − N H ( − z −1 )
is called a conjugate quadratic filter of
H(z) and vice-versa
+ (1 + 2 z −1 + 6 z − 2 − 3 z −3 )(1 + 2 z + 6 z 2 − 3 z 3 )
= (3 z 3 + 4 z + 50 + 4 z −1 + 3 z −3 )
+ ( −3 z 3 − 4 z + 50 − 4 z −1 − 3 z −3 ) = 100
H(z) is a power-symmetric transfer
function
Copyright © 2010, S. K. Mitra
(ejω)|
• It can be shown that the gain function G(ω)
of a power-symmetric transfer function at ω
= π is given by
10 log10 K − 3 dB
• If we define G ( z ) = H (− z ) , then it follows
from the definition of the power-symmetric
filter that H(z) and G(z) are powercomplementary as
H ( z ) H ( z −1 ) + G ( z )G ( z −1 ) = a constant
• Example - Let H ( z ) = 1 − 2 z −1 + 6 z − 2 + 3 z −3
• We form
H ( z ) H ( z −1 ) + H (− z ) H ( − z −1 )
•
HP
jω
0.4
LP
1
|H
Power-Symmetric Filters
Power-Symmetric Filters
23
|H (ejω)|2 + |H
(ejω)|
0.6
0
0
• A real-coefficient causal digital filter with a
transfer function H(z) is said to be a powersymmetric filter if it satisfies the condition
H ( z ) H ( z −1 ) + H ( − z ) H ( − z −1 ) = K
where K > 0 is a constant
Copyright © 2010, S. K. Mitra
HP
0.2
Power-Symmetric Filters
21
LP
0.8
Magnitude
• Its power-complementary highpass transfer
function is thus given by
−1
H HP ( z ) = 1 [ A 0( z ) − A 1( z )] = 1 ⎛⎜ 1 − −α + z−1 ⎞⎟
2
2⎝
1−α z ⎠
−1
= 1+ α ⎛⎜ 1− z −1 ⎞⎟
2 ⎝ 1− α z ⎠
• The above expression is precisely the firstorder highpass transfer function described
earlier
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Copyright © 2010, S. K. Mitra
4
Conjugate Quadratic Filters
• 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
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Copyright © 2010, S. K. Mitra
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