Realization Using Multiplier
Extraction Approach
Realization of Allpass Filters
• An M-th order real-coefficient allpass
transfer function AM (z ) is characterized by
M unique coefficients as here the numerator
is the mirror-image polynomial of the
denominator
• A direct form realization of AM (z ) requires
2M multipliers
• Objective - Develop realizations of AM (z )
requiring only M multipliers
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Copyright © 2010, S. K. Mitra
• Now, an arbitrary allpass transfer function
can be expressed as a product of 2nd-order
and/or 1st-order allpass transfer functions
• We consider first the minimum multiplier
realization of a 1st-order and a 2nd-order
allpass transfer functions
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First-Order Allpass Structures
First-Order Allpass Structures
• Consider first the 1st-order allpass transfer
function given by
d + z −1
A 1( z ) = 1
1 + d1z −1
• We shall realize the above transfer function
in the form a structure containing a single
multiplier d1 as shown below
X1
Y1
3
• We express the transfer function
A1( z ) = Y1 / X1 in terms of the transfer
parameters of the two-pair as
t t d
t − d (t t − t t )
A 1( z ) = t11 + 12 21 1 = 11 1 11 22 12 21
1− d1t22
X2
d1
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1− d1t22
• A comparison of the above with
d + z −1
A 1( z ) = 1
1 + d1z −1
yields
Y2
Multiplier-less
two-pair
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4
First-Order Allpass Structures
t11 = z −1, t22 = − z −1, t11t22 − t12t21 = −1
Copyright © 2010, S. K. Mitra
First-Order Allpass Structures
• Substituting t11 = z −1 and t22 = − z −1 in
t11t22 − t12t21 = −1 we get
t12t21 = 1 − z − 2
• Type 1A t : t11 = z −1, t22 = − z −1, t12 = 1, t21 = 1 − z −2
• Type 1B t :
• There are 4 possible solutions to the above
equation:
Type 1A: t11 = z −1, t22 = − z −1, t12 = 1 − z − 2 , t21 = 1
Type 1B:
• We now develop the two-pair structure for
the Type 1A allpass transfer function
t11 = z −1, t22 = − z −1, t12 = 1 − z −1, t21 = 1 + z −1
t11 = z −1, t22 = − z −1, t12 = 1 + z −1, t21 = 1 − z −1
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Copyright © 2010, S. K. Mitra
1
First-Order Allpass Structures
First-Order Allpass Structures
• From the transfer parameters of this allpass
we arrive at the input-output relations:
Y2 = X1 − z −1 X 2
• By constraining the X 2 , Y2 terminal-pair
with the multiplier d1 , we arrive at the
Type 1A allpass filter structure shown
below
Y1 = z −1 X1 + (1 − z − 2 ) X 2 = z −1Y2 + X 2
• A realization of the above two-pair is
sketched below
_1
+
X1
+
z
Y2
_1
X1
_
z 1
Copyright © 2010, S. K. Mitra
_1
_
z 1
_
z 1
X1
+
+
d1
d1
+
Y1
+
_
z 1
+
_
d1
+
9
_
z 1
Type 1A t
1
d1
X2
Type 1B t
Copyright © 2010, S. K. Mitra
• A 2nd-order allpass transfer function is
characterized by 2 unique coefficients
• Hence, it can be realized using only 2
multipliers
• Type 2 allpass transfer function:
d d + d z −1+ z − 2
A 2( z ) = 1 2 −11
1 + d1z + d1d 2 z −2
_1
+
Y1
Type 1B
X1
Y1
Second-Order Allpass
Structures
• In a similar fashion, the other three singlemultiplier first-order allpass filter structures
can be developed as shown below
+
+
8
First-Order Allpass Structures
X1
_
z 1
Type 1A
X2
7
Y2
Y1
_1
z
+
_1
Y1
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10
Type 2 Allpass Structures
Copyright © 2010, S. K. Mitra
Type 3 Allpass Structures
• Type 3 allpass transfer function:
d + d z −1+ z − 2
A 3( z ) = 2 1−1
1 + d1z + d 2 z − 2
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Copyright © 2010, S. K. Mitra
2
Realization Using Multiplier
Extraction Approach
Type 3 Allpass Structures
• Example - Realize
A 3( z ) = −
0.2+ 0.18 z −1 + 0.4 z − 2 + z −3
1+ 0.4 z −1 + 0.18 z − 2 −0.2 z −3
( −0.4+ z −1 )(0.5+ 0.8 z −1 + z − 2 )
=
(1−0.4 z −1 )(1+ 0.8 z −1 + 0.5 z − 2 )
• A 3-multiplier cascade realization of the
above allpass transfer function is shown
below
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Copyright © 2010, S. K. Mitra
14
Realization Using Two-Pair
Extraction Approach
Realization Using Two-Pair
Extraction Approach
• The stability test algorithm described earlier
in the course also leads to an elegant
realization of an Mth-order allpass transfer
function
• The algorithm is based on the development
of a series of (m − 1)th-order allpass transfer
functions A m−1 ( z ) from an mth-order allpass
transfer function Am (z ) for m =M , M −1,... ,1
• Let
d + d z −1 + d
z − 2 + ... + d1z −( m −1) + z − m
A m ( z ) = m m−−11 −2m−.2..
− ( m −1)
−m
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1+ d1z + d 2 z
16
where km = A m(∞) = d m
• It has been shown earlier that AM ( z ) is
stable if and only if
km2 < 1 for m = M , M − 1, ...,1
Copyright © 2010, S. K. Mitra
Realization Using Two-Pair
Extraction Approach
• To develop the realization method we
express A m (z ) in terms of A m−1( z ):
' −1 + d m
' − 2 z −1 + ...+ d1' z −( m− 2 ) + z −( m −1)
dm
−1 ...
'
' − 2 z −( m − 2 ) + d m
' −1z −( m −1)
1+ d1 z + + d m
then the coefficients of A m−1 ( z ) are simply
related to the coefficients of A m (z ) through
d − d m d m −i
di' = i
, 1 ≤ i ≤ m −1
1 − d m2
Copyright © 2010, S. K. Mitra
+dm z
mA m
• If the allpass transfer function A m−1 ( z ) is
expressed in the form
A m −1( z ) =
+ + d m −1z
• We use the recursion
m ( z ) − k m m = M , M − 1, . . . ,1
],
A m −1 ( z ) = z[1A
−k
( z)
Realization Using Two-Pair
Extraction Approach
17
Copyright © 2010, S. K. Mitra
A m ( z) =
k m + z −1A m−1( z )
1+ k m z −1A m −1( z )
• We realize A m (z ) in the form shown below
X1
18
A m ( z)
Y1
Y2
t11 t12
t21 t22
A m−1 ( z )
X2
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3
Realization Using Two-Pair
Extraction Approach
Realization Using Two-Pair
Extraction Approach
• The transfer function A m( z ) = Y1 / X1 of the
constrained two-pair can be expressed as
A m( z ) =
t11 = km , t22 = − km z −1
t11t22 − t12t21 = − z −1
• Substituting t11 = km and t22 = − km z −1 in the
equation above we get
t12t21 = (1 − km2 ) z −1
• There are a number of solutions for t12 and
t21
t11 − (t11t22 − t12t21 ) A m−1( z )
1− t22 A m−1( z )
• Comparing the above with
A m ( z) =
k m + z −1A m−1( z )
1+ k m z −1A m −1( z )
we arrive at the two-pair transfer parameters
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Copyright © 2010, S. K. Mitra
20
Realization Using Two-Pair
Extraction Approach
Realization Using Two-Pair
Extraction Approach
• Some possible solutions are given below:
• Consider the solution
t11 = km , t22 = − km z −1, t12 = (1 − km2 ) z −1, t21 = 1
t11 = km , t22 = − km z −1, t12 = z −1, t21 = 1 − km2
• Corresponding input-output relations are
Y1 = km X1 + (1 − km2 ) z −1 X 2
t11 = km , t22 = − km z −1, t12 = (1 − km ) z −1, t21 = 1 + km
t11 = km , t22 = − km z −1, t12 = 1 − km2 z −1, t21 = 1 − km2
t11 = km , t22 = − km z −1, t12 = (1 − km2 ) z −1, t21 = 1
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Copyright © 2010, S. K. Mitra
Copyright © 2010, S. K. Mitra
22
Realization Using Two-Pair
Extraction Approach
Y2 = X1 − km z −1 X 2
• A direct realization of the above equations
leads to the 3-multiplier two-pair shown on
the next slide
Copyright © 2010, S. K. Mitra
Realization Using Two-Pair
Extraction Approach
• Likewise, the transfer parameters
t11 = km , t22 = − km z −1, t12 = 1 − km2 z −1, t21 = 1 − km2
• The transfer parameters
t11 = km , t22 = − km z −1, t12 = (1 − km ) z −1, t21 = 1 + km
lead to the 4-multiplier two-pair structure
shown below
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Copyright © 2010, S. K. Mitra
lead to the 4-multiplier two-pair structure
shown below
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Copyright © 2010, S. K. Mitra
4
Realization Using Two-Pair
Extraction Approach
Realization Using Two-Pair
Extraction Approach
• A 2-multiplier realization can be derived by
manipulating the input-output relations:
2 −1
Y1 = km X1 + (1 − km
)z X 2
−1
Y2 = X1 − km z X 2
• A direct realization of
Y1 = kmY2 + z −1 X 2
Y2 = X1 − km z −1 X 2
lead to the 2-multiplier two-pair structure,
known as the lattice structure, shown below
• Making use of the second equation, we can
rewrite the first equation as
Y1 = kmY2 + z −1 X 2
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Copyright © 2010, S. K. Mitra
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Copyright © 2010, S. K. Mitra
Realization Using Two-Pair
Extraction Approach
Realization Using Two-Pair
Extraction Approach
• Consider the two-pair described by
t11 = km , t22 = − km z −1, t12 = (1 − km ) z −1, t21 = 1 + km
• We can then rewrite the input-output
relations as Y1 = V1 + z −1 X 2 and Y2 = X1 + V1
• The corresponding 1-multiplier realization
is shown below
• Its input-output relations are given by
Y1 = km X1 + (1 − km ) z −1 X 2
Y2 = (1 + km ) X1 − km z −1 X 2
• Define
V1 = km ( X1 − z −1X 2 )
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Copyright © 2010, S. K. Mitra
V1
28
Realization Using Two-Pair
Extraction Approach
Copyright © 2010, S. K. Mitra
Realization Using Two-Pair
Extraction Approach
• The process is repeated until the
constraining transfer function is A 0 ( z ) = 1
• The complete realization of A M (z ) based on
the extraction of the two-pair lattice is
shown below
• An mth-order allpass transfer function A m (z )
is then realized by constraining any one of
the two-pairs developed earlier by the
(m − 1) th-order allpass transfer function
A m−1 ( z )
A m (z )
A m (z )
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Copyright © 2010, S. K. Mitra
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Copyright © 2010, S. K. Mitra
5
Realization Using Two-Pair
Extraction Approach
Realization Using Two-Pair
Extraction Approach
• It follows from our earlier discussion that
A M (z ) is stable if the magnitudes of all
multiplier coefficients in the realization are
less than 1, i.e., | km | < 1 for m = M , M − 1, ...,1
• The cascaded lattice allpass filter structure
requires 2M multipliers
• A realization with M multipliers is obtained if
instead the single multiplier two-pair is used
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Copyright © 2010, S. K. Mitra
• Example - Realize
A 3( z ) =
=
Copyright © 2010, S. K. Mitra
Realization Using Two-Pair
Extraction Approach
• We first realize A 3( z ) in the form of a
lattice two-pair characterized by the
multiplier coefficient k3 = d3 = −0.2
and constrained by a 2nd-order allpass
A 2 ( z ) as indicated below
• The allpass transfer function A 2 ( z ) is of the
form
' ' −1 − 2
A 2 ( z ) = d 2 + d1z + z
' −1 ' − 2
1+ d1 z + d 2 z
• Its coefficients are given by
d1' =
A 3( z )
33
d 2' =
Copyright © 2010, S. K. Mitra
d3 + d 2 z −1 + d1z − 2 + z −3
1 + d1z −1 + d 2 z − 2 + d3 z −3
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Realization Using Two-Pair
Extraction Approach
k 3 = − 0 .2
− 0.2 + 0.18 z −1 + 0.4 z − 2 + z −3
1 + 0.4 z −1 + 0.18 z − 2 − 0.2 z −3
d1 − d 3d 2
1− d 32
d 2 − d3d1
1− d 32
=
=
0.4 −( −0.2)(0.18)
= 0.4541667
1−( −0.2) 2
0.18−( −0.2)(0.4)
= 0.2708333
1−( −0.2) 2
34
Realization Using Two-Pair
Extraction Approach
Copyright © 2010, S. K. Mitra
Realization Using Two-Pair
Extraction Approach
• Next, the allpass A 2 ( z ) is realized as a
lattice two-pair characterized by the
multiplier coefficient k2 = d 2' = 0.2708333
and constrained by an allpass A1 ( z ) as
indicated below
• The allpass transfer function A1 ( z ) is of the
form
d" + z −1
A1 ( z ) = 1 " −1
1 + d1 z
• It coefficient is given by
d1" =
A 3( z )
d1' − d 2' d1'
1−( d 2' ) 2
=
d1'
1+ d 2'
=
0.4541667
1.2708333
= 0.3573771
k3 = − 0.2, k 2 =0.2708333
35
Copyright © 2010, S. K. Mitra
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Copyright © 2010, S. K. Mitra
6
Realization Using Two-Pair
Extraction Approach
Cascaded Lattice Realization
Using MATLAB
• Finally, the allpass A1 ( z ) is realized as a
lattice two-pair characterized by the
multiplier coefficient k1 = d1" = 0.3573771
and constrained by an allpass A 0 ( z ) = 1 as
indicated below
• The M-file poly2rc can be used to realize
an allpass transfer function in the cascaded
lattice form
• To this end Program 8_4 can be employed
A 3( z )
37
k3 = − 0.2,
k 2 = 0.2708333,
A2 ( z )
k1 = 0.3573771
A1 ( z )
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38
Copyright © 2010, S. K. Mitra
7