Tunable IIR Digital Filters
Tunable Lowpass and
Highpass Digital Filters
• We have described earlier two 1st-order and
two 2nd-order IIR digital transfer functions
with tunable frequency response
characteristics
• We shall show now that these transfer
functions can be realized easily using
allpass structures providing independent
tuning of the filter parameters
• We have shown earlier that the 1st-order
lowpass transfer function
1 − α  1 + z −1 


H L P ( z) =
2  1 − α z −1 
and the 1st-order highpass transfer function
1 + α  1 − z −1 


H HP ( z) =
2 1 − α z−1 
are doubly-complementary pair
1
Copyright © 2001, S. K. Mitra
2
Copyright © 2001, S. K. Mitra
Tunable Lowpass and
Highpass Digital Filters
Tunable Lowpass and
Highpass Digital Filters
• A realization ofH LP (z) and H HP (z) based
on the allpass-based decomposition is
shown below
H HP (z)
• Moreover, they can be expressed as
H LP ( z ) = 1 [1 + A1( z )]
2
H HP ( z) = 1 [1 − A1( z )]
2
where
• The 1st-order allpass filter can be realized
using any one of the 4 single-multiplier
allpass structures described earlier
− α + z−1
1 − α z−1
is a 1st-order allpass transfer function
A1( z ) =
3
Copyright © 2001, S. K. Mitra
4
Tunable Lowpass and
Highpass Digital Filters
Copyright © 2001, S. K. Mitra
Tunable Lowpass and
Highpass Digital Filters
• One such realization is shown below in
which the 3-dB cutoff frequency of both
lowpass and highpass filters can be varied
simultaneously by changing the multiplier
coefficient α
• Figure below shows the composite
magnitude responses of the two filters for
two different values of α
1
α = 0.4
α = 0.05
Magnitude
0.8
0.6
0.4
0.2
0
0
5
Copyright © 2001, S. K. Mitra
6
0.2
0.4
ω/ π
0.6
0.8
1
Copyright © 2001, S. K. Mitra
1
Tunable Bandpass and
Bandstop Digital Filters
Tunable Bandpass and
Bandstop Digital Filters
• The 2nd-order bandpass transfer function
1− α 
1 − z− 2

H BP ( z) =


2 1 − β(1 + α) z −1 + α z − 2 
and the 2nd-order bandstop transfer
function
1+ α 
1 − β z−1 + z − 2

H BS ( z ) =


2 1 − β(1 + α) z −1 + α z − 2 
also form a doubly-complementary pair
7
• Thus, they can be expressed in the form
H BP ( z) = 1 [1 − A2 ( z )]
2
H BS ( z ) = 1 [1 + A2 ( z )]
where
A2( z ) =
8
• The final structure is as shown below
• A realization ofH BP (z) and H BS (z) based
on the allpass-based decomposition is
shown below
• In the above structure, the multiplier β
controls the center frequency and the
multiplier α controls the 3-dB bandwidth
• The 2nd-order allpass filter is realized using
a cascaded single-multiplier lattice structure
10
Copyright © 2001, S. K. Mitra
Tunable Bandpass and
Bandstop Digital Filters
Magnitude
Magnitude
0.6
0.4
0.4
0.2
0.2
11
β = 0.8
β = 0.1
0.8
0.6
0
0
Realization of an All-pole IIR Transfer
Function
• Consider the cascaded lattice structure
derived earlier for the realization of an
allpass transfer function
α = 0.8
1
α = 0.4
α = 0.05
0.8
Copyright © 2001, S. K. Mitra
IIR Tapped Cascaded Lattice
Structures
• Figure below illustrates the parametric
tuning property of the overall structure
β = 0.5
Copyright © 2001, S. K. Mitra
Tunable Bandpass and
Bandstop Digital Filters
Tunable Bandpass and
Bandstop Digital Filters
1
α − β(1 + α) z−1 + z − 2
1 − β(1 + α ) z −1 + α z− 2
is a 2nd-order allpass transfer function
Copyright © 2001, S. K. Mitra
9
2
0.2
0.4
ω/ π
0.6
0.8
1
0
0
0.2
0.4
0.6
0.8
ω/π
Copyright © 2001, S. K. Mitra
1
12
Copyright © 2001, S. K. Mitra
2
IIR Tapped Cascaded Lattice
Structures
IIR Tapped Cascaded Lattice
Structures
• A typical lattice two -pair here is as shown
below
• From the input-output relations we derive
the chain matrix description of the two -pair:
Wi +1( z)  =  1 k i z −1 Wi ( z )
 Si +1( z )  ki z −1  Si ( z ) 
• The chain matrix description of the
cascaded lattice structure is therefore
Wm+ 1 ( z )
Wm ( z )
Sm+ 1( z )
Sm ( z )
• Its input-output relations are given by
Wm ( z) = Wm +1 ( z) − k m z −1S m ( z)
Sm +1( z ) = k m Wm ( z) + z−1Sm ( z)
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Copyright © 2001, S. K. Mitra
 X 1( z) =  1
 Y1( z )  k3
14
IIR Tapped Cascaded Lattice
Structures
Copyright © 2001, S. K. Mitra
Copyright © 2001, S. K. Mitra
• The transfer functionW1 ( z) / X1 ( z) is thus an
all-pole function with the same denominator
as that of the 3rd-order allpass function A3( z):
W1( z)
1
=
X1( z ) 1 + d1z−1 + d 2z − 2 + d3 z− 3
16
Copyright © 2001, S. K. Mitra
IIR Tapped Cascaded Lattice
Structures
IIR Tapped Cascaded Lattice
Structures
Gray-Markel Method
• A two -step method to realize an Mth-order
arbitrary IIR transfer function
• Step 2: A set of independent variables are
summed with appropriate weights to yield
the desired numerator PM (z )
• To illustrate the method, consider the
realization of a 3rd-order transfer function
H ( z) = PM ( z) / DM ( z )
• Step 1: An intermediate allpass transfer
function AM ( z ) = z− M DM ( z −1) / DM ( z ) is
realized in the form of a cascaded lattice
structure
17
k 2 z−1  1 k1z −1 W1( z) 
z −1  k1 z −1  S1( z) 
IIR Tapped Cascaded Lattice
Structures
• From the above equation we arrive at
X 1( z) = {1 + [k1(1 + k2 ) + k 2k 3] z −1
+ [k 2 + k1k 2 (1 + k 2 )] z− 2 + k 3 z− 2}W1( z)
= (1 + d1 z−1 + d2 z − 2 + d 3z −3 )W1 ( z)
using the relation S1 ( z) = W1( z) and the
relations
k1 = d1" , k 2 = d 2' , k3 = d 3
15
k 3z −1  1
z −1 k 2
Copyright © 2001, S. K. Mitra
H ( z) =
18
P3 ( z) p 0 + p1z−1 + p2 z − 2 + p3z −3
=
D3( z ) 1 + d1z −1 + d 2 z −2 + d3 z −3
Copyright © 2001, S. K. Mitra
3
IIR Tapped Cascaded Lattice
Structures
IIR Tapped Cascaded Lattice
Structures
• In the first step, we form a 3rd-order allpass
transfer function
• Objective: Sum the independent signal
variables Y1 , S1 , S2 , and S 3with weights {αi}
as shown below to realize the desired
numerator P3(z )
A3( z) = Y1( z ) / X1 ( z) = z −3 D3 ( z −1 ) / D3 ( z)
• Realization of A3(z) has been illustrated
earlier resulting in the structure shown below
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Copyright © 2001, S. K. Mitra
20
IIR Tapped Cascaded Lattice
Structures
Copyright © 2001, S. K. Mitra
IIR Tapped Cascaded Lattice
Structures
• To this end, we first analyze the cascaded
lattice structure realizing and determine the
transfer functions S1( z) / X 1( z) , S2 ( z ) / X 1( z ),
and S3 ( z ) / X 1 ( z )
• From the figure it follows that
S2 ( z ) = (k1 + z −1 )S1( z) = ( d1" + z −1)S1( z )
and hence
S2 ( z ) d1" + z−1
=
X1( z ) D3( z )
X1
Y1
• We have already shown
S1( z )
1
=
X 1 ( z ) D3 ( z )
21
Copyright © 2001, S. K. Mitra
22
IIR Tapped Cascaded Lattice
Structures
IIR Tapped Cascaded Lattice
Structures
• In a similar manner it can be shown that
S 3( z)
• Thus,
23
= ( d2'
• We now form
+ d1' z−1 + z − 2 )S1( z)
Yo ( z)
Y (z )
S ( z)
S ( z)
S (z )
= α1 1 + α 2 3 + α3 2 + α 4 1
X1( z)
X 1( z )
X1( z )
X 1( z )
X1( z )
S 3( z ) d '2 + d1' z −1 + z− 2
=
X 1( z )
D3 ( z)
• Note: The numerator of Si ( z ) / X1( z) is
precisely the numerator of the allpass
transfer function Ai( z ) = S i ( z) / Wi ( z)
Copyright © 2001, S. K. Mitra
Copyright © 2001, S. K. Mitra
24
Copyright © 2001, S. K. Mitra
4
IIR Tapped Cascaded Lattice
Structures
IIR Tapped Cascaded Lattice
Structures
• Substituting the expressions for the various
transfer functions in the above equation we
arrive at
α1( d3 + d 2 z −1 + d1z− 2 + z −3 )
−1
−2
−1
Yo ( z) + α2 ( d2' + d1' z + z ) + α3 (d1" + z ) + α 4
=
X 1( z )
D3 ( z)
25
Copyright © 2001, S. K. Mitra
• Comparing the numerator of Yo ( z) / X1( z )
with the desired numeratorP3 (z ) and
equating like powers of z −1 we obtain
α1d 3 + α2 d2' + α3d1" + α 4 = p0
α1d2 + α 2d1' + α 3 = p1
α1d1 +α2 = p2
α1 = p3
26
IIR Tapped Cascaded Lattice
Structures
Copyright © 2001, S. K. Mitra
IIR Tapped Cascaded Lattice
Structures
• Solving the above equations we arrive at
α1 = p3
• Example - Consider
P3( z ) 0 .44 z −1 + 0.362 z − 2 + 0.02 z − 3
=
D3 ( z) 1 + 0 .4 z −1 + 0. 18 z− 2 − 0.2 z− 3
• The corresponding intermediate allpass
transfer function is given by
H ( z) =
α 2 = p2 − α1d1
α 3 = p1 −α1d 2 − α 2d1'
α 4 = p0 − α1d3 − α 2d '2 −α 3d1"
A3( z) =
27
Copyright © 2001, S. K. Mitra
28
Copyright © 2001, S. K. Mitra
IIR Tapped Cascaded Lattice
Structures
IIR Tapped Cascaded Lattice
Structures
• The allpass transfer function A3( z) was
realized earlier in the cascaded lattice form
as shown below
• Other pertinent coefficients are:
d1 = 0.4, d2 = 0.18, d3 = − 0. 2, d1' = 0 .4541667
p0 = 0, p1 = 0.44, p 2 = 0.36, p3 = 0.02,
X1
• Substituting these coefficients in
α 1 = p3
α 2 = p2 − α1d1
α 3 = p1 −α1d 2 − α 2d1'
Y1
• In the figure,
k3 = d 3 = −0 .2, k 2 = d 2' = 0.2708333
k1 = d1" = 0.3573771
29
z −3D3( z −1) − 0.2 + 0.18 z −1 + 0. 0.4 z − 2 + z − 3
=
D3 ( z )
1 + 0 .4 z −1 + 0.18 z − 2 − 0.2 z− 3
Copyright © 2001, S. K. Mitra
30
α 4 = p0 − α1d3 − α 2d '2 −α 3d1"
Copyright © 2001, S. K. Mitra
5
IIR Tapped Cascaded Lattice
Structures
Tapped Cascaded Lattice
Realization Using MATLAB
α1 = 0.02 , α 2 = 0.352
α 3 = 0 .2765333 , α 4 = − 0.19016
• The final realization is as shown below
• Both the pole-zero and the all-pole IIR
cascaded lattice structures can be developed
from their prescribed transfer functions
using the M-file tf2latc
• To this end, Program 6_4 can be employed
k1 = 0.3573771 , k 2 = 0 .2708333 , k3 = − 0. 2
31
33
Copyright © 2001, S. K. Mitra
32
Tapped Cascaded Lattice
Realization Using MATLAB
FIR Cascaded Lattice
Structures
• The M-file latc2tf implements the
reverse process and can be used to verify
the structure developed using tf2latc
• To this end, Program 6_5 can be employed
• An arbitrary Nth-order FIR transfer function
of the form
−n
HN (z ) = 1+ ∑N
n=1 pn z
can be realized as a cascaded lattice structure
as shown below
Copyright © 2001, S. K. Mitra
34
FIR Cascaded Lattice
Structures
35
Copyright © 2001, S. K. Mitra
• From figure, it follows that
X m ( z ) = X m−1( z ) + k m z −1Ym −1( z)
Ym ( z) = k m X m−1( z ) + z −1Ym −1 ( z)
• In matrix form the above equations can be
written as
 X m ( z ) =  1 k m z −1  X m−1( z )
 Ym ( z )  k m
z −1   Ym−1( z ) 
where m = 1, 2, ... , N
Copyright © 2001, S. K. Mitra
Copyright © 2001, S. K. Mitra
FIR Cascaded Lattice
Structures
• Denote
Hm ( z) =
Xm ( z )
,
X 0 (z )
Gm ( z ) =
Ym ( z)
X 0 ( z)
• Then it follows from the input-output
relations of the m-th two-pair that
Hm ( z ) = Hm−1( z) + k m z −1Gm −1 ( z)
Gm ( z ) = k m Hm−1( z ) + z −1Gm −1 ( z)
36
Copyright © 2001, S. K. Mitra
6
FIR Cascaded Lattice
Structures
FIR Cascaded Lattice
Structures
• From the previous equation we observe
• From the input-output relations of the m-th
two-pair we obtain for m = 2:
H1( z ) = 1 + k1z −1, G1( z ) = k1 + z −1
where we have used the facts
H 2 ( z ) = H1 ( z ) + k2 z −1G1( z )
H0 ( z) = X 0 ( z ) / X 0 ( z) = 1
G2 ( z ) = k 2 H1 ( z) + z −1G1( z)
G0 ( z) = Y0 ( z ) / X 0 ( z ) = X 0 ( z ) / X0 ( z) = 1
• Since H1( z) and G1( z) are 1st-order
polynomials, it follows from the above that
H 2 ( z ) and G2 ( z) are 2nd-order polynomials
• It follows from the above that
G1( z ) = z −1 ( z k1 + 1) = z −1H 1( z −1)
•
G1( z) is the mirror-image of H1( z)
37
Copyright © 2001, S. K. Mitra
38
FIR Cascaded Lattice
Structures
FIR Cascaded Lattice
Structures
• SubstitutingG1( z ) = z −1H 1 ( z−1) in the two
previous equations we get
H2 ( z) = H1( z) + k2 z − 2H1 ( z−1)
G2 ( z ) = k 2 H1 ( z) + z − 2H1 ( z−1)
• Now we can write
G2 ( z ) = k 2 H1 ( z) + z − 2H1 ( z−1)
= z −2 [k 2 z 2 H1 ( z) + H1( z−1)] = z − 2H 2 ( z −1)
•
• In the general case, from the input-output
relations of the m-th two-pair we obtain
Hm ( z ) = Hm−1( z) + km z −1Gm −1 ( z)
Gm ( z) = km Hm −1( z) + z−1Gm −1 ( z)
• It can be easily shown by induction that
Gm ( z ) = z −m Hm ( z−1), m = 1, 2,..., N − 1, N
•
Gm (z ) is the mirror-image of Hm (z)
G2 ( z ) is the mirror-image of H 2 ( z )
39
Copyright © 2001, S. K. Mitra
40
FIR Cascaded Lattice
Structures
GN −1( z ) =
41
HN ( z ) + GN ( z)}
Copyright © 2001, S. K. Mitra
Copyright © 2001, S. K. Mitra
FIR Cascaded Lattice
Structures
• To develop the synthesis algorithm, we
express Hm−1( z) and G m −1 ( z ) in terms of
H m(z ) and G m (z) for m = N, N − 1,... ,2,1
arriving at
HN −1( z ) = 1 2 { HN ( z) − k N GN ( z)}
(1−k N )
1
{−k N
(1− k 2N ) z −1
Copyright © 2001, S. K. Mitra
• Substituting the expressions for
H N ( z) = 1 + ∑nN=1 pn z −n
and
GN ( z ) = z − N H N ( z−1) = ∑nN=−01pn z − n + z − N
in the first equation we get
H N −1 ( z ) =
42
1
−1
−n
{(1 − k N pN ) + ∑ N
n =1 ( pn − kn p N − n ) z
1 − kN2
+ ( pN − k N ) z − N }
Copyright © 2001, S. K. Mitra
7
FIR Cascaded Lattice
Structures
FIR Cascaded Lattice
Structures
• If we choose k N = p N , then HN −1( z)
reduces to an FIR transfer function of order
N − 1 and can be written in the form
HN −1( z) = 1 + ∑nN=−11 p'n z −n
where pn' = pn −k N p2 N −n , 1 ≤ n ≤ N −1
1− k N
• Continuing the above recursion algorithm,
all multiplier coefficients of the cascaded
lattice structure can be computed
43
Copyright © 2001, S. K. Mitra
• Example - Consider
H4 ( z ) = 1 + 1.2 z −1 + 1.12 z− 2 + 0. 12 z −3 − 0.08 z − 4
• From the above, we observe k4 = p4 = − 0.08
• Using
pn' = pn −k 4 p24 −n , 1 ≤ n ≤ 3
1− k4
44
we determine the coefficients of H3( z ):
p3' = 0.2173913 , p2' = 1. 2173913
p1' = 1. 2173913
Copyright © 2001, S. K. Mitra
FIR Cascaded Lattice
Structures
FIR Cascaded Lattice
Structures
• As a result,
H3 ( z) = 1 + 1.2173913 z −1 + 1.2173913 z − 2
+ 0.2173913 z− 3
• Thus, k3 = p3' = 0.2173913
• Using
'
'
pn" = pn − k 3 p22−n , 1 ≤ n ≤ 2
• As a result, H2 ( z ) = 1 + z −1 + z − 2
• From the above, we get k2 = p2" = 1
• The final recursion yields the last multiplier
coefficient k1 = p1" /(1 + k 2 ) = 0 .5
• The complete realization is shown below
1− k3
45
we determine the coefficients of H2 ( z):
p2" = 1 .0, p1" = 1.0
Copyright © 2001, S. K. Mitra
46
k1 = 0.5, k 2 = 1, k3 = 0.2173913 , k 4 = − 0.08
Copyright © 2001, S. K. Mitra
FIR Cascaded Lattice
Realization Using MATLAB
• The M-file tf2latc can be used to
compute the multiplier coefficients of the
FIR cascaded lattice structure
• To this end Program 6_6 can be employed
• The multiplier coefficients can also be
determined using the M-file poly2rc
47
Copyright © 2001, S. K. Mitra
8