Lattice and Lattice-Ladder
Structures for IIR Systems

Lattice IIR Structure
Consider a system with finite poles and all its zeros at z
=
0 whose transfer function is of the form
H
=
Α
N
1
( ) where
Α
N k
N
=
1 a
N
[ ] − k
N z
=
0.

Lattice IIR Structure
A Direct Form II system with
N finite poles and N zeros at z = 0.

Lattice IIR Structure
Compare the Direct Form II structures for FIR and IIR systems.
If, in the FIR system, we exchange the roles of X
( )
and Y
( )
, b
− a 's (with b
0
=
1) and let N
=
M
−
1, we get the
IIR system.
FIR IIR

Lattice-Ladder IIR Structure
Modify the FIR lattice structure as illustrated below. Reverse the arrows on all the "f" signals. Reverse the lattice and apply x
[ ] to the previous output and take y
[ ]
from the previous input.
Also re verse the signs of the signals arriving from the bottom. This is now a recursive or feedback structure which can implement an
IIR filter.

Lattice-Ladder IIR Structure
Take the case N
=
1. x
[ ] = f
1
[ ]
, f
0
[ ] = f
1
[ ] −
K g
1 0
[ n
− ]
1
[ ] =
K f
1 0 n
+ g
0
[ n
−
1
] and y
[ ] = f
0
[ ] = g
0
[ ] = f n
−
K g
1 0
[ n
−
1
]
= x
[ ]
= y
[ ] z transforming
Y
( ) +
K z
1
−
1
Y
( ) =
X
( ) ⇒
H
1
( ) =
1
+
1
−
1 z K
1
= z
+ z
K
1
Single pole at z
= −
K
1
and a zero at z
=
0.

Lattice-Ladder IIR Structure z
Also, for the N
=
1 case
g
1
G
1
[ ] =
K
1 y
[ ] [ n
−
1
]
( ) =
K
1
Y
( ) + z
−
1 Y
⇒
G
Y
1
( )
( )
G
1
Y
( )
( )
=
K z 1 K
1
+ − =
1
is the transfer function of a system with a single zero at z
+
1/ K
1 z
= −
1/ K
1
and a pole at zero.

Lattice-Ladder IIR Structure
For the N
=
2 case, it can be shown that
H
2
=
Y
X
( )
( )
=
1
+
K K
1 2
+
1
1
) z
−
1 +
K z
2
−
2
= z
2 +
1
( z
2
2
+
1
) z
+
K
2 and
G
Y
2
( )
( )
=
K
2
+
1
(
2
+
1
) z
−
1 + z
−
2 =
K
2 z
2 +
K
1
( z
2
K
2
) z
+
1/ K
2
Notice that the coefficients for the FIR and IIR systems occur in rever se order as before.

Lattice-Ladder IIR Structure
H m
=
Y
X
( )
( )
=
Α m
1
( ) and
G
Y m
( )
( )
= Β m
( ) = z
− m Α m and the previous relations for FIR lattices still hold.
Α
0
( ) = Β
0
( ) =
1 ,
Α m
= Α m
−
1
( ) +
K z m
−
1 Β m
−
1
Β m
( ) = z
− m Α m
Α m
−
1
=
( )
, K m
Α m
( ) −
K
Β m m
1
−
K m
2
= α m

Lattice-Ladder IIR Structure
If we want to add finite zeros to H m
we can add a ladder network to the lattice. Then the transfer function will be of the form
H
=
1
+
γ
α
N
N
( )
( ) z
+
−
1
γ
+
N
α
( )
N z
−
1
( ) z
−
2
γ
N
( )
α
N
−
( )
N
−
N
=
Γ
N
Α
N
( )
( )
y
=
N
m
=
0 v g m m
[ ] ⇒
Y
( ) = m
N
=
0 v G m m

Lattice-Ladder Example
Synthesize the transfer function
H
=
1
− z
−
1 +
0.5
z
−
+ z
−
1 −
0.15
2
−
2
= z z
2 z
2 − +
0.5
+
0.2
z
−
0.15
using a lattice-ladder network.
Α
2
( ) = + z
−
1 −
0.15
z
−
2
K
2
= −
0.15 and
Β
2
( ) = − + z
−
1 + z
−
2
Γ
2
( ) = m
2
=
0 v
Β m m z
−
1 +
0.5
z
−
2
Γ
1
( ) = Γ
2
( ) v
2 2
( ) = − z
−
1 +
0.5
⇒ v
2
= z
−
2 −
0.5
0.5
(
− +
Γ
1
( ) = − z
−
1 ⇒ = −
1
1.1
Γ
0
( ) = Γ
1
( ) v
1 1
( ) z
−
1 + z
−
2
)

Using
Α m
Α
1
( ) =
−
1
Lattice-Ladder Example
Α m
( ) −
K m
Β m =
1
−
K
2 m
+ z
−
1 −
0.15
z
−
2
(
0.15
) (
− + z
−
1 + z
−
2
)
1
(
0.15
)
2
K
1
−
1
Α
1
( ) =
= z
0.9775
0.23529 and
Β
1
( ) =
Γ
0
( ) = − z
−
1
0.23529
( ) (
+ z
−
1 z
−
1
+ z
−
1
)
=
1.3382
⇒ v
0
=
1.3382