Digital Signal Processing Reference
In-Depth Information
From the second section (cascaded with the first), using (4.28) and (4.29),
()
=
()
+
(
)
yn yn ken
xn
-
1
2
1
2
1
()
+
(
)
+
(
)
+
(
)
=
kxn
-
1
kkxn
-
1
kxn
-
2
1
2
1
2
()
++
(
)
(
)
+
(
)
=
xn
k
kk xn
-
1
kxn
-
2
(4.30)
1
1
2
2
and
()
=
()
+-
(
)
en kyn en
kxn
1
2
2
1
1
()
+
(
)
+
(
)
+-
(
)
=
kkxn
-
1
kxn
-
1
xn
2
2
2
1
1
()
++
(
)
(
)
+-
(
)
=
kxn
k
kk xn
-
1
xn
2
(4.31)
2
1
1
2
For a specific section
i
,
()
=
()
+
(
)
yn
y
n ke
n
-
1
(4.32)
i
i
-
1
i
i
-
1
()
=
()
+
(
)
en
ky n e
n
-
1
(4.33)
i
i
i
-
1
i
-
1
It is instructive to see that (4.30) and (4.31) have the same coefficients but in
reversed order. It can be shown that this property also holds true for a higher-order
structure. In general, for an
N
th-order FIR lattice system, (4.30) and (4.31) become
N
=
Â
0
()
=
(
)
yn
axni
-
(4.34)
N
i
i
and
N
=
Â
0
()
=
(
)
en
a xni
-
(4.35)
N
N
-
i
i
with
a
0
=
1. If we take the ZT of (4.34) and (4.35) and find their impulse responses,
N
=
Â
0
()
=
Yz
az
-
i
(4.36)
N
i
i
N
=
Â
0
()
=
Ez
a z
-
i
(4.37)
N
N
-
i
i
It is interesting to note that
()
=
(
)
Ez zY
-
N
1
z
(4.38)
N
N
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