Digital Signal Processing Reference
In-Depth Information
FIGURE 4.2.
FIR filter structure showing delays.
FIGURE 4.3.
FIR lattice structure.
components are delayed by the same amount. Otherwise, harmonic distortion can
occur. Linear phase filters are FIR filters; however, not all FIR filters have linear
phase.
The Fourier transform of a delayed input sample
x
(
n
-
k
) is
e
-
j
w
kT
X
(
j
w
), yielding
a phase of
. Note that the group
delay function, defined as the derivative of the phase, is a constant, or
d
q=-w
kT
, which is a linear function in terms of
w
q
/
d
w=-
kT
.
4.4 FIR LATTICE STRUCTURE
The lattice structure is commonly used for applications in adaptive filtering and
speech processing [48,49], such as in a linear predictive coding (LPC) application.
An
N
th-order lattice structure is shown in Figure 4.3. The coefficients
k
1
,
k
2
,...,
k
N
are commonly referred to as
reflection coefficients
(or
k
-parameters). An advantage
of this structure is that the frequency response is not as sensitive as the previous
structure to small changes in the coefficients. From the first section in Figure 4.3,
with
N
=
1, we have
()
=
()
+
(
)
yn
xn kxn
-
1
(4.28)
1
1
()
=
()
+-
(
)
en kxn xn
1
1
(4.29)
1
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