48
D. Rocchesso: Sound Processing
reflection coefficient
In particular, the IIR lattice filters are interesting because they have physical
analogues that can be considered as physical sound processing systems. The
lattice structure can be defined in a recursive fashion as indicated in fig. 28,
where H
aM-1
is an order M - 1 allpass filter, k
M
is called reflection coefficient
and it is a real number not exceeding one. Between the signals x and y there is
z
-1
x
y
-k
k
y
a
H
a M-1
H
a M
M
M
Figure 28: Lattice filter
an all-pole transfer function 1/A(z), while between the points x and y
a
there is
an allpass transfer function H
aM
(z) having the same denominator A(z). More
precisely, it can be shown that, if H
aM-1
is an allpass stable transfer function
and |k
M
| < 1, then H
aM
is an allpass stable transfer function. Proceeding with
the recursion, the allpass filter H
aM-1
can be realized as a lattice structure, and
so on. The recursion termination is obtained by replacing H
a1
with a short cir-
cuit. The lattice section having coefficient k
M
can be interpreted as the junction
between two cylindrical lossless tubes, where k
M
is the ratio between the two
cross-sectional areas. This number is also the scaling factor that an incoming
wave is subject to when it hits the junction, so that the name reflection coef-
ficient is justified. To have a physical understanding of lattice filters, think of
modeling the human vocal tract. The lattice realization of the transfer function
that relates the signals produced by the vocal folds to the pressure waves in the
mouth can be interpreted as a piecewise cylindrical approximation of the vocal
tract. In this book, we do not show how to derive the reflection coefficients from
a given transfer function [65]. We just give the result that, for a second-order
filter, a denominator such as A(z) = 1 + a
1
z
-1
+ a
2
z
-2
gives the reflection
k
1
= a
1
/(1 + a
2
)
(51)
k
2
= a
2
.
9
Verify that the filter is stable if and only if |k
1
| < 1 and |k
2
| < 1.