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.