66
D. Rocchesso: Sound Processing
be structurally divided into three parts:
· Head Shadow and ITD
· Shoulder Echo
· Pinna Reflections
Starting from the approximation of the head as a rigid sphere that diffracts a
plane wave, the shadowing effect can be effectively approximated by a first-order
continuous-time system, i.e., a pole-zero couple in the Laplace complex plane:
s
z
=
-2
0
()
(28)
s
p
= -2
0
,
(29)
where
0
is related to the effective radius a of the head and the speed of sound
c by
0
=
c
a
.
(30)
The position of the zero varies with the azimuth (see fig. 10 of the appendix C))
according to the function
() = 1.05 + 0.95 cos
- ear
150
180
,
(31)
where ear is the angle of the ear that is being considered, typically 100
for
the right ear and -100
for the left ear. The pole-zero couple can be directly
translated into a stable IIR digital filter by bilinear transformation, and the
resulting filter (with proper scaling) is
Hhs =
(
0
+ F
s
) + (
0
- F
s
)z
-1
(
0
+ F
s
) + (
0
- F
s
)z
-1
.
(32)
The ITD can be obtained by means of a first-order allpass filter [65, 100] whose
group delay in seconds is the following function of the azimuth angle :
h() =
a
c
+
-
a
c
cos ( - ear)
if 0 | - ear| <
2
a
c
| - ear| -
2
if
2
| - ear| <
.
(33)
Actually, the group delay provided by the allpass filter varies with frequency,
but for these purposes such variability can be neglected. Instead, the filter (32)
gives an excess delay at DC that is about 50% that given by (33). This increase
of the group delay at DC is exactly what one observes for the real head [49], and
it has already been outlined in fig. 9. The overall magnitude and group delay
responses of the block responsible for head shadowing and ITD are reported in
fig. 10.
In a rough approximation, the shoulder and torso effects are synthesized in
a single echo. An approximate expression of the time delay can be deduced by
the measurements reported in [19, fig. 8]
sh = 1.2
180
-
180
1 - 0.00004 ( - 80
)
180
180
+
2
[msec] ,
(34)