Digital Signal Processing Reference
In-Depth Information
Assuming that we choose an ordinary realization of white noise
1
for the synthesis
filter input with power
σ
Y
=
σ
Y
, we can observe the following property at the level
of the power spectral densities:
σ
Y
S
X
(
f
)=
|
2
=
S
X
(
f
)
|
A
(
f
)
The reconstructed signal has the same power distribution as a function of
frequency but the waveforms are different. As the ear is relatively insensitive to phase
changes, this technique can be used to reconstruct a signal which is perceived to
be approximately identical to the original signal. This type of coder is known as a
vocoder
, short for voice coder.
This whole explanation is valid assuming that the speech signal can be considered
as the realization of a random process. This hypothesis is quite realistic only for
unvoiced sounds.
6.2.3.
Voiced sounds
The graphs in Figure 6.2 are of a voiced sound in both the time domain (on the
left) and the frequency domain (on the right), showing both the original signal
x
(
n
)
and the prediction error
y
(
n
). The filter
A
(
z
) is obviously not entirely whitening. A
noticeable periodicity remains in the signal
y
(
n
), visible in the time domain as well as
in the frequency domain. In a 32 ms time interval, there are approximately 7.5 periods.
The fundamental frequency is therefore of the order of
f
0
≈
250 Hz.
We can readily observe a line spectrum in the frequency domain with a fundamental
frequency of 250 Hz (corresponding to a female speaker) and the different harmonics.
7
.
5
/
0
.
032
≈
The problem which now presents itself is to find a model
y
(
n
) for
y
(
n
) that will
enable us to obtain
S
X
(
f
)
≈
S
X
(
f
) through filtering and which has a very economic
bit rate. A comb of the form:
+
∞
y
(
n
)=
α
λ
(
n
−
mT
0
+
ϕ
)
m
=
−∞
is a good candidate. In this expression,
λ
(
n
) is the Kronecker symbol which takes
the value 1 if
n
= 0, 0 otherwise,
T
0
=
f
e
/f
0
is the fundamental period expressed
as a number of samples, and
ϕ
is a value from
which conveys the
uncertainty in the phase. The signal
y
(
n
) can perhaps be interpreted as the realization
of a random process
Y
(
n
) whose properties we are now interested in determining.
{
0
,...,T
0
−
1
}
1. Using the Matlab
function randn, for example.
