Digital Signal Processing Reference
In-Depth Information
the normalized TA for a pitch candidate
τ
is given by
N
−
τ
−
1
s(n)s(n
+
τ)
n
=
0
R
T
(τ )
=
(6.18)
N
−
τ
−
1
N
−
τ
−
1
s
2
(n)
s
2
(n
+
τ)
n
=
0
n
=
0
which differs from the autocorrelation method discussed earlier in the limits
of the summations (the earlier method was more like a cross-correlation). The
TA has been widely used for PDAs due to its relatively good performance
especially over noisy speech signals [2]. Autocorrelation can also be used in
the frequency domain to bring out spectral similarities which are mainly due
to the pitch frequency spacing of the harmonics. If the spectrum of windowed
speech is given by
S(m)
A(m)e
jθ(m)
1, where
A(m)
and
θ(m)
are the magnitude and phase of the normalized spectral autocorrelation
(SA),
R
S
(τ )
can be defined as
=
for 0
≤
m
≤
M
−
−
ω
τ
M/
2
+
A
z
(m)A
z
(m
ω
τ
)
m
=
0
,
for
T
(l)
T
(u)
R
S
(τ )
=
≤
τ
≤
(6.19)
0
0
M/
2
−
ω
τ
A
z
(m)
M/
2
−
ω
τ
A
z
(m
+
ω
τ
)
m
=
0
m
=
0
0
and
T
(u
0
are the lower and upper limits
for the pitch search. In equation (6.19), the zero-crossing spectrum
A
z
(m)
is
given by
,and
T
(l)
where
ω
τ
=
M/τ
+
0
.
5
A
z
(m)
=
A(m)
−
gA(m)
(6.20)
where
A(m)
is the spectral envelope of
A(m)
. The envelope may be estimated
using the peak-picking method [17, 18]. The magnitude spectrum,
A(m)
,is
converted into the zero-crossing spectrum
A
z
(m)
to make it feasible for the
autocorrelation defined in equation (6.19). The gain,
g
, is calculated as:
M/
2
A(m)A(m)/
M/
2
g
=
A(m)A(m)
(6.21)
m
=
0
m
=
0
In equation (6.20), the logarithmic spectrum can also be considered to obtain
a zero-crossing spectrum. However, the SA with the logarithmic spectrum
produces a high correlation ratio for large lags,
τ
,closeto
T
(u)
(small
0
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