Digital Signal Processing Reference
In-Depth Information
where
ω
0
is the normalized fundamental frequency,
K
=
π
/
ω
0
,
κ
=
N
n
=
0
−
1
w (n)
,
w(n)
is the window function,
N
is the length of the window,
and
S
w
(m)
, the windowed speech spectrum, is given by,
N
−
1
s (n) w (n) e
−
j
2
N
mn
=
=
S
w
(m)
for
m
0
,
1
,
2
, ... ,N
(8.10)
n
=
0
Spectral Correlation
Harmonic amplitudes may be estimated by computing the normalized cross-
correlation between the harmonic lobes of the speech spectrum and the main
lobe of the window spectrum. This method is based on the fact that the
spectrum of the windowed speech is equivalent to the convolution between
the speech spectrum and the window spectrum. It is also assumed that the
speech signal is stationary during the windowed segment and the spectral
leakage due to the side lobes of the window spectrum is negligible.
b
k
−
1
S
w
(m)W
∗
(
2
πm
/
N
−
kω
0
)
m
=
a
k
A
k
=
for
k
=
1
,
2
, ... ,K
(8.11)
b
k
−
1
W
2
(
2
πm/N
−
kω
0
)
m
=
a
k
max
2
π
k
2
ω
0
,
0
and
b
k
1
where
a
k
=
−
=
min [
a
k
+
1
,N
/2],and
W(ω)
is
the spectrum of the window function, given by,
N
−
1
w (n) e
−
jωn
W (ω)
=
(8.12)
n
=
0
In practice,
W(ω)
is computed with a high-resolution FFT, e.g. 2
14
samples,
by zero-padding the window function, and stored in a lookup table. The
high-resolution FFT is required because, in general, the spectral samples
m
of
S
w
(m)
do not coincide with the harmonic locations,
kω
0
,ofthefundamental
frequency. Hence
W(ω)
is shifted to the harmonic frequency and down-
sampled to coincide with the corresponding spectral samples of
S
w
(m)
,as
shown in equation (8.11).
W(ω)
is pre-computed and stored in order to reduce
the computational complexity.
The spectral cross-correlation-based amplitude estimation gives the opti-
mum gain of the harmonic lobes with respect to the main lobe of the window
spectrum, hence it is a more accurate estimate than the simple peak-picking.
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