基于正弦模型的语音处理分析与合成技术
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《基于正弦模型的语音处理》是一篇发表于1988年的论文，由R.J. McAulay和T.F. Quatieri共同撰写。该研究提出了一种利用正弦模型进行语音分析与合成的技术。在语音信号的处理中，这种方法将语音分解为一系列正弦波，每个正弦波由其幅度、频率和相位参数来描述。这些参数的估计是通过短时傅立叶变换（STFT）对输入语音进行简单峰值检测算法实现的。
论文的核心思想是，通过高频分辨率的谱成分分析，使用频率匹配算法跟踪快速变化的特征。当一个特定频率的轨迹被追踪到时，会应用一个三次相位函数到一个正弦波发生器上，这个发生器的输出会被幅度调制，并与其他频率轨道的正弦波合成。这种合成后的语音输出保持了原始语音的基本波形形状，且几乎难以与原信号区分。这种技术在一定程度上保留了语音的感知特性，即使在噪声环境中也能保持清晰度。
令人欣喜的是，研究发现这一方法在广泛的输入条件下都能实现高质量的语音再现，包括各种语速、音调和口音。这种基于正弦模型的处理方式在当时的数字信号处理领域具有重要意义，因为它提供了一种有效且灵活的方法来理解和生成语音信号，对于语音识别、语音合成和降噪等应用场景具有实用价值。它标志着在语音处理技术中对复杂信号的深入理解以及对简洁数学模型的有效应用的里程碑。
McAulay
et
aI. - Speech Processing
Based
on
a
Sinusoidal
Model
where
from
which
it
follows
that
the
optimal
estimate
for
the
amplitude
and
phase
is
(16)
(17)
if
l = i
(13)
if
l ~ i
L
k
+(N+
1)
~
[IYlw~)-'Y~12-IYlW~)12I.
1=1
sine
(w~
-
w~)
=
sine
[(l-
i)
w~l
={6
.y~
=
Y(lw;)
•
which
reduces
the
error
to
where
w
k
=
lw
k
Then
the
error
expression
re-
I
o'
duces
to
fk
=
~
Iy(n)
1
2
-
2(N
+ 1) Re I
~
(')'k)
*
y(w
k
)
I
n
~1
I 1
~
(14)
+
(N+
1)
~
1'Y~12
1=1
Ylw) =
N:
1
~
y(n)
exp
(-jnw)
(15)
n
is
the
STFT of
the
measurement
signal. By
completing
the
square
in
Eq. 14.
the
error
can
be
written
as
From
this
calculation
it
follows.
therefore,
that
the
error
is
minimized
by
selecting
all
of
the
har-
monic
frequencies
in
the
speech
bandwidth,
fl
(ie, L
k
=
fl/w~).
Equations
15
and
17
completely
specify
the
structure
of
the
ideal
estimator
and
show
that
the
optimal
estimator
depends
on
the
speech
data
through
the
STFT (Eq. 15).
Although
these
results
are
eqUivalent
to
a
Fourier-series
repre-
sentation
of
a periodic waveform,
the
results
lead
to
an
intuitive
generalization
to
the
more
practical
case.
This
is
done
by
considering
the
functionlY
(w)
1
2
to
be
a
continuous
function
of
w.
For
the
idealized
voiced-speech
case,
this
function
(called a periodogram) will
be
pulse-
L
k
s(n) =
~
'Y~
exp
Unlw~)
(12)
1=1
represents
the
1
th
complex
amplitude
for
the
1
th
component
of
the
L
k
sine
waves.
Since
the
measurements
are
made
on
digitized
speech,
sampled-data
notation
[s(n)]
is
used.
In
this
respect
the
time
index
n
corresponds
to
the
uniform
samples
of
t- t
k
;
therefore n
ranges
from
-N/2
to
N/2.
with
n = 0
reset
to
the
center
of
the
analysis
window
for every
frame
and
where
N+1
is
the
duration
of
the
analysis
window.
The
problem
now
is
to
fit
the
synthetic
speech
wave-
form
in
Eq. 9
to
the
measured
waveform.
de-
noted
by
y(n).
A
useful
criterion
for
judging
the
quality
of
fit
is
the
mean-squared
error
fk
=
~
Iy(n) _ s(n)
,2
n
(10)
=
~
Iy(n)
,2
_ 2
Re
~
y(n)
s*(n)
+
~
Is(n)
1
2
.
n n n
Substituting
the
speech
model
of
Eq. 9
into
Eq.
10
leads
to
the
error
expression
L
k
fk
=
~
Iy(n) 1
2
-
2 Re
~
('Y~)*
~
y(n) exp
(-jnw~)
n
1=1
n
L
k
L
k
(11)
+
(N
+ 1)
~ ~
'Y
k
('Y
k
)*
sine
(w
k
-
w
k
)
~
~
I 1 I 1
1=1 1=1
where
sinc(x)
=
sin
[(N+
l)x/21/[(N
+
1)
sin
(x/2)].
The
task
of
the
estimator
is
to
identifya
set
of
sine
waves
that
minimizes
Eq. 11.
Insights
into
the
development
of
a
suitable
estimator
can
be
obtained
by
restricting
the
class
of
input
signals
to
the
idealization
of
perfectly voiced
speech,
ie,
speech
that
is
periodic,
hence
having
compo-
nent
sine
waves
that
are
harmonically
related.
In
this
case
the
synthetic
speech
waveform
can
be
written
as
where
w~
=
21T/T~
and
where
T~
is
the
pitch
period
assumed
to
be
constant
over
the
duration
of
the
k
th
frame.
For
the
purpose
of
establishing
the
structure
of
the
ideal
estimator.
it
is
further
assumed
that
the
pitch
period
is
known
and
that
the
width
of
the
analysis
window
is
a
multiple
of
T~
•
Under
these
highly
idealized
conditions.
the
sinc
(.)
function
in
the
last
term
of
Eq. 11
reduces
to
156
The
Lincoln
Laboratory
Journal.
Volume
1.
Number
2
(l988)
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