维特比算法在离散时间马尔可夫过程中的最优状态估计
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收藏 1.03MB PDF 举报 "这篇资源是一篇关于维特比算法的英文论文，主要探讨了该算法在估计离散时间有限状态马尔科夫过程在无记忆噪声中的状态序列问题上的递归最优解法。论文提到了该算法在卷积码解码中的应用，并引用了一系列与电磁学和相对论相关的文献，可能在解释或对比维特比算法在非惯性参考系中的电磁现象处理方面的理论基础。"
维特比算法（Viterbi Algorithm，简称VA）是由Andrew Viterbi在1967年提出的一种动态规划方法，主要用于在有噪声的通信系统中寻找最可能的数据序列。它是解决 Hidden Markov Model (HMM) 最优解的高效算法，尤其在数字通信、语音识别、图像处理等领域有着广泛的应用。
在通信系统中，数据经常被编码成二进制序列进行传输，而卷积码是一种常见的前向纠错编码方式，可以提高信息传输的可靠性。维特比算法在解码卷积码时，通过计算每一步的“路径概率”和“后向概率”，找到一条具有最高总体概率的路径，即最有可能的原始信息序列。这个过程涉及到计算每个状态到当前时刻的概率以及从当前状态到后续状态的转移概率。
描述中提到的文献引用涵盖了从经典电磁学（如E. J. Post的《形式电磁学》和A. Sommerfeld的《电动力学》）到加速系统中的电磁辐射（例如J. L. Anderson和J. W. Ryon的工作）等多个方面。这些引用可能在论文中用于建立维特比算法在处理动态系统中的电磁问题时的理论框架，或者比较不同参考系下的解码效果。
在非惯性参考系中的电磁理论，如R. M. Fano, L. J. Chu, 和 R. B. Adler的工作，以及T. C. Mo的理论，可能与维特比算法在处理高速移动或加速系统中的信号解析问题有关。这些理论可能被用来解释在非惯性参考系下，如何修正由相对论效应引起的时间膨胀和长度收缩对解码过程的影响。
这篇论文不仅深入探讨了维特比算法在状态估计中的应用，还可能涉及了算法在处理与相对论和电磁学相关问题时的特殊考虑，这对于理解在复杂物理环境下的信息处理技术至关重要。
270
PROCEEDINGS
OF
THE
IEEE,
MARCH
1973
B.
Intersymbol Interference
In digital transmission through analog channels, we fre-
quently encounter the following situation. The input se-
quence
u,
discrete-time and discrete-valued
as
in the shift-
register model, is used to modulate some continuous waveform
which is transmitted through
a
channel and then sampled.
Ideally, samples
zk
would equal the corresponding
Uk,
or some
simple function thereof; in fact, however, the samples
,zk
are
perturbed both by noise and by neighboring inputs
Up.
The
latter effect is called intersymbol interference. Sometimes
intersymbol interference is introduced deliberately for pur-
poses
of
spectral shaping, in so-called partial-response sys-
tems.
In such cases the output samples can often be modeled
as
zk
=
yk
+
nk
where
yk
is
a
deterministic function of
a
finite number of in-
puts, say,
yk
=f(Uk,
* *
.
,
Uk-y),
and
nk
is
a
white Gaussian
noise sequence. This is precisely Fig.
2.
To
be still more specific, in pulse-amplitude modulation
(PAM)
the signal sequence
y
may be taken
as
the convolu-
tion
of
the input sequence
u
with some discrete-time channel
impulse-response sequence
(ho, hl,
*
*
):
yk
=
hi24k-i.
i
If
h,
=O
for
i>v
(finite impulse response), then we obtain our
shift-register model.
An
illustration
of
such
a
model in which
intersymbol interference spans three time units
(v=
2)
ap-
pears in Fig.
4.
It was shown in
[29]
that even problems where time is
actually continuous-Le., the received signal
r(t)
has the form
r(t)
=
Ukh(1
-
KT)
+
n(1)
for some impulse response
h(t),
signaling interval
T,
and reali-
zation
n(t)
of
a
white Gaussian noise process-can be reduced
without
loss
of optimality to the aforementioned discrete-time
form (via
a
"whitened matched filter").
K
k=O
C.
Continuous-Phase
FSK
This example is cited not for its practical importance, but
because, first, it leads to
a
simple model we shall later use in
an example, and, second, it shows how the
VA
may lead to
fresh insight even in the most traditional situations.
In FSK,
a
digital input sequence
u
selects one of
m
fre-
quencies (if
z&
is m-ary) in each signaling interval of length
T;
that is, the transmitted signal
q(t)
is
where
O(Uk)
is the frequency selected by
Uk,
and
ek
is some
phase angle. It is desirable for reasons both
of
spectral shap-
ing and
of
modulator simplicity that the phase be continuous
at
the transition interval; that
is,
that
w(Uk-1)kT
+
ek-1
E
w(uk)kT
+
ek
modulo
27r.
This is called continuous-phase FSK.
The continuity
of
the phase introduces memory into the
modulation process; i.e., it makes the signal actually trans-
mitted in the kth interval dependent
on
previous signals. To
take the simplest possible case ("deviation ratio"
=a),
let
the
JEi-J-E+
Yh
+
h0
Fig.
4.
Model
of
PAM
system subject to intersymbol
interference and white Gaussian noise.
"k
L9-jw-g
SIGNAL
+
"lk
Fig.
5.
Model
for
binary continuous-phase
FSK
with deviation ratio
3
and coherent detection in white Gaussian noise.
input sequence
u
be binary and let
w(0)
and
w(1)
be chosen
so
that
w(0)
goes through an integer number of cycles in
T
seconds and
w(1)
through an odd half-integer number; i.e.,
w(O)T=Oandw(l)T=7rmodulo27r.
Thenif
eo=O,
O1=Oorr,
according to whether
uo
equals zero
or
one, and similarly
&=O
or
r,
according to whether an even or odd number of
ones has been transmitted.
Here we have
a
two-state process, with
X
=
(0,
7r
1.
The
transmitted signal
yk
is
a
function
of
both the current input
uk
and the state
xk:
yk
=
cos
[W(Uk)t
+
Xk]
=
cos
cos
w(Uk)f,
KT
t
<
(R
+
1)T.
Since transitions
&
=
(xk+l,
%$
are one-to-one functions
of
the
current state
xk
and input
Uk,
we may alternately regard
yk
as
being determined by
(k.
If we take
qo(t)
&os
w(0)t
and
vl(t)
&os
w(1)t
as
bases
of
the signal space, we may write
yk
=
yOk'?O(l)
+
ylkvl(t)
where the coordinates
(yo, ylk)
are given by
I
((0,
-I),
if
Uk
=
1,
Xk
=
Finally, if the received signal
((t)
is
q(t)
plus white Gaussian
noise
v(t),
then by correlating the received signal against both
qo(t)
and
ql(t)
in each signal interval (coherent detection), we
may arrive without
loss
of information
at
a
discrete-time out-
put signal
zk
=
(ZW,
Zlk)
=
(yak,
ylk)
+
(nokt
flu)
where
no
and
n1
are independent equal-variance white Gaus-
sian noise sequences. This model appears in Fig.
5,
where the
signal generator generates
(y~,
ya)
according to the aforemen-
tioned rules.
D.
Text Recognition
\Ve include this example to show that the
VA
is not limited
to digital communication. In optical-character-recognition
(OCR)
readers, individual characters are scanned, salient
Authorized licensed use limited to: Universitatsbibliothek der TU Wien. Downloaded on March 03,2010 at 07:23:58 EST from IEEE Xplore. Restrictions apply.
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