IEEE SPECTRUM - August, 1969
Vol. 6, No. 3
The State-V aria We Approach to Continuous Estimation with Applications to
Analog Communication Theory, Donald
L. Snyderâ€”The MIT. Press, 50 Ames
St., Cambridge, Mass. 02142, 1969;
It4 pages, itlus , $7.50. This monograph
is a useful contribution to the engineering literature on estimation theory and
of particular interest to communication
and control theorists. In brief summary,
it begins with a standard exposition of
state-space representation of Markov
processes, proceeds with a derivation
of Kushner's diffusion equation for^he
conditional density function, and from
this develops the principal result, which
is the derivation of approximate differential equations for the conditional
mean estimator and the resulting error
variance for the nonlinear filtering
problem. These represent approximate
generalizations of the Kalman Bucy
equations to which they specialize in the
linear case. The remainder of the book
treats applications, primarily to analog
communication including phase and
frequency demodulation systems. A
final chapter treats the performance of
these quasi-optimum demodulators.
As a research monograph, this work
suffers from two drawbacks: virtually
no new results are obtained, and the
approximations are never really justified
except by limited experimental evidence.
On the other hand, these quasi-optimum
nonlinear estimators (phase-locked loop
demodulators) had previously been derived using a maximum a posteriori
probability (MAP) criterion, which is
both less general and less satisfying
than the conditional mean estimator, for
the latter is optimum for a large class of
performance criteria, including the most
useful minimum variance criterion.
Also, the same set of approximations are
required in the MAP derivation and
equally lack substantiation except at
high signal-to-noise ratios.
In the final chapter, besides pre
viously known performance results for
quasi-optimum PM and FM demodulsv
tors, one new and interesting result it
obtained. This is an expression for the
probability density function of the
phase error for the quasi-optimum estimator of a first-order Markov phase
process. Although the functional form
[Eq. (6.9)] of the result is exact, the
parameters are approximate, since the
derivation begins with an expression
for the minimum phase variance [Eq.
(6.2)], derived from a linearized model
and thus valid only at high signal-to-
noise ratios. The result could be made
exact, though considerably more complex, by leaving the variance unspecified, obtaining Eq. (6.9) in terms of
oV and finally obtaining a transcendental
equation in this parameter by calculating the second moment of p(+) from the
revised Eq. (6.9). From this the unknown variance could be obtained by
I numerical methods.
Generally, this monograph i$ useful
in that it presents concisely a conoep-
i luaiiy pleasing and seemingly elegant
I approach to nonlinear estimation with
: particular emphasis on important analog
{ communication applications. Unfortunately, it shares die weakness of all
1 other efforts in nonlinear estimation and
1 analog demodulation that concise, sim-
, pie, and practical results can only be
i obtained through approximations whose
I validity is a monotomc function of the
| signal-to-noise ratio.
A. 7. Viterbi
University of California
Los Angeles, Calif.