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designs to measure the temperature of a
CPU core. Because devices are usually
noisy, the measurements are likely to
differ from the actual temperature of the
core. As the devices are of different de-
signs, let us assume that noise affects
the two devices in unrelated ways (this is
formalized here using the notion of cor-
relation). Therefore, the measurements
x1 and x2 are likely to be different from
each other and from the actual core tem-
perature xc. A natural question is the fol-
lowing: is there a way to combine the in-
formation in the noisy measurements x1
and x2 to obtain a good approximation of
the actual temperature xc?
One ad hoc solution is to use the for-
mula 0.5*x1+0.5*x2 to take the average
of the two measurements, giving them
equal weight. Formulas of this sort are
called linear estimators because they use
a weighted sum to fuse values; for our
temperature problem, their general form
is β*x1+α*x2. In this presentation, we use
the term estimate to refer to both a noisy
measurement and a value computed by
an estimator, as both are approxima-
tions of unknown values of interest.
Suppose we have additional infor-
mation about the two devices, say the
second one uses more advanced tem-
perature sensing. Because we would
have more confidence in the second
measurement, it seems reasonable
that we should discard the first one,
which is equivalent to using the linear
estimator 0.0*x1 + 1.0*x2. Kalman filter-
ing tells us that in general, this intui-
tively reasonable linear estimator is not
“optimal;” paradoxically, there is useful
information even in the measurement
from the lower quality device, and the
optimal estimator is one in which the
weight given to each measurement is
proportional to the confidence we have
in the device producing that measure-
ment. Only if we have no confidence
whatever in the first device should we
discard its measurement.
The goal of this articlea is to present
the abstract concepts behind Kalman
filtering in a way that is accessible to
most computer scientists while clarify-
ing the key assumptions, and then show
how the problem of state estimation in
linear systems can be solved as an
a An extended version of this article that in-
cludes additional background material and
proofs is available. 30