``You Have No Control...''-Bad Religion song lyric
We'll represent our system as the following Gauss-Markov sequence.
Where 
and 
are are, respectively, 
and 
vectors and 
and 
are 
and 
dimensional Gaussian
White Noise sequences. We may trade a great deal of hassle for a little bit of
generality by assuming them to be mutually independant random variables, so that 
. 
is our Transfer Function and 
is our measurement function, and both are assumed to be known.
Note that this is
a single sequence in 
corresponding to an (uncountably) infinite
number of sample sequences in 
. That is, each variable at each time step is a single vector in ,
but may represent any number of possible values in the Sample (State) Space.Note also that we have no control
in this model. That
is, there is no control function, only a transfer function, a measurement function, and some noise
terms. This is because the control has no probabalistic importance. It is entirely deterministic,
so we can simply subtract out its effects to make the development simpler. Of course, in any real
world application, we'd need it in our system, and it is easily incorporated. For now, we are only
concerned with the stochastic behavior of the system,
and the control only affects the mean.
For some examples of the type of system and behavior we're discussing, please refer to the ``Implementations and Examples'' section below.
It is important to make explicit the following simple property of the above general system: the dimension
of need not equal that of . If it is greater, of course, this is trivial. If smaller, however,
this is one of the most profound advantages and fascinating aspects of the Kalman Filter 
.
This is what sets the Kalman Filter apart from existing filtering techniques. Under suitable conditions
(think observability from basic linear control theory...) we may estimate and control many more
state variables than we have observations.
We can imagine some Dark Cabal of Control Theorists in Pynchon's ``Gravity's Rainbow'' designing the circuitry which will be the Nervous System, the very Pavlovian Complex of Responses, for their mythic V2 rocket. How can they deal with the fact that their lovely Rocket has over 29 state Variables : Pitch, Yaw, Velocity, Temperature, Weight, a veritable litany of Relevant Factors. But they must, back on Earth, be content with the two variables they are passed back along some static-churned tenuous link of electro-magnetic spectra. How can they, given only, say, Yaw and Velocity, know when Brennschluss has been reached?
That is, given only a noisy measurement of two quantities, the complex, dynamic state of the Rocket must be estimated to exacting precision, so that the Engine may be turned off at that Threshold, Aeolian Moment at the Apex of Arc in order to turn Ascent to Descent, to bring the (from that moment on) mute, dumb Rocket crashing in Super-Sonic Fire into huddled, blacked-out Wartime London. Fortunately for England, this all took place before the advent of Kalman Filtering, and the Germans were forced to use sub-optimal methods. Had they had access to this Grail of Stochastic Control Systems, we might all be learning German in school.
Well, Hyperbole is all well and good, but it cannot be emphasized enough how powerful this property of the filter truly is. This is what makes it truly a step forward, and has made its influence felt far beyond the confines of Linear Quadratic Gaussian Control theory.