I must be missing something. I passed 128 records of 1300 samples each (1300x128) through a Kalman Filter (This is basically a laser return signal - often times quite small- buried in solar noise generated by an APD - Avalanche Photo Diode) and the output appears as noisy as any one of the record inputs. Simple Averaging of these 128 records yields a signal from the noise, but I was under the impression that Kalman Filter is superior to simple averaging for lifting signal from noise. I can only conclude that I am doing something wrong. I had set my State Transition Matrix (A) and Observation Matrix (H) to an Identity Matrix, and I played around with process noise covariance(Q) and measurement noise covariance (R) values but to no avail. This is a stationary process so I employed a Simple Discrete Kalman Filter. Are there any limitations to the use of Kalman filters? Sampling Rates? Any suggestions why a filter may fail you? Thanks.

Hi,

I am not entirely sure what you are doing. Do you use 128 states, i.e., the matrix A is 128x128? Assume that is

what you are doing. I think the "problem" stems from an over-expectation for Kalman Filter (KF). KF is only as good as

its model (A,H,Q,R) can provide. It is optimal within the linear estimation methods, assuming the matrics A,H,Q,R do properly

desribe the underline physical phenomenon. Intuitively, assuming A=I (the identical matrix) expects that the next

state x(k+1) is the same as the current x(k) when there is no noise. This means that any difference x(k+1)-x(k) between

those 2 states come from the process noise (Q). This kind of model does not seem to be much helpful, unless

you are convenced that the data you are analysing should obey such a simple model.

When the process model and observation model are assumed to be such simple (but physically very restrictive model in

the sense that x(k+1) has to be the same as x(k)), one way to see why KF may not be better than a simple average

(a low pass filter) is to look at the Riccati equation.

The steady-state covariance maxtix of KF is the solution of the (algebraic) Riccati equation:

P=A(P-PH'(HPH'+R)^-1HP]A'+Q

In our case, A=H=I, so the above equation is simplified into

P=P-P(P+R)^-1P+Q

0=-P(P+R)^-1P+Q

P^-1QP^-1= -(P+R)^-1

PQ^-1P= -(P+R)

To get some intuition, assume Q=qI, R=rI, where q,r are positive scalar (not a vector), we have

qP^2+P+rI=0

This is a matrix quardratic equation. In the extreme case, when P is a scalar, i.e., when demention of x is only 1

(this is not our case, but we just want to get some intuition), the equation has solution

P=(-1 +/- sqrt(1-4qr))/2

Since sqrt(1-4qr)<1, this solution P<=0, which is not a valid solution. This means that KF will not converge to any steady-state,

i.e., the covariant matrix P(k|k) obtained at each time instance k will vary even when time goes to infinity.