This is a follow up to the question How to Pick a Journal for Publishing a Research Work? question. Suppose we take discrete samples of a low frequency signal corrupted by high frequency noise. We know the signal has no frequencies higher than 20% of the Nyquist rate. We also know all the noise is at or above 50% of the Nyquist rate, and below the Nyquist rate. I will use the following as a specific example to experiment with.
\begin{align}&\textrm{measurement}(t) = \textrm{signal}(t) + \textrm{noise}(t)\\ &\textrm{signal}(t) = 4.4+\sin(0.06\pi t-0.1) + \sin(0.14\pi t+0.07) + \sin(0.2\pi t+0.4)\\ &\textrm{noise}(t) = \sin(0.06+0.5\pi t) + \sin(0.1-0.68\pi t) - \sin(0.04-0.74\pi t) + \sin(0.03-0.93\pi t) \end{align}
The measurement is sampled at integer values of ($t$). A plot of the measurement samples and the continuous signal is below.
How well can we estimate $\textrm{signal}(t)$ at integer values of ($t$) using the current and past measurements? I was able to filter it using the Kalman filter below.
A plot of the signal and Kalman filter output is in the next plot. The final plot shows the error in the Kalman filter output which has a maximum magnitude of 1.865 and the RMS error is 0.7895. Can we make a Kalman filter that does better than the Kalman filter above? What known filter method would give the best estimates, and how good would the estimates be?
I learned about discrete lowpass filters 25 years ago, and I always assumed a problem such as the one above was a common application for them. At that time I didn't know this problem is called estimation, and I didn't know about the Kalman filter. Then I recently took this course, and another course Georgia Tech provided that goes into the details of estimation algorithms. However, the Georgia Tech courses mention nothing about using a discrete lowpass filter for estimation. I check several books on estimation, and found nothing about discrete lowpass filters. I work with a computer scientist who has been doing research in data fusion for 20 years, and he knows nothing about z-transforms, transfer functions, etc. Why is it that the estimation community ignores the use of a discrete lowpass filter for estimation? As far as I can tell it's the best approach to estimate the signal above.