442.012 Mystery Cache

This cache draws you into the exciting world of adaptive linear filtering - a tough task, but manageable with a little linear algebra and probability theory. I hope you enjoy it!

Information about the cache (and ways to solve it) can be found in the link below or in the book "Adaptive Filter Theory" by Simon Haykin.

The first goal is to solve a mathematical problem: Consider the figure above, where c is an adaptive filter with three coefficients, i.e., c=[C,D,E]. h is a linear, time-invariant filter with two coefficients, h=[5,5]. x[n] is a unit-variance white noise process with zero mean. v[n] is jointly stationary with x[n], with cross-correlation

E(x[n-k]v[n]) = 2 if k=2 and 0 else.

The filter coefficient vector minimizing the mean-squared error E(e²[n]) is given by the following equation:

c = R_xx^-1 p,

where R_xx is the autocorrelation matrix of the input process x[n], and p is the vector of cross-correlations between x[n] and d[n].

-- Let AB be the rank of the matrix R_xx (e.g., AB=07 or AB=15).

-- Compute the coefficient vector c=[C,D,E].

-- This theory behind this problem is intricately linked to a famous mathematician, who also investigated the mathematical aspects of Brownian motion and is considered the founder of cybernetics. He was born on DD-MM-YYYY; FG=DD+1.

-- It is also possible to get the solution without knowing the process statistics, by letting an algorithm run. This algorithm just takes measurements of the input process x[n], the desired output d[n], and the error e[n], and changes the coefficient vector c such that the squared error is minimized. The algorithm was developed by a professor and his PhD student. H is the second letter of the professor’s first name (e.g.: Sigurd, second letter = i, H=9). Hint: The professor also contributed greatly to the theory of quantization.

-- To which of the following applications of adaptive filters does the example above belong?

I=1: inverse modeling
I=2: prediction
I=3: system identification
I=4: interference cancellation

-- The matrix equation solving the problem above is also linked to an Austrian mathematician. He died in 1983. J = digit sum of his age - 3.

  Update 28.09.2017: Since the original cache location is being deconstructed, I had to move the cache a bit. It is now approx. 10 m to the north of the final coordinates. The cache container had to be downsized to a petling.