3 ms·
That was on the final of my Stochastic Processes class back in the Pleistocene. I just took the easy way and used system theory: find a system with a transfer f
by bandrami 10y ago
That was on the final of my Stochastic Processes class back in the Pleistocene. I just took the easy way and used system theory: find a system with a transfer function that is the inverse of the power spectral density of the autocorrelation function for the biased coin (Mason's rule guarantees this is possible, if sometimes impractical even in theory), and run the output through that system.
Interestingly the prof emailed me the next semester because that had caught his attention and he had worked out that the two processes are ultimately identical, just using very different notations.
- brian_cloutier 10y ago> find a system with a transfer function that is the inverse of the power spectral density of the autocorrelation function for the biased coin For those of us who don't know systems theory, is there a simpler explanation of this? It sounds interesting
- bandrami 10y agoSure. A "system" (which is an incredibly broad term; in practice a circuit composed of resistors, capacitors, and inductors can model any linear system so we stick with that) takes the signal x(t) and maps it to something else, f(x(t)). The autocorrelation function r(t) defines the bias of the coin (r(t) is generally given in the problem, or in real life found empirically). All of these functions are difficult to work with directly. However, their Laplace transforms are much easier: X(s) and R(s), and because of how Laplace transforms work, F(s)X(s) (convolution becomes multiplication after a Laplace transform). So if F(s) is 1 / R(s), F(s)X(s) = X(s) / R(s), which means all of the correlation in X(s) is divided out, and you're left with an uncorrelated function, or fair coin in this case.