Hi Ross, Just spotted this. I don't have an answer for you, but a possible helpful literature connection...?
The system you describe is a simple Markov model. It's ergodic and time-homogeneous and reversible, and has no hidden state, so I'd guess that there must be results from the Markov model literature that can help. In particular MCMC work, which uses reversible Markov chains and their stationary distributions. In fact they often consider the autocorrelation of the converged processes, so they can work out how to take uncorrelated samples. Best Dan On 03/11/15 17:42, Ross Bencina wrote: > Hi Everyone, > > Suppose that I generate a time series x[n] as follows: > >>>> > P is a constant value between 0 and 1 > > At each time step n (n is an integer): > > r[n] = uniform_random(0, 1) > x[n] = (r[n] <= P) ? uniform_random(-1, 1) : x[n-1] > > Where "(a) ? b : c" is the C ternary operator that takes on the value b > if a is true, and c otherwise. > <<< > > What would be a good way to derive a closed-form expression for the > spectrum of x? (Assuming that the series is infinite.) > > > I'm guessing that the answer is an integral over the spectra of shifted > step functions, but I don't know how to deal with the random magnitude > of each step, or the random onsets. Please assume that I barely know how > to take the Fourier transform of a step function. > > Maybe the spectrum of a train of randomly spaced, random amplitude > pulses is easier to model (i.e. w[n] = x[n] - x[n-1]). Either way, any > hints would be appreciated. > > Thanks in advance, > > Ross. > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp -- Dan Stowell EPSRC Research Fellow Centre for Digital Music Queen Mary, University of London Mile End Road, London E1 4NS http://www.mcld.co.uk/research/ _______________________________________________ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
