Do you mean the literal Fourier spectrum of some realization of this process, or the power spectral density? I don't think you're going to get a closed-form expression for the former (it has a random component). For the latter what you need to do is work out an expression for the autocorrelation function of the process.
As far as the autocorrelation function goes you can get some hints by thinking about what happens for different values of P. For P=1 you get an IID uniform noise process, which will have autocorrelation equal to a kronecker delta, and so psd equal to 1. For P=0 you get a constant signal. If that's the zero signal, then the autocorrelation and psd are both zero. If it's a non-zero signal (depends on your initial condition at n=-inf) then the autocorrelation is a constant and the psd is a dirac delta. Those are the extreme cases. For P in the middle, you have a piecewise-constant signal where the length of each segment is given by a stopping time criterion on the uniform process (and P). If you grind through the math, you should end up with an autocorrelation that decays down to zero, with a rate of decay related to P (the larger P, the longer the decay). The FFT of that will have a similar shape, but with the rate of decay inversely proportional to P (ala Heisenberg Uncertainty principle). So in broad strokes, what you should see is a lowpass spectrum parameterized by P - for P very small, you approach a flat spectrum, and for P close to 1 you approach a spectrum that's all DC. Deriving the exact expression for the autocorrelation/spectrum is left as an exercise for the reader :] E On Tue, Nov 3, 2015 at 9:42 AM, Ross Bencina <rossb-li...@audiomulch.com> 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 >
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