Wait, just realized I wrote that last part backwards. It should be: So in broad strokes, what you should see is a lowpass spectrum parameterized by P - for P very small, you approach a DC spectrum, and for P close to 1 you approach a spectrum that's flat.
On Tue, Nov 3, 2015 at 10:26 AM, Ethan Duni <ethan.d...@gmail.com> wrote: > 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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