I need to generate a sequence of pulses at around 1 Hz with a 1/f
characteristic (human heartbeat, as it happens). I'd like to do this
using software and a timer, so I'm looking for a clever algorithm using
a random number generator to do it.
I could take the phase noise spectrum and turn that into some form of
cumulative probability distribution for the period, then generate a
random number from 0-1 and use the inverse of the CPD to determine the
period.
But I was wondering if there's some clever way that just happens to
generate what I'm looking for. Sort of like how you sum up 12 random
numbers to generate a Gaussian with variance 1. and then, the
Box-Muller algorithm as an alternate way.
A generalized approach for the exponent between 0.5 and 1.5 would be
useful.
I found some techniques such as building a filter with the required
power spectral density and then running white noise through it. There's
a matlab (& C) package out there called cnoise, as well. cnoise builds
an array of samples and uses a FFT to do the filtering efficiently.
I'd rather have some sort of difference/recursion equation that I can
just call each time I need the next interval. I did find some code based
on a paper by Higham that does what's called the Ornstein-Uhlenbeck
stochastic difference equation.
dt = tmax / n;
x(1) = x0;
for j=1:n
dw = sqrt ( dt ) * randn;
x(j+1) = x(j) + dt*theta*(mu-x(j)) + sigma * dw
But I don't trust it, because the matlab and c versions do not agree.
1/f^2 (brownian) is easy by taking the random number sequence and
integrating.
x(j+1) = x(j) + randn(1);
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