Random-process Simulation and Monte Carlo Studies with Noisy Signals in Visual Studio .NET

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Random-process Simulation and Monte Carlo Studies with Noisy Signals
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following an OUT or SAMPLE m statement (Sections 1-8 and 2-3) This ensures periodic sampling at the sampling rate
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SR = (NN 1)/TMAX or SR = (NN 1)/(m * TMAX)
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(5-4)
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SR is determined by the values of NN, TMAX, and m set by the experiment
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protocol (Section 1-6)2 Numerical integration updates differential-equation system variables in small steps to model continuous or analog functions of the time t (Sections 1-6 and 1-7) Noisy sampled-data variables fed to a differential-equation system are sample/hold state variables (Section 2-3)3 and thus discontinuous step functions of the time t Such variables are read at every derivative call, but they change only at sampling times, so that numerical integration will be correct As we noted in Section 2-3, sample-hold state variables require initialization at t = t0 To model continuous noise Noise = Noise(t), we feed a noisy sampleddata state variable y to a differential equation system representing a low-pass or band-pass filter, as in
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d/dt Noise = w * Noise + y | -- one-stage low-pass noise filter OUT y = a * (ran()+ran()+ran()+ran()) + b | -- y is roughly Gaussian
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The noise frequency spectrum is determined by the noise-sampling rate (5-4) and the filter transfer function [2] Many kinds of random processes can be derived from such simulated analog noise, for example,
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q = A * sin(w * t) + c * p (sinusoid with additive noise) q = A * p * sin(w * t) (random-amplitude sinusoid) q = A * sin(w * t + p) (random-phase sinusoid)
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Note that multiple independent noise generators require separate calls of
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ran() To get partially correlated noise samples y, z, one can use assignments
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such as
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y = ran() z = ran() + b * y with E{y * z} = b/3
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2 This should be noted if TMAX is to be changed, for noise spectra will change unless NN is changed as well If noise sampling faster than the input/output sampling rate is required, set MM > 1 (Section 1-6) 3 Just as in Figure 2-2, one cannot observe the sample-hold action on a display unless the variable is created following a SAMPLE m statement with m > 1
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Monte Carlo Simulation with Noisy Signals
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(b) Continuous Time Averages
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To produce the time average
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xavg = (1/t)
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(5-5)
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we program the DYNAMIC segment line
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d/dt xxx = x | xavg = xxx/t with xxx(0) = 0 (default value)
(5-6a)
d/dt xavg = (x xavg)/t with xavg(0) = x(0)
(5-6b)
Instead of using the default value t = t0 = 0, we set t = t0 = 10e 275 to eliminate the singularity (see also Section 5-2c) 5-4 Problems with Simulated Noise Simulation programs assume that different calls on a pseudorandom-noise generator such as ran() produce statistically independent samples of simulated random parameters and noise functions [3 8] But this is really not true Pseudorandom-noise samples, although usually guaranteed to be uncorrelated, are generated by a deterministic program Model outputs can depend on higher-order joint probability distributions of many random-noise samples [6], and it is conceivable that hidden periodicities or correlations might produce strange unforeseen effects In 4, our Monte Carlo studies involved only random parameters and/or random initial values; that requires relatively few noise samples and is usually safe But simulations involving wideband time-variable noise may require enormous numbers of independent noise samples [9] One thousand simulation runs with, say, 5 noise sources might need 5 500 million independent samples4 References [3] and [5] list a number of tests for the quality of pseudorandom noise, but we usually assume statistical independence and then compare results obtained with different pseudorandom-noise generators Section 5-10 describes a simple method that completely rescrambles an existing noise sequence for such tests MONTE CARLO SIMULATION WITH NOISY SIGNALS 5-5 Gambling Returns The vectorized Monte Carlo program in Figure 5-1 takes statistics on a sample of n gambling sprees Each gambling spree consists of t = N successive
ran(), which is based on the GNU library routine drand48, repeats after 248 1 samples and normally produces good results If desired, it would not be difficult to implement ran() with a
pseudorandom-noise generator having a longer repetition period [3, 4]