NoisePlots

February 4, 2018 | Author: Anonymous | Category: Math, Statistics And Probability, Normal Distribution
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Statistical properties of Random time series (“noise”)

Normal (Gaussian) distribution Probability density:

A realization (ensemble element) as a 50 point “time series”

Another realization with 500 points (or 10 elements of an ensemble)

From time series to Gaussian parameters • N=50: =5.57 (11%); =3.10 • N=500: =6.23 (4%); =3.03 • N=104: =6.05 (0.8%); =3.06

Divide and conquer • Treat N=104 points as 20 sets of 500 points • Calculate: – mean of means: E{m}==5.97 – std of means: sm==0.13

• Compare with – N=500: =6.23; =3.03 – N=104: =6.05; =3.06 – 1/√500=0.04; 2sm/E{m}=0.04

Generic definitions (for any kind of ergodic, stationary noise) • Auto-correlation function For normal distributions:

Autocorrelation function of a normal distribution (boring)

Autocorrelation function of a normal distribution (boring)

Frequency domain • Fourier transform (“FFT” nowadays): IF

• Not true for random noise! • Define (two sided) power spectral density using autocorrelation function: • One sided psd: only for f >0, twice as above.

Discrete and finite time series



• • • • •

Take a time series of total time T, with sampling Dt Divide it in N segments of length T/N Calculate FT of each segment, for Df=N/T Calculate S(f) the average of the ensemble of FTs We can have few long segments (more uncertainty, more frequency resolution), or many short segments (less uncertainty, coarser frequency resolution)

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