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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: <z(t)>=5.57 (11%); <(z(t)-<z>)2>=3.10 • N=500: <z(t)>=6.23 (4%); <(z(t)-<z>)2>=3.03 • N=104: <z(t)>=6.05 (0.8%); <(z(t)-<z>)2>=3.06 Divide and conquer • Treat N=104 points as 20 sets of 500 points • Calculate: – mean of means: E{m}=<mk>=5.97 – std of means: sm=<(m-E{m})2k>=0.13 • Compare with – N=500: <z(t)>=6.23; <z2(t)>=3.03 – N=104: <z(t)>=6.05; <z2(t)>=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)