A search for a weak oscillation in a laser interferometer's output averages the…
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Scientific Computing for Physicists, Ch Spectra with error bars: the fast transform, convolution and spectral estimation
Problem
A search for a weak oscillation in a laser interferometer's output averages the periodograms of K=16 non-overlapping segments of M=2048 samples each (Welch's method), from a record of N=KM=32768 samples of stationary Gaussian noise. The segments are long compared with the noise's memory, so treat them as independent. Away from the ends the averaged spectrum is flat, at a level S known well from the many bins around it. One interior bin stands at 2.5S.
((a)) Using Sk∼Sχ2K2/(2K), find the probability that a bin of pure noise reaches 2.5S or more. For an even number of degrees of freedom the tail has the closed form P(χ2K2>x)=e−x/2∑j=0K−1(x/2)j/j!.
((b)) How many of the M/2−1=1023 interior bins of the averaged spectrum does pure noise put at 2.5S or above, on average? Should the bin be reported as a candidate line?
((c)) Transformed whole, without averaging, the same record gives a raw periodogram with N/2−1=16383 interior bins. Find the probability that one raw ordinate reaches 2.5S and the number of raw bins pure noise puts there, and say what the comparison with parts (a) and (b) shows.