Geology 6600/7600 Signal Analysis 15 Oct 2015 © A.R. Lowry 2015 Last time(s): PSE The Kirby approach to wavelet transformation (the fan wavelet) preserves.

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Geology 6600/7600 Signal Analysis 15 Oct 2015 © A.R. Lowry 2015 Last time(s): PSE The Kirby approach to wavelet transformation (the fan wavelet) preserves phase and so can be used to evaluate complex coherency The imaginary part of complex coherency, if non-zero, implies a statistical correlation of gravity and topography that is not a result of (zero-phase) flexure (hence implies correlated loading) However, may not help to identify correlated loading in situations where the load correlation is zero-phase…

+ Frequency Response Estimation with Random Inputs Consider a single-input system with additive noise: Here our data consist of one realization each of x(t) and y(t). Ideally to estimate H(  ), the transfer function or “admittance”, we would estimate from the power spectra: However we can only estimate S xx, S yy, and S xy : The noise n(t) is unknown! ~~ ~

Thus the best we can do to approximate the transfer function is to estimate it as: In this circumstance however the Coherence Function can be used to assess the effect of the noise on the transfer function estimate. Coherence is given by: For the case of no noise ( n(t) = 0 ), Given uncorrelated noise, S xy = S xv + S xn = S xv. Then ~

Thus, Note: This implies S vv < S yy. We can show this as follows: Fourier transforming, given v and n uncorrelated!

Our coherent output spectrum (I.e. what it looked like before addition of additive noise) can be estimated as: The output noise spectrum S nn = S yy – S vv : or equivalently Thus for this type of system, the coherence function provides a measure of the ratio of the noise power to the total power at a given frequency  :