Geology 6600/7600 Signal Analysis Last time: Linear Systems The Frequency Response or Transfer Function of a linear SISO system can be estimated as (Note this differs from deconvolution, in which an unknown signal or response function is found by dividing amplitudes by a known convolved function or signal) Frequency response estimated in this way is unaffected by additive noise in the output, y = v + n, because S xy = S xv The output power spectrum is S yy = S vv + S nn, and the coherence is: So the signal power and the noise power can be estimated… 19 Oct 2015 © A.R. Lowry 2015

Now consider a case where there is additive noise on both the measured input and output signals: We measure x (corrupted by input noise m ) and y ( + n ). We’ll again assume uncorrelated noise (with each other and with the signals). Then + +

The coherence in this case is: Recalling that: Then: The estimate of the transfer function will be: (And for this approach cannot independently derive input noise!)

Consider however that there are two different ways we could estimate the transfer function: (1)From the autospectrum : Note that this estimate biases upward or downward depending on relative SNR… Autospectra always include noise spectra. (2) From the cross-spectrum : This estimate is biased only by the input noise, and has the property of minimizing the mean-square error between measured and predicted output. Note the implication that different combinations can be used to bound H and the noise spectra!

We have: These relations tell us that we can also bound the signal-to-noise ratios of both the input and the output signals! (But since these are not independent equations, we can’t estimate them). (This estimate can be biased upward or downward) (This estimate is ≤ H, so a lower- bound) ( ≥ H, so an upper-bound!)

+ Now consider a multiple input, single output system: Here: (in vector notation:)

To find the spectrum of the output, multiply by the complex conjugate: (vector notation) (here superscript H denotes the Hermitian, or transpose of the complex conjugate) Recall the power spectrum represents The expected value (assuming that x i and n are uncorrelated): where S ij = S xixj

How can we calculate the frequency responses in this case? If we multiply Y by the complex conjugates of the input frequencies, X i *, This gives us a set of N equations in which we can solve for the N unknown frequency responses using the measured cross-power spectra! In matrix form: Note that this is equivalent to the minimum least-squares solution. Further note that a solution cannot be found if any of the inputs are perfectly correlated (the matrix determinant in that case will be zero because the rank of the matrix will be < N ). Thus, one should examine the coherence of the individual input signals to identify possible problem frequencies…. or