Geology 6600/7600 Signal Analysis 23 Oct 2015 Last time: Linear Systems • The Multiple Coherence Function measures coherence of combined inputs to output: giving it similar utility to the SISO coherence in characterizing additive noise that is unrelated to the inputs! • Digital Wiener Filtering seeks to design a filter h for a linear SISO system producing an optimal output y that “looks like” a desired signal d • Applications include smoothing, prediction… © A.R. Lowry 2015

Reading for Friday (30 Oct): Becker, T.W., et al., Static and dynamic support of western United States topography, Earth Planet. Sci. Lett., 2014. (I will prep discussion materials… You’ll need to be prepared to discuss specifically the SIGNAL ANALYSIS methods used in the paper!) Two “background” items to discuss: • An early draft examined only global cross-correlations, but an associate editor wanted to see wavelength-dependence of cross-correlation. How was this addressed, and how does that approach relate to other topics in this class? • The associate editor asked whether the revised approach acts as a zero-phase filter in the frequency domain. How could you test this?

where s[n] is desired signal and v[n] is Digital Wiener Filtering: Suppose we have the linear SISO digital system: And we would like to design an “optimum h” so that , some “desired” output. Possible applications for such a transfer function might include: (1) Smoothing: where s[n] is desired signal and v[n] is noise, in which case: (2) Prediction: ~ ~

To accomplish our goal, we will seek to find the h that minimizes the mean-squared error (MSE): for all m. That is, we want to set the error orthogonal to the input signal! The Orthogonality Principle (as commonly applied in linear estimation theory) states that the error vector of the optimal estimator (in a mean square error sense) is orthogonal to any possible estimator. Expanding: Thus or

So, our time domain solution is: Obviously if we “know” Rxd a priori, this problem can be solved in the frequency domain, where for  = t: But in the time-domain, the 2N + 1 length filter, expanding via the Yule-Walker equations, can be written: for all n = –N, –N+1, …, 0, 1, …, N This system of linear equations can be written in matrix form: for which output would then be:

So now let’s briefly return to our applications mentioned earlier: (1) Smoothing: where s[n] is desired signal and v[n] is noise. Recall that Rsx[m] = Rxx[m]h[m] in the time domain, while in the frequency domain

If we assume that s and v are uncorrelated then: As a reminder, we can inverse Fourier transform spectra to correlation functions and write out the expectation operators to show that: and If we assume that s and v are uncorrelated then: Note, however, that this approach assumes a priori that we know something independently about the statistics of the “noise” signal, or the statistics of the desired signal, or both! This is the general knock on “Bayesian” approaches… ~ ~

Grokking the Fourier Transform (HW): 1) Autocorrelation of a zero-mean discrete-time WSS white noise process (use 2 = 3) 2) A constant (use a = 3) 3) A cosine function (use amplitude 1; ) 4) A sine function (as above) 5) A box function (0 on [–,–/2] & [/2,]; 1 on [–/2,/2]) Important: Defns of forward/inverse Fourier transform Useful: Euler’s eqn, function product relations

Important: Defns of forward/inverse Fourier transform Useful: Euler’s eqn, function product relations Some key things to recognize: • Integral of a sinusoid on [–,] (hence [–∞,∞]) is always zero unless the sinusoid is cos(0). • Ergo, integral of a product of sin’s/cos’s is nonzero only when . In the case of FT, the fourier transform Acos(t) and Asin(t) is thus nonzero only at 0. • By definition this means that sinusoids are a class of orthogonal functions. (Their inner product is zero unless f = g!) • This makes them a very important class of functions…

HW1 • First note autocorrelation of a discrete white noise process is Kroneker rather than Dirac delta function • Top left, autocorrelation of a random (Gaussian, white) noise sequence 10000 samples long • Replace that with a vector that is three in the first element, zero all others, fft then fftshift… Get the result on bottom left

HW2 HW3 • FFT of a constant is the constant times the number of samples N at  = 0; zero elsewhere • FFT of a cosine is AN/2 at ±0. (This calc used a single sinusoid on [0,2) with 10000 samples.) Note the small nonzero values in the imaginary component; numerical error in Matlab is double precision = O(10–16). Real HW3 Imag

HW4 HW5 Real • FFT of a sine is iAN/2 at –0, –iAN/2 at +0. Again note numerical error! • FFT of a box function. Note the sinc is pretty ratty when we FFT only [–,); this would look much smoother if zeros were extended out to greater distances. (Zero-padding of a signal results in interpolation in the spectral domain!) Imag Real HW5 Imag