1. Geology 6600/7600 Signal Analysis 30 Sep 2015 © A.R. Lowry 2015 Last time: The transfer function relating linear SISO input & output signals is given by h(  ) = S xy /S xx. The power of the transfer function, i.e. without phase information, is given by |h(  )| 2 = S yy /S xx. Examples of Fourier transforms: constant  a delta function; sinusoid  a pair of delta functions; box function  sinc function, …

2. Assignment for Mon, 23 Oct: Go to the UNAVCO website (www.unavco.org), click thewww.unavco.org “Data” tab, and visit the “Data Archive Interface” Draw a box around the Cascadia region and search “Stations”. Choose a pair of sites to evaluate that (1) have long time series; (2) are ~ 50 km from the coast; (3) ~ on a north-south line; & (4) “near” each other, and download their time-series. (BLYN & KTBW would be good). Click the time-series image lower- right, click CSV data, and save to a file. Download time-series for another more distant pair (e.g., BLYN & KELS)

3. Assignment for Mon, 23 Oct (continued): To port into matlab, you will need to either write matlab script to read in the file information or edit the data file by deleting header lines & using global-search-and-replace in the vi editor to make it Matlab-friendly. (If you’re not sure how to do either, come see me!) In matlab, write a script to cross-correlate the time series over a range of possible time and lags (using the relation given earlier, repeated next slide). Plot the correlation coefficients as a function of windowed center-time & lag (similar to Bonneville elevation-lag plots shown Sep 16). Note that Cascadia events take ~two weeks and travel of order 10s-100s of km/day, so cross-correlations within ~ two-month windows would be plenty. From the cross-correlations, (1) when did the events occur? (2) Which direction and about how fast did they travel? (3) Is there anything else going on in these data?

4. Some practical applications of cross-correlation in geophysics: If we assume ergodicity of WSS random processes that are sampled at N regular intervals in time (latter is commonly the case for geophysical data), the cross-correlation R xy [l] at lag l can be estimated as: Note that for zero-mean signals, this simplifies to:

5. Now, let’s consider a Moving Average (or Integrator ) of the type commonly used to smooth noisy data: Which we can denote y(t) : h has Fourier transform: T 1/T  h(t-  ) x()x() This is really just a convolution: t

6. Thus: Assume the usual (zero mean white noise; S xx (  ) = S 0 ): Here we can use Parseval’s identity (i.e., that the integral of the squared modulus of Fourier amplitudes equals the integral of the square of the function): Hence: So, White Noise Reduction by a moving integrator is

7. Discrete Systems: Assume that we have a zero-mean, stationary process: The digital convolution (in discrete time) is: The Discrete Time Fourier Transform (DTFT) for a sample interval T ; frequencies  =  T = 2  fT is: with inverse: Note we are limited to the interval [– ,  ] by the Nyquist frequency, f N = 1/2T … Because  N = 2  T(1/2T) = .

8. Types of Fourier Pairs: Type Time Domain Frequency Domain 1) General F.T. Continuous, Continuous, not periodic aperiodic 2) Fourier series Continuous, Discrete, periodic aperiodic 3) DTFT Discrete, Continuous, aperiodic periodic 4) Discrete F.T. Discrete, Discrete, (the FFT ) periodic periodic Thus, any function that is discrete in one domain will be periodic in the other domain. (Practically, this means that when we FFT a time- or space-limited chunk of data, the resulting amplitudes implicitly impose periodicity on the data).

9. Aside on aliasing & the Nyquist frequency: If a signal contains information at frequencies  =  N + , that information will introduce spurious amplitude at the sampled (lower) frequency  N – . This aliasing effect can be removed if higher frequencies are filtered/averaged out of the signal prior to sampling… 75 Hz signal sampled at 125 Hz 175 Hz signal sampled at 125 Hz