1. Geology 6600/7600 Signal Analysis 28 Sep 2015 © A.R. Lowry 2015 Last time: Energy Spectral Density; Linear Systems given (deterministic) finite-energy signals The Energy Density Spectrum of a process x is: This is not the same as Power Spectral Density, but it is sometimes used as a proxy (e.g., called the “ Periodogram spectral estimate ”) The power spectral density on a frequency band can be approximated by frequency-domain integration of the ESD:

2. Geology 6600/7600 Signal Analysis 28 Sep 2015 © A.R. Lowry 2015 Last time continued: Linear Systems for Deterministic Signals Given a deterministic input signal x(t) and impulse response h(t), the output y(t) for a linear SISO system can be found: y(t) = h(t)  x(t)  Y(  ) = H(  )X(  ) The Energy Density Spectra are related by: The Autocorrelation for a deterministic finite-energy signal has: ; Using the correlation theorem, can find two ways to estimate the energy spectral density:

3. Geology 6600/7600 Signal Analysis 28 Sep 2015 © A.R. Lowry 2015 Last time continued: Linear Systems for Random Signals The cross-correlation of the input and output from a SISO linear system is:

4. We may recall from properties of the cross- and auto- correlations: Thus: Consider for example a zero-mean, white noise input: The delta function is even, so  –  –  : (Let  –  ):

5. The output variance for the system is: By Parseval’s relation, i.e., the output variance is the input variance multiplied by the energy spectral density of the impulse response!

6. Recall from earlier the transform relations: (autocorrelation) (autopower spectrum) (crosscorrelation) (crosspower spectrum) Then we can also express the autocorrelation & autopower spectrum of the output signal as: Note the output autopower has no phase information, so cannot be used to fully characterize the system h(  ) !

7. Example : Consider the low-pass filter h(t) = e –at u(t) (Can get this e.g. from a resistor/capacitor circuit): h(t)h(t) t u(t)u(t) t 11 Here, given direct current I get voltages (in the frequency domain): for a = 1/RC R x(t)x(t) Cy(t)y(t) ~ ~

8. The modulus-squared of H is then: Let’s assume we have a zero-mean, white noise input to the circuit: Then: Variance of the output will be: S0S0 Input spectrum: S0S0 Output spectrum:

9. Examples of correlation functions & power spectra: Random sine wave Autocorrelation:  Power Spectrum: t x  R  S  

10. Examples of correlation functions & power spectra: White noise with zero mean, variance    : Autocorrelation:  Power Spectrum: t x  R  S

11. Examples of correlation functions & power spectra: Low-pass filter (i.e. applied in the frequency domain): Band-Pass filter (in the frequency domain): Autocorrelation:  Power Spectrum:  R  S Autocorrelation:  Power Spectrum:  R  S     (sinc fn!)

12. Examples of correlation functions & power spectra: Pseudo-random Binary Sequence (PRBS): Markov Process (may be useful to describe noise in potential field data): Autocorrelation:  Power Spectrum:  R  S Autocorrelation:  Power Spectrum:  R  S t x “Similar” to white noise, but not, because it’s deterministic