Detecting Harmonic Signals in A Noisy Time-Series: the Z-Domain Autoregressive (AR-z) Spectrum

ding, hao ; chao, benjamin f.;

We develop a new method referred to as the AR-z spectrum for detecting harmonic signals with exponential decay/growth contained in a noisy time-series by extending the auto regressive (AR) method of Chao & Gilbert. The method consists of (i) ‘blindly’ forcing one 2nd-order AR fit to the signal content in the frequency domain for any chosen frequency whether or not there is truly a signal; (ii) finding the corresponding AR (complex-conjugate pair of) poles in the complex z-domain; (iii) converting the pole locations into the corresponding complex frequencies of the harmonic signals via the Prony’s relation and (iv) constructing the Lorentzian power spectrum in the z-domain, conceptually constituting the analytical continuation of the spectrum from the (real) frequency domain to the complex z-domain, where a true harmonic signal is manifested as a Lorentzian peak. The AR-z spectrum can be further enhanced by forming the product spectrum from multiple records as available. We apply the AR-z spectral
method to detect and to estimate the complex frequencies of the Earth’s normal-modes of free oscillation using superconducting gravimeter records after recent large earthquakes. Specifically we show examples of detection and precise estimation of the frequencies and Q values of the split singlets of the spheroidal modes 0S2, 2S1, 1S2 and 0S0, and report the mode couplings manifested by the gravimeter recording of the toroidal modes 0T2, 0T3 and 0T4. The AR-z spectrum proves to be highly sensitive for harmonic signal of decaying sinusoids
in comparison to the conventional Fourier-based spectrum, particularly when the signal in question is weak and where high spectral resolution is desired.