Wavelet denoising for signals in quadrature

The idea of forming a complex-valued (analytic) signal from a real-valued one by creating an imaginary part equal to the Hilbert transform of the real part is well known in exploration geophysics for seismic character mapping via instantaneous attributes. However in this paper we consider the denoising of bivariate signals (time series) where the two real-valued components become the real and imaginary parts of a single complex-valued signal, and concentrate on the case where the two real-valued components are 'in quadrature' and also the complex signal is analytic. The Hilbert transform is applied to the noisy complex-valued signal to produce a new analytic noisy complex-valued signal with a useful noise structure. Numerical calculations show that our proposed 'complex analytic denoising' is superior to two other approaches for (i) a synthetic signal which is both in quadrature and analytic, and (ii) phase estimation for a Rayleigh wave signal which is close to analytic.

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