Wavelet thresholding for recovery of active sub-signals of a composite signal from its discrete samples

The Haar function is extended to a family of minimum-supported cardinal spline-wavelets psi(m,n), with any desired polynomial order m and arbitrarily high order n of vanishing moments, for the purpose of carrying out our strategy of continuous wavelet transform (CWT) thresholding to recover all "active" sub-signals, along with their instantaneous frequencies (IFs), from a blind-source composite signal they constitute. In this regard, the commonly used "adaptive harmonic model (AHM)" for governing the composite signals is extended to the "realistic adaptive harmonic model (RAHM)" to allow the time-varying continuous phase functions of the sub-signals to be non-differentiable or to have negative derivatives in arbitrary (unknown) sub-intervals of the time-domain. The objective of this paper is to develop a rigorous theory based on spline-wavelets and CWT thresholding, along with effective methods and efficient computational schemes, to resolve the inverse problem of determining the unknown number L-t of active sub-signals of a blind-source composite signal f (t) governed by RAHM, at any time instant t in the time domain, computing its active sub-signals along with their instantaneous frequencies (IFs), and the trend function, by using only discrete samples {f(t(j))} of f , where the set {t : ... < t(j) < t(j+1) < ...} of time instants may be non-uniformly spaced. Let Sf := S-f(;s) be a B-spline series representation of f with the normalized B-splines N-s,N-t,N-k of order s >= 1 on the knot sequence t and supported on [t(k), t(s+k)) as basis functions, obtained by using the discrete samples {f (t(j))}. Let psi = psi(m,n) := M-2r((n)) be the spline-wavelet of polynomial order m = 2r - n and vanishing moment of order n, with s <= n <= 2r - 1, where M-2r denotes the (2r)-th order centered Cardinal B-spline (with integer knots). The CWT, W-psi, is applied to the B-splines N-s,N-t,N-k to generate a one-parameter family {B-m,B-n,B-k(.; a) = B-m,B-n,B-k(.; a) := (W psi Ns,t,k(.; a)} of basis functions. This yields a series representation Pf(.; a) := P-f,P-m,P-n,P-s(.; a) of an approximate CWT (W(psi)f )(., a) of the blind-source composite signal f, by changing the basis {N-s,N-t,N-k} of S-f to the basis family {B-m,B-n,B-k(.; a)}. Let rho(m,n) denote the maximum magnitude of the Fourier transform (FT) (psi) over cap of psi, attained at k(m,n) in the interval (0, 2 pi), on which vertical bar(psi) over cap (m,n)(omega)vertical bar > 0. Then thresholding of P-f (.; a) with appropriately large order n of vanishing moments (that depends on the lower bound of the sub-signal magnitudes, upper and lower bounds of the IF, and minimum separation of the reciprocals of the IFs of the sub-signals), divides the thresholded sum P-f(.; a) into a sum of L-t "disjoint" summands for any time instant t, so that maxima estimation of the thresholded P-f(.; a) over the scale a yields the optimal scales a(l)* = a(l)*(t) for each active sub-signal f(l), from which the active sub-signals themselves are recovered simply by dividing each summand by (-i)(n)p(m,n), and the IFs phi(l)'(t) are also obtained by phi(l)'(t) = k(m,n)/a(l)*(t). The wavelets psi(m,n) = M-2r((n)) allow not only easy computation of rho(m,n) and K-m,K-n, but also simple derivation of the explicit formula of the basis family B-m,B-n,B-k(.; a) by applying the B-spline recursive formula. (C) 2020 Elsevier Inc. All rights reserved.