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 ψm,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 Lt 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(tj)} of f, where the set {t:⋯<tj<tj+1<⋯} of time instants may be non-uniformly spaced. Let Sf:=Sf;s be a B-spline series representation of f with the normalized B-splines Ns,t,k of order s≥1 on the knot sequence t and supported on [tk,ts+k) as basis functions, obtained by using the discrete samples {f(tj)}. Let ψ=ψm,n:=M2r(n) be the spline-wavelet of polynomial order m=2r−n and vanishing moment of order n, with s≤n≤2r−1, where M2r denotes the (2r)-th order centered Cardinal B-spline (with integer knots). The CWT, Wψ, is applied to the B-splines Ns,t,k to generate a one-parameter family {Bm,n,k(⋅;a)=Bm,n,s,k(⋅;a):=(WψNs,t,k)(⋅;a)} of basis functions. This yields a series representation Pf(⋅;a):=Pf;m,n,s(⋅;a) of an approximate CWT (Wψf)(⋅,a) of the blind-source composite signal f, by changing the basis {Ns,t,k} of Sf to the basis family {Bm,n,k(⋅;a)}. Let ρm,n denote the maximum magnitude of the Fourier transform (FT) ψˆ of ψ, attained at κm,n in the interval (0,2π), on which |ψˆm,n(ω)|>0. Then thresholding of Pf(⋅;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 Pf(⋅;a) into a sum of Lt “disjoint” summands for any time instant t, so that maxima estimation of the thresholded Pf(⋅;a) over the scale a yields the optimal scales aℓ⁎=aℓ⁎(t) for each active sub-signal fℓ, from which the active sub-signals themselves are recovered simply by dividing each summand by (−i)nρm,n, and the IFs ϕℓ′(t) are also obtained by ϕℓ′(t)=κm,n/aℓ⁎(t). The wavelets ψm,n=M2r(n) allow not only easy computation of ρm,n and κm,n, but also simple derivation of the explicit formula of the basis family Bm,n,k(⋅;a) by applying the B-spline recursive formula.