## Abstract

The Haar function is extended to a family of minimum-supported cardinal spline-wavelets ψ_{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 S_{f}:=S_{f;s} be a B-spline series representation of f with the normalized B-splines N_{s,t,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 ψ=ψ_{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_{ψ}, is applied to the B-splines N_{s,t,k} to generate a one-parameter family {B_{m,n,k}(⋅;a)=B_{m,n,s,k}(⋅;a):=(W_{ψ}N_{s,t,k})(⋅;a)} of basis functions. This yields a series representation P_{f}(⋅;a):=P_{f;m,n,s}(⋅;a) of an approximate CWT (W_{ψ}f)(⋅,a) of the blind-source composite signal f, by changing the basis {N_{s,t,k}} of S_{f} to the basis family {B_{m,n,k}(⋅;a)}. Let ρ_{m,n} denote the maximum magnitude of the Fourier transform (FT) ψˆ of ψ, attained at κ_{m,n} in the interval (0,2π), on which |ψˆ_{m,n}(ω)|>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}ρ_{m,n}, and the IFs ϕ_{ℓ}^{′}(t) are also obtained by ϕ_{ℓ}^{′}(t)=κ_{m,n}/a_{ℓ}^{⁎}(t). The wavelets ψ_{m,n}=M_{2r}^{(n)} allow not only easy computation of ρ_{m,n} and κ_{m,n}, but also simple derivation of the explicit formula of the basis family B_{m,n,k}(⋅;a) by applying the B-spline recursive formula.

Original language | English |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 52 |

Early online date | 26 Nov 2020 |

DOIs | |

Publication status | Published - May 2021 |

## Scopus Subject Areas

- Applied Mathematics

## User-Defined Keywords

- Composite signal separation
- Instantaneous frequencies
- Inverse problem
- Trends
- Wavelet thresholding