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 $\{\mathbf{t}:\cdots <t_j<t_{j+1}<\cdots \}$ 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 $N_{s,\mathbf{t},k}$ of order $s\ge 1$ 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 $s\le n\le 2r-1$, 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 $N_{s,\mathbf{t},k}$ to generate a one-parameter family $\{B_{m,n,k}(\cdot ;a)=B_{m,n,s,k}(\cdot ;a):=(W_{\psi }{N}_{s,\mathbf{t},k})(\cdot ;a)\}$ of basis functions. This yields a series representation $P_f(\cdot ;a):=P_{f;m,n,s}(\cdot ;a)$ of an approximate CWT $(W_{\psi }f)(\cdot ,a)$ of the blind-source composite signal f, by changing the basis $\{N_{s,\mathbf{t},k}\}$ of $S_f$ to the basis family $\{B_{m,n,k}(\cdot ;a)\}$. Let ${\rho }_{m,n}$ denote the maximum magnitude of the Fourier transform (FT) $\hat{\psi }$ of ψ, attained at ${\kappa }_{m,n}$ in the interval $(0,2\pi )$, on which $|{\hat{\psi }}_{m,n}(\omega )|>0$. Then thresholding of $P_f(\cdot ;a)$ 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 $P_f(\cdot ;a)$ into a sum of $L_t$ “disjoint” summands for any time instant t, so that maxima estimation of the thresholded $P_f(\cdot ;a)$ over the scale a yields the optimal scales $a_{\ell }^{⁎}=a_{\ell }^{⁎}(t)$ for each active sub-signal $f_{\ell }$, from which the active sub-signals themselves are recovered simply by dividing each summand by ${(-i)}^n{\rho }_{m,n}$, and the IFs ${\varphi }_{\ell }^{\prime }(t)$ are also obtained by ${\varphi }_{\ell }^{\prime }(t)={\kappa }_{m,n}/a_{\ell }^{⁎}(t)$. The wavelets ${\psi }_{m,n}=M_{2r}^{(n)}$ allow not only easy computation of ${\rho }_{m,n}$ and ${\kappa }_{m,n}$, but also simple derivation of the explicit formula of the basis family $B_{m,n,k}(\cdot ;a)$ by applying the B-spline recursive formula.

Haar函数扩展到最低支持的基数样条小波族 ${\psi }_{米，ñ}$，具有任意期望的多项式阶次m和任意高阶阶次n为了实现我们的连续小波变换（CWT）阈值化策略，目的是从它们构成的盲源复合信号中恢复所有“活动”子信号及其瞬时频率（IF）。在这方面，用于控制复合信号的常用“自适应谐波模型（AHM）”被扩展为“现实自适应谐波模型（RAHM）”，以允许子信号的时变连续相位函数不为零。可微分或在时域的任意（未知）子间隔中具有负导数。本文的目的是建立基于样条小波和CWT阈值的严格理论，以及有效的方法和有效的计算方案，以解决确定未知数的逆问题${\mathrm{大号}}_{Ť}$ 盲源复合信号的活动子信号的变化 $F（Ť）$由RAHM控制，在时域中的任何时刻t，仅使用离散样本即可计算其活动子信号及其瞬时频率（IF）和趋势函数$\{F（{Ť}_{Ĵ}）\}$的˚F，该组，其中$\{\mathbf{Ť}：\cdots <{Ť}_{Ĵ}<{Ť}_{Ĵ+\mathrm{1个}}<\cdots \}$瞬间的时间间隔可能不均匀。让${\mathrm{小号}}_F：={\mathrm{小号}}_{F;s}$是乙的-spline级数表示˚F与归一化的乙-splines${ñ}_{s，\mathbf{Ť}，ķ}$ 顺序 $s\ge \mathrm{1个}$在结序列t上并受支持$[{Ť}_{ķ}，{Ť}_{s+ķ}）$ 作为基础函数，通过使用离散样本获得 $\{F（{Ť}_{Ĵ}）\}$。让$\psi ={\psi }_{米，ñ}：={\mathrm{中号}}_{2\mathrm{[R}}^{（ñ）}$ 是多项式的样条小波 $米=2\mathrm{[R}-ñ$阶n消失，$s\le ñ\le 2\mathrm{[R}-\mathrm{1个}$，在哪里 ${\mathrm{中号}}_{2\mathrm{[R}}$ 表示 $（2\mathrm{[R}）$阶居中的基数B样条曲线（具有整数节）。CWT，${\mathrm{w\; ^}}_{\psi }$，应用于B样条曲线 ${ñ}_{s，\mathbf{Ť}，ķ}$ 生成一个单参数族 $\{{乙}_{米，ñ，ķ}（\cdot ;\mathrm{一种}）={乙}_{米，ñ，s，ķ}（\cdot ;\mathrm{一种}）：=（{\mathrm{w\; ^}}_{\psi }{ñ}_{s，\mathbf{Ť}，ķ}）（\cdot ;\mathrm{一种}）\}$基本功能。这产生了系列表示$P_F（\cdot ;\mathrm{一种}）：=P_{F;米，ñ，s}（\cdot ;\mathrm{一种}）$ 大约CWT $（{\mathrm{w\; ^}}_{\psi }F）（\cdot ，\mathrm{一种}）$通过改变基数来计算盲源复合信号f$\{{ñ}_{s，\mathbf{Ť}，ķ}\}$ 的 ${\mathrm{小号}}_F$ 到基本家庭 $\{{乙}_{米，ñ，ķ}（\cdot ;\mathrm{一种}）\}$。让${\rho }_{米，ñ}$ 表示傅立叶变换（FT）的最大幅度 $\hat{\psi }$的ψ，达到${\kappa }_{米，ñ}$ 在间隔 $（0，2\pi ）$，在 $|{\hat{\psi }}_{米，ñ}（\omega ）|>0$。然后阈值化$P_F（\cdot ;\mathrm{一种}）$具有n个消失矩的适当大阶数（取决于子信号幅度的下限，IF的上下边界以及子信号IF的倒数的最小间隔），将阈值和除以$P_F（\cdot ;\mathrm{一种}）$ 变成 ${\mathrm{大号}}_{Ť}$对于任何时刻t，“不相交”求和，因此阈值的最大值估计$P_F（\cdot ;\mathrm{一种}）$在刻度一个产率的最佳尺度${\mathrm{一种}}_{\ell }^{⁎}={\mathrm{一种}}_{\ell }^{⁎}（Ť）$ 对于每个活动的子信号 $F_{\ell }$，只需将每个被求和除以，即可从中恢复活动子信号本身 ${（-\mathrm{一世}）}^{ñ}{\rho }_{米，ñ}$以及IF ${\varphi }_{\ell }^{\prime }（Ť）$ 也可以通过 ${\varphi }_{\ell }^{\prime }（Ť）={\kappa }_{米，ñ}/{\mathrm{一种}}_{\ell }^{⁎}（Ť）$。小波${\psi }_{米，ñ}={\mathrm{中号}}_{2\mathrm{[R}}^{（ñ）}$ 不仅可以轻松计算 ${\rho }_{米，ñ}$ 和 ${\kappa }_{米，ñ}$，也可以简单推导基族的显式 ${乙}_{米，ñ，ķ}（\cdot ;\mathrm{一种}）$ 通过应用B样条递归公式。