Semi-classical signal analysis

Abstract

This study introduces a new signal analysis method, based on a semi-classical approach. The main idea in this method is to interpret a pulse-shaped signal as a potential of a Schrödinger operator and then to use the discrete spectrum of this operator for the analysis of the signal. We present some numerical examples and the first results obtained with this method on the analysis of arterial blood pressure waveforms.

Appendices

Appendix A: Direct and inverse scattering transforms

These appendices recall some known concepts on direct and inverse scattering transforms of a one-dimensional Schrödinger operator. For more details, the reader can refer to the large number of references on this subject for instance [1, 3, 6, 8, 9]. Note that in this appendix we used the usual notations, in particular the variable here is $x$ and not $t$.

We consider here the spectral problem of a Schrödinger operator $H_1(-V)$, given by

$$\begin{aligned} -\frac{d^2 \psi }{dx^2}+ V(x,t) \psi = k^2 \psi ,\quad k\in \overline{\mathbb{C }}^+,\quad x\in \mathbb R , \end{aligned}$$

(46)

where the potential $V$ such that $V\in \mathcal B$. For simplicity, we will omit the indice $1$ of the spectral parameters in the following.

For $k^2 > 0$, we introduce the solutions $\psi _{\pm }$ of Eq. (46) such that

$$\begin{aligned} \psi _{-}(k,x)&= \left\{ \begin{array}{lc} T(k)e^{-ikx}& x\rightarrow -\infty , \\ e^{-ikx} + R_r(k)e^{+ikx}& x\rightarrow +\infty , \\ \end{array} \right.\end{aligned}$$

(47)

$$\begin{aligned} \psi _+(k,x)&= \left\{ \begin{array}{lc} T(k)e^{+ikx}& x\rightarrow +\infty , \\ e^{+ikx} + R_l(k)e^{-ikx}& x\rightarrow -\infty , \\ \end{array} \right. \end{aligned}$$

(48)

where $T(k)$ is called the transmission coefficient and $R_{l(r)}(k)$ are the reflection coefficients from the left and the right, respectively. The solution $\psi _-$ for example describes the scattering phenomenon for a wave $e^{-ikx}$ of amplitude $1$, sent from $+\infty$. This wave hit an obstacle which is the potential so that a part of the wave is transmitted $T(k) e^{-ikx}$ and the other part is reflected $R_r(k) e^{+ikx}$. $\psi _+$ describes the scattering phenomenon for a wave $e^{+ikx}$ sent from $-\infty$.

For $k^2 <0$, the Schrödinger operator spectrum has $N$ negative eigenvalues denoted $-\kappa _n^2$, $n=1,\ldots , N$. The associated $L^2$-normalized eigenfunctions are such that

$$\begin{aligned} \psi _n(x)&= c_{ln} e^{-\kappa _n x}, \quad \quad \quad \quad \quad x\rightarrow +\infty ,\end{aligned}$$

(49)

$$\begin{aligned} \psi _n(x)&= (-1)^{N-n} c_{rn} e^{+\kappa _n x}, \quad x\rightarrow -\infty , \end{aligned}$$

(50)

$c_{ln}$ and $c_{rn}$ are the normalizing constants from the left and the right, respectively.

The spectral analysis of the Schrödinger operator introduces two transforms:

• The direct scattering transform (DST) which consists in determining the so-called scattering data for a given potential. Let us denote $\mathcal S _l(V)$ and $\mathcal S _r(V)$ the scattering data from the left and the right, respectively:

  $$\begin{aligned} \mathcal S _j(V) : = \{R_{\overline{j}}(k), \; \kappa _n, \; c_{jn}, \;\; n=1,\ldots ,N \},\quad \quad j=l,r, \end{aligned}$$

  (51)

  where $\overline{j}=r$ if $j=l$ and $\overline{j}=l$ if $j=r$.

• The inverse scattering transform (IST) that aims at reconstructing a potential $V$ using the scattering data.

The scattering transforms have been proposed to solve some partial derivative equations for instance the KdV equation [10].

Appendix B: Reflectionless potentials

Deift and Trubowitz [6] showed that when the Schrödinger operator potential $V$ satisfies hypothesis (2), then it can be reconstructed using an explicit formula given by

$$\begin{aligned} V(x)=-4\sum _{n=1}^N{\kappa _n\psi _n^2(x)}+\frac{2i}{\pi }\int _{-\infty }^{+\infty }{kR_{r(l)}(k) \, f_{\pm }^2(k,x) dk},\quad x\in \mathbb R . \end{aligned}$$

(52)

This formula is called the Deift-Trubowitz trace formula. It is given by the sum of two terms: a sum of $\kappa _n \psi _n^2$ that characterizes the discrete spectrum, and an integral term that characterizes the contribution of the continuous spectrum.

There is a special class of potentials called reflectionless potentials for which the problem is simplified. A reflectionless potential is defined by $R_{l(r)}(k) = 0$, $\forall k\in \mathbb R$. According to the Deift-Trubowitz formula, a reflectionless potential can be written using the discrete spectrum only,

$$\begin{aligned} V(x,t)= - 4 \sum _{n=1}^{N}{\kappa _n\psi _n^2(x,t)}, \end{aligned}$$

(53)

Appendix C: An infinite number of invariants

There is an infinite number of time invariants for the KdV equation given by the conserved quantities [10, 12, 26]. Let us denote these invariants $I_m(V)$, $m=0,1,2,\ldots$. They are of the form

$$\begin{aligned} I_m(V)=(-1)^{m+1}\frac{2m+1}{2^{2m+2}}\!\!\int _{-\infty }^{+\infty }{P_m \left(V,\frac{\partial V}{\partial x},\frac{\partial ^2 V}{\partial x^2}, \cdots \right)dx}, \end{aligned}$$

(54)

where $P_m$, $m=0,1,2,\ldots$ are known polynomials in $V$ and its successive derivatives with respect to $x\in \mathbb R$ [3].

A general formula relates $I_m(V)$ to the scattering data of $H_1(-V)$ [3, 12, 26] as follows:

$$\begin{aligned} I_m(V) = \sum _{n=1}^{N} {\kappa _{n}^{2m+1}}+\frac{2m+1}{2\pi }\int _{-\infty }^{+\infty }{{(-k^2)^m \ln {(1-|R_{r(l)}(k)|^2)}dk}}, \end{aligned}$$

(55)

$m=0,1,2,\ldots$. So, for $m=0$, $P_0(V,\cdots )=V$, we get with (54) and (55):

$$\begin{aligned} \int _{-\infty }^{+\infty }{V(x) dx}= -4\sum _{n=1}^{N}{\kappa _{n}} - \frac{1}{\pi } \int _{-\infty }^{+\infty }{\ln {(1-|R_{r(l)}(k)|^2)} dk}. \end{aligned}$$

(56)