Description of Partly Correlated Random Sequences: Replacement of Random Sequences by the Generalised Prony Spectrum

Abstract

Any young researcher is accustomed to the conventional Fourier transform (F-transform). However, it would be useful to think about the meaning of this transformation and its purpose. Is it just for detecting the harmonic components in the input signal? This chapter tries to answer this question, such that the reader will be acquainted with not only periodic signals but also a different class of signals defined as quasi-periodic. Based on this information, one can enter another world of processes that can be used in practical applications. In this chapter, a general algorithm to describe these processes is provided along the methods to detect them from real, available data. Two nontrivial examples should confirm that this “quasi-periodic world” exists, and keen students could use the newly introduced tools and methods to find quasi-periodic processes from data at his/her disposal. This chapter also illustrates some open problems that can encourage and invite the reader to a new way of thinking and modelling. This way should increase the limits of the conventional F-transform and should probably set on thinking about the physical meaning of Laplace-transform (L-transform), which is a powerful tool from the mathematical point of view, but with a still uncertain physical meaning.

Appendices

5.A Appendix: Generalization of the Model for Quasi-Periodic Processes to Consider Incommensurable Periods

This Appendix provides a possible generalization of the functional eq. (5.5), which was introduced for a wide class of quasi-periodical processes, for incommensurable periods. Namely, consider the following generalization of (5.5):

$$ {\displaystyle \begin{array}{l}F\left(x+{\upalpha}_L{T}_x\right)=\sum \limits_{l=0}^{L-1}{a}_lF\left(x+{\upalpha}_l{T}_x\right)+b,\\ {}{\upalpha}_l=0<{\upalpha}_1<\dots <{\upalpha}_L,\kern0.36em \\ {}{\upalpha}_m/{\upalpha}_n\ne m/n= irrational\kern0.34em number\end{array}} $$

(5.34)

The solution of eq. (5.34) is searched in the form

$$ {\displaystyle \begin{array}{l}F(x)=\sum \limits_{k=0}^K\left[{Ac}_k{yc}_k(x)+{As}_k{ys}_k(x)\right]\\ {}{yc}_k(x)={\left(\upkappa \right)}^{x/{T}_x}\cos \left(2\uppi k\frac{x}{T_x}\right),{ys}_k(x)={\left(\upkappa \right)}^{x/{T}_x}\sin \left(2\uppi k\frac{x}{T_x}\right).\end{array}} $$

(5.35)

Here the parameter κ is not known, the period T_x can be found from the fitting procedure described earlier. After simple algebra, it is easy to establish the following relationship

$$ {\displaystyle \begin{array}{l}F\left(x+\Delta \cdot {T}_x\right)={\left(\upkappa \right)}^{\Delta}\sum \limits_{k=0}^K\left[{Ac}_k\cos \left(2\uppi k\Delta \right)+{As}_k\sin \left(2\uppi k\Delta \right)\right]\cdot {yc}_k(x)\\ {}\kern3.359999em +{\left(\upkappa \right)}^{\Delta}\sum \limits_{k=0}^K\left[-{Ac}_k\sin \left(2\uppi k\Delta \right)+{As}_k\cos \left(2\uppi k\Delta \right)\right]\cdot {ys}_k(x).\end{array}} $$

(5.36)

Taking into account the fact that functions yc_k(x) and ys_k(x) figuring in (5.35) are linearly independent, substitution of (5.36) into (5.34) yields

$$ {\displaystyle \begin{array}{l}{\left(\upkappa \right)}^{\upalpha_L}\left[{Ac}_k\cos \left(2\uppi k{\upalpha}_L\right)+{As}_k\sin \left(2\uppi k{\upalpha}_L\right)\right]=\\ {}\sum \limits_{l=0}^{L-1}{a}_l{\left(\kappa \right)}^{\upalpha_l}\left[{Ac}_k\cos \left(2\uppi k{\upalpha}_l\right)+{As}_k\sin \left(2\uppi k{\upalpha}_l\right)\right].\end{array}} $$

(5.37)

But the decomposition coefficients Ac_k and As_k form also a couple of linearly independent sets, and for any k = 0, 1,…, K two independent relationships are obtained:

$$ {\displaystyle \begin{array}{l}{\upkappa}^{\upalpha_L}\cos \left(2\uppi k{\upalpha}_L\right)=\sum \limits_{l=0}^{L-1}{a}_l{\left(\upkappa \right)}^{\upalpha_l}\cos \left(2\uppi k{\upalpha}_l\right),\\ {}{\upkappa}^{\upalpha_L}\sin \left(2\uppi k{\upalpha}_L\right)=\sum \limits_{l=0}^{L-1}{a}_l{\left(\upkappa \right)}^{\upalpha_l}\sin \left(2\uppi k{\upalpha}_l\right).\end{array}} $$

(5.38)

Multiplying the second relationship by the complex unit $i=\sqrt{-1}$ yields:

$$ {\displaystyle \begin{array}{l}{\upkappa}^{\upalpha_L}\exp \left(i2\uppi k{\upalpha}_L\right)=\sum \limits_{l=0}^{L-1}{a}_l{\left(\upkappa \right)}^{\upalpha_l}\exp \left(i2\uppi k{\upalpha}_l\right),\\ {}{\upkappa}^{\upalpha_L}{\left[\exp \left(i2\uppi k\right)\right]}^{\upalpha_L}=\sum \limits_{l=0}^{L-1}{a}_l{\left(\upkappa \right)}^{\upalpha_l}{\left[\exp \left(i2\uppi k\right)\right]}^{\upalpha_l}.\end{array}} $$

(5.39)

From expression (5.39) it follows for any k = 0, 1,…, K

$$ {\upkappa}^{\upalpha_L}=\sum \limits_{l=0}^{L-1}{a}_l{\left(\upkappa \right)}^{\upalpha_l}. $$

(5.40)

The last equation coincides with eq. (5.7) for the case of proportional periods (α_L → L, α_l → l) and, thereby, generalizes the case considered in Sect. 5.2. The last generalization (5.40) makes the approach considered in this chapter very flexible and general and deserves further research. Equation (5.40) represents itself a posinomial containing different power-law functions and it can be solved numerically for the given set of {α_l, l = 0, 1,…, L}.

5.B Questions for Self-Testing

• 1. In what cases the conventional decomposition to the Fourier series is valid? What kind of suppositions are made? What are the limits of applicability of the Fourier transform?

• 2. In what cases the Prony’s decomposition can replace the Fourier transform?

• 3. Find the solution of the functional equation

$$ F\left(x+2T\right)=F\left(x+T\right)+F(x)+c $$

• 4. Find the difference between the following equations

$$ {\displaystyle \begin{array}{l}{F}_{k+2}={a}_1{F}_{k+1}+{a}_0{F}_k\\ {}F\left(x+2T\right)={a}_1F\left(x+T\right)+{a}_0F(x)\end{array}} $$

• 5. What is the difference between the solutions of the eqs. (5.5) and (5.24)?

• 6. In what case the algorithm outlined in this chapter is applicable for analysis of data measured in any complex system?