2. Recovering the matrix equations

Another current tool is time reversal, or phase conjugation in the frequency domain, where a source at one location transmits sound or electromagnetic waves, which are received at another place, time reversed (or phase conjugated), and

Telecommunication Systems – Principles and Applications of Wireless-Optical Technologies

Despite the existence of the mentioned resources and others like multiple-input

Through the present chapter, we introduce for convenience a hypothetical dis-

In the same manner, because the formalism we have developed for the study of time reversibility refers to acoustic systems, we recall that the scalar wave equation

We now describe the quantities appearing in Eq. (1), represents the mass density and the compressibility of the propagation medium, while is

Because the wave equation is of second order in time, we can talk about time reversibility, and then allows solutions, which travel toward the future or the past. An efficient time reversal requires to ensure that the system be ergodic, making possible that the signal may travel both senses in time. To improve focusing, we must describe the signal propagation towards the future or past by means of equations of the same type [18, 22, 27] that is both directions inhomogeneous or both homogeneous. Linearity permits that a signal traveling toward the past can be

<sup>U</sup> <sup>∗</sup> <sup>r</sup><sup>0</sup> ð ÞGð Þ<sup>∘</sup> <sup>∗</sup> <sup>r</sup><sup>0</sup>

complex dispersion coefficients, and fð Þ r; T � t is the returning signal that has

term and makes both the outgoing and returning equations inhomogeneous integral equations. In Eq. (2), the parameter represents the time during which the outgoing signal (the one traveling toward the future) is being considered and recording. It is observed experimentally [9] that the time-reversed signal has a definition of a 14th of , the wavelength of the used signal for acoustic signals but this is also true for electromagnetic waves. On several experiments [9, 10], Lerosey, de Rosny, Tourin, and Fink have shown that when such a source term is included, the apparent cross section is increased in two ways: first, the multiple scattering also multiplies the available phase space so when the time is reversed, the information is increased, and second, in the electromagnetic case, the sink term stimulates and triggers the braking of the confinement of the evanescent waves that also raise the information and in consequence the definition to level of about <sup>λ</sup>=14. In acoustics,

; r; T � t <sup>0</sup> ð Þ ; t f r<sup>0</sup>

<sup>0</sup> ð Þ ; <sup>t</sup> is the free Green function, <sup>U</sup> <sup>∗</sup> <sup>r</sup><sup>0</sup> ð Þ depicts the

; T � t <sup>0</sup> ð Þdt<sup>0</sup>

ð Þ<sup>∘</sup> ð Þ <sup>r</sup>; <sup>T</sup> � <sup>t</sup> is known as a sink

dV<sup>0</sup> (2)

ð1Þ

multiple-output (MIMO), many problems survive, but fortunately, we have proposed some additional ways to improve the broadcasting by diminishing the information loss. Some of our results are based on communication theory and others in the mathematical properties of particular integral equations and their

crete system in order to write finite matrices. But we can certainly extend the validity of our expressions as we will see, even for both discrete and continuum systems provided the involved potentials fulfill very general conditions not

retransmitted. The effect is to eliminate noise pollution.

solutions.

discussed in the present work.

the acoustic signal.

fð Þ¼ r; T � t f

22

In Eq. (2), Gð Þ<sup>∘</sup> <sup>∗</sup> r<sup>0</sup>

for acoustic signals can be written as:

written with the aid of the integral equation:

ð Þ<sup>∘</sup> ð Þþ <sup>r</sup>; <sup>T</sup> � <sup>t</sup>

; r; T � t

traveled toward the past. The inhomogeneous term f

ð

∞ð

�∞

V

As we have said above and considering that from a strictly mathematical point of view, both the acoustic and electromagnetic waves achieve the same wave equation type (with a vector version in the electromagnetic case). Then, we can regain, without further ado, the vector matrix formalism [1–7, 11–14] which generalizes the discrete scalar time reversal acoustic model and includes an original model for discrete broadcasting systems that we have called the plasma sandwich model (PSM) [8, 16–18] and we put some associated parameters appeared on it into the named vector matrix formalism (VMF) [8, 20, 24]. But we must underline that is the resonant behavior the one must be considered for increasing efficiency on communications and to achieve extraordinary resolution. To this end, we remember that a three-dimensional version of Eq. (1) can be written as the Fourier transform of an integral generalized homogeneous Fredholm's equation (GHFE) [21–24] for resonances, and does not matter if for acoustic or electromagnetic ones. To analyze the resonant behavior, we must eliminate the inhomogeneous term so we can write the following algebraic equation satisfied by the Fourier transform of the resonant waves:

$$[1 - \eta\_{\underline{n}}(\phi)\mathbf{K}^{(\circ)}(\phi)]\_{\underline{n}}^{\mathfrak{m}}\mathbf{w}\_{\underline{n}}^{\mathfrak{n}}(\phi) = 0 \tag{3}$$

where the kernel is the product of the Fourier transform of the free Green function with the interaction U (without loss of generality we can suppose that U does not depend on ), so this can be written explicitly as:

$$\left[\mathbf{1} - \eta\_R(\boldsymbol{\alpha})\mathbf{G}^{(\circ)}(\boldsymbol{\alpha})\mathbf{U}\right]\_n^m \mathbf{w}\_R^n(\boldsymbol{\alpha}) = \mathbf{0} \tag{4}$$

At this point, we must say that we could obtain a transfer matrix description [16–18] instead Eq. (4), but our last equation represents the core of the VMF version. The fact is there are important differences between the two formalisms; for example, VMF makes the time-reversal process easy. Of course, we are moving over a frequency domain and not over a time-dependent one, the former the appropriate in agreement with information theory applications. And certainly, the most important difference is that VMF formalism includes the concept of the resonant solutions.
