Abstract

This chapter is devoted to review a set of new technologies that we have developed and to show how they can improve the process of broadcasting in two principal ways: that is, one of these avoiding the loss of transmission signals due to abrupt changes in sign of the diffraction index and the other, preventing the mutual perturbation between signals generating information leak. In this manner, we propose the join of several of the mentioned technologies to get an optimum efficiency on the process of broadcasting communications showing the theoretical foundations and discussing some experiments that bring us to create the plasma sandwich model and others. Despite our very innovative technology, we underline that a complete recipe must include other currently in use like multiple-input multiple-output (MIMO) simultaneously. We include some mathematical proofs and also give an academic example.

Keywords: wireless communications, optimal broadcasting, information packs, negative refraction index, communication theory, wave propagation through plasma

## 1. Introduction

Nowadays, one of the most innovative procedures to improve communications is the random scattering of microwave or radio signals that may enhance the amount of information that can be transmitted over a channel. This fact, from a mathematical point of view, is due to the growth of the phase space available for that channel, which provides a more rich mathematical base to define every single signal. In many recent papers, a common subject is the use of a broad range of base functions to span each signal. The hope is that every single collision of the initial signals will be scattered and reaches another phase space region providing additional information, but the increase of phase space involves a more complicated set of describing functions. A multiple scattering of the obstacles enlarges the effective aperture in a time-reversed process for acoustic or electromagnetic signals when they are placed in random manner.

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 retransmitted. The effect is to eliminate noise pollution.

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

Through the present chapter, we introduce for convenience a hypothetical discrete 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 discussed in the present work.

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 for acoustic signals can be written as:

$$k(\mathbf{r})\frac{\partial^2 f(\mathbf{r},t)}{\partial t^2} = \nabla^2 (f(\mathbf{r},t)/\rho(\mathbf{r})) \tag{1}$$

the sink term consists in the operation of the source in reverse order; in the electromagnetic case, the sink term can be implemented with a crest of fine wires

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

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

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

One of the methods we have proposed is based on experiments executed by Xiang-kun Kong, Shao-bin Liu, Hai-feng Zhang, Bo-rui Bian, Hai-ming Li et al. [8] in which they put three layers of plasma joined and alternated with one of them magnetized in the core and the other two unmagnetized in the extremes of the device; when this plasma sandwich is submitted to an external electric potential, it is observed that for a range of values of the external potential, the refraction index is negative [15, 19]. When we analyzed those experiments, we conclude that for this range of the electric potential, the plasma sandwich brakes the confinement of the evanescent waves as occurs in a left-hand material and we proposed a model named

n w<sup>n</sup>

<sup>R</sup>ð Þ¼ ω 0 (4)

suppose that U does not depend on ), so this can be written explicitly as:

<sup>1</sup> � <sup>η</sup>Rð Þ <sup>ω</sup> <sup>G</sup>ð Þ<sup>∘</sup> ð Þ <sup>ω</sup> <sup>U</sup> h i<sup>m</sup> ð3Þ

around the antennas.

waves:

resonant solutions.

23

3. Introducing the PSM parameters

2. Recovering the matrix equations

Optimum Efficiency on Broadcasting Communications DOI: http://dx.doi.org/10.5772/intechopen.84954

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

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 written with the aid of the integral equation:

$$f(\mathbf{r}; T - t) = f^{(\ast)}(\mathbf{r}; T - t) + \int\limits\_{V} \int\limits\_{-\infty}^{\infty} U^{\ast}(\mathbf{r}') G^{(\ast)\ast}(\mathbf{r}', \mathbf{r}; T - t', t) f(\mathbf{r}'; T - t') dt' dV' \tag{2}$$

In Eq. (2), Gð Þ<sup>∘</sup> <sup>∗</sup> r<sup>0</sup> ; r; T � t <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 complex dispersion coefficients, and fð Þ r; T � t is the returning signal that has traveled toward the past. The inhomogeneous term f ð Þ<sup>∘</sup> ð Þ <sup>r</sup>; <sup>T</sup> � <sup>t</sup> is known as a sink 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,

the sink term consists in the operation of the source in reverse order; in the electromagnetic case, the sink term can be implemented with a crest of fine wires around the antennas.
