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1Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México.
Communication theory and resonances on electromagnetic systems J.M. Velázquez-Arcos1 J. Granados-Samaniego2 and C.A. Vargas3 Abstract – We have shown [1] that some electromagnetic

But convergence difficulties emerge because the Gamow states are unbounded. Then we can define a well behaved Fredholm’s auxiliary problem whose solutions can be analytically continued up to the Gamow states. Returning to the electromagnetic waves, we have shown that we can obtain similar equations and then similar solutions to those described above for the quantum mechanics case. But since we are now in the frequency domain and also, we suppose that the time runs backwards. We must then make a proposal for a physical interpretation of electromagnetic resonances. Of course they are not the more typical Gamow states. We recall that the assigned interpretation is the breaking of the confinement of the evanescent waves that become travelling waves. With this parallelism, we can proceed to use the known tools in order to both great physical areas to solve the problem of improve communications. We then build packs of information labeling them with the electromagnetic resonance frequencies. Now we describe the issues treated in each section: On section 2 we write the spectral representation of the Fourier transform of the complete Green’ function for discrete electromagnetic systems and roughly we indicate how we can deform the path of integration with the goal of including the resonant solutions of the homogeneous Fredholm’s equation as part of the mathematical base. On section 3 we review the phenomenon of superoscillation for the case in which only five terms are enough to describe not only an arbitrary oscillatory function but one with a resultant frequency larger than any other belonging to the five members of the base that is used to express it. Also we propose that if we make use of the few resonant solutions as the base (really as a part of the complete base) we can recover any solution, particularly any which can voyage with little loss of

systems allow specific frequencies to satisfy special equivalent conditions to make use of a left hand material. We called these frequencies resonances because of their properties, similar to those solutions of the homogeneous Fredholm's equation for nuclear physics. But the process of information transmission requires not only the envoy of a limited number of frequencies but a very broad set of them. To answer the question of how we can use the knowledge of the resonances on a real system, we make use of the communication theory and the other properties, like the super-oscillating addition of functions and the mathematical deformation of the contour of the spectral representation of signals in order to create a set of frequency packs labeled with the final effect to optimize the information broadcast. In this paper we also suggest how we can preserve the convenience of this resonant point of view even when we cannot guaranty that the resonant conditions can be sustained permanently. Key Words – Communications, superoscillations, Fredholm´s equation.

resonances,

1 INTRODUCTION On quantum mechanics the concept of resonance is associated with unstable states whose life is related to his thickness on the energy. So a very accurate localization signifies a longer life. Physically we observe that the so called wave function grows up exponentially when we leave the initial position. This phenomenon then represents a situation in which an unstable particle decays into another (or others). The mathematical description can be made with the help of a homogeneous Lippmann-Schwinger equation whose solutions are precisely the resonant or Gamow states [2, 3].

_______________________________________ 1

Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: [email protected], Tel: +52 5553189026, Fax: +52 5553189540. 2 Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: [email protected] , Tel: +52 5553182054, Fax: +52 5553189540. 3 Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: [email protected], Tel: +52 5553189020, Fax: +52 5553189540.

978-1-4673-0335-4/12/$31.00 ©2012 IEEE 392

information, as we observe in the superoscillation phenomenon. On section 4 we review the communication theory to build the best packs of information to optimize the broadcasting of a signal and blueprint how we can apply them to the signals expressed as a spectral representation in terms of the resonant frequencies (in principle we use the resonances plus the non-resonant solutions necessary to complete the base). On section 5 five we give an example for the building of a pack. On section 6 we justify in general the use of packs labeled by the resonant frequencies, even when the physical conditions change and the resonances are no more solutions of the original homogeneous equation.

frequencies





S

0

m ³ u ( r k ; Z '; Z )

Z Z' 2

C

m

2

1

Z Z 2

u ( r j ; Z ; Z ) dZ n

2

(r k ; Z ) .

3 THE SUPEROSCILLATION PHENOMENON

u ( r j ; Z '; Z ) d Z ' n

2

Z  Ze

m

resonances we

frequencies

1

S

³ u (r k ; Z ;Z )

we ( r j ; Z ) n

2

(2) It is clear that our base now is formed by three different classes of electromagnetic solutions: confined field, travelling waves and resonant waves. We can explain the role of each set in the following manner: The continuum confined field can be expressed by linear combinations of only the discrete so called confined frequencies. Travelling waves now have two sources; the first is the common physical broadcasting; but the other is the transformation of the confined field into travelling waves. More precisely, under certain physical conditions in which the media behaves like a right-hand material, the complete field is separated in two regions: the near and far regions described mostly by the two first terms in equation (2) but if there are left-hand material conditions, the middle and the final terms dominate. Then, we have reached our first goal of extending our mathematical base to include the

Because we have shown the similar properties of the quantum mechanics formalism for scattering with the vector matrix one [4] (VMF) we have developed for discrete electromagnetic systems, we can extend other results like the spectral representation of the Green's function which appears explicitly in our equations. So by taking as a model the quantum spectral representation we can write the VMF spectral representation of the Green's function as follows: 1 m n ¦ GZm ,n ( r k , r j ) ve ( r k ; Z ) 2 ve ( r jZ ) 2 confined Z  Ze f

1

1

m

resonant

2 SPECTRAL REPRESENTATION OF THE GREEN'S FUNCTION

1

we ( r k ; Z )

 ¦

Considering now that we are interested in the representation of an oscillating function with frequency ZF in terms of a little number of functions

(1) In this equation the complete mathematical base is composed by two sets, first the field that belongs to some specific frequencies in the near electromagnetic

with frequencies Z n and n

1, 2, 3, 4, 5 that comply

the condition Z F ! Z n that is [5]:

zone ve ( r k ; Z ) that is, those that support the m

5

f ( x)

confined field and the second set, that is the

¦

an cos(2S u nx)

(3)

n 0

continuums u ( r k ; Z '; Z ) which are the travelling solutions of the vector Fredholm’s equation. But if we want to take advantage of the resonant solutions of the homogeneous vector Fredholm’s equation we can follow the recipe for quantum Gamow (resonant) states. That is we deform the path of integration over the real axis of equation (1) to include the resonances on the fourth complex quadrant in the spectral representation. Then we have the new spectral representation of the Green's function as follows: 1 m n ¦ GZm ,n ( r k , r j ) ve ( r k ; Z ) 2 ve ( r jZ ) 2 confined Z  Ze m

But if we chose the coefficients a0

1; a1 =13295000

a4 =-10836909; a5

a2 =-30802818; a3 =26581909

1762818

(4)

We can observe in Figure 1 that the resultant function f ( x) has at the origin x 0 a frequency nine times higher than the highest frequency of the components. The fact that we can obtain a more rapidly oscillating function is known as superoscillation. It is enough for our purposes to have this phenomenon as a reference because now we are sure that it is possible to build not only a slowly oscillating function as a linear combination of few functions with equal or largest frequencies, but we

frequencies

393

we apply the operator T which gives for each member (7) gD (t ) TfD (t )

can build a function with arbitrary oscillating speed from a limited number of base functions. This is our strongest argument to trust on the use of resonant solutions as a fundamental part of the mathematical base when we need to represent an arbitrary electromagnetic signal in terms of this augmented base. In other words, the presence of the set of resonances on the base is not a mathematical game but a physical brick when we make an arbitrary representation of the electromagnetic signals for any region (near or far regions). In the general case we expect to need a little more than the resonant waves to complete the correct representation of a signal. So we actually have a little more loss of information. 4 COMMUNICATION THEORIES RESONANT BROADCASTING

It implies that

gD (t  t1 )

for communication purposes the operator T which could be a modulation process is not invariant because of the phase carrier that gives certain time structure, but if the translations are multiples of the periods of the carrier, then the modulation will be invariant. At this stage it is important to remember that Wiener [6] has pointed out that if a device is linear as well as invariant (in the sense of the last definition), then the Fourier analysis is the appropriate mathematical tool for dealing with the problem. Now suppose in addition that we are interested on functions that are limited to the band from 0 to W cycles per second then we have the following theorem [7]: Let f (t ) contain no frequencies over W . Then

On this section we first recall some important properties of the ensembles of functions (dependent on time) defined on the communication theory because we will use them to build information packs. First a member of an ensemble transforms into another member of the same ensemble when we shift the function at any fixed amount of time. We take as an example the shift t1 in the argument of all the members of the following ensemble

f (t ) (5)

sin(t  t1  T ) sin(t  M )

f

S (2Wt  n)

(9)

f(

n 2W

)

(10)

In this expansion f (t ) is represented as a sum of

(6)

orthogonal (basis) functions. The coefficients X n of the various terms can be considered as coordinates in an infinite dimensional "functions space". We will take theorem (9) as a very suggestive rule to take into account the recently obtained resonant frequencies. As we have stated in section 3 the superoscillation evidence that real systems do not need an infinite number of terms in the spectral representation of an oscillatory function despite of the size of its frequency. If we use physical arguments about the reasons of the presence of a resonance, we can be sure that channels available for broadcasting are also limited in number. Then we can make the hypothesis that the series on equation (9) for the case of resonant conditions really have a little number of terms.

f ( x)

0

sin S (2Wt  n)

Xn

Where M is distributed uniformly from 0 to 2S So each function has changed but the ensemble as a whole is invariant under the transformation. Also if

-0.02

f

¦ Xn

Where

Where T is distributed uniformly from 0 to 2S Then we have

fT (t  t1 )

(8)

the output ensemble g D (t ) is also stationary. Now,

ON

fT (t ) sin(t  T )

TfD (t  t1 )

It is possible to prove that if T is an invariant operator and the input ensemble fD (t ) is stationary

5 THE BUILDING OF COMMUNICATION PACKS

0.02 x

Let's build communication packs that are functions which represent a part of the signal we want to send

Figure 1

394

with the minimum loss of information. First we do not need all the members of the basis that appears on equation (2) but with only a few terms around the resonant frequencies. The resultant expression is

f e (t )

f

sin S (2Ze t  n)

f

S (2Ze t  n)

¦ X n ,e

With

X n ,2 Of course if W

(11)

to zero is X 0,1

Where

X n ,e Every

fe (

n 2Ze

survives is X 0,2

)

(12)

but we expect that only a few terms are necessary for

we need for broadcasting. To receive the signal we .

A very important feature is that because of the properties of the modulation process stated in equations (7) and (8), we can recover, for any arbitrary signal, the behavior under spectral representation and under separated packs representation. So we can either talk about fe (t ) in equation (11) as the representation of some element of the basis function for the spectral representation (2) or directly as the " e "component of an arbitrary signal g (t ) Tf (t ) . Now, we recall the two resonances founded in another work [1]:

And

Z2 Suppose that

S

4d 3S

 Z0  Z0

4d

g (t ) is the signal sin S (2Wt ) g (t ) S (2Wt ) f

sin S (2Z1t  n)

f

S (2Z1t  n)

¦ X n ,1

g1 (

n 2Z1

)

(14)

(15)

(16)

(17)

And also we have the second pack

g 2 (t )

f

sin S (2Z2 t  n)

f

S (2Z2t  n)

¦ X n ,2

Z1 then the only coordinate distinct 1 and if W

Z2 the only term that

1 . These two late cases are not only

[1] J. M. Velázquez-Arcos et al., Electromagnetics in Advanced Applications (ICEAA), 2011 International Conference pp.167-170, 12-16 Sept. 2011 doi: 10.1109/ICEAA.2011.604629. [2] J. M. Velázquez-Arcos et al., J. Math. Phys. Vol. 49, 103508 (2008) doi: 10.1063/1.3003062. [3] A. Mondragón et al., Ann. Phys. (Leipzig) Vol. 48, 503-616 (1991) doi: 10.1002/andp.19915030802. [4] J. M. Velázquez-Arcos et al., Electromagnetics in Advanced Applications (ICEAA), 2010 International Conference on. pp. 264-267, 20-24 Sept. 2010 doi: 10.1109/ICEAA.2010.5653059. [5] N. I. Zheludev, Nature Materials Vol. 7, 420-422 (2008). [6] N. Wiener, Duke Mathematical Journal, Vol. 5, pp. 1-20, 1939. [7] C. E. Shannon, The Bell System Technical Journal, Vol. 27, pp. 379-423, 623-656, July, October, 1948. [8] F. Rusek et al., arXiv:1201.3210v1 [cs.IT] 16 Jan 2012.

With

X n ,1

(19)

References

(13)

Then we have the first pack

g1 (t )

)

We have presented a way to build information packs which due to the facts exposed on sections 3 and 4 have the important properties: first, the loss of information is minimum; second, the sum in equation (11) must only have a few number of terms; third, the packs have also the minimum interference between them because they are tailored around different resonant states that are orthogonal. In addition, because the resonant solutions are only a part of the complete set of base functions, the complete set can be taken into account by the equation (11) with a little change when the resonant conditions disappear. The reason is that equation (11) does not depend on the resonant conditions. We can also combine the use of resonances with the traditional methods like for example MIMO (multiple input-multiple outputs) [8].

separately each fe (t ) by its own device and it is all

Z1

2Z2

6 CONCLUDING REMARKS

a well representation of fe (t ) . Next, we send

Ze

n

an evidence of self-consistency of the method but a real example of how the highway of the resonances really operates.

Ze allow us to build a decomposition like (11)

need a separated device for each

g2 (

(18)

395