Let us begin by stating that it is difficult to provide an adequate response that will enlighten the audience if the correspondent neither provides th... Channel capacity - antenna theory - channel capacity - Gaussian channels - Gaussian distribution - Gaussian noise - channel capacity - Maxwellian viewpoint - noise statistics - Gaussian distribution - antenna analysis - wireless communication

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et us begin by stating that it is difficult to provide an adequate response that will enlighten the audience if the correspondent neither provides the necessary references nor the details of his computations that are in disagreement with the authors. Since science is evidence based, it is difficult to respond intelligently where evidence of a disagreement is missing. We will respond to Dr. Kimber's comments by addressing each paragraph separately. In the first paragraph, the statement that we, the authors, are not sure when to use the fonnulas of Shannon and Hartley is surprising, as we often noted their differences and the conditions under which they should be used in the article. Quite emphatically, we have no uncertainty about their differences, about appropriate usage, or about our convictions on this subject! For example, on page 26 we state the following in no uncertain terms:

The Hartley capacity can be shown to be equivalent to the Shannon's definition for the capacity, when both the transmitting and the receiving antenna systems are conjugately matched....Using Equation (9) for the channel capacity is simpler and may provide a different value for the channel capacity than that obtained when it is computed with the power relationship. In Equation (9), we need only the voltage. Furthermore, we reiterated our opinions, again in no uncertain terms, in the epilogue under points 2-6, as to when the Hartley and the Shannon capacity formulas should be used. In short, our paper is quite definite and clear as to what are our intentions. In our paper, we further demonstrated that the two expressions for channel capacity, as defined by Hartley and Shannon, may yield similar numbers under similar background noise only when the antenna systems are conjugately matched, because at a resonance the transmitted and received powers are maximized for the given environment. Dr. Kimber states that such a coincidence is accidental, which may stem from his belief that Maxwellian principles somehow do not apply to this situation, as illustrated next. Our paper is based on Maxwellian physics and Poynting's theorem, and these appear to be at odds with statements such as "Electric and Magnetic Fields Might not Exist" and "The Poynting Vector Field Might not Exist," which have been made by Dr. Kimber [1]. The vector principles of electrical engineering lead to the important conclusion that complex power is not additive but that voltages and currents (manifested through the electric and magnetic fields) are additive: this is why Maxwell's theory is also called field theory. In short, Maxwellian physics deals with vectors. In electromagnetic field theory, scalar techniques thus cannot be applied, and as a consequence, power is not additive. Scalar probability theory, which defines the Fourier transform of the autocorrelation of the voltage to be the power spectral density, is not directly applicable to a vector field problem, and thus completely misses the fundamental principles of electrical engineering, namely that power cannot be computed only from the square of the voltage, except in a purely resistive circuit, where all the quantities of interest are real. In an arbitrary circuit consisting of resistors, inductors, and capacitors, one needs both the voltage and the current to calculate the total input power. Equivalently, across a resistor, the power is evaluated from the squared magnitude of the voltage, as it is a complex quantity. In all the examples presented in the paper, the transmit164

ting and receiving antennas were conjugately matched so as to guarantee that the maximum amount of power was transmitted and received. Hence, under these circumstances, the two different fonns of the channel capacity reflect similar performance even though their starting points are different. As stated by Dr. Kimber, this is trivial; however, this simple point does not appear to have been considered in Dr. Kimber's analysis, as we illustrate next. Following this statement, Dr. Kimber then asks us to compute the voltages across the resistive part of the load, and not across the entire matched load. What this misses is that we indeed computed the voltages across the resistive part of the load, as illustrated on page 30 of our paper. Dr. Kimber also offers some numbers in support of his objections, without a derivation. In such a situation, it is absolutely necessary to provide the details of those calculations as evidence, so that one can assess the validity of the argument. For example, in his statements, no mention is made about the excitations, the loads, or the surrounding physics of his calculations. We thus do not know if his simulations accounted for the underlying physics. The goal of a scientific discourse is not to make conjectures but to provide documentary evidence, or to prove or disprove a statement through scientific means. Finally, Dr. Kimber suggests that the two similar numbers obtained from the two expressions for channel capacity (namely Shannon and Hartley et al.) in our paper were just a numerical coincidence. Dr. Kimber further states that one should choose the value that more closely matches the physical situation. What kind of scientific logic is that, since the mathematical expressions need to specify the physical situation and hence are already built into the fonnulation? The two numerical values representing channel capacity are similar at resonance even though their starting philosophies are different, and that is what we demonstrated in the paper. The Shannon form of the channel capacity computes the channel capacity by using noise power; whereas the Hartley (HNT) method calculates the capacity using the quantization voltage noise. In all of our examples, we treated a resonant system, and thus calculated the maximum power exchange that was possible under a given physical constraint. This relationship is one of the fundamental facts in electrical engineering, and we think Dr. Kimber has missed this fundamental point. We also point out in our paper when the two expressions will agree, even though they have different philosophical starting points. It is rather incredible that Dr. Kimber defends the position that measuring capacity at the front-end of the receiver is not an interesting alternative to the classical formula, which basically looks at the system from a baseband point of view. This implies that the transmitter/receiver front-ends have nothing to do with the channel capacity. Furthennore, claiming that the noise statistics are the signature of the underlying physics is not correct. Shannon's formula contains several stronger hypotheses concerning not only the noise statistics, but also implies a Gaussian distribution of the transmitted signal (again, at the baseband level). Capacity is strongly linked to the "resulting" channel, transmitter, and receiver, and our paper provides evidence that some aspects of the system, which are currently taken for granted, are relevant to the measured capacity in an experimental set-up. IEEE Antennas and Propagation Magazine, Vol. 51. No.1, February 2009

In the second paragraph, Dr. Kimber makes several statements ending with the conclusion that the effect of polarization and the use of an imperfect ground will provide different results. Unquestionably, the results for imperfect and perfect ground planes will be different. The fundamental question is the degree of the difference, which will depend on the material parameters of the ground. For this reason, we advocate using a numerical electromagnetics code to generate solutions to such problems via the exact Maxwellian physics, described by the Sommerfeld formulation for treating antennas and propagation over an imperfect ground plane. The Sommerfeld formulation for the analysis of antennas over imperfect ground planes excludes fallacious and naive arguments when investigating a particular physical scenario. The point we tried to make in our pap~r was that the presence of the ground plane will generate an image, which produces an interference pattern. Of course, the nature of this interference pattern will be different, as we presented in an earlier paper (reference [35] in our paper). As we showed in reference [35], even though the presence of an imperfect ground will change the nature of the interference pattern, the basic premise of a reduced capacity remains the same.

Finally, by reference to the height-gain of an antenna, Dr. Kimber concludes that we have missed the well-known phenomenon of height-gain. It is true that a plethora of papers exist in the mobile-communication literature on this topic, and that one can find many more by using a search engine like Google Scholar. Typically, these papers owe their existence to the classic experiment of Okamura et aI., published in [2]. In that paper, Okamura et aI. first showed experimentally that when the height of the trans-

mitting antenna changed, the field strength away from the transmitter for ranges between 600 m and 100 km in the city of Tokyo also increased. This fact led mobile-communication practitioners to assume that the signal strength increases as the height of the transmitting antenna is increased. However, a careful critical study of this environment, using correct Maxwellian physics incorporated in tools such as the commercially available AWAS code (which includes the Sommerfeld formulation for the analysis of propagation over an imperfect ground plane), reveals something quite interesting, as discussed in the next paragraph. The details of this mathematical formulation and the computational methodology are available in reference [35] of our paper. Here, we address the fact noted in the last paragraph by computing the field strength with the data in Okamura's classic paper. Consider the problem of a vertical dipole above a flat imperfect ground plane, with a relative permittivity of Gr = 4.0 and a conductivity of (J" = 2 x 10-4 mhos/m for a representative urban ground. We calculated the vertical component of the electrical field that is parallel to both the transmitting and receiving antennas as a function of their distance of separation. The receiving antenna was located at a height of 2 m above the ground, and the operating frequency (1 GHz) of the simulation was quite close to the value used in the Okamura experiment. The transmitter, a half-wavelength dipole, was excited centrally with 1 V, and was conjugately matched. In this computation, we varied the height of the transmitting antenna, and calculated the received fields at a second antenna for different values of the separation distance. In Figure 1, the separation distance was varied from 1 m to 100 km with high resolution in the range (x axis), as seen by the smooth plot. The

Figure 1. A plot of the field strength as a function of distance over an urban soil, with the antenna's height as a parameter. IEEE Antennas and Propagation MagaZine, Vol. 51, No.1, February 2009

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magnitudes of the calculated values of the electric field were plotted along the y axis of Figure 1. The five different graphs (blue, red, green, brown, indigo) corresponded to different heights of the transmitting antenna (5 m, 10 m, 20 m, 100 m, 500 m). For ranges between 10 km and 100 km (x axis), the field strength increased with the height of the transmitting antenna from the ground (although three of the plots are not displayed over the entire range), confirming Okamura's experimental results that the propagating field strength increases with antenna height. However, in mobile communication, one is interested in the variation of the near fields for distances up to 3-5 km from the transmitting antenna, and what happens beyond 5 km is of little interest, since the purpose of a base-station antenna is to serve a particular cell, which seldom extends beyond 3-5 km. The five plots of the near fields (ranges of 1 m to 1-3 km) in Figure 1 displayed the opposite of Okamura's results: that is, the emanating field strength essentially decreased with the transmitter's antenna height! The field strength near the transmitting antenna was high when the transmitting antenna was closer to the ground, and was diminished when the antenna was deployed at a higher height. Therefore, it is incorrect to conclude that in mobile communications, height gain actually takes place!

In summary, we strongly disagree with Dr. Kimber's conclusions. We think that he does not provide sufficient technical details to justify his statements, and that his claims are not supported by Maxwellian physics. Moreover, our conclusions are similar to that of the Nobel laureate Gabor, who, as mentioned before, made similar remarks over 55 years ago. In addition, the goal of our appendix on entropy was to emphasize the notion that the channel capacity is a mathematical number that has no meaning unless it is related to Maxwellian physics. Representative examples were presented in the paper to illustrate these points.

As Figure 1 was generated using an appropriate formulation that captures the true Maxwellian physics, we can only conclude that dabbling in mobile communication without an appreciation of the proper analysis tools is not very useful. We feel that this phenomenon is what motivated the Noble laureate, Dennis Gabor, 55 years ago to state in the first issue of the Transactions on Information Theory:

2. T. Okamura, E. Ohmori, and K. Fukuda, "Field Strength and Its Variability in VHF and UHF Land Mobile Service," Review Electrical Communication Laboratory, 16, 9-10, 1968, pp. 825-873.

...wireless communication systems are due to the generation, reception and transmission of electromagnetic signals. Therefore all wireless systems are subject to the general laws of radiation. Communication theory has up to now been developed mainly along mathematical lines, taking for granted the physical significance of the quantities which are fundamental in its formalism. But communication is the transmission of physical effects from one system to another. Hence communication theory should be considered as a branch of physics. Thus it is necessary to embody in its foundation such physical data. Hence we can apply to our problem the well known results of the theory of radiation by the Maxwell-Poynting theory. To emphasize Gabor's points, we started our paper with this quote. It is high time that we take this statement to heart, and study wireless communication using Maxwellian physics if we are to grasp its fundamentals. The simulations in our paper demonstrated interesting results for practical scenarios that are contrary to the accepted beliefs of the wireless community. If Dr. Kimber is aware of any publications providing results contradicting our conclusions, then we would appreciate seeing some references to them.

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In view of the above discussion, we would like to thank Dr. Kimber for his interest and for taking the time to read our paper. We appreciate his efforts in attempting to understand our thinking.

References 1. http://myweb.tiscali.co.uk/g8hqp/radio/smallant.html, accessed on December 8, 2008.

Tapan K. Sarkar, Santana Burintramart, Nuri Yilmazer, Yu Zhang, Arijit De Department of Electrical Engineering and Computer Science Syracuse University, Syracuse, New York 13244-1240 USA E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Magdalena Salazar-Palma Departmento de Teoria de la Senal y Comunicaciones Universidad Carlos III de Madrid, 28911 Leganes, Madrid, Spain E-mail: [email protected] Miguel A. Lagunas CTTC Avda. Canal Olimpic, sin 08860 Castelldefels (Barcelona), Spain E-mail: [email protected] Eric L. Mokole Radar Division, Naval Research Laboratory 4555 Overlook Avenue; Washington, DC 20375 USA E-mail: [email protected] Michael C. Wicks AFRLISN 26 Electronic Parkway, Rome, New York 13441-4514 USA E-mail: [email protected]mil E~

IEEE Antennas and Propagation Magazine, Vol. 51, No.1, February 2009