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Naftali (Tuli) HerscoviciLincoln Laboratory - Group 61Massachusetts Institute of Technology244 Wood StreetLexington, MA 02420-9108 USATel: +1 (781) 981-0801Fax: +1 (928) 832-4025Skype/AOL: tuliOlE-mail: [email protected]
Christos ChristodoulouDepartment of Electrical andComputer EngineeringUniversity of New MexicoAlbuquerque, NM 871 31-1 356 USATel: +1 (505) 277 6580Fax: +1 (505) 277 1439E-mail: [email protected]
MIMO Antennas Explained Using theWoodward-Lawson Synthesis Method
Marco Donald Migliore
DAEIMI, University of CassinoVia Di Biasio 43, 03043 Cassino, Italy
E-mail: [email protected]
Abstract
The way MIMO antennas work is explained in a simple and intuitive manner, starting from the Woodward-Lawson synthesismethod for classic antennas. The approach clarifies the strong similarities between MIMO and classic antennas, showing thatthe differences between them are mainly due to a different way in which the available degrees of freedom of theelectromagnetic field are used.
Keywords: Antenna radiation patterns; array antennas; MIMO systems; degrees of freedom
1. Introduction
In the last few decades, novel approaches in communicationsystems have pushed toward the development of a new class of
antennas, called MIMO (Multiple-Input Multiple-Output) antennas[1]. MIM40 antennas have a number of peculiarities that make theiranalysis difficult using standard approaches. In fact, quantities likethe radiation pattern, directivity, sidelobe levels, half-power aper-ture, and so on are of no importance for these kind of antennas.The difficulty in describing these antennas using the classicapproach is testified to by the fact that in antenna books adopted inthe electromagnetic community, these kinds of antennas are oftenabsent, in spite of their importance in current and future wirelesscommunication systems. When they are introduced, MIMO anten-nas are also treated as a special kind of antenna, with little or norelationship to the classic antenna-synthesis approaches.
The interesting feature of the approach followed in this paperis that the discussion of MIM40 antennas is based on a straightfor-
IEEE Antennas and Propagation Magazine, Vol. 49, No. 5, October 2007
ward extension of the well-known Woodward-Lawson antenna-synthesis method. The approach allows explaining the way MIM40antennas work in a simple and intuitive manner, by investigatingthe subtle relationship between energy and information. In thisinvestigation, antennas are characterized by their ability to conveyinformation in a communication system. This new point of view inantenna analysis has a number of conceptual advantages, allowingclarification of the similarities and the differences between classicantennas and MIMO antennas.
2. Woodward-Lawson Synthesis andMIMO Antennas
The Woodward-Lawson synthesis procedure [2, 3] is basedon the Fourier-transform relationship between the field in a planaraperture and the far-field pattern. The method is described in manyclassic books on antennas, e.g. [4- 6]. For the sake of simplicity,
175
Wireless Corner
z
a
NF c~sinc(2W4:-n),
n=-N
(3)
-a
Figure 1. The geometry of the problem.
we consider (Figure 1) a monochromatic electromagnetic source,placed on the plane y = 0, operating at wavelength 2. The tangen-
tial electric field on the plane y = 0 is null except in a thin slit,
having a length equal to 2a2 > along z, and a width b << 2 alongx. On this slit, the tangential electric field, E(z), has only a z
component. We will limit our attention to the far field in the z-yplane.
wherein 2N+1l= 2a/(2/2) is the number of samples falling inside
the interval 14:1•!1, and c,~ = Ed (4:,). An example of a synthesized
pattern in the case of a source 2a = 4A long is plotted in Figure 2.The role of the sinc functions (dashed curves) in obtaining the pat-tern is clearly visible. The samples falling in the visible domain areplotted as filled circles, while the samples in the invisible domainare indicated as empty circles.
in order to transmit information, the electromagnetic sourceand, consequently, the radiated field are varied according to a time-
domain signal, xQt), encoding information, which is usually mod-
eled as a (time-domain) bandlimited signal. If the signal has band-width B and is observed on a temporal interval T, it can be repre-sented as
2BT
wherein dr =x(tm) t, = m At, and At= - 1(2B). Consequently,
from a practical standpoint, representation of the signal requires2BT as the number of samples. This number is the number of
Let us consider the Fourier transform [2-4],
a/A
F(4) f E (T)exp (j2yr4:Y) dz-a/A
(1)
wherein Y z/2. In the range 14:1 :51, the function 1I e: F (4) is
proportional to the far-field complex pattern evaluated along the
observation angles S9= arccos(4:). The range 11:1•1 is conse-
quently associated with the visible domain, while the range 14:1 >1I
is associated with the invisible domain. Since the invisible domainis related to reactive power, a high value of F in the invisibledomain gives superdirective sources.
According to Equation (1), the radiating system can be mod-eled as a filter operating in the spatial domain (or, more precisely,in the angular domain) 4:, while the variable T represents a spatial
frequency. Since E (z) is null in jzi > a, the bandwidth of the filter
is finite. In particular, the one-sided bandwidth is equal to
W = a/2 . The field pattern can be represented by the Whittaker
Kotel'nikov-Shannon (WK(S) expansion for bandlimited signals,e.g.,
176
F
1.2
1
0.8
0.6
0.4
0.2
0
-1.5 -1 -0.5 0 0.5 1 1.5 2
Figure 2. An example of the pattern obtained by superpositionof sinc functions (filled circles: samples falling inside the visibledomain; empty circles: samples falling inside the invisibledomain). The dotted vertical lines limit the visible domain.
IEEE Antennas and Propagation Magazine, Vol. 49, No. 5, October 2007
0IF(4:n)sinc(2W4:-n), (2)n=--m
wherein 4:~=nA4:, A4: = I1(2W) =2/(2a) is the Nyquist interval,
and sinc (u) -sin (7ru)/;ru.
In the Woodward-Lawson synthesis method, the desired pat-
tern (in amplitude and phase), Ed (4:), is sampled in the 4:, samn-
pling positions falling inside the visible domain, while the samplesin the invisible domain are kept at a negligible value to avoidsuperdirectivity. The synthesized pattern is consequently equal to
R- oo'
y
degrees of freedom of the temporal signal [71, and will be denotedin the following as TNVDF. Generally speaking, the TNVDF coeffi-cients dm in Equation (4) allow unique identification of a particu-
lar funiction, x (t), and consequently information encoded in x (t).I
For further discussion, it is interesting to note the similaritybetween the representation of the time-domain signal, Equation (4),and the representation of the space-domain signal in Equation (3).In particular, the number of samples required in Equation (3) torepresent the field on an observation interval is the number ofdegrees of freedom of the space-domain signal (SND-F) on theobservation range [8, 91. If the length of the observation domain is= I we have that the SNDF is =_2WEE. In particular, in the case ofan observation interval covering the whole visible range (e.g.,-1•!ýý• 1), the SNDF is _=4W =4a/A, and will be called SNDFA.
As a first approximation, the SNDF coefficients in Equation (3)allow unique identification of the spatial distribution of the pattern.A rigorous discussion of the concept of the number of degrees offreedom of the electromagnetic field was reported in [8]. A simpleintroduction is also reported in the Appendix.
Coming back to the time-domain signal, x(t), a fundamental
result of information theory is that in the case of a (time-domain)received signal affected by additive white Gaussian noise, thesupremumi of the number of bits that we can send with a vanishingnumber of erroneously received bits -under an average receivedpower constraint, P, and noise power, X, is 0.51log 2 (1 + P/!2V) bits
per degree of freedom [7]. This give a total number of
(TATDF/2)og 2 (I + P/9v) bits. Note that the number of bits
increases only logarithmically with the signal/noise ratio, P/X~,but linearly with the number of degrees of freedom.
In order to send information, the coefficients, c~, in the
series of Equation (3) are varied in the time domain according tox t.In this way, information encoded in x (t) is transmitted in all
the angular directions. A receiving antenna placed in the far-fieldcondition along a direction -for example, in the broadside direc-tion of the transmitting antenna -will receive a signal having anaverage power, P, proportional to the power density radiatedtoward this direction. In order to maximize the amount of informa-tion received, we need to maximize the power received by thereceiving antenna, e.g., (for a fixed receiving antenna) to maximizethe directivity of the transmitting antenna toward the receivingantenna. By choosing a constant amplitude and phase distributionof the field on the radiating slit placed on the plane y - 0, we
obtain the sinc-shaped far field shown in Figure 3. This is the high-est directivity that it is possible to obtain, keeping the samples inthe invisible domain equal to zero. If we try to increase the direc-tivity also using the samples in the invisible domain, we obtain asuperdirective source. This situation must be avoided from thepractical point of view. Consequently, even if the available numberof samples for synthesizing the antenna pattern is theoreticallyinfinite, only SNDFA can be used.
It is interesting to note the different conceptual role of theseries of Equations (3) and (4). The coefficients of Equation (4) arechosen according to the information to be sent, and are not knownby the receiver before the reception of the signal. The "uncertainty"associated with these coefficients allows information to be trans-mitted. Instead, after the synthesis process, the values of the coeffi-cients in Equation (3) are fixed, and, consequently, informationcannot be associated with them. Their role is simply to assure asuitable angular distribution of the radiated power density in order
IEEE Antennas and Propagation Magazine, Vol. 49, No. 5, October 2007
to maximize the signal-to-noise ratio at the receiver. However, thisallows only a logarithmic increase in the number of bits that wecan send. This suggests using the series of Equation (3) in the sameway as the series of Equation (4), associating information also withthe spatial samples.
Let us suppose that the observation domain, E, wherein thereceiving antenna is placed, covers more than one spatial Nyquistinterval, e.g., more than one sample falls inside the observationdomain. With reference to Figure 4, three samples (e.g., three spa-tial degrees of freedom) fall inside the observation interval indi-cated by the two vertical dashed-dotted lines in the figure. We canassociate information with each of these samples. For example, thepattern in Figure 4 could encode the bits 1ll, while the pattern inFigure 5 could encode the bits 011.
0.8-
0.6-
F0 .4 -
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Figure 3. The pattern associated with the maximum directivityin the case of a non-superdirective source. The two verticaldashed-dotted lines are plotted to simplify the comparison withFigure 4.
F
Figure 4. An example of the pattern using the three samplesfalling inside the observation interval to send the string 111,using the spatial distribution of the field. The two verticaldashed-dotted lines limit the observation interval.
177
F
Figure 5. An example of the pattern using the three samplesfalling inside the observation interval to send the string 011,using the spatial distribution of the field.
2.5
2
i .1
F- I
i liii
-2 5
-2 -15 -1 -0.5 0 0.5 1 1.5
Figure 6. An example of the pattern using four samples, one ofthem falling outside the observation interval, to send thestrings 1011 (solid line) and 0011 (dotted line), using the spatialdistribution of the field in the presence of noise. The two verti-cal dashed-dotted lines limit the observation interval.
It is interesting to note that even if the samples falling insidethe observation domain are only three, the samples outside theobservation domain also contribute to the field inside the domainby means of their tails. This suggests also using the samples fallingoutside the observation domain to send information. However, thispossibility is strictly related to the noise level. Generally, noisemakes it impossible to identify the small variations in the patterndue to variations of samples placed outside the observationdomain. For example, the pattern in Figure 6 encodes the bits 10 11(solid line) and the bits 00 11 (dotted line) in the presence of noise.Information on the first bit is associated with a sample that doesnot fall inside the observation domain (limited by the two dotted-dashed vertical lines). The presence of noise covers the variationsof the observable part of the sinc function conveying informationon the first bit, causing the loss of information associated with it.
178
The example shows the strict similarity between this problem andthe problem of superdirectivity in classical antenna synthesis [10].Indeed, in both cases we use samples placed outside the interval ofinterest (e.g., the visible domain in classic antennas, and the obser-vation domain in MIM40 antennas) to modify the pattern inside theinterval.
Paralleling the discussion in the time domain, we can say thatthe observation of the spatial distribution of a field having a spatialbandwidth Won a spatial interval 7-7gives 2WEE (spatial) degreesof freedom that can be used to send independent information [9].The use of the spatial degrees of freedom increases the overallnumber of degrees of freedom that carry independent information.The supremumn of the number of bits increases from
(TNDF/2) log 2 (1 +P/J) to (ST2VDF/2) log 2 (1 +P'/5) [11],
wherein STNDF is the space-time number of degrees of freedom,while P'= P/SNDF is the average power associated with each
spatial degree of freedom (e.g., with each spatial sample fallinginside the observation domain). The spatial bandwidth, W, dependson the wavelength, and consequently on the frequency. Under thehypothesis that the bandwidth, B, is sufficiently narrow, we canconsider the spatial bandwidth, W, to be constant in the bandwidth,B, and the STNDF is equal to the product between the spatial num-ber of degrees of freedom, 2WE, and the temporal number ofdegrees of freedom, 2BT [11 ].
Summarizing, even if the use of more than one spatial degreeof freedom to send information gives a lower signal-to-noise ratiofor each spatial degree of freedom, the overall number of bits thatit is possible to send increases, thanks to the use of STNVDF degreesof freedom instead of only TNDF. This is basically the way MIMOantennas work. It is also interesting to note that with reference tothe observation range limited by the vertical dashed-dotted linesplotted in both Figures 3 and 4, the fraction of the total energy thatfalls inside the observation range in the case of Figure 4 is smallerthan in the pattern of Figure 3. This shows that when we are able touse more than one spatial degree of freedom to send information,maximization of the received energy in the observation domaindoes not assure maximization of the bit rate. This is the reason forthe failure of standard antenna parameters -based on energeticconcepts -for MIMIO antenna characterization.
Clearly, the use of more than one spatial degree of freedom tosend independent information is strictly related to the extension ofthe observation area. In order to increase the SNDF inside a fixedobservation area, let us consider the scheme plotted in Figure 7.This consists of a primary source having length 2a that illuminatesa secondary source, e.g., a "scattering line," having length2a' > 2a. The SNDF of the field radiated by the secondary source,SNDF2 , is consequently larger than the SNDF of the field radiated
by the primary source, SNDF1 .
The field on the observation domain can be represented as
(5)SNDF
2
wherein SNDF 2 is equal to 2W'-',:, and W' = a'/2. However, the
field distribution on the secondary source cannot be freely chosen,but depends on the field distribution on the primary source. In par-
ticular, the SNDF2 coefficients, F2 (~,,are a linear combination
of the SNDF, coefficients, c,,, of the series regarding the primary
source. Consequently, only SNDF1 coefficients of Equation (5) are
IEEE Antennas and Propagation Magazine, Vol. 49, No. 5, October 2007
I
a!
a
-a
-a!
Y
Figure 7. The geometry of the problem in the case of a pri-mary-source aperture (thick black line) and a secondary scat-tering source (thick gray line).
independent, and the number of degrees of freedom of the fieldradiated by the system composed of the primary and the secondarysource is SNDF1 . The practical consequence is that the amount ofinformation that we can encode in the spatial domain is limited toSNDF1 , and is fixed by the primary source.
This is an example of the fact that the degrees of freedomdepend on the a priori information that we have on the system. Ifthe existence of the primary source is not known (e.g., only theexistence of the radiating system represented by the secondarysource is known), we need SNDF2 coefficients to represent thefield. In. practice, such a number of parameters is redundant, sincenot all of the field configurations are possible on the secondarysource, but only the subset determined by the field radiated by theprimary source. If the presence of the primary source is known, it ispossible to identify' the degrees of freedom associated with theradiating system composed of the primary and secondary sources.Of course, in this case, we have more-detailed a priori informationon the radiating system, and this decreases the number of parame-ters required to identify' the field configuration. The dependence ofthe number of degrees of freedom on the a priori informationavailable on the system is true for general physical systems.Broadly speaking, the more a priori information we have, the lessdegrees of freedom we need to describe the system. From the pointof view of information theory, the more a priori information wehave, the less information we can send.
While we do not have an advantage in terms of the number ofdegrees of freedom available to transmit information, we have agreat advantage with respect to the dimension of the observationarea required to retrieve (or, more precisely, to stably recover) thetransmitted information [12]. In fact, the value of the SNDF,parameters identifying the state of the primary source (and of theinformation encoded by those parameters) can be retrieved bymeasuring SNDF1 coefficients of Equation (5). If the spatial exten-sion of the secondary source is very large compared to the primary
IEEE Antennas and Propagation Magazine, Vol. 49, No. 5, October 2007 179
source (a'>> a), we have that SNDF2 >> SDNFl , and we need amuch shorter observation domain to collect SNDFj spatial degreesof freedom.
In MIM40 communication systems, scattering objects placedin the environmnent operate as secondary sources, allowing identifi-cation of spatial variations of the field in a smaller observationdomain. Indeed, the physical reason of the use of MIM40 antennasin a rich scattering environment is that the presence of the scatter-ers makes it possible to position the receivers on a limitedextended surface.
The last observation is in regard to the relationship betweenthe simple approach followed in this paper, and the statisticalapproach used in MIMIO communication systems [1]. Let us con-sider our antenna in reception. The antenna is able to receive a sig-nal conveying independent information in each spatial Nyquistinterval Aý, e.g., a potential number of SNDFA spatiallyindependent pieces of information (of course, if the angular rangeof the impinging signal is smaller, and covers, for example, only arange E, this number decreases to 2WE). Let us now suppose thata large number of waves impinges in a single spatial Nyquist inter-val. In this case, the antenna is not able to discriminate among thedifferent contributions due to its finite angular resolution, and sim-ply receives their sum. If the contributions are statistically inde-pendent, and their number is sufficiently large, the sum can bemodeled as a random variable with a Gaussian distribution, and thefield incident on the antenna can be properly modeled as a stochas-tic process.
3. Conclusions
Given an electromagnetic source and an observation domain,the electromagnetic field conveys an amount of information equalto the number of different configurations that can be distinguishedon the observation curve. If the (spatial) number of degrees of free-dom of the field on the observation curve is one, the best solutionis to use all the available degrees of freedom of the field radiatedby the antenna to maximize the energy on the observation curve. Inclassic antenna synthesis, the receiving antenna is supposed to bein the far field. In this condition, the co-polar component of thefield has only one spatial degree of freedom. Consequently, theclassic "energetic approach" for antenna synthesis, based on theradiation patterns - e.g., on the energy distribution in the spacealso allows the maximization of the bit rate. More generally, the"energetic point of view" allows maximizing the bit rate in all thecases in which the electromagnetic field associated with the desiredsignal has only one degree of freedom, since, in this case, themaximization of the information conveyed by the system requiressimply the maximization of the signal-over-noise ratio.
Things dramatically change if the field carrying the desiredinformation has more than one degree of freedom on the observa-tion surface. This is because the use of each available degree offreedom to carry independent information can be more advanta-geous than the maximization of the energy on the observation sur-face.
A simple example of the difference between the energetic andinformational "points of view" is offered by doubly polarizedantennas. In fact, the far field has two degrees of freedom associ-ated with the two orthogonal polarizations. In doubly polarizedantennas, we use both of the degrees of freedom to send independ-
ent information. Let us suppose that the received signal's averagepower associated with the two components are different, let us sayby 2 dB. If we want to maximize the average power at the receiverwith a constraint on the total average power at the transmitter, thesolution is to associate all the power with the stronger field com-ponent. However, a quite natural heuristic solution is to use both ofthe components to send information, eventually associating ahigher power with the "stronger" component. This solution reducesthe average received power (for a fixed transmitted power), butassures a higher bit rate. The exact evaluation of the power distri-bution on the two components can be obtained using the so called"water-filling" solution [7].
The use of the classic point of view using energetic conceptsis at the base of a number of misleading and unclear explanationsof MIMO0 systems in general, and MIN40 antennas, in particular.Instead, if we accept an informational point of view, no significantconceptual differences exist between classic and MIMO antennas.According to this point of view, the question, "What is a MIMO0antenna?" has a very simple answer: "A MIIMO antenna is anantenna that conveys statistically independent information on morethan one degree of freedom of the field,"
4. Appendix:System Representation and Information
As preliminary step, let us recall some basic concepts regard-ing physical systems. In general, given a physical system, we areinterested in describing the "state" of the system, e.g., to uniquelyidentify the system's configuration. For example, let us consider afixed amount (e.g., a fixed number of moles) of an ideal gas. Themacroscopic identification of the state of the system requires a verysmall number of variables (temperature, pressure, and volume). Asis well known, there is a funictional relationship among volume,temperature, and pressure, expressed by the Clapeyron ideal gaslaw. Only two of these three parameters are free. Consequently, theminimum number of variables required to identify the state of thesystem at a macroscopic level is only two. Of course, we can useany two of the three parameters as state variables, or we can alsointroduce other state variables, but in any case, the number of inde-pendent state variables is always two.
More generally, a physical system could require an infinitenumber of variables to describe its state. However, in practice,physical quantities can be known only within an approximationerror fixed by the noise and by the measurement accuracy. Conse-quently, in practice we can know the state of a physical systemonly within an approximation level.
The minimum number of parameters required to describe thestate of the system (at the desired degree of accuracy) is the num-ber of degrees of freedom (NDF) of the system [8]. Clearly, theNDF has a relevant role in the representation of the physical sys-tem. For further observations, it is important to stress that at therequired degree of accuracy, only NDF parameters can be inde-pendently chosen.
Now, let us suppose that we want to send information bymeans of a fixed number of moles of an ideal gas. Let us supposethat we can change the volume of the gas in adiabatic conditions.We can encode information by means of the volume of the gas. Itis interesting to note that since there is no heat exchange throughthe wall of the container of the gas, the external work is wholly
180
transformed into internal energy of the gas, modifying the averagekinetic energy of the molecules. Consequently, variation of thevolume and variation of the temperature (and of the pressure) aredirectly related, and it is not possible to further encode (independ-ent) information into the temperature, and neither into the pressure.
Then, the gas container is sent to the receiver, and informra-tion is extracted by measuring the physical quantity in whichinformation has been encoded, e.g., the volume. Uncertainties inthe measurements, and the unavoidable presence of noise, make itimpossible to identify volume variations that are too small. Conse-quently, the number of different volume values that we can iden-tify, e.g., the amount of information that we can encode in a finitevolume interval, is always finite. The number of different distin-guishable volume values measured in base 2 numbers representsthe number of bits that we can "transmit" by means of volumevariations.
Let us now suppose that we are able to not only vary the vol-ume of the gas, but also to control the heat exchange on the wallsof the container. In this case, we are free to choose both the volumeand the temperature of the gas. These two state variables can carryindependent information. Finally, if we try to also change the pres-sure, the temperature and the volume change. This example showsthat it is possible to encode independent information into differentstate variables. Furthermore, the number of degrees of freedomlimits the amount of information that we can send by means of anyphysical system.
Summarizing, the number of degrees of freedom has animportant role both in the representation of the system and in thelimitation of the amount of information that the system can convey.This observation is true in any physical system, including electro-magnetic systems, and is the reason for the importance of the roleof the number of degrees of freedom of the field in limiting theperformance of MIM40 systems [9].
It must be also stressed that the approach followed in thispaper is not rigorous with reference to the concept of the number ofdegrees of freedom. In the following, a more rigorous approach isbriefly outlined, paralleling the approach introduced in [8].
As discussed in the above-reported discussion, given a physi-cal system about which we have some a priori information, thenumber of degrees of freedom of the system is the minimium num-ber of parameters required to definite the state of the system,within a given accuracy. In this paper, the physical system consistsof the field distribution radiated by an electromagnetic sourcehaving finite spatial extension, and observed on an observationmanifold consisting of a portion of the half circumference placed atinfinite distance. Furthermore, we limit our interest to linear repre-sentations of the field, for the sake of the practical usefulness of therepresentation. Consequently, the definition of the degrees of free-dom in our case is slightly modified as the minimum number ofcoefficients required to represent the field on a given observationcurve by means of a linear representation [8].
Since the field is an element of an infinite-dimensional func-tional space (we will consider in particular L2 space), the number
of parameters to represent the field is theoretically infinite. Thismeans that the number of parameters required to represent the farfield with infinite accuracy is infinite. However, as a consequenceof the compactness of the radiation operator [13], only a finitenumber of coefficients are required to represent the field withinany desired finite degree of accuracy [8, 9]. The minimum number
IEEE Antennas and Propagation Magazine, Vol. 49, No. 5, October 2007
0.16
0.14
0.12k
0.1
0.08 -
0.06k
0.04
0.02-
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Figure 8. The amplitude of the first three prolate spheroidalfunctions, plotted in the range ý of the observation domain.They are orthogonal both in the observation domain (4' rang-ing over the values in the figure) and on the unboundeddomain (-00 < ý < oo).
of coefficients required to represent the field in Equation (3) withinthe desired degree of accuracy also fixes the number of degrees offreedom of the field at such a degree of accuracy. The field can berepresented as
NDF
wherein the funictions T.', are the basis function of the representa-tion. Any representation that requires a number of functions equalto the NDF is called an optimal representation. The sine; basis is anoptimal basis (e.g., gives an optimal representation) only in thecase of unbounded observation domain, while in the case of abounded domain, an optimal basis is given by the prolate spher-oidal functions [14]. For example, in Figure 8, the three functionsthat should be used in the case discussed in Figures 4 and 5instead of the three sinc functions associated with the samples fal-ling inside the observation domain - are plotted. These functionsare both orthogonal on the observation domain and on theunbounded domain, e.g., for -oo < ý < oo, while the sinc: functionsare orthogonal only on the unbounded domain. However, the coef-ficients in Equation (5) are given by the inner product between thenth basis function and the field, and are consequently related to thefield in a complex way. For this reason, the rigorous approach losesthe clarity and simplicity of the less-rigorous but much-more-intui-tive approach using sinc functions followed in this paper.
Identification of the relationships between electromagneticsand information theory is a very recent field of research. The inter-ested reader can refer to the classic book of Brillouin [15], whichdiscussed the linkage between physics and information theory.Other interesting references related specifically to electromagneticsand information theory are, for example, [ 16-19].
5. References
IEEE Antennas and Propagation Magazine, Vol. 49, No. 5, October 200718
1. M. A. Jensen and J. Wallace, "A Review of Antennas andPropagation for MIMO Wireless Communications," IEEE Trans-actions on Antennas and Propagation, AP-52, 11, November2004, pp. 28 10-2824
2. P. M. Woodward, "A Method of Calculating the Field Over aPlanar Aperture Required to Produce a Given Polar Diagram," JIEE, Part lIlA, 93, 1946, pp. 1554-1558.
3. P. M. Woodward and J. D. Lawson, "The Theoretical Precisionwith Which an Arbitrary Radiation Pattern may be Obtained from aSource of Finite Size," J1 lEE, Part III, 95, 1948, pp. 363-370.
4. C. A. Balanis, Antenna Theory, New York, John Wiley & Sons,1982,
5. M. 1. Skolnik, Introduction to RADAR systems, Third Edition,New York, McGraw-Hill, 2001.
6. S. Deabowitch, A. Papiernik, H. Griffiths, and J. Encinas, Mod-ern Antennas, New York, Chapman & Hall, 1998.
7. C. E. Shannon and W. Weaver, The Mathematical Theory ofCommunication, University of Illinois Press, 1963.
8. 0. M. Bucci and G. Franceschetti, "On the Degrees of Freedomof Scattered Fields," IEEE Transactions on Antennas and Propa-gation, AP-38, 1989, pp. 918-926.
9. M. D. Migliore "On the Role of the Number of Degrees of Free-dom of the Field in MIMO Channel," IEEE Transactions onAntennas and Propagation, AIP-54, 2, February 2006, pp. 620-628.
10. M. L. Morris, M. A. Jensen, and J. W. Wallace, "Superdirec-tivity in MM10 Systems," IEEE Transactions on Antennas andPropagation, AP-53, 9, September 2005, pp. 2850-2857.
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CorrectionIn the August 2007 contribution to the Wireless Corner by
Frid~ric Broyd6 and Evelyne Clavelier [1], the first sentence ofSection 8 (p. 215) should have read as follows:
This section applies to a circular array of four parallel half-wave dipole antennas (side-by-side configuration) for 1880 MHz,presenting a 0.424A - 68 mm spacing between the nearest arrayelements.
1. Fr~d~ric Broyd6 and Evelyne Clavelier, "Taking Advantage ofMutual Coupling in Radio-Communication Systems Using aMulti-Port Antenna Array," IEEE Antennas and PropagationMagazine, 49, 4, August 2007, pp. 208-220. r11
182 IEEE Antennas and Propagation Magazine, Vol. 49, No. 5, October 2007182