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Transmit and Receive Array Gain of Uniform Linear Arrays of Isotrops
Michel T. Ivrlač and Josef A. Nossek Institute for Circuit Theory and Signal Processing Technische Universität München, D–þììì Munich, Germany email: {ivrlac, nossek}@tum.de
Abstract—Both in the signal processing and in the information II. T§Z«± A§§Zí theory literature, it is common to assume that the array gain of antenna arrays grows linearly with the number of antennas. How- Let us start by focusing on a communication system, where ever, such an assertion is valid only when the antennas are uncou- the transmitter is equipped with N isotropic antennas, lined pled. When antenna coupling is present and properly taken into up into a uniform linear array (¶Z). The receiver shall pos- account by the signal processing algorithms, we show that both sess a single isotropic antenna. The electric currents that ow the transmit and the receive array gain can grow super-linearly through the transmit antennas excite an electric eld which is with the number of antennas. In special cases, the receive array gain can even grow exponentially with the antenna number. These turned into an electric voltage by the receive antenna. In the results are exciting, for they imply that the potential of radio absence of receiver noise, the input-output relationship can communication systems which use more than one antenna at the therefore be written as receiver or the transmitter, is likely to be much higher than pre- v zT i, (×) viously reported. = I. I±§o¶h± where v C V, is the narrow-band complex envelope of the ∈ When it comes to the analytical treatment of antenna arrays it receive voltage, while i CN×Ô A, denotes the vector of the ⋅ ∈ is common practice, both in the signal processing and in the narrow-band complex envelopes⋅ of the electric currents ow- information theory literature, to assume that the individual an- ing through the transmit antennas. The vector z CN×Ô Ω, is ∈ tennas of the antenna array are uncoupled isotropic radiators the transimpedance vector, which depends on the wave⋅ prop- (see, e.g. [Õ], chapter ß). A consequence of this assumption is agation between transmitter and receiver. Herein, the super- that the array gain grows linearly with the number of antennas script T denotes transposition, while V, A, and Ω, denote the [Õ]. It has therefore become common knowledge in the infor- physical units »Volt«, »Ampere«, and »Ohm«, respectively. mation theory literature, that the channel capacity of wireless multi-input single-output («), or single-input multi-output («) communication systems grows proportional to the log- A. Transmit Power arithm of the number of antennas, when the signal to noise We dene as transmit power, the power that is radiated by the ratio («§) is large. antenna array. In the following we assume that the antenna On the other hand, closely spaced antennas will, in general, array is lossless, such that the electric input power is equal exhibit some electro-magnetic coupling [ó]. It is an interest- to the radiated power. As antennas are linear circuit elements, ing and highly relevant question, how such antenna coupling the relationship between the vector u CN×Ô V, of voltage en- ∈ inuences the achievable array gain, and hence, the channel velopes at the transmit antennas, and the current⋅ envelope vec- capacity of multi-antenna radio communication systems, es- tor i, is a linear one: u Zi, where Z CN×N Ω, is the so- = ∈ pecially when coupling eects are taken into account by the called impedance matrix of the antenna array. Since⋅ antennas signal processing algorithms. are reciprocal, Z ZT holds true. The transmit power is given = In this paper, we demonstrate that both the transmit and by the expression P E Re uH i , which can also be writ- Tx = the receive array gain may be substantially increased when an- ten as: tenna coupling is present and taken into account when com- P E iH Re Z i . (ö) Tx = puting the respective beamforming vectors. For the case of H uniform linear antenna arrays of isotropic radiators, we show The symbol E, and the superscript , hereby denote the ex- that the array gains can grow super-linearly with the number pectation operation, and the complex conjugate transpose, re- of antennas. In case that the receiver noise is of purely thermal spectively. For a ¶Z of isotropic radiators, the real-part of origin, it turns out that the receive array gain even grows ex- the impedance matrix can be computed in closed form [ì]: ponentially with the antenna number. Consequently, there are Re Z Rý C (î) cases where the channel capacity of a wireless « communi- = ⋅ ∆ cation system can grow linearly with the number of antennas. C sinc òπ m n , (îa) ( )m,n = λ ( ) This is a property that has previously been attributed only to − multi-input multi-output ( ) communication systems. where sinc x sin x x, λ is the wavelength, ∆ is the dis- ( ) = ( )~ These results are exciting, for they show that the potential tance between neighboring antennas, and R R Ω, is the ý ∈ + of multi-antenna wireless communication systems might well radiation resistance. For convenience, a derivation⋅ of (ì) and be substantially higher than previously reported. Key to this (ìa) is given in the appendix. As we will see, the transmit ar- increased potential are the physical coupling eects of antenna ray gain does not depend on the value of Rý. What matters elements in the array. is the relative antenna coupling matrix: C RN×N. As can be ∈
Authorized licensed use limited to: T U MUENCHEN. Downloaded on May 17,2010 at 08:57:44 UTC from IEEE Xplore. Restrictions apply. ó observed from (ìa), only when the element spacing ∆ happens 25 to be an integer multiple of λ ò, the antennas are uncoupled, N θ ý, (»end-re«) ~ = = because the C–matrix becomes the identity. For all other spac- Tx « 20
A N ¥ ings, the antennas are coupled. =
B. Optimum Beamforming 15 N ç = The transmitter uses its transmit antennas for beamforming: i w s, (®) 10 N ò = = where w CN×Ô A is the beamforming⋅ vector, while s C, ∈ ∈ is a zero-mean, unity× variance, random variable, which mod- 5 els the information signal that should be transfered to the re- Transmit Array Gain, ceiver. The receive antenna is terminated by a resistive load. 0 Consequently, the receive power is proportional to the average 0 0.5 1 1.5 2 2.5 3 squared magnitude of the receive voltage envelope: Antenna spacing, ∆ λ ò ~ PRx γ E v , (£) Figure Õ: Transmit array gain in »end-re« (θ ý), as = S S « = function of antenna spacing for dierent antenna numbers. γ wH z∗zTw, (£a) = where γ R Ω−Ô, is a positive constant, and (¢a) follows by D. Scaling Law ∈ + substituting (¦)⋅ into (Õ), and the latter into (¢). Herein, z is Let us now look at a line of sight («) propagation scenario. assumed to be constant. The optimum beamforming vector, We assume that the antenna array is centered in the origin for a given z, is the one which maximizes the received signal and aligned with the z-axis of a Cartesian coordinate system. power for a given transmit power PTx: Because there is just one path connecting receiver and trans- H ∗ T H mitter, and the receive voltage v, is proportional to the electric wopt arg max w z z w, s.t. Rý w Cw PTx, (å) = w ≤ ⋅ eld strength at the receiver, we see from (Õ), and (ìþ): where the constraint follows from putting (ì) and (¦) into (ó). z γ′ a, (××) It can be shown [ì], that the receive power under optimum = beamforming is given by: where γ′, is a constant, and max ò P γ E v w wopt T Rx = S S T = a Ô e−jµ e−j(N−Ô)µ CN×Ô , (×ö) = ∈ PTx γ ⋯ zTC−Ôz∗. (à) is the array steering vector, where µ òπ ∆ λ cos θ. Herein, = R = ( ~ ) ý θ is the elevation angle of departure – that is, the angle be- C. Transmit Array Gain tween wave propagation towards the receiver, and the array If only one transmit antenna, say the n-th one, is excited by axis (the z-axis of the coordinate system). When we substitute a non-zero electric current, the receive power becomes: (Õó) into (ÕÕ), and the latter into (Õþ), we see that the array gain in a « scenario becomes: PTx γ ò PRx,n zn , « T −Ô ∗ = Rý S S A a C a . (×î) Tx = where zn is the n-th component of z. The average received power over all possible transmit antennas then becomes: 10 N single PTx γ ò N â θ π ò, PRx,avr zn , () N = = ~
= S S Tx NRý nQ=Ô « 8 = (»front-re«) A The transmit array gain is dened as: max N ç N ¥ P 6 = = A Rx . (Ì) Tx = single PRx,avr When we substitute (ß), and () into (É), it follows that 4 zTC−Ôz∗ A N . (×ÿ) Tx = zTz∗ 2
Recall that only when the antennas are spaced an integer mul- Transmit Array Gain, N ò tiple of half the wavelength apart, the antennas are uncoupled, 0 = that is, C I. For these special spacings only, the transmit ar- 0 0.5 1 1.5 2 2.5 3 = ray gain equals the number N of antennas. All other antenna Antenna spacing, ∆ λ spacings make the transmit array gain also depend on the an- ~ Figure ó: Transmit array gain in the »front-re« (θ π ò), tenna spacing (via C), and on the transimpedance vector z, « = ~ that is, on the propagation scenario! as function of antenna spacing for dierent antenna numbers.
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uS,Ô u˜N,Ô uN,Ô iA,Ô iB,Ô
i N,Ô uA,Ô uB,Ô R uÔ
Z ZM
uS,N u˜N,N uN,N iA,N iB,N
i N,N u A,N uB,N R uN
noisy receive antenna array matching network noisy amplier
Figure ì: Circuit-theoretic model of a receive antenna array cascaded with an impedance matching network and the inputs of the low-noise ampliers.
In Figure Õ, the array gain that is achievable in the direction Every electronic Z generates two kinds of noise: voltage θ ý, is depicted for dierent N, as a function of the antenna noise, and current noise [¦]. It turns out that current noise = spacing. We can observe that the array gain raises quickly as contributes quite dierently to the noise covariance than does we reduce the antenna spacing below λ ò. The maximum ar- voltage noise. Therefore, the noise covariance depends on the ~ ray gain happens to be equal to N ò. That is, the transmit array relative intensity of these two kinds of noise contributions. It gain can grow super-linearly with the number of antennas! Fig- is specied by the so-called noise-resistance, which is dened ure ó shows the achievable array gain in the direction θ π ò. as the ratio of the root-mean-square (§«) noise voltage and = ~ For beamforming in this direction, one has to put the anten- the §« noise current [¦]. We show that the lower the noise- nas considerably apart, in order to eectively use the array for resistance, the higher the receive array gain becomes. A low beamforming. The optimum antenna spacing is larger than noise resistance may therefore be even more important than λ ò, but smaller than λ. In this setup, the array gain grows a low noise gure! For the case of noise which is purely of ~ linearly with the number N of antennas, yet its peak value is intrinsic origin, a noise-resistance of zero turns out to let the larger than N. receive array gain grow exponentially with the number of an- tennas. This result is exciting, because it demonstrates the fact « III. Ruhuêu A±uZ A§§Zí that it is possible for a radio communication system, to have a channel capacity that grows linearly with the number The receive array gain achievable by optimum beamforming, of antennas, in the high «§ regime! Such a linear growth of critically depends on the spatial covariance of receiver noise. channel capacity has previously been reported only for In radio communication systems, it makes sense to distinguish communications. between intrinsic noise, and extrinsic noise. While the latter comes from background radiation received by the antennas, the intrinsic noise originates from components of the radio A. Circuit-Theoretic System Model frequency front end, most importantly, from the rst stage of Let us start from the circuit theoretic system model that is the low noise amplier (Z). When intrinsic noise is the sole shown in Figure ì. It consists of the following three blocks: source of noise, it is the state of the art to conjecture that the the antenna array, the impedance matching network, and the receiver noise is spatially white, or at least uncorrelated. How- inputs of the Zs. The N–antenna receive array is described ever, such a conjecture is, in general, not correct! While it is as a passive, linear N–port. The received signals are taken care true that the noise sources inside the Zs do generate noise of by two voltage sources per port: the voltages uS,i , model the independently, the individual noise contributions may super- desired received signal, while u˜N,i model the received back- impose because of coupling eects. Coupling occurs in the ground noise (extrinsic noise). The impedance matching net- antenna array because of the proximity of closely spaced an- work is described by: tennas, and because of an impedance matching network that ⎡ uA jIm Z jRe Z iA can frequently be found connected between the antenna array ⎢ ⎡ ⎤ . (×®) ⎢ ⎤ = ⎢ − ⎥ ⎡ − ⎤ and the input of the Zs. uB ⎥ ⎢ jRe Z O ⎥ ⎢ iB ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
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Herein, Z CN×N Ω, is the impedance matrix of the antenna where the noise-resistance is dened as: ∈ array. The two vectors u CN×Ô V, and u CN×Ô V, con- ⋅ A B ò ∈ ∈ ¿ ′ tain the complex envelopes of the⋅ voltages at the input⋅ and E uN,i ¿ β R Á R . (ö×) the output of the matching network, respectively, while the N Á S Sò Á = Á E i = ÀÁ β vectors i CN×Ô A, and i CN×Ô A, contain the respective ÀÁ N,i A ∈ B ∈ S S complex current envelopes.⋅ The rationale⋅ behind the specic choice of matching network in (Õ¦) becomes apparent from: C. Optimum Receive Beamforming u Re Z i j u u˜ , (×£) The receiver performs linear beamforming, according to: B = B S N + ( + ) s wHu wH h u wHη, (öö) where we have stacked the desired signal voltage envelopes = = S ⋅ + uS,i , and the extrinsic noise voltage envelopes u˜N,i into the where w CN×Ô is the beamforming vector, and s C V is vectors u CN×Ô V, and u˜ CN×Ô V, respectively. As we ∈ ∈ S ∈ N ∈ the resulting scalar voltage envelope. The «§ becomes:⋅ can see from (Õ¢),⋅ the coupling of the⋅ matched antenna array wH hhHw now depends only on the real-part of the impedance matrix «§ E u ò . (öî) = S wHR w of the antenna array. In this way, we can directly make use of S S η (ì), and (ìa). The voltage envelopes in uB are then fed to the It is not too dicult to derive the maximum achievable «§: inputs of the Zs. The latter are modeled by a noiseless re- «§ E u ò hHR−Ôh. (ö®) sistor R, and two sources which model the amplier’s current max = S η and voltage noise envelopes, u , and i , respectively. Col- S S N,i N,i When we substitute (ÕÉ) and (óþ) into (ó¦) we arrive at: lecting these envelopes into respective vectors u CN×Ô V, N ∈ and i CN×Ô A, we nd the following system equation: aH Υ−Ôa N ∈ ⋅ «§ E u ò , (ö£) ⋅ max = S βRò u Q ju Re Z i u ju˜ . (×å) S S = S N N N where the matrix Υ CN×N is dened in (óþ). +T + + ∈ where u u u u , is the vector of the output voltage = Ô ò N envelopes, and Q ⋯ I Re Z R −Ô. Clearly, the amplier = N D. Receive Array Gain i u noise current N contributes( + { to}~ the) noise portion of quite If only a single isotropic radiator is present, we see by setting dierently than the amplier noise voltage uN, or the back- N Ô in (ó¢), that the «§ becomes ground noise u˜N. = ò E uS «§ , (öå) B. Signal Model single = òS S ò β Rý RN « ( + ) Let us focus again on propagation. The desired received as from (óþ), Υ Rò Rò Rò, for N Ô. The receive array signal is given by: u u a, where a, is the array steering = ý N = S = S gain A , quanties( how+ much)~ more «§ we can obtain by vector from (Õó), and u C V, is the information bearing Rx S ∈⋅ using all antennas of the array, compared to a single antenna: signal voltage envelope. By dening⋅ the receiver noise voltage «§ vector as the vector of output voltage envelopes for u ý: max S = ARx (öà) = «§single N×Ô η Q Re ZA iN uN ju˜N C V, (×à) = ∈ Rò Rò + + ⋅ aH Υ−Ôa ý N . (öàa) we can rewrite the system equation (Õä) as: = R+ò u h u η, (×) In the following, we set the input resistance R of the Zs = S ⋅ + equal to the radiation resistance Rý of the isotropic antenna. where h jQa CN×Ô. (×Ì) As furthermore aHa N, we nd for the receive array gain: = ∈ = In this paper, we assume that all Zs have the same current aH Υ−Ôa Rò A N Ô N . (ö) noise intensity, but the current noises of dierent Zs are Rx = aHa Rò + ý uncorrelated, such that E i iH β I , where β R Aò, is N N = N ∈ + Note that Υ depends on the antenna spacing ∆ λ, by virtue a constant, specic to the Z, and proportional to the⋅ noise of the matrix Z. It also depends on the noise resistance~ R . bandwidth. Similarly, E u uH β′Rò I , where β′ R Aò, N N N = N ∈ + is a constant, that is specic to the Z, and proportional⋅ to the noise bandwidth. For the reason of brevity, we limit the E. The Two-Antenna Array discussion to the case where there is no background noise, Let us now have a look how much receive array gain can be hence: E u˜ u˜H O. When we assume that all noise sources ¶Z N N = obtained by a of isotropic radiators. Let us x the an- are uncorrelated, we nd that the receiver noise covariance gle θ of the beamforming to the value of θ ý, which cor- = R E ηηH , can be written as: ¶Z η = responds to the direction of the line-up – the so-called ò ò »end-re« direction. For a small array of N ò isotrops, we ò Ô RN = Rη βR Q Re Z I Q, (öÿ) see in Figure ¦, the receive array gain in dB, as function of = Rò Rò N + antenna separation. As was pointed out earlier, current noise contributes dierently to the overall receiver noise than does ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹Υ ¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
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21 60 R ý ò ò N = 18 RN Rý ý = 50 Rò Rò Ôý− ~ N ý = 15 ò ò −ò ~ RN Rý Ôý 40 = Rò Rò Ôý−ç ~ N ý = 12 ~ Rò Rò Ôý−Ô 30 N ý = ò ò −Ô 9 RN Rý Ôý ~ ò ò = RN Rý Ô ~ = 20 RN → 6 ~ ∞ 10 Receive Array Gain in dB 3 Receive Array Gain in dB
0 0 0 0.25 0.5 0.75 1 1.25 1.5 0 2 4 6 8 10 12 14 16 18 20 Antenna spacing, ∆ λ Antenna number, N Figure ¦: Receive array gain in dB in »end-re«~ direction, for Figure ¢: Receive array gain in dB in »end-re« direction, as two antennas, as function of antenna spacing. function of the antenna number, for ∆ ý.¥λ. = voltage noise. Figure ¦ therefore contains four dierent curves, «§, the exponential growth of the receive array gain with which correspond to dierent values of the noise resistance. A the number of antennas leads to a linear growth of the chan- noise resistance of zero, means that the noise originates solely nel capacity with the antenna number. Such a linear growth from current noise. This is the case for thermal noise of a re- has previously been attributed only to communication sistor. One can therefore argue that zero noise resistance corre- systems, which can exploit spatial multiplexing, that is, can sponds to the case of having an ideal amplier, which does not transfer several data streams at the same time using the same introduce any additional noise besides the inescapable ther- band of frequencies. However, as we have demonstrated, it is mal noise of the real-part of its input impedance. Hence, as possible to achieve a linear growth of channel capacity with can be seen from the top most curve in Figure ¦, with ideal respect to the antenna number, already with a wireless « ampliers, the receive array gain grows unboundedly, as the system, that is, a radio communication system which employs distance of the antennas is reduced towards zero. However, as multiple antennas only at the receiver. soon as the noise resistance is even slightly greater than zero, When the noise-resistance is increased, we observe from the receive array gain rst grows towards a maximum value Figure ¢, that the exponential growth of the receive array gain as the antenna separation is reduced, and then begins to drop. with respect to N, starts to atten out as N is increased over We can see from Figure ¦, that the peak receive array gain a certain limit, which depends on the noise-resistance. In the increases with decreasing noise resistance. For high array gain limit of an innite noise-resistance – which just means that it is therefore important, that the Zs are designed such that there is only voltage noise present at the receiver –, there is current noise dominates over voltage noise. A low noise resis- no longer any exponential growth, but the receive array gain tance may be become more important than a low noise g- merely increases linearly with N, just as if the antennas were ure! Note that for an antenna separation of half of the wave- uncoupled. In order to enjoy high performance it is necessary length, or integer multiples thereof, the receive array gain al- to make sure that we have both a low noise-resistance, and ways equals the number of antennas. This eect comes about, coupled antennas. The former is a matter of high-frequency because the antennas are uncoupled for these separations, as engineering of the Z, while the latter can be achieved by we have pointed out already in Section II-C. using antenna spacings below half the wave-length. As can also be observed from Figure ¢, it is by no means necessary to space the antennas very closely, since a rather moderate F. The Scaling Law spacing of ∆ ý.¥λ, already shows signicant increase of the = The noise-resistance also plays a key role in how the receive receive array gain compared to the »vanilla«, half-wavelength array gain scales with the number of antennas. In Figure ¢, we spaced antenna array. see the receive array gain in dB for the direction of beamform- For nite noise-resistance which is larger than zero, it turns ing xed to θ ý. For a xed antenna spacing of ∆ ý.¥λ, we out that the receive array gain does not increase monotoni- = = have drawn ve dierent curves, each of which corresponds cally with the number of antennas, as can be observed from to a dierent noise-resistance. The best and most remarkable two of the curves in Figure ¢. That is, it can happen that in- performance is achieved as the noise-resistance is reduced to creasing the number of antennas actually leads to a lower re- zero. As we have discussed in Section III-E, this case models ceive array gain. At rst glance, this eect may look surprising, the ideal amplier, which only generates thermal noise inside or even strange. However, it can be understood from the fact the real-part of its input impedance. As we can see from the that adding one antenna to an array, changes the way how the top-most curve in Figure ¢, the receive array gain grows expo- other antennas perceive the channel. To be more specic, the nentially with the number of antennas. Because at high «§ channel vector from (ÕÉ), usually changes in all of its compo- the channel capacity grows proportional to the logarithm of nents, when we add one more antenna to the array, because
Authorized licensed use limited to: T U MUENCHEN. Downloaded on May 17,2010 at 08:57:44 UTC from IEEE Xplore. Restrictions apply. ä the new antenna is located in the near-eld of its neighbors, A££uoì and therefore, usually changes the electro-magnetic eld per- The complex envelope of the electric eld strength in a dis- ceived by the remaining antennas. This is the reason behind tance r and elevation angle θ, in the far eld of a ¶Z of the fact that adding one antenna may not necessarily improve isotropic radiators, located in the origin and aligned with the performance. z-axis, can be written as (see e.g., [ä] on pp ó¢þ and ó¢): It is interesting to note that there is a fundamental dier- −jòπr~λ N e −jòπ ∆ (n−Ô) cos θ ence between receive and transmit array gain, since the trans- E α e λ i , (öÌ) = r Q n mit array gain only may grow quadratic with N, in a line of n=Ô ⋅ sight propagation scenario. Therefore, it usually does matter where in is the complex envelope of the current owing into whether the multiple antennas are located at the transmitting the n-th radiator, λ is the wavelength, and α is a constant. T or the receiving end of the link. Especially, note that when Dening the array steering vector a Ô e−jµ e−j(N−Ô)µ , R ý, we have from (óþ) that Υ Cò, for R R , such that = N = = = ý where µ òπ ∆ λ cos θ, we can rewrite (óÉ) as:⋯ from (ó) and aH a N, follows = = ( ~ ) −jòπr~λ e T A aHC−ò a. E α a i, (îÿ) Rx = = r T Comparison with (Õì) shows the fundamental dierence in the where i i i i . The power density is then given by = Ô ò N two array gains explicitly, because the coupling matrix C ap- (see e.g., [ß], eq.⋯ () on page ¢ßÕ): pears in the second inverse power for the receive array gain, S α′ E E ò but only in its rst inverse power for the transmit array gain. = Only when C I, there is no dierence. S S = α′ α ò E iH a∗ θ aT θ i , (î×) h¶« = rSò S IV. C ( ) ( ) where α′ ý is another constant. The radiated power is then Receive array gain depends on four factors: Õ) the number of > antennas, their radiation characteristics and spacing, ó) the obtained by integrating the power density over a sphere in the noise-resistance of the receive ampliers, ì) the direction of far-eld around the array: beamforming, and ¦) the properties of background radiation. Prad S dA In order to obtain large array gain, the amplier current noise = ∫∫sphere π should signicantly dominate over its voltage noise and the re- ò Ô ¥πα′ α E ⎡iH a∗ θ aT θ sin θ dθ i⎤ . = ⎢ ⎥ ceived background radiation. Therefore, a low noise-resistance ⎢ ò ∫ý ⎥ S S ⎢ ( ) ( ) ( ) ⎥ may be even more important than low noise-gure! Large re- Rý ⎢ ⎥ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ⎣ ⎦ ceive array gains are possible, even increasing exponentially ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹C ¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ (îö) with the number of antennas. In theory, the channel capacity Assuming that the radiators are lossless, the radiated power of wireless « systems can therefore grow linearly with the equals the electric input power from (ó), such that the real- number of antennas – a property which is usually ascribed part of the array impedance matrix can be readily obtained only to multi-input multi-output () systems. from (ìó): Re Z R C, (îî) On the other hand, the transmit array gain depends on two = ý factors: Õ) the number of antennas, their radiation character- where the radiation resistance R ⋅ R Ω, and the coupling ý ∈ + istics and spacing, and ó) the direction of beamforming. In a matrix C RN×N , are dened in (ìó). The result (ìa) then ∈ ⋅ line-of-sight scenario, the transmit array gain can come arbi- follows from (ìó) and (Õó) by standard integration: trarily close to the square of the number of transmit antennas. π Ô j(m−n)òπ ∆ cos θ For this to happen, it is essential that the antenna spacing is C e λ sin θ dθ (î®) m,n = ò ∫ below half the wavelength. Exponential growth of transmit ar- ( ) ý ( ) ∆ ray gain is, however, not possible in a line of sight propagation sinc òπ m n . (î®a) = λ scenario. Consequently, there is, in general, a fundamental dif- ( − ) ference between wireless « and « communication sys- Ruu§uhu« tems, which only vanishes in case that the antennas that form the array are uncoupled. In case of uniform linear arrays of [×] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, óþþ¢. isotropic radiators, this happens when the antenna spacing is [ö] P. Russer, Electromagnetics, Microwave Circuits and Antenna Design for chosen to be an integer multiple of half of the wavelength. Communications Engineering, ónd ed. Nordwood, MA: Artec House, Antenna array based wireless communication systems may óþþä. [î] M. T. Ivrlač and J. A. Nossek, “The Maximum Achievable Array Gain possess a much higher potential than previously reported. The under Physical Transmit Power Constraint,” in Proc. IEEE International key to this increased potential lies in the antenna coupling of Symposium on Information Theory and its Applications, Dec. óþþ, pp. closely spaced antenna arrays, that has to be known and taken Õìì–Õì¦ì. [®] M. J. Buckingham, Noise in Electronic Devices and Systems. New York: into account by transmit and receive signal processing. John Wiley & Sons Ltd., ÕÉì. It should be noted, that even though we have used isotropic [£] H. Yordanov, M. T. Ivrlač, P. Russer, and J. A. Nossek, “Arrays of Isotropic radiators for ease of development, the results also hold quali- Radiators – A Field-theoretic Justication,” in Proc. ITG/IEEE Workshop on Smart Antennas, Berlin, Germany, Feb. óþþÉ. tatively for more realistic antennas, as is demonstrated by the [å] A. Balanis, Antenna Theory. Second Edition, John Wiley & Sons, ÕÉÉß. authors in [¢], for arrays of Hertzian dipoles. [à] D. Kraus, Electromagnetics. Fourth Edition, McGraw-Hill, ÕÉÉó.
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