This is the accepted version of the following article: T. Laas, J. A. Nossek, and W. Xu, “Limits of transmit and receive array gain in massive MIMO,” in Proc. IEEE Commun. Netw. Conf. (WCNC), May 2020, doi:10.1109/WCNC45663.2020.9120590. Limits of Transmit and Receive Array Gain in Massive MIMO

Tobias Laas∗†, Josef A. Nossek†‡, Wen Xu∗ ∗German Research Center, Huawei Technologies Duesseldorf GmbH, Munich, Germany †Department of Electrical and Computer Engineering, Technical University of Munich, Munich, Germany ‡Department of Teleinformatics Engineering, Federal University of Ceará, Fortaleza, Brazil Emails: {tobias.laas & josef.a.nossek}@tum.de, [email protected]

Abstract—In this paper, we consider the transmit and receive properties [3], as we define array gain as the ratio of SNRs array gain of massive MIMO systems. In particular, instead of powers. Another contribution of this paper is to we look at their dependence on the number of antennas in look at how the array gain changes if both antenna spacing the array, and the antenna spacing for uniform linear and uniform circular arrays. It is known that the transmit array and array size vary and derive design recommendations. The gain saturates at a certain antenna spacing, but the receive array influence of mutual coupling on transmit array gain has already gain had not been considered. With our physically consistent been investigated early [5]. Experimental results were provided analysis based on the Multiport Communication Theory, we in [6], but only for small arrays and without investigating the show that the receive array gain does not saturate, but that difference between transmit and receive array gain. there is a peak at a certain antenna spacing when there is no decoupling network at the receiver. As implementing a decoupling Notation: lowercase bold letters denote vectors, uppercase T ∗ network for massive MIMO would be almost impossible, this is a bold letters matrices. am denotes the mth element of a. A , A reasonable assumption. Furthermore, we analyze how the array and AH , correspond to the transpose, the complex conjugate gain changes depending on the antenna spacing and the size of and the Hermitian. 0 and I denote zero vector and identity the antenna array and derive design recommendations. N (µ, R) Index Terms—Array gain, massive MIMO, uniform linear matrix. C denotes a circularly-symmetric complex array, uniform circular array. Gaussian distribution with mean µ and covariance R. E[X] denotes the expectation of the random variable X. I.INTRODUCTION II.THEORY Massive MIMO is an important building block of future wireless systems, as, depending on the scenario, a larger number Similarly to [2], we consider a multi-antenna of base station antennas is believed to increase the achievable and a single antenna receiver, and do not use the unilateral transmit and receive array gain, i.e., it allows for a larger approximation because the currents in the receive antennas SNR at the same transmit power, a lower bit error ratio by do influence the transmit antennas, so that the near fieldis exploiting diversity, or to serve more mobiles at the same time. important to the analysis. See [3], [4], for more details on Indeed, the seminal paper [1] that introduced massive MIMO the unilateral approximation. In addition, we also consider is based on the assumption that there is an unlimited number the reverse link with a single antenna transmitter and a multi- of base station antennas. However for realistic systems, does antenna receiver. increasing the number of base station antennas always improve Consider the circuit model for a setup with N transmit and performance? M receive antennas, see Fig. 1. The lossless decoupling and In [2], it has been shown for a uniform circular array (UCA) (impedance) matching networks (DMNs) are omitted because at the base station transmitting to a mobile over a line-of- in massive MIMO systems, they would be almost impossible sight (LOS) channel (without reflections) that in general the to implement. minimum energy per bit Eb,min, which is inversely proportional to the transmit array gain, decreases as the number of antennas The transmit amplifiers are modeled as linear amplifiers. Let ZG be their internal resistance and uG their open load voltage. arXiv:2002.06107v2 [cs.IT] 26 Jun 2020 at the base station increases, but at a certain number of base Power matching is employed at the transmitter, i.e., station antennas, Eb,min saturates. The analysis is based on the Multiport Communication Theory [3], [4], which is in Z = Z∗ ,R := Re(Z ),R = Re(Z ), turn based on circuit theory and ensures that the analysis is G A r A G G (1) physically consistent. where Z is the self-impedance of the antennas and R their One contribution of this paper is to extend the analysis to A r . Let Z be the input impedance of the low the receive array gain. We also want to extend the analysis to L noise amplifier (LNA) in each RF chain, u the load voltage antenna arrays, where the antenna separation is fixed rather L and than the array size. Note that transmit and receive array gain are different, unless the noise at the receiver fulfills certain RL = Re(ZL). (2)

© 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. uA,1 uN,1 i1,1 i2,1

ZG γRr iN,1 γRr uG,1 u1,1 u2,1 ZL uL,1 Zin Zout

uA,M uN,M

Z i2,M i1,N iN,M γRr ZG γRr u2,M ZL uL,M uG,N u1,N

Transmitter Receiver

Fig. 1. Circuit model (modified from [7]).

The impedance matrix Z can be partitioned into four where kB is the Boltzmann constant, ∆f is the noise bandwidth, blocks [3] TA is the noise temperature of the antennas and σu, σi, ρ [ ] describe the zero-mean circularly-symmetric complex Gaussian Z11 Z12 (N+M)×(N+M) Z = ∈ C , (3) distribution of uN and iN in the equivalent two-port model at Z21 Z22 the receiver, similarly to [7]. N×N the transmit and receive impedance matrices Z11 ∈ C , The corresponding information-theoretic model ([2], [7] M×M and Z22 ∈ C and the mutual impedance matrices Z21 ∈ combined) is M×N N×M T C and Z12 ∈ C , where Z = Z due to reciprocity. √ y = Hx + ϑ, ϑ ∼ N (0 W, σ2 I),P = E[∥x∥2], Let √ C ϑ T 2 = + γR , = + γR , RG −1/2 −H/2 Z11,r Z11 rI Z22,r Z22 rI (4) H = σϑ √ Rη DB , (13) RL where the dissipation resistance γRr, which is connected in series, is used to model the losses in the antennas, see Fig. 1. where y is the received signal, x is the transmitted signal, H The impedance matrices seen at the input and the output are is the information-theoretic channel and ϑ is additive white −1 Gaussian noise. = − (Z + ) , 1/2 1/2 Zin Z11,r Z12 LI Z22,r Z21 (5) By choosing specific B and Rη to transform between −1 Zout =Z22,r − Z21(ZGI + Z11,r) Z12. (6) the physical model (7) and the model (13) analogously to [7], 1/2 √ −H 1/2 The physical model ([2], [7] combined) is B = RG(Zin + ZGI) Re(Zin) √ (14) √ s.t Re( ) = Re( )1/2 Re( )1/2, uL = DuG + RLη, η ∼ NC(0 W, Rη), Zin Zin Zin H (7) ZL E[uG BuG] 1/2 √ −1 1/2 PT = , Rη = (Zout + ZLI) Q , (15) RG RL 2 |ZL| −1 −H we can write Rη = (Zout + ZLI) Q(Zout + ZLI) (8) RL −1/2 −1/2 −H −1 H = σϑQ Z21,eff Re(Zin) , (16) B = RG(Zin + ZGI) Re(Zin)(Zin + ZGI) , (9) Z21,eff =Z21 (17) = Z ( + Z )−1 ( + Z )−1, D L Z22 LI Z21 Zin GI (10) −1 −1 − Z21(ZGI + Z11,r) Z12(ZLI + Z22,r) Z21, where D describes the noiseless relation between uG and uL, η describes the noise, B is the power-coupling matrix, PT where Z21,eff is the effective mutual impedance matrix between is the transmit power and Q is a noise covariance matrix. Q transmitter and receiver. comes from intrinsic noise sources uN and iN and the antenna A. Receive Array Gain noise uA, and is defined as [7] 2 2 H ∗ For the receive array gain, we consider an uplink scenario Q = σ I + σ ZoutZ − 2σuσi Re(ρ Zout) + RA, (11) u i out with a mobile with one antenna transmitting to a base station RA = 4kBTA∆f Re(Zout), (12) with NBS antennas, i.e., N = 1,M = NBS. This implies that Considering the transmit array gain is equivalent to consid- r ering the minimum transmitted energy per bit Eb,min as in [2]. mobile This can be shown as follows: by adding the losses in the d bm antennas to the model in [2] and for the more general noise distribution assumed in this paper, (a) UCA with fixed radius. 2 σq ln 2 E = . (25) b,min H −1 ∆fz21,eff Re(Zin) z21,eff array size mobile d This means bm −1 ATx ∝ Eb,min. (26)

(b) ULA transmitting into frontfire direction. Note that transmitting with Eb,min leads to the well-known 2 minimum received energy per bit σϑ ln(2)/(∆f), see [8]. Fig. 2. Scenarios. C. Channel In the following we assume that all antennas at the base : : H becomes a vector h = H and similarly z21,eff = Z21,eff station and the mobile are parallel infinitely thin but lossless and Zin := Zin. The receive array gain is defined as [3] λ/2-dipoles in series with the dissipation resistance γRr. Then max SNR ⏐ for a line of sight channel, the entries of Z can be computed A := ⏐ , Rx ⏐ (18) according to the analytical formulas using the sinusoidal current SNR|M=1,γ=0 ⏐P =const. T approximation [2], [9] because they are canonical minimum where scattering antennas [10], [11]. 2 H −1 ∥h∥ z21,eff Q z21,eff For a receiver located in the far field in direction (θ, φ), max SNR = 2 P = P 2 T T (19) θ φ σϑ Re(Zin) where is the zenith angle and the azimuth angle, is obtained by use of a matched filter at the receiver and the z21 = Rra(θ, φ), (27) SNR for M = 1 lossless receive antennas is obtained in a similar way. Then where a is the steering vector, i.e., H −1 2 ⎡ ⎤ Re(Zin,0) z21,eff Q z21,eff σq,0 cos(φ) sin(θ) ARx = , (20) j 2π rT r 2 a (θ, φ) = e λ i , r = sin(φ) sin(θ) Re(Zin) |z21,eff,0| i ⎣ ⎦ (28) cos(θ) where 2 and ri is the position vector of the i-th antenna. We choose Zin,0 := Zin|M=1,γ=0, σq,0 := Q|M=1,γ=0, (21) the coordinate system such that the origin coincides with the : z21,eff,0 = z21,eff |M=1,γ=0. center of the array. The λ/2-dipoles are oriented parallel to B. Transmit Array Gain θ = 0, the ULAs are oriented such that they lie in φ = π/2 For the transmit array gain, we consider a downlink scenario and the UCAs are oriented such that one antenna lies in φ = 0. Extending the consideration to the far field, with a base station with NBS antennas transmitting to a mobile H with one antenna, i.e., N = NBS,M = 1. Here, h := H Zin,0 → ZA, z21,eff → z21, Zin → Z11,r, H and z21,eff := Z21,eff to make them column vectors and Q 2 2 (29) 2 σq → σq,0, z21,eff,0 → z21,0, Zout → Z22,r, becomes a scalar σq . The transmit array gain is defined as [3] ⏐ where z21 and z21,0 are defined analogously to z21,eff and max SNR ⏐ ATx := ⏐ , (22) z21,eff,0. That means, ATx in the far field is [3] SNR|N=1,γ=0 ⏐P =const. T H −1 z21 Re(Z11,r) z21 where ATx = Rr 2 . (30) 2 H −1 |z21,0| ∥h∥ z21,eff Re(Zin) z21,eff max SNR = 2 P = P (23) σ2 T σ2 T Different arrays vary in Re(Z11,r) and in a(θ, φ). The former ϑ q is Toeplitz for ULAs and circulant for UCAs. is obtained by using a matched filter at the transmitter and the SNR for N = 1 lossless transmit antennas is obtained in a III.NUMERICAL RESULTS similar way. Then, Consider the distance dbm between the base station and the H −1 2 mobile, see Fig. 2. In the following section, Re(Zin,0) z21,eff Re(Zin) z21,eff σq,0 ATx = 2 2 , (24) i/2 σq |z21,eff,0| dbm ∈ {10 λ | i = 4,..., 8}. (31) 2 where Zin,0, σq,0 and z21,eff,0 are defined as in (21), but for Table I shows the value of these distances for the following N = 1, γ = 0. frequency bands: TABLE I OVERVIEW OF THE DISTANCES BETWEEN THE BASE STATION AND THE MOBILE AT THE DIFFERENT FREQUENCIES.

dbm @ fc = 680.5 MHz (n71 uplink) @ fc = 3.55 GHz (n78) @ fc = 27.925 GHz (n261) 102λ 44.1 m 8.44 m 1.07 m 102.5λ 139 m 26.7 m 3.39 m 103λ 441 m 84.4 m 10.7 m 103.5λ 1.39 km 267 m 33.9 m 104λ 4.41 km 844 m 107 m

TABLE II 104 OVERVIEW OF 3GPP 38.901 CHANNELMODELPARAMETERS. A N Tx ∝ BS 2 dbm = 10 λ RMa UMa UMi 2.5 103 dbm = 10 λ Base station altitude 35 m 25 m 10 m 3 dbm = 10 λ Mobile altitude 1.5 m between 1.5 m d = 103.5λ and 22.5 m bm

Rx 4 102 dbm = 10 λ Minimum horizontal distance 35 m 10 m A Inter site distance 5000 m 500 m 200 m Minimum distance 48.4 m 35.1 m 10.0 m ⇒ 1 Maximum distance 2.89 km 290 m 116 m 10 ⇒

100 • The uplink in LTE (5G NR) band 71 (n71), which spans 100 101 102 103 104 663 MHz to 698 MHz [12], [13], i.e., fc = 680.5 MHz, NBS which is one of the lowest frequency bands currently

supported for mobile broadband. Fig. 3. ARx for a ULA in frontfire with fixed antenna separation d = λ/2. • Band n78, from 3.3 GHz to 3.8 GHz [13], i.e., fc = 3.55 GHz. 104 • 5G NR mmWave band n261, which spans 27.5 GHz to ATx NBS ∝ 2 28.35 GHz [14], i.e. fc = 27.925 GHz. dbm = 10 λ d = 102.5λ All three bands are among the first for 5G NR deployment. 103 bm 3 Let us compare these distances to the ones in the 3GPP dbm = 10 λ 3.5 38.901 channel model [15]. For band n71, consider the rural dbm = 10 λ

Tx 4 102 dbm = 10 λ macro (RMa), for band n78 the urban macro (UMa) and A for band n261 the urban micro (UMi) scenario. The base station altitude, mobile height, minimum 2D distance and inter site distance are as shown in Table II. According to these 101 parameters, a dbm between 100λ and 10000λ almost entirely covers the various scenarios. In the following, the various 100 frequency bands will be analyzed jointly with all distances 100 101 102 103 104 normalized to λ. NBS In this section, we consider ARx and ATx according to (20) and (24). We use the measured noise parameters from [16, Fig. 4. ATx for a ULA in frontfire with fixed antenna separation d = λ/2. Tables IV & VI], i.e., in particular ZL = (186 − 31.6i) Ω, and assume that the loss factor γ = 10−3. However, when NBS increases from 1, the array gain becomes A. Fixed Distance slightly larger than NBS. Furthermore for a larger NBS, the Consider a ULA with a fixed antenna separation d = λ/2 array gain starts to saturate, with saturation occurring at a N d d = 100λ and between NBS = 1 and NBS = 10000 base station antennas lower BS the smaller the bm. For bm , the onset N = 100 300 transmitting into the frontfire direction, see Fig. 2b. ARx and of saturation starts at values as low as BS to , 49.5λ 149.5λ ATx for this scenario are shown in Figs. 3 and 4. They are corresponding to an array size between and , 3.5 almost identical here. According to conventionally modeled whereas for dbm = 10 λ only the very start of the saturation 4 4 systems, which neglect mutual coupling, we expect near NBS = 10 can be seen and for dbm = 10 , the array gain only saturates for an even larger value of NBS. When ARx = NBS,ATx = NBS. (32) we compare the array size at which saturation starts to dbm, 104 104 A N A N Tx ∝ BS Tx ∝ BS 2 2 dbm = 10 λ dbm = 10 λ 3 2.5 3 2.5 10 dbm = 10 λ 10 dbm = 10 λ 3 3 dbm = 10 λ dbm = 10 λ 3.5 3.5 dbm = 10 λ dbm = 10 λ

Rx 2 4 Rx 2 4 10 dbm = 10 λ 10 dbm = 10 λ A A

101 101

d = λ d = λ/2 d = λ d = λ/2 100 100 100 101 102 103 104 100 101 102 103 104

NBS NBS

Fig. 5. ARx for a ULA in frontfire with a fixed array size of 40λ. Fig. 7. ARx for a UCA with fixed radius r = 20λ.

104 104 A N ATx NBS Tx ∝ BS ∝ 2 2 dbm = 10 λ dbm = 10 λ 3 2.5 3 2.5 10 dbm = 10 λ 10 dbm = 10 λ 3 3 dbm = 10 λ dbm = 10 λ 3.5 3.5 dbm = 10 λ dbm = 10 λ

Tx 2 4 Tx 2 4 10 dbm = 10 λ 10 dbm = 10 λ A A

101 101

d = λ d = λ/2 d = λ d = λ/2 100 100 100 101 102 103 104 100 101 102 103 104

NBS NBS

Fig. 6. ATx for a ULA in frontfire with a fixed array size of 40λ. Fig. 8. ATx for a UCA with fixed radius r = 20λ.

104 we can see that the array size there is on the same order of A N Tx ∝ BS d 2 magnitude as bm. This indicates that saturation occurs as the dbm = 10 λ additional antennas’ path-loss increases so they contribute less. 3 2.5 10 dbm = 10 λ For typical cellular systems, whose array size is significantly 3 dbm = 10 λ smaller than the distance between the base station and the 3.5 dbm = 10 λ

mobile, this is not a practical limitation. Rx 2 4 10 dbm = 10 λ A B. Fixed Radius 1 Consider a ULA with array size 40λ and a UCA with 10 radius r = 20λ, see Fig. 2. As NBS increases, the antenna separation d decreases. ARx and ATx are shown in Figs. 5 d = λ d = λ/2 100 to 8 for different distances between the base station and the 100 101 102 103 104 mobile. The curves for the ULA show that even for values NBS slightly greater than NBS = 10 antennas, ARx and ATx deviate from the linear increase expected from conventionally modeled Fig. 9. ARx for a UCA with fixed radius r = 20λ, antenna noise only. systems, see (32). For the UCA, for d > 1.26λ (corresponding to NBS ≤ 100), (32) holds approximately. Technically there is a small gap to NBS in the array gain when the receiver is However for smaller distances between the antennas, the close to the base station, see the curve for dbm = 100λ. trend for ARx and ATx is different. Considering the former, for the ULA there is a maximum at about 71 antennas (d ≈ 0.571λ) differently to the ULA. For transmission into the plane of the and for the UCA at about 275 antennas (d ≈ 0.457λ), and if UCA, ATx is independent of φ for an odd NBS, while for the number of base station antennas is increased further, ARx an even NBS it only oscillates slightly with φ. Therefore, for decreases. Considering ATx, it (almost) saturates at about 68 maximum ATx, NBS should be chosen such that ATx is at the antennas for the ULA and 275 antennas for the UCA. If the largest peak, i.e., the peak near d = λ/2, unless γ is too large; number of base station antennas is increased further, ATx only then the peak of ATx near d = λ is the largest. increases slightly. Notably, the number of antennas for which According to the expectation from conventionally modeled the maximum ARx is obtained, and for which ATx starts to systems, (32) should hold, but for the ULA there are signif- saturate, is (almost) independent of the distance to the mobile. icantly larger maximum transmit array gains for front- and There is a saturation, as the achievable array gain for a certain especially endfire (ATx ≈ 219 and 533 for NBS = 130), but array size is bounded. The sharp increase in array gain for also smaller maximum transmit array gains as seen for φ = 60◦ the ULA at about d = λ can be explained by an increasing (ATx ≈ 99 for NBS = 130). For the UCA, there is not such number of antennas, and a decreasing antenna separation at a large direction-dependent variation, but the maximum is the same time, compared to [3]. ATx ≈ 152 for NBS = 130. In Fig. 9, the receive array gain for the UCA is shown for These figures could be used for further evaluations, e.g., the case when there is only the noise of the antennas, i.e., whether using a single UCA for a base station deployment σu = 0 V, σi = 0 A. In this case, the curves for ARx are rather than non-cooperating ULAs in the typical 3 sectors leads (almost) the same as the curves for ATx in Fig. 8, i.e., the to higher performance, with the same number of antennas in coupling of the noise of the LNA and other analog components both scenarios. between the receive chain causes the decrease in ARx. V. CONCLUSIONS IV. NUMERICAL RESULTS FOR THE ARRAY GAININTHE Firstly, we have shown for two different massive MIMO FAR FIELD scenarios, one with a fixed distance between the antennas In this section, we consider the transmit array gain in the far and one with a fixed size of the array, the transmit array field for θ = π/2, see (30), for ULAs and UCAs with different gain saturates above a certain number of antennas – contrary sizes and numbers of antennas NBS. Firstly, consider the ULA to the expectation of a linear increase with the number of in three scenarios: into the endfire (φ = 90◦), antennas when mutual coupling is neglected. Similarly, the into the φ = 60◦ and into the frontfire direction (φ = 0◦), see receive array gain saturates when there is only thermal noise Figs. 10 to 12. The cut for frontfire direction with an array from the antennas at the receiver. However, if there is noise size of 40λ corresponds to Fig. 6 with dbm → ∞λ. We can from the LNA and other components in the receive chains see that ATx depends on which directions the beamforming and no decoupling network, as it would be almost impossible vector points to. Consider a fixed antenna array size. Then for to implement for massive MIMO, the coupling of this noise the frontfire direction, saturation starts slightly below d = λ, between the receive chains leads to a maximum of the receive but for the 60◦ direction, saturation only starts slightly below gain for a certain number of antennas, and a decreasing receive d = 0.54λ and for the endfire direction saturation starts slightly gain for a larger number of antennas. Therefore it is essential to below d = λ/2. Further, we can observe that the larger the take mutual coupling into account in the design of large scale NBS, the closer the saturation starts to d = λ, d ≈ 0.54λ and massive MIMO systems because otherwise, large array gains, d = λ/2 respectively. To maximize ATx the array should be which are unphysical, may be predicted. Secondly, we have positioned so that its endfire direction points to the angle of derived practical design recommendations for ULAs and UCAs: interest. If this is not possible, to optimize ATx for angles ULAs should be oriented such that their endfire direction points −φ0 ≤ φ ≤ φ0, where φ0 is the maximum angle of interest in the direction of interest, and for both ULAs and UCAs, the that the array is transmitting to, NBS should be chosen such optimum antenna spacing is slightly below λ/2. Future work that d is at a peak for φ = φ0. The maximum peak is slightly includes the analysis of the array gain in a rich scattering below d = λ/2 to d = λ depending on φ0 – unless the loss environment. factor γ is too large. On the one hand, if NBS is chosen to ACKNOWLEDGMENT be slightly smaller, ATx falls into a valley for |φ| close to φ0. On the other hand, NBS is chosen to be slightly larger, either The authors would like to thank their colleague S. Bazzi for ATx only increases slightly for d < λ/2 and ATx becomes useful discussions and critical review of the paper. more sensitive to tolerances in the position of the antennas in REFERENCES the array, or ATx decreases again (d > λ/2). Secondly, consider a UCA with beamforming into a direction [1] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless Commun., vol. 9, no. 11, that lies on the line between an antenna and the center of the pp. 3590–3600, Nov. 2010. array (w.l.o.g. φ = 0), see Fig. 13. Here, similarly the cut for [2] M. T. Ivrlacˇ and J. A. Nossek, “On physical limits of massive MISO systems,” in Proc. 20th Int. ITG Workshop Smart Antennas (WSA), an array size of 40λ corresponds to Fig. 8 with dbm → ∞λ. Munich, Germany, Mar. 2016. 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600 0.4λ

500

400 Tx

A 300 1λ

200 1.5λ 2λ 100 2.5λ

0 80 70 130 60 120 50 100 40 80 30 60 20 40 10 20 1 2 array size in λ NBS

◦ Fig. 10. ATx for a ULA transmitting into endfire directionφ ( = 90 ), where the lines for d = 0.4λ to 0.46λ are in 0.01λ increments.

0.5λ

1λ 100 0.4λ

80 1.5λ

Tx 60 2λ A 2.5λ 40

20

0 80 70 130 60 120 50 100 40 80 30 60 20 40 10 20 1 2 array size in λ NBS

◦ Fig. 11. ATx for a ULA transmitting into φ = 60 direction, where the lines for d = 0.4λ to 0.5λ are in 0.01λ increments.

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250 0.5λ

200 1λ 0.4λ

150 Tx A 100 1.5λ 2λ 2.5λ 50

0 80 70 130 60 120 50 100 40 80 30 60 20 40 10 20 1 2 array size in λ NBS

◦ Fig. 12. ATx for a ULA transmitting into frontfire directionφ ( = 0 ), where the lines for d = 0.4λ to 0.5λ are in 0.01λ increments.

1λ

300 0.44λ 0.4λ 1.5λ 250

200 2λ 2.5λ Tx 150 A 3λ 3.5λ 100

50

0 80 70 240 60 220 200 50 180 160 40 140 30 120 80 100 20 60 10 40 array size in λ 1 2 20 NBS

Fig. 13. ATx for a UCA, where the lines for d = 0.4λ to 0.44λ are in 0.02λ increments.

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