1 On Array Out-of-Band Emissions Lauri Anttila, Member, IEEE, Alberto Brihuega, Student Member, IEEE, Mikko Valkama, Senior Member, IEEE

Abstract—We substantiate and extend recent research on desired signals.” In this letter, we adopt a more general I/Q the behavior of the out-of-band emissions in antenna array modulated signal model and show that this conclusion is . Specifically, with multi-user precoding, we show largely incomplete. Particularly, we show that some of the that the emissions are always beamformed in the directions of the intended receivers, contrary to some claims in the recent inband and OOB emissions are always beamformed in the literature. Moreover, while also other spurious directions exist, we same directions as the desired signals. Moreover, we also show show that the emissions are always strongest in the directions of that the emissions are always strongest in these directions. the intended receivers. We also show that power amplifiers with Finally, we quantitatively address the impact of the phase mutually different nonlinear phase characteristics will reduce deviations in the nonlinear behavior of the involved PA units. the gain of the unwanted emissions, with the gain approaching the noncoherent combining limit as the phase deviations are increased. II.SIGNAL MODELS Index Terms—antenna array, beamforming, precoding, power amplifier, nonlinearity, distortion, out-of-band emission. A. Basic Two-tone Model To show the limitations of the results in [1], we start with a I.INTRODUCTION basic two-tone test signal written for the mth PA input

OWER amplifiers (PA) in transmitters are always m m m P nonlinear. Such nonlinearities induce both in-band and x (t) = cos(ω1t + φ1 ) + cos(ω2t + φ2 ), (1) out-of-band (OOB) emissions, which may limit the capacity where ω and ω are the tone (angular) frequencies, and φm and quality of service of the intended and neighboring channel 1 2 1 and φm are the phases that control the beamforming direction. users. The transmit signal quality and unwanted emissions are 2 Fig. 1 illustrates how the beamforming phases are related to governed by radio standardization bodies, typically through the propagation delays τ , τ and the carrier frequencies ω , ω error vector magnitude (EVM), adjacent channel leakage ratio 1 2 1 2 between two antennas in a two-user system. As in [1], the PAs (ACLR), and spurious emission limits. The problem is well are modeled with a strictly memoryless 3rd-order polynomial known in traditional single and few-antenna systems, while the behavior of the unwanted emissions in large antenna arrays, f(x) = x + αx3, (2) especially with multiple users, has been studied less thoroughly, though research on this topic is mounting [1]–[4]. which yields the PA output signal of the form [1] In [2], a rigorous analysis based on Hermite-orthogonal  9α polynomials for representing the amplified signal was con- m m y (t) = 1 + cos(ω1t + φ1 ) (3) ducted. By leveraging the spatial cross-correlation matrix of 4  9α m the nonlinear distortion, the behaviour of the OOB emissions + 1 + cos(ω2t + φ2 ) (4) was exposed under different propagation scenarios in single- 4 3α user and multi-user transmission. However, the analysis was + cos(2ω t − ω t + 2φm − φm) (5) 4 2 1 2 1 conducted considering an identical polynomial model across 3α m m the array. In this letter, we consider a more general case with + cos(ω2t − 2ω1t + φ − 2φ ), (6) 4 2 1

arXiv:1908.02982v1 [eess.SP] 8 Aug 2019 mutually different polynomial models for the parallel PAs and expound on the potential implications of such differences. where the intermodulations that are far away from the allocated Additionally, in [4], analysis similar to [2] was conducted band have been omitted for brevity. Referring to the definitions based on the spatial cross-correlation matrix of the distortion. in Fig. 1, it is easy to see that (5) and (6) add up coherently Different from [2], it was claimed that the is only in the respective directions defined by experienced within the desired band, implying that the nonlinear n n m m (2φ2 − φ1 ) − (2φ2 − φ1 ) distortion would not combine coherently. However, this claim τ˜1 = (7) 2ω2 − ω1 has been refuted in many other works [1]–[3], [5], [6], where n n m m (φ2 − 2φ1 ) − (φ2 − 2φ1 ) the nonlinear distortion has been shown to follow essentially τ˜2 = . (8) the same spatial response as the inband signal. ω2 − 2ω1 Recently, in [1], the authors used a simple two-tone signal It was stated in [1], while referring to (3) and (4), that model to argue that the PA-induced OOB emissions of antenna ”the first two terms represent the desired signal, scaled by arrays, in a multi-user scenario, are ”beamformed into distinct a constant,” and were omitted in the subsequent distortion spatial directions, which are different from those of the analysis. While this is a correct interpretation in the most simple The authors are with the Department of Electrical Engineering, Tampere single-user/two-tone signal case, we show in the following that University, Tampere, Finland; e-mail: lauri.anttila@tuni.fi. it is incorrect in the general multi-user/modulated signal case. 2

Considering then true spatial multiplexing, i.e., ω1 = ω2 =  ω, and substituting back ω (t) ← ωt + θ (t), these reduce to 1  n l l 1 m 1 3α 3α   2 3 m 1 A A + A cos(ωt + θ (t) + φ ) (18) user 2 2 1 2 4 1 1 1 n n 3α 2 3α 3 m x() t y() t A1A2 + A2 cos(ωt + θ2(t) + φ2 ) (19) m 2 4 n 2  2  3α 2 m m  2 A A cos(ωt + 2θ (t) − θ (t) + 2φ − φ ) (20) 2 4 1 2 2 1 2 1 3α m m A2A cos(ωt − θ (t) + 2θ (t) − φm + 2φm). (21) x() t y() t 4 1 2 2 1 2 1 user 1 The first two distortion terms add up coherently in the directions of the corresponding intended receivers, while the two latter Fig. 1. Relation between the beamforming phases and the propagation delays terms are beamformed into directions defined by (7) and (8). We between two antennas in a two-user system. note that in general, the spectral contents of the above distortion terms are covering both the inband and OOB frequencies. B. Narrowband I/Q Modulated Signal Model III.BEAMFORMED DISTORTION POWER ANALYSIS Consider now a general two-carrier I/Q modulated signal as We next analyze the powers of the distortion terms (18)- the PA input, defined as (21) from the beamformed channel perspective, utilizing their m m x (t) = A1(t) cos(ω1t + θ1(t) + φ1 ) baseband equivalent forms. For generality, we also consider and m allow for mutually different power amplifiers, with complex- + A2(t) cos(ω2t + θ2(t) + φ2 ). (9) valued coefficients of the form αm =α ¯ · exp(jψm). Here, α¯ The corresponding complex baseband signals are denoted as denotes the baseband equivalent PA coefficient corresponding s1(t) = A1(t)exp(jθ1(t)) and s2(t) = A2(t)exp(jθ2(t)), and to the real-valued PA coefficient α, and ψm are random, zero- we assume that the typical narrowband assumption applies to mean, phase deviations. While the distortion coefficients of them, i.e., phase steering works over the full bandwidth of the true PA’s may also exhibit amplitude differences between the signals [7]. units, we focus specifically on the phase variations, because Now, repeating the derivation for the PA output signal with they are more critical from the beamforming perspective. the I/Q modulated input signal, we obtain A. Observed Baseband Equivalent Distortion Terms  3α 3α  Now, since the beamforming coefficients are chosen such ym(t) = A + A A2 + A3 cos(ω (t) + φm) (10) 1 2 1 2 4 1 1 1 that coherent combining takes place in the directions defined  3α 3α  by τ and τ , the equivalent beamformed distortions observed + A + A2A + A3 cos(ω (t) + φm) (11) 1 2 2 2 1 2 4 2 2 2 by the intended users are given by 3α 2 m m + A1A2 cos(2ω2(t) − ω1(t) + 2φ2 − φ1 ) (12) 4 M 3α 3α  X m 2 m 3 jθ1(t) 3α 2 m m r1(t) = A1(t)A2(t) + A1(t) e + A1A2 cos(ω2(t) − 2ω1(t) + φ2 − 2φ1 ), (13) 2 4 4 m=1 (22) M where again the intermodulations that are far away from the X = z (t) ejψm allocated band have been omitted. Here, in order to have 1 a compact expression, we have used the shorthand notation m=1 M ωl(t) = ωlt + θl(t) and dropped the time argument from the X 3αm 3αm  r (t) = A2(t)A (t) + A3(t) ejθ2(t) amplitude signals. 2 2 1 2 4 2 The above expression now reveals a clearly more complex m=1 (23) M structure for the first two terms, compared to (3) and (4). X = z (t) ejψm , They do not represent only the desired signal, but also contain 2 m=1 intermodulation and crossmodulation distortions from the two 3 2 3 3 jθ1(t) amplitude signals. Thus, the complete list of relevant distortion where z1(t) = ( 2 αA¯ 1(t)A2(t) + 4 αA¯ 1(t))e and z2(t) = ( 3 αA¯ (t)2A (t) + 3 αA¯ 3(t))ejθ2(t). terms near the frequencies ω1 and ω2 is given by 2 1 2 4 2 On the other hand, potential victim receivers located in the directions defined by the delays τ˜ and τ˜ would observe 3α 3α  1 2 A A2 + A3 cos(ω (t) + φm) (14) effective beamformed distortions of the form 2 1 2 4 1 1 1 3α 2 3α 3 m A1A2 + A2 cos(ω2(t) + φ2 ) (15) M 3α  2 4 X m 2 j(2θ2(t)−θ1(t)) v1(t) = A1(t)A2(t) e 3α 2 m m 4 A A cos(2ω (t) − ω (t) + 2φ − φ ) (16) m=1 1 2 2 1 2 1 (24) 4 M 3α X A2A cos(ω (t) − 2ω (t) + φm − 2φm). (17) = u (t) ejψm 4 1 2 2 1 2 1 1 m=1 3

M 3α  sinσ X m 2 j(−θ2(t)+2θ1(t)) where sincσ = . Consequently, the beamformed distortion v2(t) = A1(t)A2(t) e σ 4 towards the intended users, with equal-power signals, has a m=1 (25) M power of X jψ m 2 = u2(t) e , 99α 2 P N = m P 3(M + (M 2 − M)e−σ ) (32) m=1 r 8 s 2 3 2 j(2θ2(t)−θ1(t)) 99α where u1(t) = 4 αA1(t)A2(t)e and u2(t) = U m 3 2 2 M Pr = Ps (M + (M − M).sinc σ) (33) 3 2 j(2θ1(t)−θ2(t)) P jψm 8 4 αA1(t)A2(t)e . The term m=1 e in (22)- (25) can be interpreted as the effective beamforming gain. Similarly, evaluating the powers of the spurious distortion terms in (24) and (25), we arrive at 2 B. Power Analysis 9αm 2 Pv,1 = Ps,1Ps,2 (34) To evaluate the powers of the distortion terms in (22)-(25), 8 9α2 we assume that the baseband waveforms s1(t) and s2(t) are m 2 Pv,2 = P Ps,2. (35) mutually independent complex-circular Gaussian signals with 8 s,1 2 zero mean and variances Ps,l = E[|sl(t)| ], while ψm is With Ps,1 = Ps,2 = Ps, these reduce to assumed to be either Gaussian distributed ψ ∼ N (0, σ2) m 9α2 or uniformly distributed ψ ∼ U(−σ, σ). We note that in this P = P = m P 3. (36) m v,1 v,2 8 s analysis, we address the total powers of the different terms while pursue the differentiation between the inband and OOB Therefore, the power of the beamformed distortions in the distortion powers along the numerical results. spurious directions is 2 The powers of the distortion terms (22)-(23), which are 9α 2 P N = m P 3(M + (M 2 − M)e−σ ) (37) received by the intended receivers, are given by v 8 s 2 ( M ) 9αm 2 X 2 P U = P 3(M + (M 2 − M).sinc σ) (38) jψm v s Pr,l = E zl(t) e 8 m=1 (26) Comparing (32) and (33) with (37) and (38), we see that ( M )  2 X 2 with equal-power user signals, the distortion terms observed in jψm = E zl(t) E e . the directions of the intended receivers are 11 times, or about m=1 10.4 dB’s, stronger than in the spurious directions. First, evaluating the left-hand side expectation operation, which represents the power of the baseband equivalents of (18)-(19), IV. NUMERICAL EXAMPLESAND RESULTS we obtain In this section, we assess and validate the above nonlinear dis- 27α2 9α2 9α2 tortion analysis with numerical simulations. First, we consider P = m P 3 + m P P 2 + m P 2 P , (27) z,l 8 s,l 2 s,1 s,2 2 s,1 s,2 the simple 3rd order polynomial model case where the adopted PA model is given as f(x) = x−0.0368x|x|2, which has been [|s (t)|4] = 2P 2 [|s (t)|6] = 6P 3 where the identities E l s,l and E l s,l obtained by fitting a 3rd order polynomial to a true measured P = [8] have been utilized. With equal-power user signals, s,1 PA response, while constraining the linear coefficient to one. P = P s,2 s, these reduce to Then, as a more practical case, we also consider 9th order 99α2 polynomial PA models with complex coefficients, obtained from P = P = m P 3. (28) z,1 z,2 8 s RF measurements on a set of PAs. These models are of the form f (x) = P5 α x|x|2p−2, while are also clipped The right-hand side expectation in (26) represents the m p=1 m,2p−1 to reflect the true saturation levels of the measured PAs. Among effective beamforming power gain. Considering first that the measured PAs, the maximum phase difference between the ψ ∼ N (0, σ2), this yields m 3rd-order coefficients is 0.55 radians while the corresponding ( M ) X 2 2 standard deviation is 0.1145 radians. The considered waveform N jψm 2 −σ Gb = E e = M + (M − M)e , (29) is an LTE-like 20 MHz OFDM waveform sampled at 120 m=1 MHz, including windowing to improve the spectral containment. which follows from applying ordinary trigonometric identities OFDM signals are known to have quasi-Gaussian distribution and the following integral identity [9] due to the central limit theorem [8], and are thus comparable to the earlier analysis assumptions. We consider a MIMO Z +∞ −ψ2 1 − m −σ2/2 √ cosψme 2σ2 dψm = e . (30) with 60 antennas and PAs serving two users 2 2πσ −∞ in the same frequency band under pure line-of-sight (LOS) On the other hand, if ψm is uniformly distributed in the propagation, and phase-only baseband precoder is utilized. interval ψm ∼ U(−σ, σ), the effective beamforming power Fig. 2 illustrates the corresponding beampatterns of the gain can be shown to read distortion terms in (18)-(21) separately, along with the linear beampattern of the array (in dashed red line), with the main ( M 2) X 2 beams being steered towards −15 and 12 degrees. The relative GU = ejψm = M + (M 2 − M)sinc σ, (31) b E power of the distortion that gets beamformed towards the users’ m=1 4

0 0 30 -30 30 -30

60 -60 60 -60

90 -90 90 -90 -80 -60 -40 -20 0 -80 -60 -40 -20 0

Fig. 2. Full-bandwidth total power based beampatterns of the linear signal Fig. 3. Inband power (black solid) and OOB emission power based (dashed-line) and the four different distortion terms stemming from the 3rd beampatterns for the 3rd order (red solid) and the clipped 9th order (blue order PA model, with 60 antenna units. The r-axis shows relative powers, with dashed) PA models with 60 antenna units. The passband power received by the passband power received by the intended receivers normalized to 0 dB. the intended receivers is normalized to 0 dB.

1 directions is 10.4 dB stronger than that of the other spurious emissions (the distortions beamformed away from the main 0.9 beam directions), which exactly matches the theory. 0.8 In Fig. 3, we focus specifically on visualizing the spatial 0.7 behaviour of the OOB distortion that is radiated from the trans- mitter, while also show the inband power based beampattern 0.6 for reference. The same time-domain signals are used, as in 0.5 the previous example, while the inband and OOB distortion 0.4 powers are extracted through FFT-based processing. As can be seen, the OOB distortion is beamformed partially towards 0.3 the same directions as the passband and partially towards 0.2 the directions defined by (7) and (8), exactly as predicted by 0.1 (14)-(17). This applies to both simple 3rd order and more elaborate clipped 9th order PA models, while in the 9th order 0 0 /5 2 /5 3 /5 4 /5 case there are also some other weaker spurious directions. These confirm the analytical results of Section II. In general, understanding the spatial behaviour of OOB distortions is key Fig. 4. Relative beamforming gain as a function of σ with M = 60 antennas.√ when developing efficient digital predistortion (DPD) solutions The dashed line corresponds to the fully noncoherent combining gain of M. for multiple antenna transmitters [5], [6]. Lastly, Fig. 4 illustrates the relative beamforming gain of REFERENCES the distortion terms, given by the square root of (29) and (31), [1] E. G. Larsson and L. Van der Perre, “Out-of-band radiation from antenna and normalized by the maximum gain M. It can be observed, arrays clarified,” IEEE Commun. Lett., vol. 7, pp. 610–613, Aug. that the phase deviation of the 3rd order PA coefficients can 2018. have a significant impact on the amount of distortion that is [2] T. Eriksson C. Mollen,´ U. Gustavsson and E. G. Larsson, “Spatial characteristics of distortion radiated from antenna arrays with transceiver observed by the different receivers. However, with small phase nonlinearities,” IEEE Trans. Wireless Commun., vol. 17, pp. 6663–6679, deviations, the impact is observed to be small. Oct. 2018. [3] L. Anttila A. Hakkarainen J. Vieira F. Tufvesson Q. Cui Y. Zou, O. Raeesi V. CONCLUSIONS and M. Valkama, “Impact of power amplifier nonlinearities in multi-user massive downlink,” 2015 IEEE Globecom Workshops (GC Wkshps), In this letter, we analyzed the nonlinear distortions stemming Dec. 2015. from multi-user precoded large array transmitters with mutually [4] A. Bourdoux L. Van der Perre S. Blandino, C. Desset and S. Pollin, “Analysis of out-of-band interference from saturated power amplifiers different PAs. It was shown that the distortions are beamformed in massive mimo,” 2017 European Conference on Networks and not only to spurious directions, but also to the main beam Communications (EuCNC), June 2016. directions, contrary to some claims made in recent literature. [5] M. Abdelaziz A. Brihuega, L. Anttila and M. Valkama, “Digital predistortion in large-array digital beamforming transmitters,” 2018 52nd Furthermore, the impact of differences in the PA phase Asilomar Conference on Signals, Systems, and Computers, Oct. 2018. characteristics on the beamformed distortion was analyzed, and [6] A. Brihuega F. Tufvesson M. Abdelaziz, L. Anttila and M. Valkama, they were shown to be potentially important in controlling the “Digital predistortion for hybrid mimo transmitters,” IEEE J. Sel. Topics in Signal Process., vol. 12, no. 3, pp. 445–454, June 2018. beamforming gain of the nonlinear distortions that are radiated [7] R.J. Mailloux, “Phased array theory and technology,” Proc. IEEE, vol. from the transmitter. Proper understanding of the unwanted 70, pp. 246–291, Mar. 1982. emissions in array transmitters is of fundamental importance [8] B. Picinbono, Random Signals and Systems, Englewood Cliffs, NJ, USA: Prentice Hall, 1993. for the development of array-based systems, and for developing [9] I. S. Gradshteyn and I. M Ryzhik, Table of Integrals, Series, and Products, efficient linearization solutions in them. Academic Press, 1980.