MASSIVE multiple-input multiple-output (MIMO) has

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1 IEEE SIGNAL PROCESSING LETTERS, VOL. 4, NO. 1, DECEMBER Spectral Efficiency uner Energy Constraint for Mixe-ADC MRC Massive MIMO Hessam Pirzaeh, Stuent Member, IEEE, an A. Lee Swinlehurst, Fellow, IEEE Abstract We consier uplink transmission for a massive multiuser multiple-input multiple-output MU-MIMO system in which the base station BS is equippe with a mixe analog-toigital converter ADC architecture. In this architecture, a portion of the antennas at the BS is connecte to low-power one-bit ADCs while the other is fe to power-hungry high-resolution ADCs. By taking into account the mixe-adc architecture, we erive a close-form expression for the sum spectral efficiency SE when maximum ratio combining MRC etection is employe at the BS. Then, we formulate an optimization problem to etermine, for a given power buget at the BS, what is the optimal istribution of one-bit an high-resolution ADCs that maximizes the sum SE. Interestingly, it is shown that in most realistic scenarios, using only one-bit ADCs provies the best spectral efficiency for a given power buget constraint. Inex Terms Massive multiple-input multiple-output MIMO, mixe-analog-to-igital converter ADC, one-bit ADC, spectral efficiency SE. I. INTRODUCTION MASSIVE multiple-input multiple-output MIMO has emerge as a promising technology for fifth- generation wireless networks 1. While there are numerous avantages in exploiting a large number of antennas at the base station BS, the corresponing harware cost an power consumption make realizing massive MIMO systems a challenging task. Since power-hungry high-resolution analog-to-igital converters ADCs are one of the primary reasons for the high harware cost an power consumption at the BS, substituting highresolution ADCs with low-resolution ones coul be a viable solution to combat this problem. The uplink achievable rate for massive MIMO systems with low-resolution ADCs is analyze in using the so-calle aitive quantization noise moel Manuscript receive August 15, 017; accepte October 4, 017. Date of publication October 9, 017; ate of current version October 7, 017. This work was supporte by the National Science Founation uner Grant ECCS , an by a Hans Fischer Senior Fellowship from the Technische Universität München Institute for Avance Stuy. The associate eitor coorinating the review of this manuscript an approving it for publication was Dr. Feifei Gao. Corresponing author: A. Lee Swinlehurst. H. Pirzaeh is with the Center for Pervasive Communications an Computing, University of California, Irvine, CA 9697 USA hpirzae@uci.eu. A. L. Swinlehurst is with the with the Center for Pervasive Communications an Computing, University of California, Irvine, CA 9697 USA, an also with the Hans Fischer Senior Fellow of the Institute for Avance Stuy, Technical University of Munich, München 80333, Germany swinle@uci.eu. Color versions of one or more of the figures in this letter are available online at Digital Object Ientifier /LSP AQNM an the effectiveness of eploying a large number of antennas to compensate for the loss ue to low-resolution ADCs is emphasize. Recently, the impact of using purely onebit ADCs on the spectral efficiency SE of massive MIMO systems has been stuie in 3 6. It is shown that the high spatial multiplexing gain owing to the use of a large number of antennas is still achievable with one-bit ADCs. However, at least..5 times as many antennas with one-bit ADCs are require to attain the same performance as in the high-resolution ADC case. To provie a compromise between SE loss an power consumption, so-calle mixe-adc architectures have been propose In a mixe-adc BS, a portion of the antenna Raio Frequency RF chains are connecte to high-resolution ADCs, while the remainers are connecte to one-bit ADCs. Prior work has shown that mixe-adc architectures can provie significant gains in SE with consierably reuce power consumption. In 7 an 9, the spectral an energy efficiency trae-off of the mixe-adc massive MIMO uplink is investigate, for both cases involving perfect an estimate channel state information CSI. Both 7 an 9 assume a roun-robin approach to obtain the estimate CSI, in which the high-resolution ADCs are connecte sequentially to each antenna over several training intervals to estimate the channel. This significantly increases the training overhea an reuces the ultimate SE, but allows one to etermine the strongest channels to select for the highresolution ADCs in the ata transmission phase. In this letter, we take a ifferent approach an assume that the high-resolution ADCs are connecte to an arbitrary subset of the antennas to reuce the training overhea, an we maximize the SE subject to a fixe power buget. We show that for a maximum ratio combiner, the optimal architecture uses either all one-bit or all full-resolution ADCs, epening on the relative power consume by each type of RF chain. We further show that for typical power consumption parameters, an all-one-bit system is likely the optimal approach in practice. Notation: We use bolface letters to enote vectors an capitals to enote matrices. The symbols.,. T,. H, an. represent conjugate, transpose, conjugate transpose, an optimal value, respectively. A circularly-symmetric complex Gaussian ranom vector with zero mean an covariance matrix R is enote v CN0, R. The symbol. represents the Eucliean norm. The operator vec. vectorizes a matrix an vec 1. performs the reverse operation. The K K ientity matrix is enote by I K, the Kronecker prouct by, an the expectation operator by E{.} IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See stanars/publications/rights/inex.html for more information.

2 1848 IEEE SIGNAL PROCESSING LETTERS, VOL. 4, NO. 1, DECEMBER 017 II. SYSTEM MODEL We moel the uplink of a single-cell multiuser MIMO system consisting of K single-antenna users an a BS equippe with M antennas. The M 1 signal receive at the BS from the K users is given by y = phx n 1 where p represents the transmit power, H C M K is the channel matrix whose elements are istribute inepenently an ientically as CN 0, 1, the symbol vector is x CN0, I K, an the aitive noise is n CN0,σn I M. We consier a mixe-adc architecture at the BS in which M 0 = μm antennas are connecte to high-resolution ADCs while M 1 =1 μ M antennas are fe to one-bit ADCs. Since M 0 an M 1 are integers, μ is restricte to certain rational values, but for the moment we ignore this constraint. As a result, by partitioning the channel matrix H, we can rewrite 1 as y0 = p x n y 1 where H 0 C M 0 K C M 1 K contains the channel coefficients from the users to the M 0 M 1 antennas connecte to high-resolution one-bit ADCs. Therefore, the receive signal at the BS after one-bit quantization is r0 y0 r = = 3 r 1 Q y 1 where the element-wise one-bit quantization operation Q replaces each input entry with the quantize value 1 ±1 ± j, epening on the sign of the real an imaginary parts. We can represent the nonlinear quantization using a statistically equivalent linear moel base on the Bussgang ecomposition as see 4 for etails Qy 1 =Ay 1 q 4 where A is a linear gain an q enotes the statistically equivalent quantization noise which is uncorrelate with y 1. As explaine in 4, the Bussgang approach provies a more accurate analysis than the AQNM in the case of one-bit quantization. We assume a block-faing moel where the channel remains constant in a coherence interval of length T symbols an changes inepenently between ifferent intervals. At the beginning of each coherence interval, the users sen their η-tuple mutually orthogonal pilot sequences K η T tothebsfor channel estimation. The remaining T η symbols are eicate to uplink ata transmission. A. Training Phase The pilot sequences are stacke into an η K matrix Φ, where the kth column of Φ, φ k,isthekth user s pilot sequence an Φ H Φ = I K. Therefore, the M η receive signal at the BS in the training phase is Yt0 Y t = = ηp Φ T N0 5 Y t1 where N 0 N 1 is an M 0 η M 1 η matrix with i.i.. CN0,σn elements. Although it has been shown that η>k N 1 can improve performance in one-bit massive MIMO systems 4, we assume η = K in the sequel for analytical simplification. To apply the Bussgang ecomposition as in 4, we first vectorize 5 vec Yt0 vec Y t1 = Φ0 vec H 0 Φ 1 vec vec N0 vec N 1 where Φ i = Φ I M i, i {0, 1}. After quantization, from 4 we have ỹt0 = Φ0 vec H 0 vec N 0 ỹ t1 A t Φ1 vec A t vec N 1 q t 7 with A t = 1 π σ I n M 1 K 4. Accoringly, the linear minimum mean square error estimate of H for a mixe-adc architecture is 4, 5 Ĥ = = This estimation leas to σ 0 = π vec 1 ΦH 0 ỹt 0 σn σn vec 1 ΦH 1 1 σ n pk, σ 1 = π 1 1 σ n pk 1 ỹt where σ 0 an σ 1 are the variances of the inepenent zero mean elements of an Ĥ1, respectively. B. Data Transmission Phase In the ata transmission phase, using 3 an 4 the receive signal after quantization is 4 r0 = n 0 p x 10 r 1 A A n 1 q where A = 1 π σ I n M 1 = α I M 1 an q is the statistically equivalent quantization noise with covariance matrix C q. For ata etection, the BS employs the MRC etector W C M K assuming that the channel estimate is the true channel. Note that the quantization moel consiere in 4 oes not preserve the power of the input of the quantizer since the output is force to be ±1. Thus, we implement MRC as follows to offset this effect W = A Therefore, the resulting signal at the BS is Ĥ 0 ˆx = W H r0 = p r 1 H n 0 n 1 A 1 q H x 9. 1

3 PIRZADEH AND SWINDLEHURST: SPECTRAL EFFICIENCY UNDER ENERGY CONSTRAINT FOR MIXED-ADC MRC MASSIVE MIMO 1849 Note that premultiplication by A 1 only makes the quantization a power-preserving operation an oes not alter the information of the quantizer output. Therefore, the kth element of ˆx is ˆx k = pĥh k h k x k K i=1,i k p ĥ H k h i x i ĥh k n ĥ k A 1 q 13 where ĥk, h k, an ĥ1 k are the kth column of Ĥ, H, an Ĥ1, respectively. The BS treats ĥh k h k as the esire channel an the other terms of 13 as worst-case Gaussian noise when ecoing the signal. Consequently, a lower boun for the ergoic achievable SE at the kth user is 11 S k = R SQINR k 14 where R θ 1 η/t log 1 θ an SQINR k is the effective signal-to-quantization-interference-an-noise ratio at the kth user given by 15 shown at the bottom of this page. The following theorem presents an expression for the sum SE of a mixe-adc architecture with MRC etection. Theorem 1: The SE of the kth user in a mixe-adc massive MIMO system with MRC etection is M S k = R K σ n p μσ 0 1 μ σ 1 μσ 0 1 μ π. 16 Proof: The proof follows by calculating the expecte values in 15 an using 9, following the reasoning in 4. In the next section, we fin the optimal values of M an μ by consiering the power consumption at the BS to maximize the sum SE in 16. We will enote the fixe power consume at the BS ue to the local oscillator, baseban processors, etc, by P FIX an the power consume by a single RF chain with high-resolution an one-bit ADCs by P 0 an P 1, respectively. Hence, the total consume power at the BS is P tot = P FIX M 0 P 0 M 1 P 1 = P FIX M μp 0 1 μ P Since P 0 P 1, a small value for μ means a larger number of antennas an hence an increase in the MIMO spatial multiplexing gain, but a loss of SE ue to increase quantization noise. Increasing μ reuces the SE loss ue to quantization, but also ecreases the size of the antenna array. The problem of fining the optimal value of μ that maximizes the SE for a given total power constraint P is aresse next. III. OPTIMAL MIXED-ADC DESIGN In this section, we etermine what fraction of highresolution/one-bit ADCs shoul be installe at the BS to maximize the sum SE, S = K k=1 S k, subject to a given power buget P. The optimization problem is expresse as maximize μ S P : subject to P tot P 18 0 μ 1. The following theorem provies the optimal solution for μ, assuming μ can take on any value in the interval 0, 1. Once μ is obtaine, M 0 an M 1 are taken to be the integers closest to μ M an 1 μ M that result in P tot P. Theorem : Assume P 0 = ρp 1 an enote ρ th = π 4. Then, the maximum SE is achieve by { μ 0 ρ ρth, = 19 1 ρ<ρ th. Proof: Denote the argument of R in 16 as γμ. Since Rγ is a continuous an strictly increasing function of γ, the first inequality constraint becomes equality, an the optimization problem is equivalent to { maximize γ μ P : μ 0 subject to 0 μ 1. The proof is carrie out by analyzing the behavior of the first erivative of μσ γ μ = 0 1 μ σ1 μp 0 1 μ P 1 μσ0 1 μ π 1 in 0 μ 1 for ifferent values of ρ>1. The first erivative of γ can be written as γ = f μ =β a b μ a μ where β is a positive constant in 0 μ 1, a = π ρσ4 1 1 πσ0 σ1, an b = π ρσ 0 σ1 σ For 1 <ρ ρ 0 = π 1σ 0 σ 0 4,wehavea<0, b> π σ 0 π 0, an a b<0. Thus, f is a line with negative slope an a root at μ>1, which amounts to γ μ > 0 an hence μ =1. In the interval of ρ 0 <ρ ρ 1 = /π σ 0,wehave a 0, b>0, an a b>0. As a result, f has a positive slope an a root at μ<0. Then, γ μ > 0, i.e., μ =1. 3 For ρ 1 <ρ<ρ = 1 σ 0 π/, a>0 an b>0. Consequently, f has a positive slope with a root at 0 <μ<1 which means either μ =0 or μ =1 in this interval. SQINR k = p { } K i=1 E ĥ H k h i p E } p E {ĥh k h k {ĥh k h k } σ n E { ĥk } } 15 α {ĥh E 1 k C q ĥ 1k

4 1850 IEEE SIGNAL PROCESSING LETTERS, VOL. 4, NO. 1, DECEMBER 017 Since, γ0 = π P 1 an γ1 = σ 0 ρp 1, Then, μ =1 if ρ< π σ 0 an μ =0if ρ> π σ 0. 4 For ρ ρ, it can be shown that b 0, a>0, an a b> 0. Therefore f has a positive slope with root at μ 1, which leas to γ μ 0. Thus, γμ is a strictly ecreasing function an μ =0. Combining the above four intervals together an π σ 0 = π 4 from 9 results in 19. Theorem states that, for a given power buget, the optimal approach is to eploy the BS with either purely high-resolution or one-bit ADCs. The proof implies that when ρ = ρ th,the performance of a system with M 0 antennas connecte to highresolution ADCs is the same as that of with M 1 = ρ th M 0 antennas connecte to one-bit ADCs. This is consistent with the results of 4, 5, where it was shown that for an MRC receiver, the SE loss ue to one-bit quantization can be offset by eploying π /4 more antennas at the BS. The question of whether to use all one-bit or all highresolution ADCs boils own to whether or not the cost of a high-resolution RF chain is more than ρ th = π /4.5 times that of an one-bit RF chain. In the next section, we show that this will likely be the case in realistic scenarios, an thus a BS equippe with only one-bit ADCs is the most cost-effective solution. IV. NUMERICAL RESULTS To fin a typical value for ρ, we use as an example the RF power consumption moel consiere in 9 P BS = P LO P BB M P LNA P H P M M 0 PAGC P H ADC M1 P L ADC 3 where P LO, P BB, P LNA, P H, P M, P AGC, P H ADC, an P L ADC enote power consumption of the local oscillator, baseban processing, low noise amplifier, π/ hybri an local oscillator buffer, mixer, aaptive gain controller, high-resolution ADCs, an one-bit ADCs, respectively. To employ the result of Theorem, we can rewrite 3 in the following form As a result, P BS P LO P BB = M 0 PLNA P H P M P AGC PADC H M 1 PLNA P H P M PADC L. 4 ρ = P 0 P 1 = P LNA P H P M P AGC PADC H P LNA P H P M PADC L where we have use the following power consumption values from 9, 1: P LO =.5 mw, P BB = 00 mw, P LNA = 5.4 mw, P H = 3mW, P M = 0.3 mw, P AGC = mw,padc H = 5.6 mw, an P ADC L = 0. mw. It is apparent from 5 that ρ>ρ th.5. The moel in 3 may even Fig. 1. Sum SE S versus the average SNR for P BS P LO P BB = 00 P 1, K = 10, an T = 00. Soli lines enote the theoritical expression in 16 an symbols enote Monte-Carlo simulation for 1000 trials. be pessimistic for the one-bit case, since one-bit quantization can simplify the low noise amplifier LNA an subsequent baseban processing. Thus, we observe that using only one-bit ADCs likely provies the highest SE for a given power buget assuming MRC processing. It is worthwhile to note that this conclusion may change for ifferent power consumption moels. However, every moel can be written in the form of 17 an, hence, our result in Theorem is general an can be evaluate base on ifferent moels for power consumption. In aition, this conclusion may change for ifferent approaches such as using a zero-forcing receiver or antenna selection for the high-resolution ADCs, as these approaches ten to better exploit such ADCs, albeit with higher cost, complexity, an training overhea. Fig. 1 shows the sum SE with respect to SNR p/σ n for various values of μ uner the power constraint P BS P LO P BB = 00P 1, an assuming η = K = 10 an T = 00. As expecte, the sum SE is maximize when the BS is equippe with a large number of antennas connecte to one-bit quantizers. Note that the simulate rates symbols match the theoretical expression in 16 lines quite accurately. V. CONCLUSION We have stuie the spectral efficiency of the mixe-adc massive MIMO uplink uner energy constraint an assuming MRC processing. We erive a close form expression for the SE assuming the channel is estimate using both high-resolution an one-bit ADCs, without using the training-intensive rounrobin approach. We then maximize the SE uner a constraint on the power buget, an showe that in fact the mixe-adc approach is not optimal. Typical power consumption moels inicate that a system with all one-bit ADCs provies the highest SE for a given uplink power buget.

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