Antenna Selection for Large-Scale MIMO Systems with Low-Resolution ADCs
AANTENNA SELECTION FORLARGE-SCALE MIMO SYSTEMS WITH LOW-RESOLUTION ADCS
Jinseok Choi † , Junmo Sung † , Brian L. Evans † , and Alan Gatherer ∗† Wireless Networking and Communications Group, The University of Texas at AustinEmail: {jinseokchoi89, junmo.sung}@utexas.edu, [email protected] ∗ Systems and Design for Wireless Communications, Huawei, Plano, Texas, USAEmail: [email protected]
ABSTRACT
One way to reduce the power consumption in large-scalemultiple-input multiple-output (MIMO) systems is to employlow-resolution analog-to-digital converters (ADCs). In thispaper, we investigate antenna selection for large-scale MIMOreceivers with low-resolution ADCs, thereby providing moreflexibility in resolution and number of ADCs. To incorpo-rate quantization effects, we generalize an existing objectivefunction for a greedy capacity-maximization antenna selec-tion approach. The derived objective function offers an op-portunity to select an antenna with the best tradeoff betweenthe additional channel gain and increase in quantization er-ror. Using the generalized objective function, we propose anantenna selection algorithm based on a conventional antennaselection algorithm without an increase in overall complexity.Simulation results show that the proposed algorithm outper-forms the conventional algorithm in achievable capacity forthe same number of antennas.
Index Terms — Antenna selection, large-scale MIMO,low-resolution ADCs, capacity-maximization algorithm.
1. INTRODUCTION
Large-scale MIMO systems have drawn considerable atten-tion as a promising technology for next generation cellularsystems because they offer orders of magnitude improvementin spectral efficiency [1–3]. Practical challenges such ashardware cost and power consumption, however, arise due tothe large number of antennas [4]. Since the power consump-tion of ADCs, P ADC , scales exponentially in the number ofquantization bits b , i.e., P ADC ∝ b [5], high-speed and high-resolution ADCs would be the primary power consumers.Consequently, receivers with low-resolution ADCs (1-3 bits)have emerged as a possible solution to this problem [6, 7].Most prior work in low-resolution ADCs has focused onthe case where the number of ADCs is same as the numberof antennas [6–9] without considering the tradeoff between J. Choi, J. Sung, and B. L. Evans were supported by gift funding fromHuawei Technologies. the number of quantization bits and the number of ADCs.Accordingly, a reduced number of low-resolution ADCshas recently been investigated by using analog processingwith phase shifters [10–12]. Analog processing with phaseshifters, however, is mostly effective in a poor scatteringchannel environment with respect to reducing the number ofADCs [13, 14]. Moreover, phase shifters require additionalhardware cost and power consumption [15].For large-scale MIMO receivers, analog processing withphase shifters was compared against analog processing withswitches, i.e., antenna selection, to reduce the number ofADCs [15]. It was shown in [15] that antenna selectionprovides similar spectral efficiencies as analog processingwith phase shifters for equal power consumption in mil-limeter wave (mmWave) channels. The antenna selectionused in [15] does not exploit the sparsity of mmWave chan-nels, whereas the analog processing with phase shifters does.Hence, antenna selection is less limited by the channel en-vironment such as low scattering than analog processingwith phase shifters when reducing the number of ADCs. Onthat account, antenna selection is more applicable to generalchannels for the massive MIMO receiver. Indeed, for chan-nels measured at . GHz, a great number of ADCs couldbe turned off by using antenna selection without a substantialperformance loss [16]. Previously proposed antenna selectionmethods [16–19], however, focused on MIMO systems with-out any quantization errors. Consequently, for low-resolutionADC receivers, a new antenna selection method that incorpo-rates coarse quantization effect needs to be developed.In this paper, we investigate antenna selection for large-scale MIMO systems with low-resolution ADCs. We derivethe tradeoff between the channel gain from selecting an an-tenna and its impact on quantization error. In particular, wederive an antenna selection objective function for greedy-based antenna selection, which is a generalized version ofthe function in [17]. The objective function measures ( i ) theeffect of quantization error from previously chosen antennasin the capacity gain that comes from selecting an additionalantenna, and ( ii ) the increase in quantization error from the a r X i v : . [ c s . I T ] A p r dditional antenna which acts as a penalty. Accordingly, thetradeoff between the additional channel gain and the increaseof the quantization error in antenna selection can be cap-tured by using the derived objective function. Leveraging theobjective function, we propose a capacity-maximization an-tenna selection algorithm based on the conventional antennaselection algorithm without increasing the overall complexity.Simulation results demonstrate that the proposed algorithmoutperforms the conventional algorithm.
2. SYSTEM MODEL
We consider a multi-user MIMO uplink system in which theBS with N r antennas serves N u users with single antenna.We assume that the number of the BS antennas is much largerthan the number of users, N r (cid:29) N u . Once the BS receivesuser signals, it selects K antennas to use. The selected an-tennas are connected to RF chains with ADCs. Assuminga narrowband channel, the received baseband analog signal r ∈ C N r can be expressed as r = √ ρ Hs + n (1)where ρ , H , s , and n denotes the transmit power, the N r × N u channel matrix, the user symbol vector, and the additive whiteGaussian noise (AWGN) vector, respectively. We assume azero mean and unit variance for s and n . The i th column of H corresponds to the channel vector for user i , given as h i = √ γ i g i . The large scale fading gain γ i includes the geometric attenu-ation and shadow fading, and g i denotes the vector of smallscale fading gains for user i . We use f Hi to denote the i th rowof H . We consider that the channel H is known at the BS andunknown at users. Since the BS selects K antennas and con-nects the selected antennas to K RF chains, after the antennaselection, the received analog signal (1) becomes r K = √ ρ H K s + n K (2)where K represents the index set of selected antennas withthe cardinality of |K| = K , r K ∈ C K denotes the receivedsignal vector for the selected antennas, H K ∈ C K × N u is thechannel matrix for the selected antennas in K , and n K ∈ C K indicates the corresponding noise vector.After the antenna selection, each real and imaginary com-ponent of the complex output r K ( i ) is quantized at the pair ofADCs, where K ( i ) is the i th selected antenna. For analyticaltractability, the additive quantization noise model (AQNM)is used to linearize the quantization process as a function ofquantization bits b . AQNM [20] shows reasonable accuracyfor b = 1, 2, and 3 in low and medium SNR ranges. Adoptingthe AQNM, the quantized signal becomes y = Q (cid:0) Re { r K } (cid:1) + j Q (cid:0) Im { r K } (cid:1) = α √ ρ H K s + α n K + q (3) Table 1 : The Values of β for Quantization Bits bb β Q ( · ) is the element-wise quantizer function. Here, α is defined as α = 1 − β and considered to be the quantiza-tion gain ( α < , and β is the normalized mean squaredquantization error β = E [ | y i − y q i | ] E [ | y i | ] . Assuming a scalar min-imum mean squared error quantizer and Gaussian signalingfor s ∼ CN ( , I ) where CN ( , I ) represents complex Gaus-sian distribution with a zero mean vector and the identitymatrix I with proper dimension for a covariance matrix, β isapproximated as β ≈ π √ − b for b > . Note that b isthe number of quantization bits for each real and imaginarypart. The values of β for b ≤ are shown in Table 1. Thevector q represents the additive quantization noise and is un-correlated with the quantization input r K . The quantizationnoise follows the complex Gaussian distribution with a zeromean q ∼ CN ( , R qq ) . The covariance matrix R qq for thechannel H K is given by R qq = α (1 − α ) diag( ρ H K H H K + I ) (4)where diag( ρ H K H H K + I ) represents the diagonal matrix ofthe diagonal entries of ρ H K H H K + I .
3. ANTENNA SELECTION3.1. Performance Measure
In this section, we examine the key difference of the antennaselection problem at the receiver with low-resolution ADCsfrom the conventional problem, and further propose an an-tenna selection method based on a capacity-maximization ap-proach. Under the considered system in (3), the capacity forthe given channel matrix H K can be expressed as C ( H K ) = log (cid:12)(cid:12)(cid:12) I + ρα (cid:0) α I + R qq (cid:1) − H K H H K (cid:12)(cid:12)(cid:12) (5)where R qq is given in (4). Note that in the system withlow-resolution ADC, the quantization noise covariance ma-trix R qq is included in the capacity expression (5) as a penaltyterm for each antenna. Remark 1
Since each diagonal entry of R qq contains an ag-gregated channel gains at each selected antenna (cid:107) f K ( i ) (cid:107) , thetradeoff between the channel gain and its influence on quan-tization error needs to be considered in antenna selection. Using the capacity expression in (5), we formulate the an-tenna selection problem as follows: K (cid:63) = argmax K⊂{ ,...,N r } : |K| = K C (cid:0) H K (cid:1) . (6)he large number of antennas N r makes it almost infeasibleto search over all possible K . Accordingly, to avoid an ex-haustive search, we propose a quantization-aware antenna se-lection algorithm based on the greedy capacity-maximizationapproach [17]. The capacity in (5) can be rewritten as C ( H K ) = log (cid:12)(cid:12)(cid:12) I + ρα D − K H K H H K (cid:12)(cid:12)(cid:12) (7)where D K = diag(1 + ρ (1 − α ) (cid:107) f K ( i ) (cid:107) ) is the diagonalmatrix with ρ (1 − α ) (cid:107) f K ( i ) (cid:107) for i = 1 , . . . , K at itsdiagonal entries.Using (7), we can express the capacity at the ( n + 1) thselection stage as follows: C ( H n +1 ) (8) = log (cid:12)(cid:12)(cid:12) I + ρα D − n +1 H n +1 H Hn +1 (cid:12)(cid:12)(cid:12) = log (cid:12)(cid:12)(cid:12) I + ρα H Hn +1 D − n +1 H n +1 (cid:12)(cid:12)(cid:12) = log (cid:12)(cid:12)(cid:12)(cid:12) I + ρα (cid:16) H Hn D − n H n + 1 d K ( n +1) f K ( n +1) f H K ( n +1) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) where H n +1 is the channel matrix that corresponds to the ( n + 1) selected antennas after the ( n + 1) th selection stage.Recall that f H K ( n +1) denotes the ( n + 1) th selected row of H ,and d K ( n +1) is the corresponding diagonal entry of D n +1 .We can decompose (8) into the capacity after the n th se-lection and the capacity increase that comes from selectingthe ( n + 1) th antenna. To this end, we use the matrix deter-minant lemma | A + uv H | = | A | (1 + v H A − u ) as follows: C ( H n +1 ) = C ( H n ) + log (cid:18) ραd K ( n +1) c K ( n +1) ,n (cid:19) where c K ( n +1) ,n = f H K ( n +1) (cid:16) I + ρα H Hn D − n H n (cid:17) − f K ( n +1) . (9)To maximize C ( H n +1 ) given the n selected antennas, weneed to find the next antenna j which maximizes c j,n /d j . Ac-cordingly, the antenna selection problem becomes J = argmax j c j,n d j . (10)Unlike the objective function of the capacity-maximizationalgorithm with no quantization error in [17], the derivedobjective function c j,n /d j incorporates ( i ) the effect of thequantization error from the previously selected n antennas tothe next antenna j in c j,n , and ( ii ) the additional quantizationerror from selecting the antenna j as a penalty for the antenna j in the form of /d j . Since d j is a function of the aggregatedchannel gain for the j th antenna, d j = 1 + ρ (1 − α ) (cid:107) f j (cid:107) ,solving the problem (10) gives the antenna J which offersthe best tradeoff between the channel gain from selecting anantenna and its influence on the increase of the quantization Algorithm 1
Quantization-Aware Fast Antenna Selection1) Initialize: T = { , . . . , N r } and Q = I .2) Initialize antenna gain and compute penalty: c j = (cid:107) f j (cid:107) and d j = 1 + ρ (1 − α ) (cid:107) f j (cid:107) for j ∈ T .3) Select antenna J using (10): J = argmax j ∈T c j /d j .4) Update candidate set: T = T \ { J } .5) Compute: a = (cid:0) c J + d J ρα (cid:1) − Qf J and Q = Q − aa H .6) Update c j = c j − | f Hj a | for j ∈ T .7) Go to step 3 and repeat until select K antennas.error. This corresponds to the intuition in Remark 1. Notethat (10) is the generalized antenna selection objective func-tion of the one in [17]; as the number of quantization bits b increases, the quantization gain α increases as α → , whichleads to d j → and D n → I .To propose a quantization-aware antenna selection algo-rithm, we modify the fast antenna selection algorithm in [17]by using the derived objective function in (10) without in-creasing the overall complexity. Unlike the algorithm for per-fect quantization, the quantization error term d j needs to becomputed in advance. Then, at the selection step, the pro-posed algorithm adopts (10) to incorporate the influence ofthe quantization error of the previously selected antennas andthe candidate antenna. To compute c j,n in (9), we define Q n = (cid:16) I + ρα H Hn D − n H n (cid:17) − . (11)Then, Q n can be updated efficiently by using the matrixinversion lemma as Q n +1 = Q n − aa H , where a = (cid:0) c J + d J ρα (cid:1) − / Qf J . Finally, c j,n can be updated as c j,n +1 = f Hj Q n +1 f j = f Hj (cid:0) Q n − aa (cid:1) H f j = c j,n − | f Hj a | . The proposed algorithm, we call quantization-aware fastantenna selection (QAFAS), is described in Algorithm 1.Note that the complexity for step 5 and 6 are O ( KN u ) and O ( KN u N r ) , respectively. Since we assume the large an-tenna arrays at the BS ( N r (cid:29) N u ) , the overall complexitybecomes O ( KN u N r ) . Thus, the proposed algorithm doesnot increase the overall complexity from the conventionalalgorithm for the perfect quantization in [17], which providesthe opportunity to be practically implemented without in-crease of computational burden. In the following section, wedemonstrate the performance of the proposed algorithm forthe large-scale MIMO systems with coarse quantization. . SIMULATION RESULTS In this section, we evaluate the proposed algorithm and com-pare the algorithm with the fast antenna selection algorithmin [17] which shows a comparable performance to the opti-mal selection case under a perfect quantization assumption.The Matlab code is available at [21]. We also include a ran-dom selection case for a reference in capacity performance.We assume Rayleigh channel with a zero mean and unit vari-ance. For the large scale fading, we adopt the log-distancepathloss model [22]. We consider randomly distributed usersover a single cell with a cell radius of 2 km . We assume theminimum distance between the BS and users to be m .Considering a . GHz carrier frequency with MHz band-width, we use . dB lognormal shadowing variance and dBnoise figure with N r = 128 BS antennas and N u = 10 users.Fig. 1 shows the average capacity versus the transmitpower ρ for b = 3 quantization bits in K ∈ { , , } an-tenna selection cases. The proposed QAFAS algorithm out-performs the conventional fast antenna selection method. Thegap between the algorithms increases as the transmit power ρ increases because the quantization error becomes more dom-inant than the AWGN as the transmit power increases. Ac-cordingly, the proposed QAFAS algorithm which incorpo-rates the quantization error when selecting antennas improvesthe capacity performance more when the quantization errorbecomes prevailing. Note that the QAFAS algorithm with K ∈ { , } achieves the capacity comparable to the ran-dom selection case with K ∈ { , } whose number of se-lected antennas K is twice as many, respectively.For the different number of quantization bits b , the aver-age capacities for ρ = 5 dBm transmit power in K ∈ { , } antenna selection cases are shown in Fig. 2. The proposed al-gorithm provides the highest capacity and improves the ca-pacity compared to the random selection case in the low-resolution regime, whereas the conventional fast antenna se-lection marginally improves the capacity. Notably, the ca-pacity of the QAFAS algorithm shows relatively larger im-provement compared to that of the fast antenna selection algo-rithm in the low-resolution regime than in the high-resolutionregime, which corresponds to the motivation of this work. Inthe low-resolution regime, the QAFAS algorithm with K ∈{ , } shows the comparable capacity to that of random se-lection with K ∈ { , } , respectively.
5. CONCLUSION
In this paper, we proposed a new antenna selection algorithmfor large-scale MIMO systems with low-resolution ADCs.For each unselected antenna, the derived objective func-tion measures the increase in capacity, like a conventionalapproach. Unlike a conventional approach, the proposedobjective function measures the effect on capacity due toquantization error from previously chosen antennas and the
Fig. 1 : Average capacity for different transmit power with N r = 128 BS antennas, N u = 10 users, b = 3 quantizationbits, and K ∈ { , , } selected antennas. Fig. 2 : Average capacity for the different number of quantiza-tion bits with N r = 128 BS antennas, N u = 10 users, ρ = 5 dBm transmit power, and K ∈ { , } selected antennas.increase in quantization error if an unselected antenna werechosen. Since the derived objective function captures thetradeoff between the additional channel gain and the increasein quantization error for antenna selection, we used the ob-jective function to propose a capacity-maximization antennaselection algorithm. The simulation results validated the pro-posed antenna selection algorithm for low-resolution ADCs.Therefore, using the proposed algorithm, the antenna se-lection for large-scale MIMO systems with low-resolutionADCs offers more flexibility in the resolution and number ofADCs with higher rates than other selection methods. . REFERENCES [1] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energyand spectral efficiency of very large multiuser MIMOsystems,” IEEE Trans. on Comm. , vol. 61, no. 4, pp.1436–1449, Apr. 2013.[2] J. Hoydis, S. ten Brink, and M. Debbah, “MassiveMIMO in the UL/DL of cellular networks: How manyantennas do we need?”
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