arXiv:1810.11584v1 [cs.IT] 27 Oct 2018 ntemnmzto fthe b of is minimization AGC with the the systems of optimization on (MU-MIMO) The output signals. quantized multiple coarsely input a multiple algorithm user (AGC) control gain linear a automatic an of optimization vrtm,a stecs nmbl cnro.Poe AGC Proper scenarios. mobile in AGC case an varie the power of is received use as the signal The time, where the [7]. applications over analog minimize in quantization the to important the of is order to range in due dynamic (ADC) distortion the converter to digital level to of signal bits analog few the with [4]. quantized [2], signals [6], detec provide with [1], resolution present to deal studies that techniques Several methods processing transmission. and, signal data distortion reliable new nonlinear require significant thus, generate resoluti which the low employ ADCs, of digital approaches function transmission to and novel complexity a energy, circuit analog reduce as To an resolution. exponentially F quantization of grows [5]. consumption circuit [4], (ADC) [3], converter and energy number [2], consumption the [1], energy the example, accordingly the as increases up, However, complexity scales rates. lin antennas wireless achievable of in and commu- improvements reliability wireless significant in provide systems Multiple- nications requirements. (MIMO) key multiple-output are input consumption power and cost 3 2, Zero- and of bits. and resolution simulations, quantization 5 filter using receive by receiver MMSE (ZF) evaluated conventional Forcing by is the (LRA-MMSE) with algorithm system MMSE compared AGC Aware MU-MIMO and perform Low-Resolution the The receiver proposed of rate. achievable the capacity the th of for the provid of expression we on Moreover, an presence deriving quantization. bound the to lower due account a effects into the takes and AGC filter receive proposed in MEreceiver MMSE tion, td fJitAtmtcGi oto n MMSE and Control Gain Automatic Joint of Study h uhr ol iet hn h AE rzla gnyfo agency Brazilian CAPES the thank to like would authors The Abstract uoai ancnrl(G)i h rcs fadjusting of process the is (AGC) control gain Automatic low links, reliable rates, data high systems, celular 5G In Terms Index iiu ensur error square mean minimum 1 etefrTlcmuiain tde CTC,Pontifical (CETUC), Studies Telecommunications for Centre Ti ae rsnstedvlpeto joint a of development the presents paper —This eevrDsg ehiusfrQuantized for Techniques Design Receiver mi:{haola,dlmr}ctcpcrob,tadeu delamare}@cetuc.puc-rio.br, {thiagoelias, Email: Cas uniain G,M-IOdetec- MU-MIMO AGC, Quantization, —Coarse 3 nttt o omnctosEgneig nvriä de Universität Engineering, Communications for Institute utue utpeAtnaSystems Multiple-Antenna Multiuser .INTRODUCTION I. 2 niern col lmnneFdrlUiest (UFF), University Federal Fluminense School, Engineering ensur error square mean .E .Cunha B. E. T. MS)rcie o multi- for receiver (MMSE) 1 .C eLamare de C. R. , ME n the and (MSE) funding. r and 4 , ance save ased tion nd on or k e e s pe aeiai odltes h operators The letters. bold italic case upper h aaiyo hssse sivsiae n nexpressio developed. an is and rates investigated achievable is the system compute on to this to both bound of lower order A capacity considers derived. is the in that AGC the AGC and receiver effects quantization the (LRA-MMSE) Low-Resolution a of MMSE Then, coefficients. th Aware presence AGC function optimal and, the cost the [1] obtain account the of in into derivative presented the takes com- receiver computing of MMSE that, MMSE after consists the modified procedure the to with The puting operating according signals. system filter quantized MU-MIMO coarsely large-scale receive a linear for criterion a and AGC AGC quantizati the the considers optimized that not detector have effects. a with used examined authors nor were system the algorithm receiver AGC. MIMO However, the an quantized [6]. at of a in filter presence on Zero-Forcing the AGC standard account an a into of take effects system MIMO not The a that do AGCs. in receiver they effects of MMSE quantization but modified the design a account the into presented takes address MIMO authors the few in [1], quantization literature, In on the articles in many systems are ADC there resolution low Although for important especially becomes design tn o rnps,Hriintasoeadtaeo matr a of trace and respectively. transpose Hermitian transpose, for stand niett arx h operator The matrix. identity an ignlmti otiigol h ignleeet of elements diagonal the only and containing matrix operator diagonal the a and variables random the to respect with and osdrd tec ieinstant time each At considered. epnst h aaadpout Finally, product. Hadamard the to responds [ ybl hc r raie noa into organized are which symbols itn fabs tto B)with (BS) station base a of sisting x 1 1 Notation the designing jointly for framework a presents work This ag-cl pikM-IOsse 1] 1]con- [14] [12], system MU-MIMO uplink large-scale A .N Ferreira N. T. , [ i nondiag( ] frer@dufb,[email protected] [email protected], K x , ahlcUiest fRod aer,Brazil Janeiro, de Rio of University Catholic 2 sr qipdwith equipped users [ i udserMnhn Germany München, Bundeswehr r ] etr n arcsaedntdb oe and lower by denoted are matrices and Vectors : .,x ..., , 1 A eoe ounvco foe and ones of vector column a denotes = ) N I S II. T 2 ieó,R,Brazil RJ, Niterói, [ A i .Hälsig T. , ]] YSTEM T − ahetyo h vector the of entry Each . diag( D N A ESCRIPTION 3 T ) . E i rnmtatna ahis each antennas transmit ahue transmits user each , [ · ] N N tnsfrexpectation for stands T R × ( · eev antennas, receive ) 1 diag( T vector , ( · ) A H x ) I and k x denotes denotes [ ⊙ k i ] [ tr( i cor- sa is = ] N ix, on A at · s. T n ) symbol taken from the modulation alphabet A. The symbol undergo nearly the same distortion factor ρq. It was shown in vector is then transmitted through flat fading channels and [8] for the uniform quantizer case, that optimal quantization b corrupted by additive white Gaussian noise (AWGN). The step ∆ for a Gaussian source decreases as √b2− and that ρq 2 received signal collected by the receive antennas at the BS ∆ is asymptotically well approximated by 12 . is given by the following equation: III. PROPOSED JOINT AGC AND LINEAR MMSE K RECEIVER DESIGN y[i]= Hkxk[i]+ n[i]= Hx[i]+ n[i], (1) The procedure for joint optimization of the AGC algorithm k=1 X and the LRA-MMSE receiver carries out alternating compu- NR 1 NR NT where y[i] C × , and Hk C × is a matrix that ∈ ∈ tations between the AGC and the LRA-MMSE receiver. The contains the complex channel gains from the NT transmit first step consists of computing an LRA-MMSE receive filter antennas of user k to the NR receive antennas of the BS. The that considers the quantization effects. Other approaches to NR 1 vector n[i] is a zero-mean complex circular symmetric computing linear MMSE or related filters [18], [19], [20], [21] × H Gaussian noise vector with covariance matrix E[n[i]n [i]] = can also be considered. After that, we compute the derivative 2 σ I. H is a NR KNT matrix that contains the coefficients n × of the cost function to obtain the optimal AGC coefficients. of the flat fading channels between the transmit antennas of Then, an updated LRA-MMSE receiver is computed. the K users and the receive antennas of the BS. The symbol T vector x[i] = [x1[i], ..., xk[i], ..., xK [i]] contains all symbols A. Linear LRA-MMSE Receive Filter Design that are transmitted by the users at time instant i. The channel In this first step we do not consider the presence of the AGC state information (CSI) is assumed to be unknown to the users in the system. Thus, the received signal after the quantizer is at the transmit side. Therefore, we assume the same symbol expressed, with the Bussgang decomposition [9], as a linear H 2 energy per user and transmit antenna, i.e. E[xx ]= σxI. model r = y + q. To develop the linear receive filter W that minimizes the MSE we use the Wiener-Hopf equations: 1 W = RxrRrr− , (3)

where the auto-correlation matrix Rrr is given by H H Rrr = E[rr ]= Ryy + Ryq + Ryq + Rqq, (4)

and the cross-correlation matrix Rxr can be expressed as H Rxr = E[xr ]= Rxy + Rxq (5) Fig. 1. An uplink quantized MU-MIMO system We get the auto-correlation matrix Ryy and the cross- As depicted in Fig.1, before to be quantized, y is jointly pre- correlation matrix Rxy directly from the MIMO model as multiplied by the clipping level factor α and the AGC matrix H H G to minimizes the granular and the overload distortions. After Ryy = E[yy ]= HRxxH + Rnn (6) that, the product αGy is quantized and the estimation of the and, transmitted symbol vector x is performed by the LRA-MMSE H H receiver, represented by L. The estimated symbol vector is Rxy = E[xy ]= RxxH (7) represented by xˆ. The real, yi,R, and imaginary, yi,I , parts To compute (4) and (5) we need to obtain the covariance of the complex received signal at each antenna are quantized matrices Ryq, Rqq and Rxq as a function of the channel separately by uniform b-bit resolution ADCs. Therefore, the parameters and the distortion factor ρq. The procedure of how resulting quantized signals read as to obtain these matrices was developed in [1] and we will

ri,l = Q(yi,l)= yi,l + qi,l, l R, I , 1 i NR, (2) use some of these results in this work. The cross-correlation ∈{ } ≤ ≤ between the received signal vector and the quantization error where Q( ) denotes the quantization operation and qi,l is the is approximated by resulting quantization· error. Ryq ρqRyy (8) The distortion factor indicates the relative amount of quan- ≈ − tization noise generated by the quantizer, and is given by The covariance matrix of the quantization error is deduced (i,l) 2 2 , where 2 is the variance of the quantizer ρq = σqi,l /σyi,l σqi,l from 2 error and, σ is the variance of the input yi,l. This factor yi,l 2 Rqq ρq diag(Ryy)+ ρ nondiag(Ryy) depends on the number of quantization bits b, the quantizer ≈ q type, and the probability density function of yi,l [1]. In this = ρqRyy (1 ρq)ρq nondiag(Ryy), (9) work the scalar uniform quantizer processes the real and − − and the cross-correlation matrix between the desired signal imaginary parts of the input signal y in a range √b . i,l 2 vector and the quantization error can be obtained by With a high number of antennas the input signals of± the quantizer are approximately Gaussian distributed and they Rxq = ρqRxy (10) − Substituting (10) in (5) we get

∂ε ∂ H ∂ H Rxr = (1 ρq)Rxy (11) = α tr(Rxy diag(g)W ) α tr(W diag(g)Rxy) − ∂g − ∂g − ∂g and substituting (6), (8) and (9) in (4) we get I II 2 ∂ H + α | tr(W{zdiag(g)Ryy}diag(g|)W ) {z } Rrr (1 ρq)(Ryy ρq nondiag(Ryy)) (12) ∂g ≈ − − III Finally, by substituting (11) and (12) in (3) we get the ∂ + α | tr(W diag(g){zR WH ) } expression of the LRA-MMSE receive filter for a quantized ∂g yq MU-MIMO system. As shown in [1], we can write this IV solution as ∂ H H + α | tr(WRyq{zdiag(g)W }) (15) 1 ∂g W = Rxy(Ryy ρq nondiag(Ryy))− (13) − V With the presence of the AGC the expression of the received We have| to take the{z derivative of} each term of Eq. (15). vector changes and can be computed by z = Gy + q. Consider the conversion between matrix notation and index With the same procedure as before, the MMSE filter with notation and the tricky case of a diag( ) operator AGC can be computed through the Wiener-Hopf equations · 1 [AB]ik = Aij Bjk (16) as L = RxzRzz− . The auto-correlation matrix Rzz and the j cross-correlation matrix Rxz can be computed similarly by X

H H Rzz = E[zz ]= GRyyG + GRyq + RyqG + Rqq f = tr[A diag(g)B]= Aij gj Bji (17) H i j Rxz = E[xz ]= RxyG + Rxq X X Taking the derivative with respect to the coefficients gj of We note that nonlinear receiver structures can also be con- the diagonal operator we have sidered following the approaches reported in [22], [23], [24], [25], [26]. ∂f T = Aij Bji = [(A B)1]j (18) ∂g ⊙ j i B. AGC Design X Therefore, we can write In [6], the authors proposed a standard AGC algorithm by using a diagonal matrix G with real coefficients. This matrix ∂ tr[A diag(g)B] is used to compensate the gain differences of the propagation = (AT B)1 (19) channel and involves a search over a transmitted symbol ∂g ⊙ alphabet. This approach is very computationally demanding With these considerations we can take the derivative of in an environment with a high number of antennas. Writing terms I , II, III, IV and V from Eq. (15). The derivatives G as diag(g), where g is a column vector with the diagonal of the terms I and II can be computed by elements of G, the proposed AGC algorithm is based on the minimization of the cost function: ∂ tr[R diag(g)WH ] I = xy = [(RT WH )1] (20) ∂g xy ⊙ ε = E[ x xˆ 2] || − || 2 = E[ x W(α diag(g)y + q) ] (14) H || − || ∂ tr[W diag(g)Rxy] H T II = = [(Rxy W )1] (21) and since G is a diagonal matrix with real coefficients we ∂g ⊙ have diag(g)H = diag(g). Then, To compute the derivative of term III we apply the chain rule H H ε = tr(Rxx αRxy diag(g)W RxqW − − H 2 H H αW diag(g)Rxy + α W diag(g)Ryy diag(g)W ∂ tr[W diag(g)A] ∂tr[B diag(g)W ] − H H III = + (22) + αW diag(g)RyqW WR ∂g ∂g − xq H H H + αWRyq diag(g)W + WRqqW ) III.1 III.2 H where A =| Ryy diag({z g)W } and| B = W{zdiag(g)R} yy. The To obtain the optimum G matrix we compute the derivative term III.1 can be computed by of the MSE cost function with respect to diag(g), equate the T H derivative terms to zero and solve for g: III.1 = [(W (Ryy diag(g)W ))1] (23) ⊙ and the term III.2 as IV. CLIP-LEVELADJUSTMENT In the following we outline the computation of the clipping T T H factor α based on the signal power. This factor conforms the III.2 = [((Ryy diag(g)W ) W )1] (24) ⊙ received signal power between the quantizer range to minimize Substituting (23) and (24) in (25) we have the overload distortion. The received signal power can be computed by T H III = [(W (Ryy diag(g)W ))1] P = tr(E[(y + q)(y + q)H ]) T ⊙ T H + [((Ryy diag(g)W ) W )1] (25) H ⊙ = tr(Ryy + Ryq + Ryq + Rqq) (33) The derivative of the term is given by IV and received symbol energy by

H ∂tr[W diag(g)C] T H tr(Ryy + Ryq + Ryq + Rqq) IV = = [(W (RyqW ))1] (26) Erx = (34) ∂g ⊙ s NR H where C = RyqW . Finally, the derivative of the term V Thus, the clipping factor α can be obtained from can be computed by H tr(Ryy + Ryq + Ryq + Rqq) H α = β , (35) ∂ tr[D diag(g)W ] T H · s NR V = = [((Ryq∗ W ) W )1] (27) ∂g ⊙ where β is a calibration factor. In our simulations the value H where D = WRyq. Substituting (20), (21), (25), (26) and of β was set to √b which corresponds to the quantizer output (27) in (15) and equating the derivatives to zero we have range, to ensure an optimized performance. T H T T H [W (Ryy diag(g)W ) + (R diag(g)W ) W ]1 = V. CAPACITY LOWER BOUND ⊙ yy ⊙ 1 ([(RT WH )1] + [(RH WT )1]+ In [1] a lower bound on the mutual information between α xy ⊙ xy ⊙ the input sequence x and the quantized output sequence r of T H T H [(W (RyqW ))1] [((R∗ W ) W )1]) (28) a quantized MIMO system was developed, based on the MSE − ⊙ − yq ⊙ To achieve the desired g we have to do some manipulations approach. We will use a similar procedure to consider a ca- with the first term of (28). To do this we will write the first pacity lower bound of our quantized MU-MIMO system with and second terms of g with the index notation and after that the optimal AGC and to derive an expression for computing we will return to the matrix notation. We can write the first the achievable rates for the proposed AGC and LRA-MMSE term as receiver. We remark that similar analyses can be considered for the downlink with the use of precoding techniques [15], KNT NR T H H [16], [17]. As described in [11] the mutual information of this [(W (Ryy diag(g)W )1]= WjiRyy,ilglW (29) ⊙ lj channel can be expressed as j=1 l X X=1 and the second term as I(x, r)= h(x) h(x r) (36) − | KNT NR H T T H Given Rxx under a power contraint tr(Rxx) PTr, we [(W (R diag(g)W )1]= W Ryy,liglWjl (30) ≤ ⊙ yy ij choose x to be Gaussian, which is not necessarily the capacity j=1 l X X=1 achieving distribution for our quantized system. Then, we can With some manipulations we can isolate the vector g obtain a lower bound for I(x, r) (in bit/transmission) as T H H T T [W (Ryy diag(g)W )+ W (Ryy diag(g)W )]1 I(x, r) = log det(Rxx) h(x r) ⊙ ⊙ 2 − | KNT NR KNT NR H H = log2 det(Rxx) h(x xˆ r) = WjiRyy,ilglW + W Ryy,liglWjl − − | lj ij log det(R ) h(x xˆ) (37) j=1 l j=1 l 2 xx X X=1 X X=1 ≥ − − NR ǫ T H T det(Rxx) = ([(W W∗) Ryy + (W W) Ryy]il)gl ⊙ ⊙ log2 | {z } (38) l=1 ≥ det(Rǫǫ) X T H T = [(W W∗) Ryy + (W W) Ryy]g (31) The second term in (37) is upper bounded by the entropy of ⊙ ⊙ Substituting (31) in (28) and solving with respect to g we a Gaussian random variable whose covariance is equal to the have error covariance matrix Rǫǫ of the LRA-MMSE estimate of x. Thus, we have to compute the expressions of Rxx and Rǫǫ T H T 1 g =[(W W∗) Ryy + (W W) Ryy]− for our system. Considering unknown CSI at the transmitter, ⊙ ⊙ 2 T H T H the autocorrelation matrix Rxx is given by (Re([(Rxy W )1]) Re([(W RyqW )1])) · α ⊙ − ⊙ 2 (32) Rxx = σxIKNT , (39) and the error covariance matrix can be computed by to the performance of the Full Resolution standard MMSE

H receiver (FR standard MMSE). Rǫǫ =E[(x xˆ)(x xˆ) ] − − H H H =Rxx RxyGW RxqW WGRxy+ − H − − H H 400 + WGRyyGW + WGRyqW WR + FR-MMSE − xq H H H Joint AGC and LRA-MMSE Design - 5 bits + WRyqGW + WRqqW (40) 350 Joint AGC and LRA-MMSE Design - 4 bits Joint AGC and LRA-MMSE Design - 3 bits Substituting (39) and (40) in (38) we obtain an expression 300 Joint AGC and LRA-MMSE Design - 2 bits to compute the achievable rates for the MU-MIMO system with coarsely quantized signals. 250 VI. RESULTS 200 To evaluate the results obtained in previous sections we consider a MU-MIMO system with K = 16 users who are 150 each equipped with NT = 2 transmit antennas and one BS with NR = 64 receive antennas. At each time instant the 100 users transmit data packets with 100 symbols using BPSK

Achievable sum-rate (bit/channel use) 50 modulation. The channels are obtained with independent and 0 5 10 15 20 identically distributed complex Gaussian random variables SNR (dB) with zero mean and unit variance. For each simulation 10000 packets are transmitted, by each transmit antenna, over a flat- Fig. 3. Achievable sum-rate of the quantized MU-MIMO system with 32 fading channel. The received signal is quantized with 2, 3, 4 users with 2 transmit antennas each and one BS with 64 receive antennas. and 5 bits. In Fig. 3 we illustrate the achievable sum rates of the MU-MIMO system with the joint AGC and LRA-MMSE 100 receiver design for different numbers of quantization bits. This result shows that, as the number of quantization bits increases, the sum-rate also increases approaching those values obtained by the FR standard MMSE receiver in an unquantized environment. 10-2 VII. CONCLUSIONS In this work we have discussed the joint design of an AGC and a LRA-MMSE receive filter for coarsely quantized MU- 10-4 MIMO systems. Simulations results have shown that the joint FR standard MMSE Joint AGC and LRA-MMSE Design - 5 bits AGC and the LRA-MMSE receiver obtained a performance Joint AGC and LRA-MMSE Design - 4 bits close to the full resolution MMSE receiver in a quantized MU- Uncoded BER Joint AGC and LRA-MMSE Design - 3 bits Joint AGC and LRA-MMSE Design - 2 bits MIMO system with 4 and 5 bits of resolution. Furthermore, MMSE and standard AGC - 5 bits ZF and standard AGC - 5bits we have derived an expression for computing the achievable 10-6 0 5 10 15 rates for the system. The results have shown that with 4 and 5 SNR (dB) quantization bits we achieve a rate very close to the capacity of an unquantized large-scale MU-MIMO system. Fig. 2. Joint AGC and LRA-MMSE receiver performance comparison REFERENCES Fig. 2 shows the BER performance of the proposed joint [1] A. 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