Channel Estimation and Data Decoding Analysis of Massive MIMO with 1-Bit ADCs
11 Channel Estimation and Data DecodingAnalysis of Massive MIMO with 1-Bit ADCs
Italo Atzeni,
Member, IEEE and Antti Tölli,
Senior Member, IEEE
Abstract
We present an analytical framework for the channel estimation and the data decoding in massive multiple-input multiple-output uplink systems with 1-bit analog-to-digital converters (ADCs). First, we provide aclosed-form expression of the mean squared error of the channel estimation for a general class of linearestimators. In addition, we propose a novel linear estimator with significantly enhanced performancecompared with existing estimators with the same structure. For the data decoding, we provide closed-formexpressions of the expected value and the variance of the estimated symbols when maximum ratiocombining is adopted, which can be exploited to efficiently implement maximum likelihood decodingand, potentially, to design the set of transmit symbols. Comprehensive numerical results are presented tostudy the performance of the channel estimation and the data decoding with 1-bit ADCs with respectto the signal-to-noise ratio (SNR), the number of user equipments, and the pilot length. The proposedanalysis highlights a fundamental SNR trade-off, according to which operating at the right noise levelsignificantly enhances the system performance.
I. I
NTRODUCTION
The migration of operating frequencies from first- to fourth-generation wireless systems, i.e.,from 800 MHz to the sub-3 GHz range, did not bring major changes in terms of signal propagation.The current fifth generation (5G) features a more pronounced transition in this respect by operatingat sub-6 GHz frequencies and, eventually, up to 30 GHz with the objective of boosting the datarates. Following this trend, beyond-5G systems will exploit the large amount of bandwidth availablein the mmWave band (i.e., 30 GHz–300 GHz) and raise the operating frequencies up to 1 THz [1].However, as the free-space pathloss scales as the square of the frequency, maintaining the samesignal-to-noise ratio (SNR) over a given distance will require increasingly sharp beamforming tospatially focus the signal power. Although the short wavelength at mmWave and THz frequenciesallows to pack many antennas into a very small area, realizing fully digital, high-resolutionmassive multiple-input multiple-output (MIMO) arrays remains prohibitive in practice [2], [3].
The authors are with the Centre for Wireless Communications, University of Oulu, Finland (emails: {italo.atzeni,antti.tolli}@oulu.fi). The work of I. Atzeni was supported by the Marie Skłodowska-Curie Actions (MSCA-IF 897938 DELIGHT).The work of A. Tölli was supported by the Academy of Finland under grant no. 318927 (6Genesis Flagship).This work has been submitted to the IEEE for possible publication.Copyright may be transferred without notice, after which this version may no longer be accessible. a r X i v : . [ c s . I T ] F e b ... ...RFRF ADCADCADCADC c h a nn e l e s ti m a ti on a nd r ece i v e r d e s i gn ... IQIQ
K M K
RRH BBU
Fig. 1: Massive MIMO uplink system model with 1-bit ADCs.
As in the system model illustrated in Fig. 1, each base station (BS) antenna is generallyequipped with a dedicated radio-frequency (RF) chain that includes complex, power-hungryanalog-to-digital/digital-to-analog converters (ADCs/DACs) [3]. In this setting, while the transmitpower can be made inversely proportional to the number of antennas, the power dissipated byeach ADC/DAC scales linearly with the sampling rate and exponentially with the number ofquantization bits [4]–[8]. Another limiting factor is the amount of raw data exchanged betweenthe remote radio head (RRH) and the base-band unit (BBU), which scales linearly with boththe sampling rate and the number of quantization bits [9]–[11]. For these reasons, adoptinglow-resolution ADCs/DACs with 1 to 4 quantization bits as opposed to the typical 10 or more [12]enables the implementation of massive MIMO arrays comprising hundreds (or even thousands) ofantennas, which are necessary to operate in the mmWave and THz bands [9]. In this regard, 1-bitADCs/DACs are particularly appealing due to their minimal power consumption and complexitysince they only evaluate the sign of the input signal [4]. Such a coarse quantization is especiallymotivated at very high frequencies, where high-order modulations are not essential.There is a large body of literature on massive MIMO with low-resolution and 1-bit ADCs/DACs,ranging from performance analysis to channel estimation and precoding design. The capacity ofthe 1-bit quantized MIMO channel is characterized in [4], whereas [13] shows that replacing evena small number of high-resolution ADCs with 1-bit ADCs entails a modest performance loss whilesignificantly reducing the power consumption. [5] studies the performance-quantization trade-offof orthogonal frequency-division multiplexing (OFDM) uplink systems and shows that using 4 to6 quantization bits involves almost no performance loss compared with infinite-resolution ADCs.The spectral efficiency of single-carrier and OFDM uplink systems with 1-bit ADCs is analyzed in[14]. The problem of multi-user detection is considered, e.g., in [15] and [16] for low-resolutionand 1-bit ADCs, respectively, whereas [17] focuses on the joint channel estimation and datadecoding. [6] proposes an efficient iterative method for near maximum likelihood decoding (MLD) with 1-bit ADCs. The work in [7] analyzes the channel estimation and the uplink achievablerate with 1-bit ADCs. In addition, it proposes a linear channel estimator based on the Bussgangdecomposition, which allows to reformulate the nonlinear quantization function as a linearfunction with identical first- and second-order statistics [18]. A similar analysis is presented in[19] for the downlink direction. [9] extends some of the results derived in [7], [14] for 1-bitADCs to the multi-bit case. Specifically, it presents a throughput analysis of uplink systems andproposes a linear channel estimator based on the Bussgang decomposition with low-resolutionADCs. The channel estimation with 1-bit ADCs when the quantization threshold is not known isstudied in [20]. The channel estimation exploiting the angular and delay structure is consideredin [21] and [22] for low-resolution and 1-bit ADCs, respectively. A recent line of works employsmachine learning techniques in scenarios where obtaining accurate channel state informationwith low-resolution ADCs is impractical (see, e.g., [23], [24]). [8] analyzes the performanceof linear precoding schemes for downlink systems with 1-bit DACs. A similar analysis withmulti-bit DACs is presented in [10] considering both linear and nonlinear precoding, and in[11] considering linear precoding with oversampling in OFDM downlink systems. Lastly, [25]proposes a general optimization framework for downlink precoding with 1-bit DACs and constantenvelope assuming quadrature amplitude modulation (QAM) transmit symbols.
A. Contribution
This paper broadens prior analytical studies on the channel estimation and the data decoding inmassive MIMO uplink systems with 1-bit ADCs. On the one hand, existing works do not provideexact expressions of the mean squared error (MSE) of the channel estimation, which makes itdifficult to analyze its performance with respect to different parameters. We fill this gap by derivingsuch an expression for a general class of linear estimators with the structure of the one proposedin [14], which can be obtained from the state-of-the-art linear estimator in [7] by ignoring thetemporal correlation of the quantization distortion. As a valuable side product, we obtain a newestimator with the same simple structure but with significantly better accuracy. On the other hand,in the context of data decoding, existing works fail to characterize the statistical properties of theestimated symbols. In this regard, an interesting SNR trade-off was observed in [9], whereby theestimated symbols resulting from transmit symbols with the same phase overlap at high SNR;however, this aspect has not been formally described in the literature. We fill this gap by analyzingthe expected value and the variance of the estimated symbols along with their asymptotic behavior at high SNR. These results ultimately impact the symbol error rate (SER) and thus provideimportant practical insights into the design and the implementation of 1-bit quantized systems.The contributions of this paper are summarized as follows: • For the channel estimation with 1-bit ADCs, we derive a general closed-form expression ofthe MSE for linear estimators with the structure of the one proposed in [14]. This enables aprecise characterization of the performance of the channel estimation with respect to theSNR, the number of user equipments (UEs), and the pilot length. • Using the above result, we propose a novel linear estimator with the same simple structure ofthe one proposed in [14] but significantly enhanced performance in terms of MSE. Throughnumerical simulations with uncorrelated channels, we show that our proposed estimatorachieves the same accuracy as the (more complex) estimator proposed in [7]. • For the data decoding with 1-bit ADCs, we characterize the statistical properties of theestimated symbols by deriving closed-form expressions of the expected value and the variancewhen maximum ratio combining (MRC) is adopted at the BS. These can be exploited toefficiently implement MLD and, potentially, to design the set of transmit symbols. • Building on the proposed analysis, we provide a thorough discussion on the effect of 1-bitquantization on both the channel estimation and the data decoding. For each of the twoaspects, we describe a fundamental SNR trade-off, according to which operating at theright noise level significantly enhances the system performance. In this respect, the optimaltransmit SNR for the channel estimation is shown to decrease as the pilot length increases.
Outline.
The rest of the paper is structured as follows. Section II introduces the system modelwith 1-bit ADCs. Sections III and IV present our performance analysis results on the channelestimation and the data decoding, respectively, each including dedicated numerical results anddiscussion. Finally, Section V summarizes our contributions and draws some concluding remarks.
Notation. A = ( A m,n ) specifies that A m,n is the ( m, n ) th entry of matrix A ; likewise, a = ( a n ) specifies that a n is the n th entry of vector a . ( · ) T , ( · ) H , and ( · ) ∗ represent the transpose, Hermitiantranspose, and conjugate operators, respectively. Re[ · ] and Im[ · ] denote the real part and imaginarypart operators, respectively, whereas j is the imaginary unit. E [ · ] and V [ · ] are the expectationand variance operators, respectively. I N , N , and N denote the N -dimensional identity matrix,all-zero vector, and all-one vector, respectively. sgn( · ) is the sign function and ( a mod b ) denotesthe modulo operator, whereas {·} is used to represent sets. vec [ · ] is the vectorization operator, whereas ⊗ denotes the Kronecker product. Lastly, N ( N , Σ ) and CN ( N , Σ ) are the real andcomplex N -variate normal distributions with zero mean and covariance matrix Σ , respectively.II. S YSTEM M ODEL
Consider the scenario depicted in Fig. 1, where a BS with M antennas serves K single-antennaUEs in the uplink. Let H (cid:44) ( H m,k ) ∈ C M × K denote the uplink channel matrix whose entriesare assumed to be distributed independently as CN (0 , (as, e.g., in [7], [9]–[11]). Each UEtransmits with power ρ and the additive white Gaussian noise (AWGN) at the BS has unitvariance: thus, ρ can be interpreted as the transmit SNR. Note that the same transmit SNR isassumed for the two phases of channel estimation and uplink data transmission. Each BS antennais connected with two 1-bit ADCs, one for the in-phase component and one for the quadraturecomponent of the receive signal. Therefore, according to [9], we introduce the 1-bit quantizationfunction Q ( · ) : C L × N → Q , with Q ( A ) (cid:44) (cid:114) ρK + 12 (cid:16) sgn (cid:0) Re[ A ] (cid:1) + j sgn (cid:0) Im[ A ] (cid:1)(cid:17) (1)and where Q (cid:44) (cid:113) ρK +12 {± ± j } L × N is the set containing the scaled symbols of the quadraturephase-shift keying (QPSK) constellation. A. Channel Estimation
In the channel estimation phase, the UEs simultaneously transmit their uplink pilots of length τ . Let P (cid:44) ( P u,k ) ∈ C τ × K denote the pilot matrix whose columns correspond to the pilots usedby the UEs, with | P u,k | = 1 , ∀ u, k : assuming τ ≥ K and orthogonal pilots among the UEs, wehave P H P = τ I K . Hence, the receive signal at the BS prior to quantization is given by Y p (cid:44) √ ρ HP H + Z p ∈ C M × τ (2)where Z p (cid:44) ( Z m,u ) ∈ C M × τ is the AWGN matrix with entries distributed as CN (0 , . Then, atthe output of the ADCs, we have R p (cid:44) Q ( Y p ) ∈ C M × τ (3)with R p = ( R m,u ) , which is used by the BS to estimate H . Some comments are in order. Firstly,correlating the quantized receive signal R p in (3) with P (as done next in (5) and (17)) results inresidual pilot contamination even when the pilots are orthogonal (see, e.g., [14]). Secondly, thepilots should be preferably chosen such that their entries span the unit circle so as to accuratelyestimate the phases, especially at high SNR. This is explained in Appendix I, which provides adetailed discussion on the channel estimation with 1-bit ADCs. A linear minimum MSE (MMSE) estimator based on the Bussgang decomposition (see [18]),which we refer to as Bussgang linear MMSE (BLM) estimator, was proposed in [7], whereby h (cid:44) vec[ H ] ∈ C MK × is estimated as ˆ h BLM (cid:44) (cid:114) π ρ ¯ P T Σ − p r p ∈ C MK × (4)with ¯ P (cid:44) P ⊗ I M ∈ C Mτ × MK and r p (cid:44) vec[ R p ] ∈ C Mτ × , and where Σ p (cid:44) E [ r p r H p ] ∈ C Mτ × Mτ denotes the covariance matrix of r p . A linear estimator with a simpler structure can be obtainedfrom (4) by ignoring the temporal correlation of the quantization distortion, which implies that theoff-diagonal entries of Σ p are zero. Such an estimator was proposed in [14] (and later extendedto the case of multi-bit ADCs in [9]), whereby H is estimated as ˆ H (cid:44) √ Ψ R p P ∈ C M × K (5)where we have defined Ψ (cid:44) π ρ (cid:18) π ρ ( τ − K ) + ρK + 1 (cid:19) − . (6)Note that, when τ = K , (5) coincides with (4) since Σ p = ( ρK +1) I MK ; otherwise, (5) accuratelyapproximates (4) at low SNR or when K is large [9]. Due to the presence of Σ − p , it is difficultto analyze the performance of the BLM estimator in (4) with respect to the different parameters.Hence, in Section III, we focus on analyzing the estimator in (5); in addition, we propose a newestimator that is characterized by the same simple structure but with significantly better accuracy. B. Uplink Data Transmission
Let x k ∈ C be the transmit symbol of UE k , with E (cid:2) | x k | (cid:3) = 1 and x (cid:44) ( x k ) ∈ C K × . Thereceive signal at the BS prior to quantization is given by y (cid:44) √ ρ Hx + z ∈ C M × (7)where z (cid:44) ( z m ) ∈ C M × is the AWGN term with entries distributed as CN (0 , . Then, at theoutput of the ADCs, we have r (cid:44) Q ( y ) ∈ C M × (8)and the BS obtains a soft estimate of x as ˆ x (cid:44) V H r ∈ C K × (9)where V ∈ C M × K is the combining matrix adopted at the BS. Finally, the decoding processassociates each estimated symbol to a transmit symbol (e.g., via MLD). In Section IV, we focus on Note that, in case of correlated channels, the channel covariance can be embedded into (4) as described in [7]. characterizing the statistical properties of the estimated symbols when MRC is adopted at the BS.III. C
HANNEL E STIMATION WITH BIT
ADC S In this section, we are interested in characterizing the performance of the channel estimationwith respect to the different parameters when 1-bit ADCs are adopted at each BS antenna(see Section II-A). In this regard, we focus on the estimator in (5) since the presence of Σ − p considerably complicates the analysis of the BLM estimator in (4). Subsequently, we propose anovel estimator with the same simple structure of (5) and significantly improved accuracy. Tothis end, we first introduce the following proposition, which will be used to prove several of ourresults in this section and in Section IV. Proposition 1.
Let ζ ∼ N (0 , α I N ) with α > . For a , a ∈ R N × , we have E (cid:2) sgn( a T1 ζ )sgn( a T2 ζ ) (cid:3) = Ω (cid:18) a T1 a (cid:107) a (cid:107) (cid:107) a (cid:107) (cid:19) (10)where we have defined Ω( w ) (cid:44) π arcsin( w ) . (11) Proof:
See Appendix II.
A. MSE of the Channel Estimation
The (normalized) MSE of the channel estimation when (5) is used is defined asMSE p (cid:44) M K E (cid:2) (cid:107) ˆ H − H (cid:107) (cid:3) . (12)In [9, Eq. (23)] and in [7, Eq. (17)], the closed-form expression of (12) was derived for the caseof τ = K , which gives MSE p = 1 − π ρKρK + 1 . (13)Note that the above expression is lower bounded by − π (cid:39) . , which is achieved in thelimit of ρK → ∞ . Hence, in realistic scenarios (especially for small values of K ), using a pilotlength that is equal to the number of UEs results in quite inaccurate channel estimates. In general, τ should be sufficiently large to compensate for the low granularity of the ADCs, as detailedin Appendix I. For this reason, we derive a general closed-form expression of (12) when anestimator with the structure of (5) is used, which is valid for any value of τ and K . Theorem 1.
Suppose that the estimator in (5) is used with arbitrary Ψ . Then, the MSE of thechannel estimation in (12) is given by MSE p = 1 + ( ρK + 1)Ψ( τ + ∆) − (cid:114) π ρ Ψ τ (14)where we have defined ∆ (cid:44) K K (cid:88) k =1 (cid:88) u (cid:54) = v (cid:18) Re[ P ∗ u,k P v,k ]Ω (cid:18) ρ (cid:80) Ki =1 Re[ P u,i P ∗ v,i ] ρK + 1 (cid:19) − Im[ P ∗ u,k P v,k ]Ω (cid:18) ρ (cid:80) Ki =1 Im[ P u,i P ∗ v,i ] ρK + 1 (cid:19)(cid:19) . (15) Proof:
See Appendix III.The result of Theorem 1 enables a precise characterization of the performance of estimators withthe structure of (5), i.e., with arbitrary Ψ , with respect to the transmit SNR ρ , the number ofUEs K , and the pilot length τ . In particular, when Ψ defined in (6) is used, (14) becomesMSE p = 1 − π ρ (cid:18) π ρ ( τ − K ) + ρK + 1 (cid:19) − (cid:18) π ρτ ( τ − K ) + ( ρK + 1)( τ − ∆) (cid:19) . (16)When τ = K , (16) recovers the expression in (13): in fact, τ = K implies ∆ = 0 since (cid:80) Ki =1 P u,i P ∗ v,i = 0 , ∀ u (cid:54) = v . Moreover, we point out that the parameter ∆ in (15) is roughlyproportional to τ ( τ − .Now, if we consider Ψ as a tuning parameter, we can minimize MSE p in (14) by optimizingover Ψ . As a result, we obtain a novel estimator that is characterized by the same simple structureof (5) and enhanced performance in terms of MSE. Theorem 2.
Suppose that the estimator ˆ H (cid:48) (cid:44) √ Ψ (cid:48) R p P ∈ C M × K (17)is used, where Ψ (cid:48) is obtained by minimizing (14) with respect to Ψ and is defined as Ψ (cid:48) (cid:44) π ρ ( ρK + 1) τ ( τ + ∆) (18)with ∆ defined in (15). Then, the MSE of the channel estimation is given byMSE (cid:48) p (cid:44) M K E (cid:2) (cid:107) ˆ H (cid:48) − H (cid:107) (cid:3) (19) = 1 − π ρτ ( ρK + 1)( τ + ∆) (20) ≤ MSE p (21)with MSE p given in (16). Proof:
Since (14) is a convex function of Ψ , Ψ (cid:48) in (18) can be obtained by setting ddΨ MSE p = 0 . Then, since (16) is a special case of (14), it follows that MSE (cid:48) p ≤ MSE p . When τ = K , our proposed estimator in (17) coincides with the estimator in (5), and, in turn, with the BLM estimator in (4): in fact, τ = K implies that Ψ (cid:48) in (18) reduces to Ψ in (6) and(20) recovers the expression in (13). Since Ψ (cid:48) results from the optimization of (14) over Ψ , ourproposed estimator shall be always preferred to the estimator in (5), and the performance gapbetween the two widens with τ − K . Furthermore, the associated MSE expression in (20) turnsout to be more tractable than its counterpart in (16). Remarkably, in Section III-C, we shownumerically that our proposed estimator achieves the same accuracy as the BLM estimator in (4).It is of particular interest to study the asymptotic behavior of the MSE of the channel estimationat high SNR. Corollary 1.
From Theorems 1 and 2, in the limit of ρ → ∞ , we have lim ρ →∞ MSE p = 1 − π (cid:18) π ( τ − K ) + K (cid:19) − (cid:18) π τ ( τ − K ) + K ( τ − ¯∆) (cid:19) (22)and lim ρ →∞ MSE (cid:48) p = 1 − π τ K ( τ + ¯∆) (23)where we have defined ¯∆ (cid:44) K K (cid:88) k =1 (cid:88) u (cid:54) = v (cid:18) Re[ P ∗ u,k P v,k ]Ω (cid:18) (cid:80) Ki =1 Re[ P u,i P ∗ v,i ] K (cid:19) − Im[ P ∗ u,k P v,k ]Ω (cid:18) (cid:80) Ki =1 Im[ P u,i P ∗ v,i ] K (cid:19)(cid:19) . (24)The results of Corollary 1 show that arbitrarily increasing the transmit SNR is detrimental forthe performance of the channel estimation, since the right amount of noise is necessary torecover the difference in amplitude between channel entries (see Appendix I). This is in sheercontrast with the case of infinite-resolution ADCs, where boosting ρ produces the same beneficialnoise-averaging effect as increasing τ . B. Tractable Upper Bounds
The MSE expressions derived so far depend on the specific pilot choice through the parameter ∆ in (15). To gain more practical insights, we now consider the single-UE case (i.e., K = 1 )where p (cid:44) ( p u ) ∈ C τ × denotes the pilot used by the UE. In this setting, tractable upper boundson the MSE expressions can be obtained by fixing p such that p u ∈ {± β, ± j β } , ∀ u , where β ∈ C and | β | = 1 ; in the rest of the paper, when referring to this case, we will simply use p = τ . In fact, as detailed in Appendix I, such a structure of p represents the worst possiblepilot choice, especially at high SNR. Hence, from (15), we have ∆ = τ ( τ − (cid:18) ρρ + 1 (cid:19) (25) and, plugging (25) into (16) and (22) yieldsMSE p = 1 − π ρτ (cid:18) π ρ ( τ − ρ +1 (cid:19) − (cid:18) π ρ ( τ − ρ +1) (cid:18) − ( τ − (cid:18) ρρ +1 (cid:19)(cid:19)(cid:19) (26)and lim ρ →∞ MSE p = 1 − π τ (cid:18) π ( τ −
1) + 1 (cid:19) − (cid:18) π ( τ − − τ + 2 (cid:19) (27)respectively. In addition, considering (26) in the limit of τ → ∞ , we have lim τ →∞ MSE p = π ρ + 1 ρ Ω (cid:18) ρρ + 1 (cid:19) − . (28)Likewise, plugging (25) into (20) and (23) yieldsMSE (cid:48) p = 1 − π ρτ ( ρ + 1) (cid:0) τ − (cid:0) ρρ +1 (cid:1)(cid:1) (29)and lim ρ →∞ MSE (cid:48) p = 1 − π (30)respectively. In addition, considering (29) in the limit of τ → ∞ , we have lim τ →∞ MSE (cid:48) p = 1 − π ρ ( ρ + 1)Ω (cid:0) ρρ +1 (cid:1) . (31)Some comments are in order. Firstly, when τ = 1 , both (26) and (29) recover the expression in(13) with K = 1 . Secondly, (30) does not depend on τ since, in the absence of noise, estimatingthe channel repeatedly over the same pilot symbol does not bring any benefit. Thirdly, it canbe shown that (26) and (29) are quasiconvex functions of ρ and, as such, they have a uniqueminimum. This defines a clear SNR trade-off, according to which operating at the right noise levelenhances the channel estimation accuracy. In particular, as discussed in Appendix I, we have that: • At low SNR, the channel estimates are corrupted by the strong noise; • At high SNR, the difference in amplitude between channel entries cannot be recovered.In general, for τ > , the value of ρ that minimizes (29), denoted by ρ (cid:63) , satisfies π ρ (cid:63) √ ρ (cid:63) − Ω (cid:18) ρ (cid:63) ρ (cid:63) + 1 (cid:19) = 1 τ − . (32)Since the left-hand side of (32) monotonically increases with the transmit SNR, ρ (cid:63) decreases asthe left-hand side of (32) decreases, i.e., as τ increases. This means that using longer pilots allowsto operate at lower SNR as the noise can be averaged out more efficiently. This interdependencebetween ρ and τ can be also observed from (31): in the limit of τ → ∞ , since lim w → w arcsin( w ) = 1 ,it follows that MSE (cid:48) p → as ρ → . Lastly, it is shown in Section III-C that the upper boundsobtained by fixing p = τ are remarkably tight at low SNR and up to the region around the optimal value of ρ . Therefore, the above observations also apply to the general case. C. Numerical Results and Discussion
We now focus on the performance evaluation of the channel estimation with 1-bit ADCswith respect to the different parameters based on the analytical results presented in Sec-tions III-A and III-B. In this regard, when
K > , we use a pilot matrix P composed ofthe first K columns of the τ -dimensional discrete Fourier transform (DFT) matrix; however, weremark that our analytical framework is valid for any choice of P satisfying P H P = τ I K and | P u,k | = 1 , ∀ u, k . On the other hand, when K = 1 , we use the pilots p = d , where d (cid:44) [1 , e − j πτ , e − j πτ , . . . , e − j ( τ − πτ ] T ∈ C τ × (33)denotes the second column of the τ -dimensional DFT matrix, and p = τ : these represent thebest and the worst possible pilot choices, respectively (see Appendix I for more details on thepilot choice). We thus consider the expressions of MSE p derived in (16), resulting from theestimator in (5) (see [14]), and MSE (cid:48) p derived in (20), resulting from our proposed estimatorin (17). In addition, we consider the MSE of the channel estimation resulting from the BLMestimator in (4) (see [7]), defined asMSE BLMp (cid:44) M K (cid:13)(cid:13) ˆ h BLM − h (cid:13)(cid:13) (34)which we compute numerically by means of Monte Carlo simulations with independentchannel realizations. To facilitate the computation of the channel estimates as in (4), we derivethe closed-form expression of Σ p in the following proposition. Proposition 2.
Suppose that the entries of H are distributed independently as CN (0 , . Then,the covariance matrix of r p can be written as Σ p = ( ρK + 1) Φ ⊗ I M (35)where we have defined Φ (cid:44) (Φ u,v ) ∈ C τ × τ , with Φ u,v (cid:44) Ω (cid:18) ρ (cid:80) Kk =1 Re[ P u,k P ∗ v,k ] ρK + 1 (cid:19) − j Ω (cid:18) ρ (cid:80) Kk =1 Im[ P u,k P ∗ v,k ] ρK + 1 (cid:19) . (36) Proof:
See Appendix IV.The result of Proposition 2 allows to avoid the numerical computation of Σ p , which significantlysimplifies the computation of the channel estimates with the BLM estimator. Nonetheless, the For instance, one can use orthogonal Zadoff-Chu sequences, which are widely adopted in the LTE air interface [26]. Note that, according to Proposition 2 and the properties of the Kronecker product, we can write Σ − p = ρK +1 Φ − ⊗ I M . −
10 0 10 20 30 40 ρ [dB]0 . . . . . . . . M S E o f t h ec h a nn e l e s t i m a t i o n × − MSE
BLMp
MSE p approx.MSE p lim ρ →∞ MSE p MSE p lim ρ →∞ MSE p
10 15 201 . . . . × − (a) K = 4 , τ = 32 . −
10 0 10 20 30 40 ρ [dB]1 . . . . . . . . M S E o f t h ec h a nn e l e s t i m a t i o n × − K = 32 K = 16 K = 8 MSE p lim ρ →∞ MSE p MSE p lim ρ →∞ MSE p (b) K ∈ , , , τ = 128 . −
10 0 10 20 30 40 ρ [dB]123456 M S E o f t h ec h a nn e l e s t i m a t i o n × − p = d p = τ MSE p lim ρ →∞ MSE p MSE p lim ρ →∞ MSE p (c) K = 1 , τ = 32 . Fig. 2: MSE of the channel estimation against the transmit SNR. complexity involved in obtaining MSE
BLMp in (34) via Monte Carlo simulations with reasonableaccuracy remains prohibitive for large values of K and τ . Fig. 2 illustrates the MSE of the channelestimation against the transmit SNR ρ , also including the asymptotic MSE expressions in (22) and(23). Fig. 2(a) considers K = 4 and τ = 32 ; here “MSE p approx.” indicates the approximated MSEexpression in [9, Eq. (23)], which is exact when τ = K . We begin by observing that our proposedestimator in (17) outperforms the estimator in (5) and reduces the MSE of the channel estimationby . at high SNR. Even more remarkably, MSE (cid:48) p exactly matches MSE BLMp , which signifiesthat our proposed estimator achieves the same accuracy as the BLM estimator in (4) (see alsoFig. 3(a)). In the following plots, we thus omit MSE
BLMp , which allows us to evaluate scenarioswith more UEs and longer pilots. A rigorous comparative analysis of our proposed estimator and τ . . . . . . . . . M S E o f t h ec h a nn e l e s t i m a t i o n × − MSE
BLMp
MSE p approx.MSE p MSE p
32 40 48 56 641 . . . × − (a) K = 4 , ρ = 10 dB. τ . . . . . . . . M S E o f t h ec h a nn e l e s t i m a t i o n × − K = 32 K = 16 K = 8 MSE p MSE p (b) ρ = 10 dB. τ . . . . . . M S E o f t h ec h a nn e l e s t i m a t i o n × − K = τ/ K = τ/ K = τ/ MSE p MSE p (c) ρ = 10 dB. O p t i m a l M S E × − MSE p
64 128 192 256 320 384 448 512 τ − − − − − O p t i m a l ρ [ d B ] ρ ? (d) K = 1 , p = τ . Fig. 3: MSE of the channel estimation against the pilot length. the BLM estimator is left for future work. On the other hand, the approximated MSE expressionin [9, Eq. (23)] proves to be highly unreliable even for ρ < dB. Lastly, we highlight the SNRtrade-off described in Sections III-A and III-B as well as in Appendix I, whereby the MSE of thechannel estimation exhibits a valley at about ρ = 3 dB. Fig. 2(b) considers τ = 128 and differentvalues of K . Here, the gap between MSE (cid:48) p and MSE p widens as the K decreases, reaching . for K = 8 at high SNR. Moreover, the SNR trade-off appears more evident for small valuesof K . Fig. 2(c) considers the single-UE case, showing that the upper bounds in (26) and (29)obtained by fixing p = τ are remarkably tight at low SNR and up to the region around theoptimal transmit SNR. Note that the optimal value of ρ with p = τ satisfies the condition in(32) and gives an accurate approximation of the optimal value of ρ with p = d . Fig. 3 plots the MSE of the channel estimation against the pilot length τ . The transmit SNRis fixed to ρ = 10 dB in Fig. 3(a)–(c), whereas Fig. 3(d) considers the optimized transmit SNR(we recall that ρ should be reduced as τ increases to enhance the channel estimation accuracy).Fig. 3(a) considers K = 4 , where such a modest number of UEs allows us to evaluate the BLMestimator in (4) as well. We first observe that our proposed estimator in (17) reduces the MSE ofthe channel estimation with respect to the estimator in (5) by for τ = 128 . Additionally, asin Fig. 2(a), MSE (cid:48) p exactly matches MSE BLMp , which means that our proposed estimator achievesthe same accuracy as the BLM estimator. On the other hand, the approximated MSE expressionin [9, Eq. (23)] proves to be highly unreliable as the gap with the exact expression increaseswith τ − K . Fig. 3(b) considers different values of K , showing that the gap between MSE (cid:48) p andMSE p widens as K decreases and reaches . for K = 8 and τ = 128 . Fig. 3(c) and examinesthe case where the number of UEs grows together with the pilot length. In this setting, the gapbetween MSE (cid:48) p and MSE p is roughly constant and increases with the ratio τK , reaching about for τK = 8 . Lastly, Fig. 3(d) considers the single-UE case and the upper bound on MSE (cid:48) p in (29)obtained by fixing p = τ , which is optimized over the transmit SNR for each τ . As discussed inSection III-B, the optimal value of ρ satisfies the condition in (32) and decreases as τ increases.IV. D ATA D ECODING WITH BIT
ADC
S AND
MRCIn this section, we are interested in characterizing the performance of the data decodingwith respect to the different parameters when 1-bit ADCs are adopted at each BS antenna (seeSection II-B). In this regard, we consider the scenario where the BS uses our proposed estimatorin (17) in the channel estimation phase and the MRC receiver in the data decoding phase. Underthese assumptions, the combining matrix is given by V = ˆ H and we can rewrite (9) as ˆ x = √ Ψ (cid:48) P H R H p r (37) = ρ + 12 √ Ψ (cid:48) P H (cid:16) sgn (cid:0) Re[ √ ρ HP H + Z p ] (cid:1) + j sgn (cid:0) Im[ √ ρ HP H + Z p ] (cid:1)(cid:17) H × (cid:16) sgn (cid:0) Re[ √ ρ Hx + z ] (cid:1) + j sgn (cid:0) Im[ √ ρ Hx + z ] (cid:1)(cid:17) (38)with Ψ (cid:48) defined in (18). In the following, we focus on the single-UE case (i.e., K = 1 ) andcharacterize the statistical properties of the estimated symbols. Note that, in a multi-UE massiveMIMO context with infinite-resolution ADCs, MRC asymptotically becomes the optimal receivestrategy as the number of BS antennas increases. However, when the MRC receiver results fromthe quantized channel estimation, it cannot be perfectly aligned with the channel matrix, resulting in residual multi-UE interference. Hence, the following analysis of the single-UE case does notconsider this interference; nonetheless, the latter can be straightforwardly included at the expenseof more involved and less insightful expressions. A. Expected Value and Variance of the Estimated Symbols
Let x ∈ S be the transmit symbol of the UE, where S (cid:44) { s , . . . , s L } denotes the set oftransmit symbols, with s (cid:96) ∈ C , ∀ (cid:96) (for instance, S may correspond to the QPSK or 16-QAMconstellation). To facilitate the data decoding process at the BS, for each transmit symbol s (cid:96) ∈ S ,we are interested in deriving the closed-form expression of the expected value of the resultingestimated symbol ˆ s (cid:96) . Theorem 3.
Assuming K = 1 and MRC, for each transmit symbol s (cid:96) ∈ S , the expected valueof the resulting estimated symbol ˆ s (cid:96) , denoted by E (cid:96) (cid:44) E [ˆ s (cid:96) ] , is given by E (cid:96) = (cid:114) π ρM ττ +∆ τ (cid:88) u =1 p ∗ u (cid:18) Ω (cid:18) ρ Re[ p u s (cid:96) ] (cid:112) ( ρ +1)( ρ | s (cid:96) | +1) (cid:19) + j Ω (cid:18) ρ Im[ p u s (cid:96) ] (cid:112) ( ρ +1)( ρ | s (cid:96) | +1) (cid:19)(cid:19) (39)with ∆ defined in (15), which can be simplified for K = 1 as ∆ = (cid:88) u (cid:54) = v (cid:18) Re[ p ∗ u p v ]Ω (cid:18) ρ Re[ p u p ∗ v ] ρ + 1 (cid:19) − Im[ p ∗ u p v ]Ω (cid:18) ρ Im[ p u p ∗ v ] ρ + 1 (cid:19)(cid:19) . (40) Proof:
See Appendix V.The result of Theorem 3 can be exploited to efficiently implement MLD. In this context, eachestimated symbol ˆ x resulting from transmitting x ∈ S can be readily mapped to one of theexpected values { E , . . . , E L } derived as in (39) without any prior Monte Carlo computationof the latter. To further simplify the process and avoid computing the distance between ˆ x andeach E (cid:96) , one can construct the Voronoi tessellation based on { E , . . . , E L } (and possibly otheravailable information) obtaining well-defined decoding regions.Now, for each transmit symbol s (cid:96) ∈ S , we derive the closed-form expression of the varianceof the resulting estimated symbol ˆ s (cid:96) . Theorem 4.
Assuming K = 1 and MRC, for each transmit symbol s (cid:96) ∈ S , the variance of theresulting estimated symbol ˆ s (cid:96) , denoted by V (cid:96) (cid:44) V [ˆ s (cid:96) ] , is given by V (cid:96) = 2 π ρM τ τ + ∆ − M | E (cid:96) | (41)with ∆ given in (40) and where E (cid:96) is derived in closed form in (39). Proof:
See Appendix VI. The result of Theorem 4 quantifies the dispersion of the estimated symbols about their expectedvalue, which results from the 1-bit quantization applied to both the channel estimation (through theMRC receiver) and the data decoding. This dispersion is not isotropic and assumes different shapesfor different transmit symbols, as illustrated in Section IV-C. Some additional comments are inorder. Firstly, V (cid:96) reduces as | s (cid:96) | increases due to the negative term on the right-hand side of (41):this is somewhat intuitive since the transmit symbols that lie further from the origin are lesssubject to noise. Secondly, although V (cid:96) increases linearly with the number of BS antennas M ,the normalized variance V (cid:96) | E (cid:96) | (which expresses the relative dispersion of the estimated symbolsabout their expected value) is inversely proportional to M . Thirdly, the combined results ofTheorems 3 and 4 can be exploited to design the set of transmit symbols S by jointly minimizingthe relative dispersion and the overlap between different symbols after the estimation (this is leftfor future work). Lastly, in the context of MLD via Voronoi tessellation described above, onecan use the variance derived as in (41) to further refine the decoding regions.It is of particular interest to study the asymptotic behavior of the expected value and varianceof the estimated symbols at high SNR. Corollary 2.
From Theorems 3 and 4, in the limit of ρ → ∞ , we have lim ρ →∞ E (cid:96) √ ρ = (cid:114) π M ττ + ¯∆ τ (cid:88) u =1 p ∗ u (cid:18) Ω (cid:18) Re[ p u s (cid:96) ] | s (cid:96) | (cid:19) + j Ω (cid:18) Im[ p u s (cid:96) ] | s (cid:96) | (cid:19)(cid:19) (42)and lim ρ →∞ V (cid:96) ρ = 2 π M τ τ + ¯∆ − M lim ρ →∞ | E (cid:96) | ρ (43)with ¯∆ defined in (24), which can be simplified for K = 1 as ¯∆ = (cid:88) u (cid:54) = v (cid:16) Re[ p ∗ u p v ]Ω (cid:0) Re[ p u p ∗ v ] (cid:1) − Im[ p ∗ u p v ]Ω (cid:0) Im[ p u p ∗ v ] (cid:1)(cid:17) . (44)Corollary 2 formalizes a behavior of the estimated symbols that was observed in [9]. From (42),it emerges that, at high SNR, all the estimated symbols lie on a circle around the origin and theiramplitude no longer conveys any information. As a consequence, the estimated symbols resultingfrom transmit symbols with the same phase become indistinguishable in terms of their expectedvalue, which depends only on Re[ s (cid:96) ] | s (cid:96) | and Im[ s (cid:96) ] | s (cid:96) | . For instance, if S corresponds to the 16-QAMconstellation as in Section IV-C, the inner estimated symbols become indistinguishable from theouter estimated symbols with the same phase. Furthermore, according to (43), these estimatedsymbols become identical also in terms of variance. In the light of this, blindly minimizing the (normalized) variance of the estimated symbols is not the key to enhancing the systemperformance. Instead, the variance (which roughly decreases with the transmit SNR) should beminimized alongside the overlap between different symbols after the estimation (which generallyincreases with the transmit SNR). This determines a clear SNR trade-off, according to whichoperating at the right noise level enhances the data decoding accuracy and thus reduces the SER. B. Tractable Upper Bounds
As done in Section III-B for the MSE of the channel estimation, tractable upper bounds onthe normalized variance of the estimated symbols (i.e., that do not depend on the specific pilotchoice) can be obtained by fixing p = τ , since such a structure of p represents the worstpossible pilot choice (see Appendix I). Hence, plugging (25) into (41) and (43) yields V (cid:96) | E (cid:96) | = 1 M τ − (cid:0) ρρ +1 (cid:1) τ (cid:18)(cid:18) Ω (cid:18) ρ Re[ s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19)(cid:19) + (cid:18) Ω (cid:18) ρ Im[ s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19)(cid:19) (cid:19) − − M (45)and lim ρ →∞ V (cid:96) | E (cid:96) | = 1 M (cid:18)(cid:18) Ω (cid:18) Re[ s (cid:96) ] | s (cid:96) | (cid:19)(cid:19) + (cid:18) Ω (cid:18) Im[ s (cid:96) ] | s (cid:96) | (cid:19)(cid:19) (cid:19) − − M (46)respectively. In addition, considering (45) in the limit of τ → ∞ , we have lim τ →∞ V (cid:96) | E (cid:96) | = 1 M Ω (cid:18) ρρ + 1 (cid:19)(cid:18)(cid:18) Ω (cid:18) ρ Re[ s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19)(cid:19) + (cid:18) Ω (cid:18) ρ Im[ s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19)(cid:19) (cid:19) − − M . (47)Some comments are in order. Firstly, the normalized variance of the estimated symbols can bemade arbitrarily close to zero by increasing the number of BS antennas M . Secondly, (46) doesnot depend on τ since, in the absence of noise, estimating the channel repeatedly over the samepilot symbol does not bring any benefit. Thirdly, it can be shown that (45) is a quasiconvexfunction of ρ and, as such, it has a unique minimum that defines a further SNR trade-off. It isshown in Section IV-C that this SNR trade-off, which is inherited from the channel estimationphase through the MRC receiver, is not as significant as that described by Corollary 2. In fact,the normalized variance of the estimated symbols roughly decreases with ρ ; on the other hand,the difference in amplitude between symbols cannot be recovered if ρ is too high. Lastly, wepoint out that the system performance can be further enhanced by optimizing the transmit SNRseparately for the two phases of channel estimation and uplink data transmission. C. Numerical Results and Discussion
We now focus on the performance evaluation of the data decoding with 1-bit ADCs with respectto the different parameters using the analytical results presented in Sections IV-A and IV-B. Inthis regard, we assume that the BS uses our proposed estimator in (17) in the channel estimationphase and the MRC receiver in the data decoding phase. We thus consider the expressions of E (cid:96) and V (cid:96) derived in (39) and (41), respectively, for the single-UE case (i.e, K = 1 ). As inSection III-C, we use the pilots p = d , with d defined in (33), and p = τ . Moreover, wespecifically analyze the scenario where the set of transmit symbols S corresponds to the 16-QAMconstellation, i.e., S = √ (cid:8) ± ± j, ± ± j , ± ± j, ± ± j (cid:9) , which is normalized such that L (cid:80) L(cid:96) =1 | s (cid:96) | = 1 ; however, we remark that our analytical framework is valid for any choice of S .Fig. 4 plots the estimated symbols for different settings, where each 16-QAM symbol istransmitted over independent channel realizations and p = d is used in the channelestimation phase. The expected value of the estimated symbols is computed as in Theorem 3:this matches the sample average of the estimated symbols for each 16-QAM transmit symboland can be used to efficiently implement MLD. Comparing Fig. 4(a)–(c), which consider thesame transmit SNR and pilot length, the relative dispersion of the estimated symbols about theirexpected value reduces as the number of BS antennas grows from M = 64 to M = 256 . In fact,a higher granularity in the antenna domain allows to sum the contribution of a larger number ofindependent channel entries. On the other hand, comparing Fig. 4(b) and (d), which consider thesame number of BS antennas and transmit SNR, the relative dispersion of the estimated symbolsabout their expected value slightly intensifies as we decrease the pilot length from τ = 32 to τ = 8 . This stems from the overall diminished accuracy of the channel estimate used to computethe MRC receiver for each channel realization. Lastly, comparing Fig. 4(b) and (e)–(f), whichconsider the same number of BS antennas and pilot length, the estimated symbols resulting fromthe 16-QAM transmit symbols with the same phase, i.e., ± √ (1 ± j ) and ± √ (3 ± j , getcloser as the transmit SNR increases from ρ = 0 dB to ρ = 10 dB and they almost fully overlapfor ρ = 20 dB. This behavior was observed in [9] and is formalized in Corollary 2, accordingto which such estimated symbols become identical in terms of both their expected value andvariance at high SNR. In this respect, the SNR trade-off described in Sections IV-A and IV-B isquite evident: while the normalized variance of the estimated symbols roughly decreases with ρ ,the difference in amplitude between symbols cannot be recovered if ρ is too high. For the 16- − − × − − Q × ˆ s ‘ E ‘ (a) M = 64 , ρ = 0 dB, τ = 32 . − . − . . . . × − . − . . . . Q × ˆ s ‘ E ‘ (b) M = 128 , ρ = 0 dB, τ = 32 . − . − . . . . × − . − . . . . Q × ˆ s ‘ E ‘ (c) M = 256 , ρ = 0 dB, τ = 32 . − . − . . . . × − . − . . . . Q × ˆ s ‘ E ‘ (d) M = 128 , ρ = 0 dB, τ = 8 . − . − . . . . × − . − . . . . Q × ˆ s ‘ E ‘ (e) M = 128 , ρ = 10 dB, τ = 32 . − . − . . . . × − . − . . . . Q × ˆ s ‘ E ‘ (f) M = 128 , ρ = 20 dB, τ = 32 . Fig. 4: Estimated symbols with the MRC receiver, with 16-QAM transmit symbols and K = 1 . The expected valueof the estimated symbols is computed as in (39). QAM, this produces a SER of about since there are four pairs of indistinguishable estimatedsymbols (see also Fig. 8). In summary, having independent phases between the channel entries andoperating at the right noise level are crucial to accurately estimate the phases and the amplitudes,respectively; we refer to Appendix I and to the related discussion in [9] for more details.Let us now examine the behavior of the variance of the estimated symbols derived in Theorem 4.Fig. 5 considers ρ = 10 dB and τ = 32 , showing how the normalized variance of the estimatedsymbols V (cid:96) | E (cid:96) | decreases with the number of BS antennas M . The transmit symbols ± √ (1 ± j ) ,having the smallest power among the 16-QAM constellation, exhibit the most severe dispersion ofthe estimated symbols about their expected value. Furthermore, the upper bound in (45) becomesmore accurate as M grows. Fig. 6 considers M = 128 and τ = 32 , showing that V (cid:96) | E (cid:96) | generallydiminishes with the transmit SNR ρ except for the SNR trade-off exhibited with p = τ atabout ρ = 10 dB; here, the asymptotic expressions in (43) and (46) are also included. Despitethis trend, we recall that the difference in amplitude between symbols cannot be recovered if
32 64 96 128 160 192 224 256 M V ‘ / | E ‘ | × − p = τ p = d (1 + j ) / √ j ) / √ j / √ Fig. 5: Normalized variance of the estimated symbolsagainst the number of BS antennas, with 16-QAMtransmit symbols, K = 1 , ρ = 10 dB, and τ = 32 . −
10 0 10 20 30 40 ρ [dB]0246810 V ‘ / | E ‘ | × − ρ →∞ V ‘ | E ‘ | (1 + j ) / √ j ) / √ j / √
100 10 20 300 . . . . × − p = τ p = d Fig. 6: Normalized variance of the estimated symbolsagainst the transmit SNR, with 16-QAM transmit sym-bols, K = 1 , M = 128 , and τ = 32 . ρ is too high, as discussed in the previous paragraph for Fig. 4: thus, arbitrarily increasing thetransmit SNR is detrimental for the system performance. Lastly, Fig. 7 considers M = 128 and ρ = 10 dB, showing how V (cid:96) | E (cid:96) | reduces with the pilot length τ .We conclude this section by investigating the combined impact of the channel estimation andthe data decoding with 1-bit ADCs on the system performance in terms of SER, which wecompute numerically via Monte Carlo simulations with independent channel realizations.In this context, the symbols are decoded by means of MLD aided by the result of Theorem 3.Fig. 8 illustrates the SER against the transmit SNR ρ , with M = 128 and τ = 32 . Here, theSNR trade-off appears quite evident, whereby the SER decreases until it reaches its minimumat about ρ = 5 dB (where the upper bound obtained with p = τ proves to be remarkablytight) before escalating again. Then, the SER asymptotically reaches at high SNR, wherethe inner estimated symbols of the 16-QAM constellation become indistinguishable from theouter estimated symbols with the same phase (see also Fig. 4(f)). We remark that the SER canbe further reduced by optimizing the transmit SNR separately for the two phases of channelestimation and uplink data transmission, which is left for future work.V. C ONCLUSIONS
This paper presents an analytical framework for the channel estimation and the data decodingin massive MIMO uplink systems with 1-bit ADCs. First, we provide a precise characterization ofthe MSE of the channel estimation with respect to different parameters and additionally propose anovel linear estimator with significantly enhanced performance compared with existing estimators τ . . . . . . V ‘ / | E ‘ | × − p = τ p = d lim τ →∞ V ‘ | E ‘ | (1 + j ) / √ j ) / √ j / √ Fig. 7: Normalized variance of the estimated symbolsagainst the pilot length, with 16-QAM transmit symbols, K = 1 , M = 128 , and ρ = 10 dB. −
10 0 10 20 30 40 ρ [dB]0 . . . . . . . . . . S E R × − p = τ p = d . . . . . × − Fig. 8: SER against the transmit SNR, with 16-QAMtransmit symbols, K = 1 , M = 128 , and τ = 32 . with the same structure. For the data decoding, we characterize the expected value and thevariance of the estimated symbols when MRC is adopted, which can be exploited to efficientlyimplement MLD and to properly design the set of transmit symbols. The proposed analysis givesimportant practical insights into the design and the implementation of 1-bit quantized systems.In particular, it highlights a fundamental SNR trade-off, according to which arbitrarily increasingthe transmit SNR is detrimental for the system performance. In this respect, the optimal transmitSNR for the channel estimation is shown to decrease as the pilot length increases. Future workwill consider a rigorous comparative analysis between our proposed estimator and the BLMestimator, extensions of the proposed analytical framework to more realistic channel models, anda SER optimal design of the set of transmit symbols capitalizing on our data decoding analysis.A PPENDIX IF UNDAMENTALS OF C HANNEL E STIMATION WITH BIT
ADC S Assuming K = 1 , let h (cid:44) ( h m ) ∈ C M × and p (cid:44) ( p u ) ∈ C τ × denote the uplink channelvector and the pilot, respectively, of the UE. When an estimator with the structure of (5) is used,the channel estimate ˆ h (cid:44) (ˆ h m ) is obtained as ˆ h = √ Ψ Q √ ρh p ∗ + Z , · · · √ ρh p ∗ τ + Z ,τ ... . . . ... √ ρh M p ∗ + Z m, · · · √ ρh M p ∗ τ + Z M,τ p ... p τ (48)with ˆ h m = (cid:114) ρ + 12 Ψ τ (cid:88) u =1 p u (cid:16) sgn (cid:0) Re[ √ ρh m p ∗ u + Z m,u ] (cid:1) + j sgn (cid:0) Im[ √ ρh m p ∗ u + Z m,u ] (cid:1)(cid:17) . (49)The phase of each channel entry h m can be estimated with improved accuracy as the pilot lengthincreases, provided that the entries of p span the unit circle as much as possible. In this regard,the best pilot choice is given by the second column of the τ -dimensional DFT matrix definedin (33), since its entries are non-repeating and equispaced on the unit circle; on the contrary,the worst pilot choice is given by fixing p such that p u ∈ {± β, ± j β } , ∀ u , where β ∈ C and | β | = 1 (a simple choice is p = τ ). On the other hand, the estimation of the amplitude of h m benefits from operating at the right noise level, as we will see next.Let h m = α m e j θ m , with ϑ m (cid:44) (cid:0) θ m mod π (cid:1) , and let p u = e j φ u (recall that | p u | = 1 , ∀ u ).Assuming ρ → ∞ , the phase of h m can be estimated from Q ( h m p H ) p , where Q ( h m p ∗ u ) p u = Q (cid:0) e j ( θ m − φ u ) (cid:1) e j φ u (50) = (cid:113) ρ +12 (cid:0) sgn (cid:0) Re[ h m ] (cid:1) + j sgn (cid:0) Im[ h m ] (cid:1)(cid:1) e j φ u if φ u ∈ (cid:2) ϑ m − π , ϑ m (cid:3) , (cid:113) ρ +12 (cid:0) sgn (cid:0) Im[ h m ] (cid:1) − j sgn (cid:0) Re[ h m ] (cid:1)(cid:1) e j φ u if φ u ∈ (cid:2) ϑ m , ϑ m + π (cid:3) , (cid:113) ρ +12 (cid:0) − sgn (cid:0) Re[ h m ] (cid:1) − j sgn (cid:0) Im[ h m ] (cid:1)(cid:1) e j φ u if φ u ∈ (cid:2) ϑ m + π , ϑ m + π (cid:3) , (cid:113) ρ +12 (cid:0) − sgn (cid:0) Im[ h m ] (cid:1) + j sgn (cid:0) Re[ h m ] (cid:1)(cid:1) e j φ u if φ u ∈ (cid:2) ϑ m + π, ϑ m + π (cid:3) (51)i.e., Q (cid:0) e j ( θ m − φ u ) (cid:1) shifts quadrant according to the phase of p u . Assuming that the entries of p span the unit circle, in the limit of τ → ∞ , we have lim τ →∞ τ τ (cid:88) u =1 Q ( h m p ∗ u ) p u = 12 π (cid:114) ρ + 12 (cid:18)(cid:16) sgn (cid:0) Re[ h m ] (cid:1) + j sgn (cid:0) Im[ h m ] (cid:1)(cid:17) (cid:90) ϑ m ϑ m − π e j φ d φ + (cid:16) sgn (cid:0) Im[ h m ] (cid:1) − j sgn (cid:0) Re[ h m ] (cid:1)(cid:17) (cid:90) ϑ m + π ϑ m e j φ d φ + (cid:16) − sgn (cid:0) Re[ h m ] (cid:1) − j sgn (cid:0) Im[ h m ] (cid:1)(cid:17) (cid:90) ϑ m + πϑ m + π e j φ d φ + (cid:16) − sgn (cid:0) Im[ h m ] (cid:1) + j sgn (cid:0) Re[ h m ] (cid:1)(cid:17) (cid:90) ϑ m + π ϑ m + π e j φ d φ (cid:19) (52) = 2 π (cid:114) ρ + 12 (1 − j ) (cid:16) sgn (cid:0) Re[ h m ] (cid:1) + j sgn (cid:0) Im[ h m ] (cid:1)(cid:17) e j ϑ m (53)where (53) follows from (cid:90) ϑ m ϑ m − π e j φ d φ = (1 − j ) e j ϑ m , (cid:90) ϑ m + π ϑ m e j φ d φ = (1 + j ) e j ϑ m , (54) (cid:90) ϑ m + πϑ m + π e j φ d φ = ( − j ) e j ϑ m , (cid:90) ϑ m + π ϑ m + π e j φ d φ = ( − − j ) e j ϑ m . (55) Finally, from (53), we have (1 − j ) (cid:16) sgn (cid:0) Re[ h m ] (cid:1) + j sgn (cid:0) Im[ h m ] (cid:1)(cid:17) e j ϑ m = e j ϑ m if θ m ∈ (cid:2) , π (cid:3) ( i.e., if θ m = ϑ m ) , j e j ϑ m if θ m ∈ (cid:2) π , π (cid:3) ( i.e., if θ m = ϑ m + π ) , − e j ϑ m if θ m ∈ (cid:2) π, π (cid:3) ( i.e., if θ m = ϑ m + π ) , − j e j ϑ m if θ m ∈ (cid:2) π , π (cid:3) ( i.e., if θ m = ϑ m + π ) (56) = 2 e j θ m (57)which gives lim τ →∞ τ τ (cid:88) u =1 Q ( h m p ∗ u ) p u = 4 π (cid:114) ρ + 12 e j θ m . (58)Hence, the phase of h m can be estimated accurately if the pilot symbols span the unit circle and τ is sufficiently large. However, (58) does not include any information about the amplitude dueto the assumption that ρ → ∞ .Assuming now finite ρ and, for simplicity, p = τ , the amplitude of h m can be estimated from Q (cid:0) √ ρh m T τ + [ Z m, , . . . , Z m,τ ] (cid:1) τ , where Q ( √ ρh m + Z m,u ) = (cid:114) ρ +12 (cid:16) sgn (cid:0) √ ρ Re[ h m ]+Re[ Z m,u ] (cid:1) + j sgn (cid:0) √ ρ Im[ h m ]+Im[ Z m,u ] (cid:1)(cid:17) . (59)In the limit of τ → ∞ , we have lim τ →∞ τ τ (cid:88) u =1 Q ( √ ρh m + Z m,u ) = (cid:114) ρ + 12 (cid:16) erf (cid:0) √ ρ Re[ h m ] (cid:1) + j erf (cid:0) √ ρ Im[ h m ] (cid:1)(cid:17) (60)where erf ( w ) (cid:44) √ π (cid:82) w e − t d t denotes the error function. Since erf ( w ) is approximately linear for w ∈ [ − , , the difference in amplitude between channel entries can be estimated accurately iftheir real and imaginary parts lie in (cid:2) − √ ρ , √ ρ (cid:3) and τ is sufficiently large (a similar conclusioncan be drawn for p (cid:54) = τ ). On the other hand, if τ is not sufficiently large at low SNR, thechannel estimates are corrupted by the strong noise. Hence, the SNR plays an important rolesince operating at the right noise level significantly enhances the channel estimation accuracy.A PPENDIX
IIP
ROOF OF P ROPOSITION E (cid:2) sgn( X )sgn( X ) (cid:3) = P [ X > , X >
0] + P [ X < , X < − P [ X > , X < − P [ X < , X > . (61) For X = a T1 ζ and X = a T2 ζ , the first term on the right-hand side of (61) can be obtainedbuilding on [27] as P (cid:2) a T1 ζ > , a T2 ζ > (cid:3) = 14 (cid:18) (cid:18) a T1 a (cid:107) a (cid:107) (cid:107) a (cid:107) (cid:19)(cid:19) (62)where a T1 a (cid:107) a (cid:107) (cid:107) a (cid:107) represents the correlation coefficient between X and X , and the other termscan be obtained following similar steps. A PPENDIX
IIIP
ROOF OF T HEOREM p = 1 M K (cid:16) E (cid:2) tr( H H H ) (cid:3) + E (cid:2) tr( ˆ H H ˆ H ) (cid:3) − E (cid:104) Re (cid:2) tr( ˆ H H H ) (cid:3)(cid:105)(cid:17) (63)with E (cid:2) tr( H H H ) (cid:3) = M K . Next, we derive the second and third expectation terms of (63). Tothis end, we introduce the following definitions: A m,u (cid:44) sgn (cid:18) Re (cid:20) √ ρ K (cid:88) k =1 H m,k P ∗ u,k + Z m,u (cid:21)(cid:19) (64) = sgn (cid:18) √ ρ K (cid:88) k =1 (cid:0) Re[ H m,k ]Re[ P u,k ] + Im[ H m,k ]Im[ P u,k ] (cid:1) + Re[ Z m,u ] (cid:19) , (65) B m,u (cid:44) sgn (cid:18) Im (cid:20) √ ρ K (cid:88) k =1 H m,k P ∗ u,k + Z m,u (cid:21)(cid:19) (66) = sgn (cid:18) √ ρ K (cid:88) k =1 (cid:0) − Re[ H m,k ]Im[ P u,k ] + Im[ H m,k ]Re[ P u,k ] (cid:1) + Im[ Z m,u ] (cid:19) . (67) • Considering the second expectation term in (63), we have tr( ˆ H H ˆ H ) = ρK + 12 Ψ M (cid:88) m =1 K (cid:88) k =1 (cid:18) τ (cid:88) u =1 P u,k ( A m,u + j B m,u ) (cid:19) ∗ (cid:18) τ (cid:88) u =1 P u,k ( A m,u + j B m,u ) (cid:19) (68) = ρK + 12 Ψ M (cid:88) m =1 K (cid:88) k =1 τ (cid:88) u =1 τ (cid:88) v =1 P ∗ u,k P v,k (cid:0) A m,u A m,v + B m,u B m,v + j ( A m,u B m,v − B m,u A m,v ) (cid:1) (69) = ρK + 12 Ψ (cid:18) M Kτ + M (cid:88) m =1 K (cid:88) k =1 (cid:88) u (cid:54) = v P ∗ u,k P v,k (cid:0) A m,u A m,v + B m,u B m,v + j ( A m,u B m,v − B m,u A m,v ) (cid:1)(cid:19) . (70)Hence, the expected value of (70) is given by E (cid:2) tr( ˆ H H ˆ H ) (cid:3) = ( ρK + 1)Ψ M (cid:18) Kτ + K (cid:88) k =1 (cid:88) u (cid:54) = v P ∗ u,k P v,k (cid:18) Ω (cid:18) ρ (cid:80) Ki =1 Re[ P u,i P ∗ v,i ] ρK + 1 (cid:19) + j Ω (cid:18) ρ (cid:80) Ki =1 Im[ P u,i P ∗ v,i ] ρK + 1 (cid:19)(cid:19)(cid:19) (71) = ( ρK + 1)Ψ M (cid:18) Kτ + K (cid:88) k =1 (cid:88) u (cid:54) = v (cid:18) Re[ P ∗ u,k P v,k ]Ω (cid:18) ρ (cid:80) Ki =1 Re[ P u,i P ∗ v,i ] ρK + 1 (cid:19) − Im[ P ∗ u,k P v,k ]Ω (cid:18) ρ (cid:80) Ki =1 Im[ P u,i P ∗ v,i ] ρK + 1 (cid:19)(cid:19)(cid:19) (72)where (71) follows from E [ A m,u A m,v ] = E [ B m,u B m,v ] (73) = Ω (cid:18) ρ (cid:80) Kk =1 Re[ P u,k P ∗ v,k ] ρK + 1 (cid:19) , (74) E [ A m,u B m,v ] = Ω (cid:18) ρ (cid:80) Kk =1 Im[ P u,k P ∗ v,k ] ρK + 1 (cid:19) (75)which are derived by applying Proposition 1. For instance, E [ A m,u A m,v ] in (73)–(74) canbe obtained by plugging ζ = (cid:2) Re[ H m, ] , . . . , Re[ H m,K ] , Im[ H m, ] , . . . , Im[ H m,K ] , Re[ Z m,u ] , Re[ Z m,v ] (cid:3) T , (76) a = (cid:2) √ ρ Re[ P u, ] , . . . , √ ρ Re[ P u,K ] , √ ρ Im[ P u, ] , . . . , √ ρ Im[ P u,K ] , , (cid:3) T , (77) a = (cid:2) √ ρ Re[ P v, ] , . . . , √ ρ Re[ P v,K ] , √ ρ Im[ P v, ] , . . . , √ ρ Im[ P v,K ] , , (cid:3) T (78)with ζ ∼ N ( (2 K +2) , I (2 K +2) ) , into (10), which gives E (cid:20) sgn (cid:18) √ ρ K (cid:88) i =1 (cid:0) Re[ H m,i ]Re[ P u,i ] + Im[ H m,i ]Im[ P u,i ] (cid:1) + Re[ Z m,u ] (cid:19) × sgn (cid:18) √ ρ K (cid:88) i =1 (cid:0) Re[ H m,i ]Re[ P v,i ] + Im[ H m,i ]Im[ P v,i ] (cid:1) + Re[ Z m,v ] (cid:19)(cid:21) = Ω (cid:18) ρ (cid:80) Kk =1 (Re[ P u,k ]Re[ p v,k ] + Im[ P u,k ]Im[ P v,k ])( ρ (cid:80) Kk =1 (Re[ P u,k ] + Im[ P u,k ] ) + 1) ( ρ (cid:80) Kk =1 (Re[ P v,k ] + Im[ P v,k ] ) + 1) (cid:19) . (79) • Considering the third expectation term in (63), we have Re (cid:2) tr( ˆ H H H ) (cid:3) = (cid:114) ρK + 12 ΨRe (cid:20) M (cid:88) m =1 K (cid:88) k =1 (cid:18) τ (cid:88) u =1 P u,k ( A m,u + j B m,u ) (cid:19) ∗ H m,k (cid:21) (80) = (cid:114) ρK + 12 Ψ M (cid:88) m =1 K (cid:88) k =1 τ (cid:88) u =1 (cid:0) Re[ P u,k ]Re[ H m,k ] A m,u + Im[ P u,k ]Im[ H m,k ] A m,u + Re[ P u,k ]Im[ H m,k ] B m,u − Im[ P u,k ]Re[ H m,k ] B m,u (cid:1) . (81) Hence, the expected value of (81) is given by E (cid:104) Re (cid:2) tr( ˆ H H H ) (cid:3)(cid:105) = (cid:114) π ρ Ψ M Kτ (82)where (82) follows from E (cid:2) Re[ H m,k ] A m,u (cid:3) = E (cid:2) Im[ H m,k ] B m,u (cid:3) (83) = 1 √ π (cid:114) ρρK + 1 Re[ p u ] (84) E (cid:2) Im[ H m,k ] A m,u (cid:3) = − E (cid:2) Re[ H m,k ] B m,u (cid:3) (85) = 1 √ π (cid:114) ρρK + 1 Im[ p u ] . (86)Finally, the expression in (14) is obtained by plugging (72) and (82) into (63).A PPENDIX
IVP
ROOF OF P ROPOSITION (cid:0) ( u − M + m, ( v − M + n (cid:1) thentry of r p r H p as R m,u R ∗ n,v = ρK + 12 ( A m,u + j B m,u )( A n,v + j B n,v ) ∗ (87) = ρK + 12 (cid:0) A m,u A n,v + B m,u B n,v + j ( B m,u A n,v − A m,u B n,v ) (cid:1) (88)with R m,u R ∗ m,u = ρK + 1 . Hence, from (73)–(75), we obtain E [ R m,u R ∗ n,v ] = ( ρK + 1) (cid:16) Ω (cid:16) ρ (cid:80) Kk =1 Re[ P u,k P ∗ v,k ] ρK +1 (cid:17) − j Ω (cid:16) ρ (cid:80) Kk =1 Im[ P u,k P ∗ v,k ] ρK +1 (cid:17)(cid:17) if m = n, u (cid:54) = v, if m (cid:54) = n (89)and the expression in (35) readily follows.A PPENDIX VP ROOF OF T HEOREM a m,u (cid:44) sgn (cid:0) Re[ √ ρh m p ∗ u + Z m,u ] (cid:1) (90) = sgn (cid:16) √ ρ (cid:0) Re[ h m ]Re[ p u ] + Im[ h m ]Im[ p u ] (cid:1) + Re[ Z m,u ] (cid:17) , (91) b m,u (cid:44) sgn (cid:0) Im[ √ ρh m p ∗ u + Z m,u ] (cid:1) (92) = sgn (cid:16) √ ρ (cid:0) − Re[ h m ]Im[ p u ] + Im[ h m ]Re[ p u ] (cid:1) + Im[ Z m,u ] (cid:17) (93) Note that (90)–(93) are equivalent to (64)–(67) for K = 1 . and c m (cid:44) sgn (cid:0) Re[ √ ρh m s (cid:96) + z m ] (cid:1) (94) = sgn (cid:16) √ ρ (cid:0) Re[ h m ]Re[ s (cid:96) ] − Im[ h m ]Im[ s (cid:96) ] (cid:1) + Re[ z m ] (cid:17) , (95) d m (cid:44) sgn (cid:0) Im[ √ ρh m s (cid:96) + z m ] (cid:1) (96) = sgn (cid:16) √ ρ (cid:0) Re[ h m ]Im[ s (cid:96) ] + Im[ h m ]Re[ s (cid:96) ] (cid:1) + Im[ z m ] (cid:17) . (97)From (38), we can write the estimated symbol as ˆ s (cid:96) = ρ + 12 √ Ψ (cid:48) M (cid:88) m =1 τ (cid:88) u =1 p ∗ u ( a m,u + j b m,u ) ∗ ( c m + j d m ) (98) = ρ + 12 √ Ψ (cid:48) M (cid:88) m =1 τ (cid:88) u =1 p ∗ u (cid:0) a m,u c m + b m,u d m + j ( a m,u d m − b m,u c m ) (cid:1) . (99)Hence, the expression in (39) follows from E [ a m,u c m ] = E [ b m,u d m ] (100) = Ω (cid:18) ρ Re[ p u s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) , (101) E [ a m,u d m ] = − E [ b m,u c m ] (102) = Ω (cid:18) ρ Im[ p u s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) (103)which are derived again by applying Proposition 1. For instance, E [ a m,u c m ] in (100)–(101) canbe obtained by plugging ζ = (cid:2) Re[ h m ] , Im[ h m ] , Re[ Z m,u ] , Re[ z m ] (cid:3) T ∼ N ( , I ) , a = (cid:2) √ ρ Re[ p u ] , √ ρ Im[ p u ] , , (cid:3) T , and a = (cid:2) √ ρ Re[ s (cid:96) ] , −√ ρ Im[ s (cid:96) ] , , (cid:3) T into (10), which gives E (cid:104) sgn (cid:16) √ ρ (cid:0) Re[ h m ]Re[ p u ] + Im[ h m ]Im[ p u ] (cid:1) + Re[ Z m,u ] (cid:17) × sgn (cid:16) √ ρ (cid:0) Re[ h m ]Re[ s (cid:96) ] − Im[ h m ]Im[ s (cid:96) ] (cid:1) + Re[ z m ] (cid:17)(cid:105) = Ω (cid:18) ρ (Re[ p u ]Re[ s (cid:96) ] − Im[ p u ]Im[ s (cid:96) ]) (cid:112) ρ (Re[ p u ] + Im[ p u ] ) + 1 (cid:112) ρ (Re[ s (cid:96) ] + Im[ s (cid:96) ] ) + 1 (cid:19) . (104)A PPENDIX
VIP
ROOF OF T HEOREM ˆ s (cid:96) can be written as V (cid:96) = E (cid:2) | ˆ s (cid:96) | (cid:3) − | E (cid:96) | (105)where E (cid:2) | ˆ s (cid:96) | (cid:3) = E (cid:2) Re[ˆ s (cid:96) ] (cid:3) + E (cid:2) Im[ˆ s (cid:96) ] (cid:3) and (cf. (39)) | E (cid:96) | = ( ρ +1) Ψ (cid:48) M τ (cid:88) u =1 τ (cid:88) v =1 (cid:18) Re[ p ∗ u p v ] (cid:18) Ω (cid:18) ρ Re[ p u s (cid:96) ] (cid:112) ( ρ +1)( ρ | s (cid:96) | +1) (cid:19) Ω (cid:18) ρ Re[ p v s (cid:96) ] (cid:112) ( ρ +1)( ρ | s (cid:96) | +1) (cid:19) + Ω (cid:18) ρ Im[ p u s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) Ω (cid:18) ρ Im[ p v s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19)(cid:19) + Im[ p ∗ u p v ] (cid:18) Ω (cid:18) ρ Re[ p u s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) Ω (cid:18) ρ Im[ p v s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) − Ω (cid:18) ρ Im[ p u s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) Ω (cid:18) ρ Re[ p v s (cid:96) ] (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19)(cid:19)(cid:19) . (106)Furthermore, recalling the definitions in (90)–(97) and building on (98)–(99), we obtain Re[ˆ s (cid:96) ] = ( ρ +1) (cid:48) (cid:18) M (cid:88) m =1 τ (cid:88) u =1 (cid:0) Re[ p u ]( a m,u c m + b m,u d m )+Im[ p u ]( a m,u d m − b m,u c m ) (cid:1)(cid:19) (107) = ( ρ + 1) (cid:48) (cid:18) M (cid:88) m =1 τ (cid:88) u =1 (cid:0) Re[ p u ]( a m,u c m + b m,u d m ) + Im[ p u ]( a m,u d m − b m,u c m ) (cid:1) + M (cid:88) m =1 (cid:88) u (cid:54) = v (cid:0) Re[ p u ]( a m,u c m + b m,u d m ) + Im[ p u ]( a m,u d m − b m,u c m ) (cid:1) × (cid:0) Re[ p v ]( a m,v c m + b m,v d m ) + Im[ p v ]( a m,v d m − b m,v c m ) (cid:1) + (cid:88) m (cid:54) = n τ (cid:88) u =1 τ (cid:88) v =1 (cid:0) Re[ p u ]( a m,u c m + b m,u d m ) + Im[ p u ]( a m,u d m − b m,u c m ) (cid:1) × (cid:0) Re[ p v ]( a n,v c n + b n,v d n ) + Im[ p v ]( a n,v d n − b n,v c n ) (cid:1)(cid:19) (108)and Im[ˆ s (cid:96) ] can be obtained following similar steps. Then, summing up (108) and Im[ˆ s (cid:96) ] yields | ˆ s (cid:96) | = ( ρ + 1) (cid:48) (cid:18) M τ + 2 M (cid:88) m =1 (cid:88) u (cid:54) = v (cid:0) Re[ p ∗ u p v ]( a m,u a m,v + b m,u b m,v ) − Im[ p ∗ u p v ]( a m,u b m,v − b m,u a m,v ) (cid:1) + (cid:88) m (cid:54) = n τ (cid:88) u =1 τ (cid:88) v =1 (cid:16) Re[ p ∗ u p v ] (cid:0) ( a m,u c m + b m,u d m )( a n,v c n + b n,v d n )+ ( a m,u d m − b m,u c m )( a n,v d n − b n,v c n ) (cid:1) + Im[ p ∗ u p v ] (cid:0) ( a m,u c m + b m,u d m )( a n,v d n − b n,v c n ) − ( a m,u d m − b m,u c m )( a n,v c n + b n,v d n ) (cid:1)(cid:17)(cid:19) . (109)Now, we have that ξ m,u ∈ { a m,u , b m,u , c m , d m } and ξ n,v ∈ { a n,v , b n,v , c n , d n } are independentrandom variables if m (cid:54) = n regardless of the indices u, v . This implies that E [ a m,u c m b n,v d n ] = E [ a m,u c m ] E [ b n,v d n ] and the same holds for the other products in the second summation of (109). Hence, the expected value of (109) is given by E (cid:2) | ˆ s (cid:96) | (cid:3) = ( ρ + 1) Ψ (cid:48) M (cid:18) τ + (cid:88) u (cid:54) = v (cid:18) Re[ p ∗ u p v ]Ω (cid:18) ρ Re[ p u p ∗ v ]) ρ + 1 (cid:19) − Im[ p ∗ u p v ]Ω (cid:18) ρ Im[ p u p ∗ v ]) ρ + 1 (cid:19)(cid:19) + ( M − τ (cid:88) u =1 τ (cid:88) v =1 Re[ p ∗ u p v ] (cid:18) Ω (cid:18) ρ Re[ p u s (cid:96) ]) (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) Ω (cid:18) ρ Re[ p v s (cid:96) ]) (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) + Ω (cid:18) ρ Im[ p u s (cid:96) ]) (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) Ω (cid:18) ρ Im[ p v s (cid:96) ]) (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19)(cid:19) + ( M − τ (cid:88) u =1 τ (cid:88) v =1 Im[ p ∗ u p v ] (cid:18) Ω (cid:18) ρ Re[ p u s (cid:96) ]) (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) Ω (cid:18) ρ Im[ p v s (cid:96) ]) (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) − Ω (cid:18) ρ Im[ p u s (cid:96) ]) (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19) Ω (cid:18) ρ Re[ p v s (cid:96) ]) (cid:112) ( ρ + 1)( ρ | s (cid:96) | + 1) (cid:19)(cid:19)(cid:19) (110)where (110) follows from (100)–(103) and from E [ a m,u a m,v ] = E [ b m,u b m,v ] (111) = Ω (cid:18) ρ Re[ p u p ∗ v ] ρ + 1 (cid:19) , (112) E [ a m,u b m,v ] = Ω (cid:18) ρ Im[ p u p ∗ v ] ρ + 1 (cid:19) . (113)Finally, the expression in (41) is obtained by plugging (106) and (110) into (105).R EFERENCES [1] N. Rajatheva, I. Atzeni, E. Björnson et al. , “White paper on broadband connectivity in 6G,” June 2020. [Online]. Available:http://jultika.oulu.fi/files/isbn9789526226798.pdf[2] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO:Opportunities and challenges with very large arrays,”
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