I Introduction
Equipping the base station (BS) with lowresolution analogtodigital converters (ADCs) or digitaltoanalog converters (DACs) is highly appealing in massive multipleinput multipleoutput (MIMO) system, owing to the substantial reduction in hardware complexity, energy consumption and the amount of baseband data generated at the BS [1, 2]. These three considerations are important in massive MIMO, specifically in enabling the new emerging mmWave MIMO technology [3, 4, 5], where a much broader bandwidth than the traditional sub6 GHz band is included and a particularly large number of antennas is employed at the BS. The 1bit ADC, which is our interest in this work, has a simple structure which boils down to a single comparator with a relatively negligible energy consumption [6]. Also, the compensation for variations in the received signal level by automatic gain control is needless. Promoted by such simplicity, there has been an increasing interest in replacing the highresolution ADCs at the BS by the 1bit ADCs [7, 8, 9, 10, 11, 12, 13]. According to this new architecture, each antenna at the BS is equipped with a pair of 1bit ADCs, for both the inphase and quadrature components of the received complexbaseband signal (see the illustration in Fig. 2(a)).
The impact of the nonlinear distortion of quantization on channel capacity is studied in the literature under various assumptions on the channel state information at both transmitter (CSIT) or receiver (CSIR). With perfect CSIT and CSIR, the capacity of the 1bit quantized realvalued singleinput singleoutput (SISO) Gaussian channel is studied in [14], where the results show that antipodal signaling is capacity achieving. For fading SISO channel, it turns out that quadrature phaseshift keying (QPSK) signaling is optimal [15] [16] [8]. A more general transmitreceive antenna configurations with the 1bit ADC is considered in [8], where it is shown that QPSK signaling combined with the maximum ratio transmission is optimal for the multipleinput singleoutput channel (MISO) channel.
In all above works on the capacity of quantized channels, perfect CSI is assumed, while in practice the channel requires being estimated beforehand at the receiver by the aid of dedicated training pilots, for instance. Since quantization process throws away some information about the channel, it turns out that gaining reliable CSI is a challenging issue in quantized systems, especially when the 1bit ADCs are used. In massive MIMO, the accuracy of CSI at the BS is one of the key requirements to harness its high spectral efficiency [17, 18]. Thus understanding the performance gap between the quantized and unquantized massive MIMO is important. Since determining the exact capacity of quantized MIMO channel is hard, many researchers have focused on the achievable rate in massive MIMO and the possibility of supporting highorder modulation, while assuming various channel and data estimation techniques at the BS.
In [7], the 1bit quantized massive MIMO with QPSK signaling is considered. With leastsquare (LS) channel estimator and maximum ratio combining (MRC) or zeroforcing (ZF) receiver, it is shown that high data rates can be attained. In [19] two efficient near maximumlikelihood channel estimator and data detector are developed which allow the support of multiuser and the use of highorder modulation transmission. As well, the authors in [19] show that LSchannel estimation combined with MRC receiver suffices for holding highorder modulation transmission in multiuser massive MIMO scenario with 1bit quantization. In [20] pilot and data are jointly utilized for channel estimation in a singlecell with 1bit ADCs, where it is indicated that this approach outperforms the pilotonly approach. Still, its performance under the multicell case is not addressed and no bounds on capacity are given. Results in [21] show that a mixed structure of 1bit ADCs and conventional ADCs can achieve higher spectral and energy efficiency than the traditional massive MIMO. In [22]
, the channel sparsity in massive MIMO channel is exploited to estimate the channel blindly using the expectationmaximization algorithm. It is demonstrated that reliable channel estimation is still possible with 1bit ADCs.
In [12], the uplink in a wideband massive MIMO channel with 1bit ADCs is considered. With the assumption of quantization noise (QN) being independent and identically distributed (i.i.d.), linear minimum mean square error (LMMSE) channel estimator is derived. Further, with employing MRC or ZF receiver, a lower bound on the achievable rate is obtained in [12]. Moreover, it is proven that the assumption of i.i.d. QN becomes increasingly accurate when working in the low signaltonoise ratio (SNR) regime or when the number of channel taps is sufficiently large. In [13], a more accurate model which captures the temporal/spatial correlation of QN is derived by utilizing the Bussgang theorem [23], while assuming a singlecell case with 1bit quantization. It is shown that considering correlation among QN samples can further improve channel estimate, especially when the input signal is correlated. In Rayleigh fading channels, it turns out that the assumption of i.i.d. QN serves as a good approximation when operating in the lowSNR regime or when the number of users is sufficiently large, validating the observations in [12]. Grounded on the approximate lower bounds on achievable rate established in [13] for MRC and ZF receivers, it is concluded that high spectral efficiency can be achieved in the singlecell scenarios. In [24], the authors generalize the 1bit quantized model in [13] to an arbitrary number of quantization bits. Therefore, they show that highorder modulation is possible even under the 1bit quantization case, while with a few bits, one can approach the rate achieved when no quantization is used.
Previous works on the 1bit quantized massive MIMO have focused on timemultiplexed pilot (TP) scheme [25, 26], where pilot and data symbols are orthogonal in the time domain. In TP scheme, pure pilot symbols of length, say symbols, are transmitted by all users at the start of transmission for channel estimation at the BS. Then, data communication takes place during symbol intervals, where is the coherence time of the channel over which the channel is assumed constant (see the illustration in Fig. 1(a)). An interesting and wellknown technique called superimposed pilot (SP) [27, 28, 29], sends pilot and data symbols sidebyside (see the illustration in Fig. 1(b)). Unlike SP scheme, with TP scheme no data is transmitted during the training phase, thus a decrease in spectral efficiency might be incurred, especially when is short. For the unquantized massive MIMO channel, it was shown in [30] that SP has a potential to achieve higher spectral efficiency than TP, especially when advanced signal processing is used at the BS.
The authors in [31] consider the conventional (unquantized) multicell massive MIMO system with SP approach and derive a closedform expression for the ergodic achievable rate in the uplink. In that respect, it is shown that when both SP and TP approaches are optimized, comparable data rates can be obtained in practical multicellular scenarios. This is because pilot contamination resulting in TP can be, in some sense, equally bad as data interference in SP due to simultaneous transmission of pilots and data. With more advanced signal processing, it is believed that SP has the potential to outperform TP. The direct extension of the work in [31] to the quantized channel is not possible due to the presence of the QN (not independent of the input signal) which enters in many places in the analysis, rendering the analysis intractable.
Our contributions: In this work, we consider a powercontrolled 1bit quantized massive MIMO system with SP scheme. We are interested in the uplink achievable rate of such a system while assuming LMMSE channel estimator and MRC receiver for data detection at the BS. According to our knowledge, this is perhaps the first attempt that focuses on superimposed pilots in quantized massive MIMO systems. An important contribution of this paper is that the quantized and unquantized systems are asymptotically equivalent as the number of BS antennas . Throughout this manuscript we shall refer to the 1bit quantized channel with SP scheme as quantized SP (QSP) and its unquantized (i.e., with infiniteresolution ADCs) counterpart as unquantized SP (UQSP).
We commence our work by considering a singlecell scenario, then we extend the results to the multicell scenario. Our theoretical findings agree with the numerical results to corroborate that regardless of the very coarse quantization, QSP provides high data rate. Due to the scope limitation of this work and the lack of an explicit formula of the achievable rate for the 1bit quantized multicell massive MIMO system with TP, the comparison with the 1bit quantization with TP case, dubbed “QTP”, is obtained numerically, where no optimization over training duration or power is considered. The simulation results indicate that QSP can outperform the nonoptimized QTP approach in most times. Despite this, a fair evaluation of QSP against the optimized QTP needs more investigations, considering optimizing QTP and using different techniques for channel and data estimation, which is a tedious task and beyond the purpose of this work and hence left for future study.
The main contributions of our work are summarized as follows:

A closedform expression of the LMMSE channel estimate and its meansquare error (MSE) are derived for the QSP. A similar closedform expression under the noquantization case is obtained as a special case of the 1bit QSP.

We obtain a closedform approximation on the achievable rate for the 1bit QSP and the optimal power allocation among pilot and data is also given, where the results turn out to be accurate when working in the lowSNR regime or when the number of users is large. A lower bound on the achievable rate for the infiniteresolution case is recovered as a special case.

It is shown that regardless the coarsely quantized signal and symbolbysymbol superposition of pilot and data, it is even possible to achieve high data rates in multicell massive MIMO.

We show numerically that, under 1bit quantization case, removing the pilot contribution utilizing the estimated channel from the quantized signal incurs a negligible loss in information in the regimes of lowSNR and a large number of users.

We show that the performance gap between the quantized and unquantized systems vanishes when the number of BS antennas is asymptotically large, i.e., QN can be averaged out asymptotically. In the multicell case, the data rates for QSP and UQSP saturate and converge to a fixed value, given by bits/s/Hz, where is the power fraction allocated to pilot and is a constant which depends on the number of users and geometry of the network.
Outline of the paper: In Sec.II the signal model for the 1bit quantized singlecell MIMO with SP is presented. In Sec.III a linear modeling of the quantizer and the details of channel estimation are given. In Sec.IV the analysis of achievable rates for the quantized and unquantized systems is shown. In Sec.V, we extend the previous results to the multicell case and some asymptotic results are established. In Sec.VI, some numerical results for a multicell massive MIMO system are presented to validate the analytical results and Sec.VII summarizes this paper.
Ii System Model for singlecell Massive MIMO
In the first part of this work, we consider the uplink of a singlecell massive MIMO system where the BS has antennas and serves () singleantenna users in the same timefrequency resource. We consider a blockflat Rayleigh fading channel which remains constant over symbol intervals, i.e., is the coherence time of the channel. The channel gain between the th BS antenna and user is represented by , where and are the largescale and smallscale fading coefficients, respectively. The coefficients are assumed to be i.i.d. . We assume that the distance between user and the BS is sufficiently large so that becomes constant over the antenna array.
In Fig. 2 (a) a simplified illustration of the uplink system is shown where the th BS antenna is equipped with a pair of 1bit ADCs for both the inphase and quadrature components of the discretetime baseband signal . With superimposed pilots, is a noisy superposition of pilot and data symbols subject to channel distortion. For ease of analysis, we assume the transmission of pilot and data symbols takes place over symbol intervals, i.e., pilot and data have the same length . So, during one coherence interval, the BS collects binary samples. We assume the BS first estimates the channel using LMMSE, then uses the channel estimate for data estimation using MRC receiver. For pilots, we consider all users use mutually orthogonal sequences.
With quantized channels, it is shown in [32] that the structure of pilot sequences affects the performance. Lately, the authors in [12] show that sparse pilot sequences (i.e., with many zero entries) give rise to degradation of channel estimation performance. Thus, under quantized channels, nonsparse pilot sequences in the time domain are more preferable. To meet such a requirement, without loss of generality, we assume that the pilot sequences are drawn randomly from a Fourier basis matrix. We recall here that such pilot sequences have nonzero entries where each entry has an absolute value of one.
Let
be a vector of
symbols sent by user during one coherence interval satisfying power constraint:(1) 
Further, let and be the pilot and data symbols associated with user during time instant , respectively. Then is written as the superposition of and :
(2) 
where and are the power fractions allocated to pilot and data symbols, respectively, such that . According to our choice of pilot sequences, we have , therefore, to meet the power constraint (1), we assume .
In cellular communication systems, power control in the uplink is one of the most important mechanisms for fairness provision among different users experiencing different fading conditions. In particular, the imbalance in received powers of different users becomes a critical issue in 1bit quantized channels due to the saturation effect of ADC. For instance, when the received signal strength of a closetoBS user overwhelms the signal of a celledge user, the output of 1bit ADC would have insufficient information (if not lost) about the weak user, making the detection of their presence hard. In this work, power control based on statistical channelinverse is assumed [13] [31]. Thus, in Rayleigh fading, the transmit power of user , denoted by , is chosen to be
(3) 
such that the average received power by the BS from each user is the same, given by , where is some fixed power requirement. As a result, we have and .
Based on the above discussion, can be written as
(4) 
where is additive white Gaussian noise (AWGN) which is assumed i.i.d. across space and time. The quantized signal of (4) with zerothreshold 1bit ADCs is:
(5) 
where and denote the real and complex parts of a complex quantity, respectively, is the signum function which returns or when its argument is greater or less than zero, respectively and is a scale normalization factor. It follows that .
Equation (4) can be written in a compact matrix form. Let and be the length vectors of data and pilot symbols, channel vector between user and all BS antennas, the received signal and AWGN during time at all BS antennas, respectively. Similarly, we denote by and the received quantized signal and QN across all BS antennas during time instant , respectively. Rearranging (4) for all and , the received signal can be written as
(6) 
and hence the corresponding quantized signal is
(7) 
where the function is applied elementwisely on a matrix, , and and are the composite channel, pilot and data matrices, defined, respectively, as and .
Iii Linear model and channel estimation
The model equation (5) is nonlinear which makes the analysis intricate. Yet, according to the Bussgang theorem [23], if the quantizer’s input is Gaussian, its output can be decomposed into a sum of a scaled version of its input and uncorrelated QN, where the QN itself is generally correlated [12]. Despite this, the decomposition involves the nonlinear arcsine law which makes the nonlinearity unavoidable, i.e., see [33] [13] for more details. To avoid such nonlinearity, we make simplifications to the model which allow us to get a closedform expression for the channel estimator and uplink achievable rate.
Iiia Assumptions
To proceed with the development of theoretical results, we allow the following assumptions:

All data symbols are i.i.d. . This allows using existing lowerbounding techniques to obtain a lower bound on capacity [25].

The signals
are Gaussian random variables, each with zeromean and variance
. This is justified by the central limit theorem (CLT) as the distribution of
tends towards Gaussian as gets larger. For being sufficiently large should come as no surprise in massive MIMO.
IiiB The linear model of 1bit quantizer
Here we apply the Bussgang decomposition to obtain a linear model for the quantizer, which will be utilized for analysis.
Let , and be the column vectors constructed from the th rows of and , respectively. Further, we define as the column vector of the channel from all users to the th BS antenna. We use bars over all symbols to distinguish them from the original columns of corresponding matrices. Thus, we can write and . Under Rayleigh fading, the rows of (and hence the rows of ) are i.i.d. with a common covariance matrix defined as , given by
(8) 
Thus it suffices to focus only on the linear model of an arbitrary row of .
Since is jointly Gaussian (see Assumption 2), therefore, according to the Bussgang theorem, can be written as , where is some matrix, chosen such that is uncorrelated with , i.e.,
(9) 
From [13], it is easy to show that admits the simple form , where is a scaling factor defined as
(10) 
and hence takes the following form:
(11) 
From (11), the covariance matrix of QN, defined as , is given by , where is the autocorrelation matrix of quantizer’s output. It is wellknown that when quantizer input is Gaussian, follows the arcsine law [34]. However, working with the nonlinear arcsine operator turns out to be intractable. However, with the assumption of i.i.d. QN (Assumption 3), reduces to a diagonal matrix given by [13]:
(12) 
where is the variance of QN samples. Figure. 2(b) shows a schematic representation of the equivalent linear model of 1bit quantizer.
IiiC Channel estimation
For QSP, we use (11) to derive the LMMSE channel estimate of the channel between user and the th BS antenna, . With UQSP, the LMMSE channel estimate can be recovered as a special case of the result of QSP when (i.e, and ). By leftmultiplying by we get the following singlevalued function:
(14) 
where is a nonGaussian effective noise defined as
(15) 
Based on (14), the channel estimate and variance of estimation error are given in the following lemma.
Lemma 2.
Consider QSP with i.i.d. QN, the LMMSE estimate of , denoted by , is
(16) 
and the variance of estimation error is
(17) 
Proof.
Using (14) and applying the standard LMMSE solution [35] for , we get . In calculating we make use of Lemma 1 and in calculating we make use of (8), (9) and (12).
∎
The channel estimate and the variance of estimation error of UQSP can be recovered from Lemma 2. Thus we are led to the following lemma:
Lemma 3.
For UQSP, the LMMSE estimate of , denoted by , is given by
(18) 
and the variance of estimation error is
(19) 
where .
Proof.
Although in Lemma 2 doesn’t coincide with the MMSE solution ( is not truly Gaussian because is not Gaussian), can still be approximated as Gaussian. This elicits from the fact that is a weighted sum of random variables (typically large), thus by the virtue of CLT, the distribution of tends towards Gaussian.
By inspecting Lemmas 2 & 3, the variance of estimation error can be arbitrarily small as increases. It is interesting to see from (17) that the effect of AWGN and QN is scaled down by a factor of , compared with a factor of with TP scheme (see [13]), where is typically much larger than . Still, with SP, there is an extra term due to data interference while there is no data interference during the training phase under the TP scheme. In both the quantized and unquantized systems, it is worth observing that there is a saturation effect in LMMSE performance as grows large. This suggests that there is a threshold after which no increase in data rate is achieved. Increasing pilot power (by increasing ) leads always to an improvement of channel estimate quality while data rate may decrease as data power decreases, even so, it turns out that this relation is not monotonic. Hence, there is a tradeoff between channel estimate quality and data rate, implying the existence of an optimal fraction of total power which gives rise to the highest data rate.
We conclude this section by making the following remark on the fundamental challenges in the analysis of QTP.
Remark 1.
In contrast with QTP where the channel estimate is independent of data, with QSP the channel estimate (and hence the estimation error) is not only dependent on data symbols transmitted from all users but also on AWGN and QN. Besides, although the QN and input signal are uncorrelated, they are not independent. The QN enters different places in the analysis, especially when calculating the variance of MRC output. Typically, this requires a complete characterization of statistical relationships with other random variables through the joint probability densities, which are hard to obtain. Altogether, this is an undesirable property which renders the capacity analysis intractable as will be discussed next.
Iv Analysis of achievable rates
This section is concerned with obtaining an approximate lower bound on the ergodic achievable rate for QSP massive MIMO where the BS uses LMMSE and MRC for channel and data estimation, respectively. We then leverage the analysis of QSP to obtain a lower bound on the ergodic achievable rate for UQSP. In the rest of this section, a comparison between QSP and UQSP is established through asymptotic analysis.
Without loss of generality, we focus on the information rate of the th user during an arbitrary time . Thus the channel model corresponds to the th column of which can be written as , where and are the column vectors of and , respectively. We stack all channel estimates of the th user as a column vector given by . Thus we have
(20) 
where .
Given the channel estimate , the contribution of pilots can be removed from as follows:
(21) 
where is the matrix of channel estimation error. In (IV) we use the notation “PR” to denote pilot removal. The basic problem of the model (IV) is that the last term is not independent of the other remaining terms (also crosscorrelated), especially data (see Remark 1), which is a primary source of intractability in the analysis of capacity.
To maintain the analysis tractable, pilot removal (PR) is not considered in this work, similar to the technique used in [31] for the analysis of the unquantized MIMO channel with SP. Instead, is utilized directly at BS for data estimation, which can be rewritten as
(22) 
where is given by
(23)  
Although PR gives rise to increase in data rate, it turns out that the loss in information resulting from using (22), instead of (IV) is not large, especially in the lowSNR and large regimes, where these two regimes come at no surprise in massive MIMO settings. From (20) and (22), the MRC output, , can be written as
(24) 
where the scaling factor is introduced here for establishing some asymptotic results later.
While pilot removal is not considered, however, the model (24) is still intractable. This elicits from the fact that the quantized signal is not independent of information . In addition, the distribution of effective noise in (24) is unknown and the effective noise is not independent of the signal part (also correlated), especially due to the presence of QN (see also Remark 1). Despite this, we make use of some mathematical simplifications and asymptotic results, owing to some statistical properties of QN to tackle the intractability of the model (24).
In the following, we leverage (24) to derive an approximate lower bound on the mutual information with the Gaussian assumption on .
Iva Quantized superimposed pilots
From the aforementioned discussion, the exact capacity of (24) is unknown. The following result holds when the number of BS antennas is sufficiently large.
Theorem 1 (approximate lower bound).
Considering QSP massive MIMO where QN is assumed to be i.i.d. (Assumption 3), if the BS employs LMMSE channel estimation and MRC data estimation, an approximate lower bound on the achievable rate in uplink is
(25) 
where is the effective SNR at the MRC output given by (26) which appears at the top of page 26.
(26)  
Proof.
The proof unfolds by rewriting (24) as a sum of two parts; desired signal with a multiplicative deterministic channel gain and uncorrelated noise, i.e., . Then the lower bound follows by assuming the worstcase noise (i.e., Gaussian noise) with the same variance. The details of the proof and hence the closedform expression for in (26) are shown in Appendix A. ∎
The lower bound (25) can be maximized w.r.t. . The result is stated in the following corollary.
Corollary 1.
Let be the optimal value of power fraction which maximizes (25). Then is given by one of the following two roots:
(27) 
where .
Proof.
By the concavity of and noticing that the first derivative of w.r.t. changes sign due to (i.e., is nonmonotonic), can be obtained by solving the quadratic equation for and choosing the positive root less than 1. ∎
IvB Unquantized superimposed pilots
As indicated earlier, the UQSP can be treated as special case of QSP, therefore, we have the following corollary.
Corollary 2 (lower bound).
a lower bound on achievable rate under UQSP system when LMMSE and MRC are used for channel and data estimation, respectively, is
(28) 
where the effective SNR, , is given by
Moreover, the optimal power fraction maximizing (28) is given by one of the following two roots:
(30) 
where .
Proof.
IvC Asymptotic analysis
So far, we have obtained the achievable rates for QSP and UQSP systems, however; it seems difficult to obtain a useful mathematical expression of the gap between QSP and UQSP when a general set of system parameters is considered. To get the feeling of this gap, we consider the asymptotic behavior of both systems when the number of BS antennas is asymptotically large.
We first observe that the common numerator, , in (26) and (LABEL:eq:SNR_UQSP_singlecell) can be seen as the unnormalized signal power at MRC output which scales with . Second, the denominators (i.e., total noise power) consist of two main kinds of noise: 1) coherent noise (first term) which results from data and pilot interferences added constructively at the MRC output and hence their variance scales with , in the same way as the signal power does, and 2) noncoherent noise (remaining terms) resulting from all crosscorrelation terms (data, pilot, AWGN and QN) which add destructively at MRC output with a variance which doesn’t scale with and hence vanishes as .
Clearly, the coherent noise in both systems is the same which is given by . As a result, we have the following conclusion on the asymptotic performance comparison between QSP and UQSP.
Corollary 3.
The asymptotic limit of data rates for QSP and UQSP when is
(31) 
Proof.
The implication of Corollary 3 is that, under a quantized channel with superimposed pilots, the QN is averaged out asymptotically. This can be explained as follows. With SP, the channel estimate is highly correlated with received signal since the channel estimate is not independent of data (i.e., see Remark 1). This gives rise to a coherentnoise term (mainly due to data interference from concurrent users) as discussed previously. On the other hand, there is a noncoherent noise which includes the effect of QN. Whenever the coherent noise is dominated by the noncoherent noise, there’s a gap between QSP and UQSP. This occurs when is not sufficiently large. However, as increases, the coherent noise starts to dominate the effect of noncoherent noise, thus the gap between QSP and UQSP becomes smaller and smaller until it vanishes asymptotically.
V Onebit quantized multicell massive MIMO
In this section, we extend the previous results of the singlecell case to a multicell case with a network comprised of cells and singleantenna users per cell.
Va Signal model
For ease of analysis, we assume that the coherence time of the channel can accommodate mutually orthogonal pilot sequences, i.e., no pilot contamination is assumed under SP [30]. We also assume that each users in each cell use power control based on statistical channel inverse as with singlecell scenario. In the following, we shall use the same notation as before, while introducing indexes of cells. The channel between user in cell and the th antenna of BS is defined as . The pilot and data symbols associated with user in cell , sent during time , are denoted by and , respectively. The transmit power of user in cell is denoted by and the power fractions allocated to pilot and data symbols are denoted by and , respectively, such that .
Thus the received complexbaseband signal at the th antenna of BS during time can be written as
(32) 
where and . According to the powercontrol policy, is chosen to be , where is some fixed power requirement. As indicated previously, with i.i.d. Rayleigh fading, the average power received at the th BS from each of its users is the same, given by .
Let be the ratio between cross (w.r.t. BS ) and direct (w.r.t. to BS ) largescale fading coefficients of user and, as a result, and reduce to
(33) 
Since the results are averaged over many realizations (rather than a single realization) of smallscale and largescale fading coefficients, the statistic of shall be independent of cell index. This is true due to the random locations of users within each cell which make this assumption reasonable and hence there is no reason to believe they are different for different cell indexes (see [36]). Therefore, without loss of generality, in the sequel, we only focus on BS as the target BS, i.e., .
From (32), the variance of received signal is given by
(34) 
where is defined by
(35) 
In addition to (35), we also need the following definition:
(36) 
Note that and are random variables (i.e., depending on users’ distances from the target BS), thus we define the following statistics (expected values), which will be used later, as follows:
(37) 
By the Bussgang decomposition as discussed in Sec. III (also see Fig. 2(b)), the 1bit quantized version of (32) is
(38) 
where is the QN and is a scaling factor given by
(39) 
Let be the vector of smallscale fading gains between user in cell and all antennas of BS . Denote by the vector of all unquantized signals received at all antennas of BS during time , its corresponding quantized version and & the associated QN and AWGN vectors, respectively. We also define and as the data and pilot sequences of user in the th cell, respectively. Moreover, we denote by the diagonal matrix accounting for largescale fading coefficients, the composite smallscale fading matrix, the composite matrix of all received unquantized signals at all antennas during one coherence time, its quantized version, the composite pilot matrix, the composite matrix of all data symbols and & are matrices of AWGN and QN, respectively.
From the above discussion, can be written as
(40) 
and its quantized version as
(41) 
VB Channel estimation
Let be the channel estimate of user in the th cell. Then by the standard results on LMMSE, we can readily show that is given by
(42) 
where
(43) 
As indicated previously, we are interested in averaging the results w.r.t. largescale fading. The following lemma gives an upper bound on the average of variance of estimation error .
Lemma 4.
When the QN is i.i.d., the expected value (w.r.t. largescale fading) of is upper bounded by
(44) 
Proof.
From the orthogonality principle of LMMSE, the variance of estimation error per single realization of is . Note that is a concave function of . Thus, taking the expectation of w.r.t. and making use of Jensen’s inequality, (44) follows. ∎
As discussed previously (see discussion of Lemma 2), the effect of QN, AWGN and data interference is scaled down by a factor of ; as . Further, there is a saturation effect as (while is fixed) as the effect of data interference and QN persists to exist, suggesting the saturation of data rate as will be seen in Sec. VD. Increasing pilot power leads always to an improvement of channel estimate quality while data rate may decrease, however, the relation between channel estimate quality and the achievable data rate is not monotonic as discussed next. This implies that there is a tradeoff between channel estimate quality and data rate dictated by power allocation strategy.
It should be noted that (44) serves as an approximate upper bound (with some artifact effect) as there will still be some correlation between QN components. It turns out that the bound (44) is nearly tight in many cases (especially in the lowSNR and large regimes) as demonstrated by numerical results, i.e., see Fig. 3.
VC Analysis of achievable rates
Now we present an approximate lower bound on the achievable rate for QSP under multicellular case with MRC implemented at BSs and no pilot removal after estimating the channel is performed. We next extend the result for the unquantized system.
Without loss of generality we can assume an arbitrary user in cell as our target user. Thus, the scaled MRC output during time is
(45) 
Based on (45), a closedform approximation on the achievable rate is given in the following theorem.
Theorem 2 (approximate lower bound).
Consider 1bit QSP multicell massive MIMO where QN is assumed to be i.i.d., and LMMSE channel estimator and MRC receiver are employed at the BS. If power control based on statistical channel inverse is applied, then a lower bound on the achievable rate in uplink is approximated by
(46) 
Proof.
See Appendix B. ∎
(47)  
A corollary of Theorem 2 which maximizes w.r.t. is the following.
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