# Massive MIMO Channel Estimation with Low-Resolution Spatial Sigma-Delta ADCs

We consider channel estimation for an uplink massive multiple input multiple output (MIMO) system where the base station (BS) uses an array with low-resolution (1-2 bit) analog-to-digital converters and a spatial Sigma-Delta (ΣΔ) architecture to shape the quantization noise away from users in some angular sector. We develop a linear minimum mean squared error (LMMSE) channel estimator based on the Bussgang decomposition that reformulates the nonlinear quantizer model using an equivalent linear model plus quantization noise. We also analyze the uplink achievable rate with maximal ratio combining (MRC) and zero-forcing (ZF) receivers and provide a closed-form expression for the achievable rate with the MRC receiver. Numerical results show superior channel estimation and sum spectral efficiency performance using the ΣΔ architecture compared to conventional 1- or 2-bit quantized massive MIMO systems.

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## I Introduction

Massive MIMO systems provide high spatial resolution and throughput, but the cost and power consumption of the associated RF hardware, particularly the analog-to-digital converters (ADCs) and digital-to-analog converters (DACs) can be prohibitive, especially at higher bandwidths and sampling rates. To save power and chip area, low-resolution quantizers have been suggested. The simple design and low power consumption of low-resolution ADCs/DACs have made them very suitable for massive MIMO.

There has been extensive research on one-bit ADCs, in particular, for uplink transmission [20, 14, 27, 5, 17, 8] and downlink precoding [26, 21, 15, 35, 16, 19]. Channel estimation with GAMP [9], near ML [5] and linear estimators [20, 13] using one-bit ADCs, MMSE channel estimation based on recursive least-squares using two-bit ADCs [25] and with dithered feedback signals [7] using 1-3 bit ADCs have been proposed. In addition to savings in energy and circuit complexity, one-bit ADCs also reduce the fronthaul throughput from the remote radio head (RRH) in cellular applications. This is a significant issue for massive MIMO implementations operating with high bandwidths, as in a millimeter wave application; a 100-element antenna array sampled at 500 Msamples/sec with a 10-bit ADC requires a 1 Tb/sec data pipeline from the RRH to the baseband processor. One-bit sampling can significantly alleviate this requirement. While it has been shown that one-bit quantization causes only a small degradation at low SNRs and offers significant advantages in terms of power consumption and implementation complexity, at medium to high SNRs the performance loss is substantial.

One method to improve the performance is to simply increase the number of quantization bits. Simulations have shown that using ADCs with 3-5 bits of resolution in massive MIMO provides performance that is very close to that achievable with infinite precision, and achieves a good trade-off between energy and spectral efficiency [34, 31]. Alternatively, the sampling rate at which the one-bit quantizers operate can also be increased, and this approach has been studied in [39, 10, 36]. A well-known technique that combines one-bit quantization and oversampling is the ADC, which to date has primarily found application in ultrasound imaging, automotive radar and pulse-coded modulation for audio encoding. The temporal converter scheme consists of an oversampled modulator, responsible for digitization of the analog signal, followed by a negative feedback loop that shapes the quantization noise with a simple high-pass filter. The quantization noise can then be removed in favor of the desired signal using a digital lowpass filter and decimation (if necessary). The temporal architecture has been extensively studied over the years [3, 12, 11, 1], and higher-order implementations exist that can provide additional frequency-selective noise shaping. The use of ADCs in parallel architectures for MIMO systems has been studied in [29, 40]. While higher-resolution ADCs and temporal oversampling can improve performance with only a moderate increase in power consumption, they significnatly increase the required fronthaul throughput compared with one-bit quantization.

A noise-shaping effect analogous to that achieved by temporal quantizers can be achieved by implementing the architecture in the spatial domain. This approach exploits oversampling in space, which can occur in massive MIMO settings with a limited array aperture, or for scenarios where the uplink signals are confined to a given angular sector, due to cell sectorization, limited multipath scattering (e.g., as at millimeter wave frequencies), or certain small-cell geometries (narrow conference halls, city streets, etc.). In a system employing spatial ADCs, the quantization error from the ADC at one antenna is fed to the input of an adjacent antenna, rather than to the input of the same antenna. In a standard implementation without phase-shifting the feedback, this has the effect of shaping the quantization noise to high spatial frequencies, in favor of user signals that arrive from angles closer to the broadside of the array, which correspond to low spatial frequencies. While spatial oversampling, i.e., antennas spaced less than one-half wavelength apart, can produce the required low spatial frequencies for the user signals, there is a limit to how close the antennas can be placed together before mutual coupling and the physical size of the antennas come into play. For this reason, a more likely scenario is that some small degree of spatial oversampling would be combined with the assumption that the users of interest are already confined to some angular sector, since one can easily steer the angular sector of low quantization noise to arbitrary directions.

While temporal systems have been studied for decades, there is relatively little work on corresponding spatial implementations. Only recently has the noise shaping characteristics of first and second-order spatial and cascaded space-time architectures been studied for a few array processing applications. In particular, applications have been considered for massive MIMO [6, 4, 33, 38, 30], phased arrays [37, 18], interference cancellation [40], and spatio-temporal circuit implementations [23, 28]. With the exception of our preliminary study in [33], there has been no prior work focused on channel estimation for spatial massive MIMO systems.

In this paper, we consider optimal linear minimum mean squared error (LMMSE) channel estimation for massive MIMO systems with first-order one-bit and two-bit spatial ADCs. Our initial results on this problem in [33]

were derived based on a vector-wise Bussgang decomposition similar to what was used for standard one-bit quantization in

[20]. However, this approach leads to a mathematical model in which the quantization error vector is defined to be uncorrelated with the input vector to the quantizer, which is not consistent with the more traditional definition of quantization noise based on the actual system architecture. Based on this observation, we have modified our analysis to incorporate a more meaningful definition of the quantization noise, using an element-wise implementation of the Bussgang decomposition as defined in [30] in order to find an equivalent linear signal-plus-quantization-noise representation. Our approach also explicitly takes into account the spatial correlation between the quantized outputs of the ADC array.

The structure of the array allows us to find a recursive solution for the covariance matrices required to compute the LMMSE channel estimate, and also to derive analytical expressions for the resulting normalized mean squared error (NMSE). After performing the analysis for the one-bit case, we show how the LMMSE channel estimate and analytical NMSE performance characterization can be extended for a array implemented with two-bit quantization. The resulting estimators in both cases have low complexity and the simulated estimation error closely matches the derived analytical expressions. Our simulation results indicate that, at low-to-medium SNRs, the LMMSE channel estimator for the array yields channel estimates that are very close to those achieved with infinite resolution ADCs. The inevitable error floor at high SNR is significantly lower than that achieved by standard one-bit and two-bit quantization; for example, the one-bit LMMSE channel estimate has an NMSE that is two orders of magnitude smaller than that achieved by standard two-bit quantization.

The spectral efficiency of one-bit arrays was recently analyzed in [30], but only for the case where the channel is known. In this paper, we also extend the analysis of [30] to derive a lower bound on the uplink achievable rate using maximal ratio combining (MRC) and zero-forcing (ZF) receivers when implemented with imperfect channel state information (CSI) based on our LMMSE channel estimate. We derive a closed-form expression for the spectral efficiency of MRC, while for the ZF receiver we present simulated results. In both cases, the results of our spectral efficiency analysis show that the architecture significantly outperforms standard low-resolution (1-2 bits) quantization. For a ZF receiver, the implementation achieves three times the spectral efficiency of standard one- and two-bit approaches. For the case of MRC, the two-bit architecture provides 99% of the spectral efficiency of system with infinite resolution ADCs.

The rest of the paper is organized as follows. In Section II, we describe the assumed signal and channel model, and then in Section III we present the first-order spatial array architecture. Section IV derives the corresponding linear Bussgang model that describes the input-output relationship of the spatial array. LMMSE channel estimators based on the output of the one-bit or two-bit spatial ADC array are then derived in Section V, and the bounds on the uplink achievable rate are calculated in Section VI. Simulation results validating our analyses are presented in Section VII.

Notation: Boldface lowercase variables denote vectors and boldface uppercase variables denote matrices. , and are the transpose, Hermitian transpose, and conjugate of , respectively. The th element of vector is represented by . The symbols and respectively represent the Kronecker product and the vectorization operation, i.e., the stacking of the columns of a matrix one below the other. Real and imaginary parts are indicated by and , respectively. is the expectation operator. is the trace of the matrix and denotes pseudo-inverse of . The matrix denotes a identity matrix. The function represents the modulo- operator, and is the largest integer smaller than . A circularly symmetric complex Gaussian vector with mean and covariance matrix is denoted by

. The cumulative distribution function (cdf) and the standard normal density are given by

and , respectively.

## Ii System Model

We consider an uplink massive MIMO system with single-antenna user terminals, and a base station (BS) equipped with a uniform linear array (ULA) of antennas and a first-order spatial array. During the training period, all users transmit their pilot sequences of length simultaneously. The received signal, , at the BS is

 X=[x1⋯xN]=√ρGΦt+N, (1)

where we assume that all the user signals are received with the same power and, therefore, is a factor that determines the SNR, is the array output for training sample , is the channel matrix, is the pilot matrix and contains additive noise with independent and identically distributed (i.i.d.) Gaussian elements: .

We consider a channel that is composed of paths for each user with a certain angular spread . More specifically, the th column of represents the channel for the th user, and is given by

 gk=1√LAhk, (2)

where the elements of are independently and identically distributed (i.i.d.) as random variables, and is a matrix whose th column is the steering vector

 al=[1e−j2πδsin(θl)⋯e−j2πδ(M−1)sin(θl)]T, (3)

where is the inter-element antenna spacing in terms of the wavelength and . Thus, the channel is given by , where , and the channel covariance matrix is given by , which we assume to be known. This is equivalent to knowing a priori the sector over which the users are distributed; the matrix can be populated by steering vectors uniformly spaced across the sector.

Vectorizing (1) and letting and , we get

 x=vec(X)=Φg+n. (4)

The covariance matrix of can be expressed as

 Cx=ΦCgΦH+IMN, (5)

where is block-diagonal. The SNR per-user per-antenna is

 SNR=ρMKTr(E[GΦtΦHtGH]). (6)

When the pilot sequences are row-wise orthogonal and the minimum number of pilots, , is used, then and the SNR is equal to . In the derivations below, we will assume that the power of the pilot signals is time-invariant, which implies that the diagonal elements of are identical.

## Iii First-Order Spatial ΣΔ Architecture

A first-order temporal modulator can be described by the block diagram depicted in Fig. 0(a). The architecture consists of a low-resolution quantizer and a negative feedback loop where the difference between the output and input of the quantizer is subtracted from the antenna input after a one-sample delay. The operation of the quantizer is defined using the following equivalent linear model with gain :

 y[n]=Q(r[n])=γr[n]+q[n]. (7)

While the gain of the quantizer is normally taken to be simply , we consider this more general case since it is relevant to our subsequent analysis. As shown in [30], this leads to the following transfer function description of the modulator:

 Y(z) = γ1−(γ−1)z−1X(z)+(1−z−1)1−(γ−1)z−1Q(z) = Ax(z)X(z)+Aq(z)Q(z),

where respectively represent the -transforms of . When , we have that is an all-pass filter and that is a first-order high-pass filter, which is the standard result, indicating the quantization noise is shaped to higher frequencies. Given that is oversampled and is concentrated at lower frequencies, the effect of the quantization noise can be substantially reduced by passing the output of the modulator through a low-pass filter. The all-pass plus high-pass structure still remains true as long as , but the modulator approaches instability as .

Quantization noise shaping can also be achieved in the spatial domain by propagating the quantization error from one antenna to the next, in effect exploiting spatial rather than temporal correlation in the desired signal. The spatial architecture is depicted in Fig. 0(b), and shows that the quantization error from one antenna is phase-shifted by prior to being added to the input of the adjacent antenna. This architecture shapes the quantization noise away from the angle of arrival (AoA) associated with the phase shift , and thus users in an angular sector surrounding this AoA experience a significantly higher signal-to-quantization-noise ratio (SQNR). The size of the high-SQNR angular sector can be increased by placing the antennas closer together than , corresponding to spatial oversampling, although in practice mutual coupling and the physical dimensions of the antennas place a limit on how much spatial oversampling can be achieved.

To generate a mathematical model for the array, define the vectors and corresponding respectively to the quantizer inputs and the array outputs that result from the received signal vector in (4). In other words, the th elements of the vectors respectively represent the signal received by antenna , the input to the quantizer at antenna , and the output of antenna for training sample . Defining , we thus have

 ym=αrm′Qm′(Re(rm))+jαim′Qm′(Im(rm)), (8)

where represents the quantization operation for antenna , and and are the output levels for the real and imaginary parts of the quantizer, respectively. Note that we assume the output level of the quantizer in general to be different for each antenna, unlike conventional one-bit quantization where the output levels for all the antennas are the same. In vector form, the output of the array can be written as

 y =Q(r) (9) =[Q1(r1),…,QM(rM),Q1(rM+1),…,QM(rMN)]T,

where

 r=Ux−Vy, (10) V=IN⊗⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣00…00e−jψ0…00e−j2ψe−jψ…00⋮e−j(M−1)ψe−j(M−2)ψ…e−jψ0⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦Vd, U=IN⊗(IM+Vd)Ud.

## Iv Equivalent Linear Model for the Spatial ΣΔ Array

As in the scalar case described by (7), we will represent the operation of the array using an equivalent linear model defined by

 y=Γr+q (11)

for a given matrix , where is the equivalent quantization noise. There are an infinite number of such models, one for every choice of , and each will result in a particular definition for the quantization noise with differing statistical properties. One possible choice for the matrix is obtained by making uncorrelated with , i.e. . This leads to , where is the cross-covariance matrix between and , and is the covariance matrix of . Assuming the elements of are jointly Gaussian, the Bussgang theorem can be applied. This approach was used to derive channel estimates in [20] for conventional one-bit quantization, and in our initial work on the array [33]. However, the resulting definition for the quantization noise does not have a physical interpretation in the context of Fig. 0(b).

Instead, we define following the approach of [30], again assuming that is Gaussian, but applying the Bussgang decomposition element-wise such that . Further assuming that the elements of are circularly symmetric, the quantizer output levels and are identical, and we can set . With this definition, becomes a diagonal matrix whose th diagonal element, , is given by

 γm=E(rmy∗m)E(|rm|2). (12)

Plugging into (11), we get

 y=(I+ΓV)−1ΓUx+(I+ΓV)−1q. (13)

The specific numerical value for will depend on the output level . Assuming that the statistics of the received data are time-invariant over the current channel coherence interval, we can choose such that , or equivalently such that , and (13) can be simplified to

 y=x+U−1q. (14)

This model is now the exact spatial analog of the temporal architecture, and is equivalent to passing through a (spatial) all-pass filter and through a filter that shapes the quantization noise away from the AoA corresponding to . In the discussion that follows, we will derive expressions for the values of necessary to achieve for the case of one-bit and two-bit quantization, as well as the resulting power of the quantization noise on each antenna. The resulting expressions for will depend in general on the power of the signal , which would have to remain time-invariant for the quantizer gains to be fixed. This can be achieved in practice using an automatic gain control at the input to the ADC.

### Iv-a One-Bit Spatial ΣΔ ADCs

When the array is implemented with one-bit ADCs, the output is given by

 ym=αm′(sign(Re(rm))+jsign(Im(rm))), (15)

and (12) can be simplified to get

 γm =αm′E(|Re(rm)|+|Im(rm)|)E(|rm|2) (16)

for circularly symmetric . Assuming that the power of the pilot symbols is time-invariant, then the statistics of are identical to those of . Consequently, we can set , which leads to

 αm′=E(|Re(rm)|2)E(|Re(rm)|).

To simplify the expression, in [30] it was assumed that is Gaussian, in which case

 αm′=√πE(|rm|2)2. (17)

The derivation of (17) relies on the fact that

 √E(|Re(rm)|2)E(|Re(rm)|)=√π2 (18)

for Gaussian random variables. However, due to the non-linear feedback structure of the array, the tails of the distribution of are slightly heavier than a Gaussian, so the ratio on the left hand side of (18) is slightly greater than . For this reason, we will choose a slightly larger value for than in (17):

 α∗m′=β√πE[|rm|2]2, (19)

where is a correction factor. This is consistent with prior work on temporal quantization, which has shown that slightly increasing the power of the input signal for fixed output levels (or equivalently, decreasing the output levels for fixed input signal power) reduces the quantization noise [1, 2, 41]. This benefit is lost if one applies too large an increase in the input power, as the input to the quantizer quickly becomes unstable. In fact, as we show below, for the array there is a very small range of values near one that are appropriate for the correction factor . While we could set as in [30], better channel estimation results and a better match with the analytical expressions are obtained when a value slightly larger than one is used, especially when the array input has high spatial correlation due to spatial oversampling (i.e., small ) or a small angular spread.

Let . Using similar definitions for and , and the fact that and are uncorrelated, we obtain the following relationship between the powers of the input and output of the array:

 σ2ym=π2β2σ2rm,σ2qm=σ2ym−σ2rm. (20)

From (20), we can see that in order to prevent the quantization noise power from becoming greater than the input power , we must ensure that , and hence that . Otherwise, the input power to each ADC grows monotonically with the antenna index, as will become apparent in the channel estimation recursion derived in Section V-A. We have found that the performance of the one-bit quantizer begins to degrade for values of close to the extremes of the interval , and that good performance can be achieved using a value near the midpoint, e.g., .

### Iv-B Two-bit Spatial ΣΔ ADCs

In this section, we extend the above analysis to the case where the quantizers in the array employ two-bits of resolution, implying four quantization levels. We use the well-known Lloyd-max condition to determine the optimum quantization levels that minimize the distortion [24, 22]

. We will denote the quantization levels and the associated intervals that minimize the distortion for unit variance Gaussian inputs by

and , respectively. That is, for a generic Gaussian random variable with unit variance, ,

 Q(x)=νi,ifx∈(νloi,νhii]. (21)

The quantization levels satisfy , , and . Since we assume that the input to the th ADC is a circularly symmetric Gaussian random variable with variance , the quantization bins should be adjusted to span the range of the input levels. Denoting by ,

 Qm′(rRem)=σrm√2νi,ifrRem∈(σrm√2νloi,σrm√2νhii], (22)

and

 ym=αm′(Qm′(Re(rm))+jQm′(Im(rm))), (23)

where as before the factor is the same for both the real and imaginary parts since is assumed to be circularly symmetric.

Assuming a linear model as before and using an element-wise Bussgang decomposition, due to the circular symmetry of the data, Eq. (12) can be written as

 γm=E[rRemyRem]E[rRem]2, (24)

where and is obtained from Price’s theorem [32] and is a function of the quantization levels and the decision thresholds. The theorem states that the correlation between the input and the output has the following derivative

 ∂E[rRemQm′(rRem)]∂rRem (25) =σrm√2∫∞−∞1√2π∂Qm′(rRem)∂rRemexp⎛⎝−(rRem)2σ2rm⎞⎠drRem.

The derivative can be computed as

 ∂Qm′(rRem)∂rRem=σrm√24∑i=2(νi−νi−1)δ(rRem−σrm√2νloi). (26)

Substituting the above equation in (25) and evaluating the integral, we get

 E[rRemyRem]=αm′σ2rm24∑i=2(νi−νi−1)√2πexp⎛⎝−(νloi)22⎞⎠. (27)

Then, the value of that yields is given by

 αm′=√2π∑4i=2(νi−νi−1)exp⎛⎝−(νloi)22⎞⎠. (28)

From (28), we can see that, unlike the one-bit case, is not dependent on the index at all for the array with two-bit quantization. Instead, it is a constant determined only by the quantization intervals and the corresponding levels. This is because the quantization levels and the intervals for the

th ADC have been scaled by the standard deviation of the input. Therefore,

corresponds to an additional correction that has to be applied to make . In the remainder of this discussion, we will drop the dependence on and use to denote . Finally, computing the expectation , the output and quantization noise powers are, respectively, given by

 σ2ym =α2∑i∣∣Qm′(rRem)∣∣2Pr(σrm√2νloi

where

is the cumulative distribution function of the normal distribution.

One distinction between the one- and two-bit operation is the correction factor . Unlike the one-bit case, the two-bit ADC shows little dependence on since increasing the number of bits results in a quantization error that is more accurately characterized as Gaussian. As a result, we omit this correction factor altogether for the two-bit case and still achieve good performance.

## V Channel Estimation with spatial ΣΔ ADCs

By describing the input-output relationship with an equivalent linear model, we were able to arrive at closed-form expressions for the quantization noise and output powers in Eqs. (20) and (29). Leveraging the results derived in the previous section, we derive below the LMMSE channel estimate based on the one-bit or two-bit outputs of the ADC array, .

### V-a LMMSE Channel Estimation

The LMMSE channel estimate is defined by

 ^g =E[gyH](E[yyH])−1y (30) =CgyC−1yy.

In Appendix A, we show that , and hence from (14), we can obtain the covariance matrix of :

 Cy =Cx+U−1CqU−H, (31)

where is the covariance matrix of . Similarly, since we have chosen in (11), it is easy to show that

 r=x−U−1Vq (32)

and hence that

 Cr =Cx+U−1VCqVHU−H. (33)

From (31), we see that is determined by , whereas (20) and (29) show that the quantization noise power is dependent on . Due to this inter-relationship between and , these matrices cannot be computed in closed form. However, they can be computed in a recursive manner.

Using and the fact that , we can show that . As a result, is approximately diagonal with elements given by . Furthermore, noting that has the structure

 U−1V=IN⊗e−jψ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣00…0010…0001…00⋮00…10⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (34)

we can generate the following recursion for , and using Eqs. (31) and (33):

 σ2rm=⎧⎪⎨⎪⎩σ2xm,m=kM+1,k=0,1,…,M−1,σ2xm+σ2qm−1,otherwise. (35) σ2qm=σ2ym−σ2rm.

Thus, the power of the th quantizer input, , depends on the quantization noise powers computed up to index . Then, the th output power, , is given by (20) for one-bit ADCs and by (29) for two-bit ADCs. This allows us to compute , and from there , and so on. Following this process for indices to allows us to compute , and finally the complete is obtained from (31).

Thus, we can obtain the LMMSE estimate of the channel, , as

 ^g=CgyC−1yy, (36)

where

 Cgy =E[gyH] (37) =E[g(Φg+n+U−1q)H] =CgΦH+E[gqH]U−H ≈CgΦH.

The final approximation results because , since and we can show that using an argument identical to that in Appendix A for Gaussian noise . The resulting algorithm for computing the LMMSE channel estimate for the array has low complexity and is summarized in Algorithm 1.

### V-B Channel Estimation Error

In order to use a performance metric that is independent of any scaling factor affecting either the real channel or its estimate, we define the normalized MSE (NMSE) as

 NMSE =minζE(∥g−ζ^g∥22)E(∥g∥22) (38)

This expression for the NMSE is valid for any estimator. For the particular case of LMMSE estimators, which necessarily satisfy , (38) can be expressed as

 (39)

where is given by

 C^g =E[^g^gH]=CgΦHC−1yΦCg (40) =CgΦ(Cx+U−1CqU−H)−1ΦHCg.

Finally, the estimation error covariance matrix is given by

 Cϵ=Cg−CgΦ(Cx+U−1CqU−H)−1ΦHCg. (41)

When the pilots are orthogonal and , the matrices , and are block-diagonal with each block being identical. As a result, is also block-diagonal and the th block corresponding to the covariance matrix of the estimated channel of the th user is equal to that of the other users.

Remark: We note here that a candidate for comparison with the above estimator is an LMMSE channel estimator based on the output of a massive MIMO system with conventional two-bit quantizers. Although, as mentioned earlier, a channel estimator based on the Bussgang decomposition for one-bit ADCs has been derived in [20], no explicit analysis exists for two-bit ADCs. Consequently, to facilitate such a comparison, we derive the LMMSE channel estimator based on standard two-bit quantization of the array output. A fundamental difference between the standard and two-bit ADC case is the factor . With the help of , we were able to control the amplitude of the input to the adjacent array element and prevent the ADC system from becoming unstable. However, in the standard implementation, the output of the th ADC is simply

 (42)

where

 Qm′(Re(xm))=σx√2νi,ifxm∈(σx√2νloi,σx√2νhii]. (43)

As in the case, we will use an element-wise Bussgang decomposition and develop a linear model for the output. More specifically,

 ystdm=γmxm+qstdm, (44)

where previous definitions hold for and , and the coefficients are not constrained to be . By carrying out an analysis similar to that in Section IV, we can solve for to get

 (45)

The resulting expression for is independent of , so we define . Finally, the output power and quantization noise powers are, respectively,

 σ2ystdm =σ2xm24∑i=1ν2i(Ψ(σxm√2νhii)−Ψ(σxm√2νloi)), (46) σ2qstdm =σ2ystdm−γ2σ2xm.

Let the ADC outputs be stacked in . Then, defining as a diagonal matrix with as its elements, the complete auto-correlation matrix of is given by

 Cystd=γCx+Cqstd. (47)

The LMMSE channel estimate is then obtained by

 ^gstd=CgΦHC−1ystdystd. (48)

## Vi Uplink Achievable Rate Analysis

In this section, we derive the uplink achievable rate with MRC and ZF receivers. In the uplink data transmission stage, the users transmit their data represented by the vector . Using a Bussgang decomposition as described previously on the -quantized received signal, , we get

 yd= Q(rd)=√ρdGs+nd+U−1dqd, (49)

where , is the data transmission power, and are the noise and quantization noise in the data phase, respectively. The matrices and are defined by taking in Eq. (10). We assume that the user symbols are i.i.d with and the noise vector consists of zero-mean circularly symmetric white Gaussian random variables with variance . In the derivation of the achievable rate, we will assume that during the training phase, the pilots are orthogonal and that . Consequently, the matrices , and will be block-diagonal.

Additionally, the analysis of the achievable rate relies on the covariance matrix of , , which is different from the quantization noise covariance matrix of the training phase. For the data transmission stage, this matrix has to be derived in a manner similar to Section V. Inspecting the recursion equations developed in the previous section, we see that initialization of the recursion will be performed with , where . In order to perform the Bussgang decomposition, the matrix should be known in advance. However, we can leverage the inherent channel hardening in massive MIMO systems to assume that . Thus,

 Cxd≈KρdCG+I. (50)

The above approximation does not require perfect CSI and this technique has been used in related prior work (e.g., see [20, 27]).

The procedure to obtain is outlined as follows. Let , and denote the powers of the th components of , and , respectively. Then, (35) is modified for the data transmission stage as

 σ2rdm=⎧⎪⎨⎪⎩σ2xd,m=0,1,…,M,σ2xd+σ2qdm−1,otherwise. (51) σ2ydm=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩π2β2σ2rdm,for 1-bit ADCsα2σ2rdm2∑4i=1ν2i(Ψ(σrdm√2νhii)−Ψ(σrdm√2νloi)),for 2-bit ADCs σ2qdm=σ2ydm−σ2rdm,

where since is equal to and its diagonal elements are equal to unity. The diagonal matrix is completed with as its diagonal elements.

The BS uses a linear receiver for symbol detection that depends on the LMMSE channel estimate. Denoting the linear receiver by , the detected symbol vector is obtained by multiplying the conjugate transpose of with the received signal vector as

 ^s=WHyd=√ρdWHGs+WHnd+WHU−1dqd. (52)

We consider the performance of two linear receivers in particular, namely the MRC and ZF receivers, where is given by

 WMRC=^G (53) WZF=^G(^GH^G)−1.

Here, is the matrix formed from using the inverse of the operation. From (52), we can re-write the various components contributing to the th detected symbol as

 ^sk= √ρdE[wHkgk]sk+√ρd(wHkgk−E[wHkgk])sk+ (54) √ρdwHk∑i≠kgisi+wHknd+w<