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Spectral Efficiency of One-Bit Sigma-Delta Massive MIMO

We examine the uplink spectral efficiency of a massive MIMO base station employing a one-bit Sigma-Delta sampling scheme implemented in the spatial rather than the temporal domain. Using spatial rather than temporal oversampling, and feedback of the quantization error between adjacent antennas, the method shapes the spatial spectrum of the quantization noise away from an angular sector where the signals of interest are assumed to lie. It is shown that, while a direct Bussgang analysis of the Sigma-Delta approach is not suitable, an alternative equivalent linear model can be formulated to facilitate an analysis of the system performance. The theoretical properties of the spatial quantization noise power spectrum are derived for the Sigma-Delta array, as well as an expression for the spectral efficiency of maximum ratio combining (MRC). Simulations verify the theoretical results and illustrate the significant performance gains offered by the Sigma-Delta approach for both MRC and zero-forcing receivers.


I. INTRODUCTION
To reduce complexity and energy consumption in large-scale MIMO systems, researchers and system designers have recently considered implementations with low-resolution analogto-digital and digital-to-analog converters (ADCs, DACs). Compared to hybrid analog/digital approaches, fully digital architectures, even with low-resolution sampling, provide increased flexibility and fully exploit the potentially large array gain promised by massive MIMO systems. The case of onebit quantization has received the most attention, both for the uplink [1]- [9] and downlink [10]- [20] scenarios.
While one-bit ADCs and DACs offer the greatest simplicity and power savings, they also suffer the greatest performance loss compared to systems with higher resolution sampling, particularly for moderate to high signal-to-noise ratios (SNRs), and in situations with strong interference. Besides simply increasing the ADC/DAC resolution, mixed-ADC architectures [21]- [24] and temporal oversampling [25]- [27] have been proposed to bridge the performance gap, with a corresponding increase in complexity and power consumption.
Oversampled one-bit quantization has a long history in digital signal processing, particularly using the so-called sigma-This work was supported by the U.S. National Science Foundation under Grants CCF-1703635 and ECCS-1824565. delta (Σ∆) approach, which quantizes the difference (∆) between the signal and its previously quantized value, and then integrates (Σ) the resulting output [28]- [30]. This has the effect of shaping the quantization noise to higher frequencies, while the signal occupies the low end of the spectrum due to the oversampling. Higher-order Σ∆ modulators can be constructed that provide increased shaping of the quantization noise from low to high frequencies. Compared with a standard one-bit ADC, a Σ∆ ADC requires additional digital circuitry to implement the integration, but very little additional RF hardware. Σ∆ ADCs have been commonly used in process control and instrumentation applications, and more recently in the implementation of multi-channel beamformers for ultrasound imaging systems.
The concept of Σ∆ modulation can also be applied in the spatial as well as the temporal domain. In a spatial Σ∆ implementation, the difference signal is formed by subtracting the quantized output of one antenna's RF chain from the signal at an adjacent antenna. Coupled with spatial oversampling (e.g., a uniform linear array with elements separated by less than one half wavelength), the quantization noise is shaped to higher spatial frequencies, and significantly reduced for signals arriving in a sector around broadside (0 • ). Applying a phase shift to the feedback signal allows one to move the band of low quantization error to different angular regions.
In this paper, we study the uplink spectral efficiency (SE) of a massive MIMO basestation (BS) that employs one-bit spatial Σ∆ quantization, and compare it with the performance achievable by systems with infinite resolution and standard one-bit quantization for maximum ratio combining (MRC) and zero-forcing (ZF) receivers. Past work on quantifying the SE for standard one-bit quantization (e.g., [5], [12]) has relied on a vectorized version of the well-known Bussgang decomposition [40], which formulates an equivalent linear vector model for the array of non-linear quantizers assuming that the inputs to the quantizers are (at least approximately) jointly Gaussian. However, the vector Bussgang solution is not appropriate for the more complicated Σ∆ architecture, since it leads to a linear model that is inconsistent with the corresponding hardware implementation. Thus, we are led to derive an alternative linear model in which we apply a scalar version of the Bussgang approach to each quanitizer individually. This model is then used in turn to determine the overall sum SE.
The results of the analysis indicate the significant gain of the Σ∆ approach compared with standard one-bit quantization for users that lie in the angular sector where the shaped quantization error spectrum is low. For MRC, the one-bit Σ∆ array performs essentially the same for such users as a BS with infinite resolution ADCs. The angular sectorization of users in the spatial domain is not necessarily a drawback in cellular implementations, where cells are typically split into 120 • regions using different arrays on the BS tower. In addition, there are many small-cell scenarios both indoors and outdoors where the targeted users are confined to relatively narrow angular sectors (auditoriums, plazas, arenas, etc.). Such situations will become even more prevalent as frequencies move to the millimeter wave band. However, the size of the sector of good performance for Σ∆ arrays depends on the amount of spatial oversampling. Unlike the temporal case, where oversampling factors of 10 or higher are not uncommon, the physical dimensions of the antenna and the loss due to increased mutual coupling for closely-spaced antennas places a limit on the amount of spatial oversampling that is possible in massive MIMO. Fortunately, our results indicate that spatial oversampling by factors of only 2-4 is sufficient to achieve good performance for angular sectors ranging from 80 • −150 • . Furthermore, the ability of the Σ∆ array to electronically steer the desired angular sector by means of the feedback phase shift provides desirable flexibility.
In the next section we outline the basic system model, and provide some background on temporal Σ∆ modulation. In Section III we describe the spatial Σ∆ architecture, and develop an equivalent linear model. The model is then applied to analyze the spectral efficiency of the Σ∆ array in Section IV. While the analysis is conducted assuming that perfect channel state information (CSI) is available, we also discuss the impact of imperfect CSI in Section V . Several simulation results are presented in Section V, followed by our conclusions.
Notation: We use boldface letters to denote vectors, and capitals to denote matrices. The symbols (.) * , (.) T , and (.) H represent conjugate, transpose, and conjugate transpose, respectively. A circularly-symmetric complex Gaussian (CSCG) random vector with zero mean and covariance matrix R is denoted n ∼ CN (0, R). The symbol . represents the Euclidean norm. The identity matrix is denoted by I, vector of all ones by 1, and the expectation operator by E [.]. We use diag (C), diag (x), and diag (x 1 , · · · , x M ) as the diagonal matrix formed from the diagonal entries of the square matrix C, elements of vector x, and scalars x 1 , · · · , x M , respectively. For a complex value, x = x r + jx i , we define x r = Re [x]

II. SYSTEM MODEL
Consider the uplink of a single-cell multi-user MIMO system consisting of K single-antenna users that send their signals simultaneously to a BS equipped with a uniform linear array (ULA) with M antennas. The M × 1 signal received at the BS from the K users is given by where G = [g 1 , · · · , g K ] ∈ C M×K is the channel matrix between the users and the BS and P is a diagonal matrix whose kth diagonal element, p k , represents the transmitted power of the kth user. The symbol vector transmitted by the users is denoted by s ∈ C K×1 where E ss H = I K and is drawn from a circularly symmetric complex Gaussian (CSCG) codebook independent of the other users, and, n ∼ CN 0, σ 2 n I M denotes additive CSCG receiver noise at the BS.
We consider a physical channel model described in the angular domain and comprised of L paths for each user with azimuth angular spread Θ [41]. In particular, for the kth user, the channel vector is modeled as where A k is an M × L matrix whose ℓth column is the array steering vector corresponding to the direction of arrival (DoA) θ kℓ ∈ θ 0 + − Θ 2 , Θ 2 , β k models geometric attenuation and shadow fading from the kth user to the BS, and the elements of h k ∈ C L×1 are assumed to be distributed identically and independently as CN (0, 1), and model the fast fading propagation. For a ULA, the steering vector for a signal with DoA θ kℓ is expressed as where u kℓ = sin (θ kl ), z kℓ = e jωs kℓ , and thus ω s kℓ = 2π d λ u kℓ represents the spatial frequency assuming antenna spacing d and wavelength λ.
In a standard implementation involving one-bit quantization, each antenna element at the BS is connected to a one-bit ADC. In such systems, the received baseband signal at the mth antenna becomes where Q m (.) denotes the one-bit quantization operation which is applied separately to the real and imaginary parts as where α m,r and α m,i represent the output voltage levels of the one-bit quantizer. We will allow these levels to be a function of the antenna index m, unlike most prior work which assumes that the output levels are the same for all antennas. The necessity for this more general approach will become apparent

A. Temporal Σ∆ Modulation
In this subsection, we elaborate on temporal Σ∆ modulation to clarify the noise shaping characteristics of this technique. Fig. 1(a) shows a block diagram representing the temporal Σ∆ modulator. To shape the quantization noise, the output signal is fed back and subtracted form the input (∆-stage), and then this error is integrated (Σ-stage). To characterize the transfer function of this non-linear system, we substitute the one-bit quantizer with the equivalent linear model depicted in Fig.  1(b). The input-output relationship of the Σ∆ quantizer can then be written as Simply stated, the objective of Σ∆ modulation is to pass the signal through an all-pass filter and the quantization noise through a high-pass filter. This objective can be realized by setting γ ≈ 1. Since commercial quantizers are provided with a built-in automatic gain control (AGC), the γ ≈ 1 condition is inherently satisfied in implementations of temporal Σ∆ modulators, and hence this issue is not generally discussed in the literature. However, as we show in the next subsection, the choice of the scaling factor is critical in the mathematical modeling of spatial Σ∆ architectures, and we derive a criterion for addressing this issue.

B. One-Bit Spatial Σ∆ Modulation
As mentioned earlier, the basic premise of temporal Σ∆ modulation can be adopted in the angle domain, in order to spatially shape the quantization noise in a desired way. Instead of forming the ∆ component using a delayed sample of the quantized input as in the temporal case, we use the quantization error signal from an adjacent antenna. A direct transfer of the temporal Σ∆ idea to the angle domain as in [36], [37] pushes the quantization noise to higher spatial frequencies, which correspond to DoAs away from the array broadside (|θ| ≫ 0 • ), while the oversampling (reduced d/λ) pushes signals of interest near broadside closer to zero spatial frequency. However, by phase-shifting the quantization error in the feedback loop prior to the ∆ stage, a Σ∆ frequency response can be obtained in which the quantization error is shaped away from a band of frequencies not centered at zero. This bandpass approach has been proposed for both the temporal (e.g., see [28]) and spatial [39] versions of the Σ∆ architecture. Fig. 2 shows the architecture of an angle-steered Σ∆ array. Using Fig. 2 and equation (8) at the top of the next page, we can formulate a compact input-output description of the spatial Σ∆ array by defining and expressing the input to the quantizers as The output of the angle-steered one-bit Σ∆ array is then defined by y = Q (r) .

1) Linear Model:
To analyze the performance of spatial Σ∆ processing, analogous to temporal Σ∆, we will represent the one-bit quantization operation in (12) with an equivalent linear model as follows: where Γ is an M × M matrix and q denotes the effective quantization noise. The value of Γ that makes the equivalent quantization noise, q, uncorrelated with r is Γ 0 = R H ry R −1 r . For the case where the elements of r are all jointly Gaussian, the computation of R ry is possible by resorting to the Bussgang theorem 2 [40]. This was the approach used in [5], [12] for a massive MIMO implementation with standard one-bit quantization, and the resulting Γ 0 was a diagonal matrix.
For the case of the Σ∆ architecture, even if the matrix Γ 0 could be computed, this decomposition would not be of interest, for at least two reasons. First, the equivalent quantization noise q that results from setting Γ = Γ 0 in (13) bears no connection to the quantization error fed from one antenna to the next as shown in Fig. 2. Setting Γ = Γ 0 would produce a model in which r m and q m−1 are uncorrelated, but it is clear from Fig. 2 that r m for the Σ∆ array directly depends on the quantization error from the (m − 1)-th stage. Second, Γ 0 cannot be a diagonal matrix 3 , unlike the standard one-bit quantization case considered in [5]. The presence of off-diagonal elements in Γ 0 implies that the model in (13) represents the output of each quantizer as a linear combination of the inputs to that quantizer as well as other quantizers in the array. Such a model does not have an apparent connection with the scheme in Fig. 2, where each quantizer produces its output depending only on its input alone. These inconsistencies between the mathematical model based on Γ = Γ 0 and the physical block diagram of the Σ∆ array in Fig. 2 are the result of attempting to force r and q to be uncorrelated, when the architecture is actually propagating the quantization error from one stage to the next.
Consequently, in order to derive an appropriate model for the analysis of the Σ∆ architecture, we propose to apply the Bussgang decomposition to each quantizer individually. In particular, we formulate the model in (13) using a matrix Γ = diag (γ 1 , . . . , γ M ) that is forced to be diagonal. This is equivalent to imposing a model in which r and q are uncorrelated component-wise: E [r m q * m ] = 0. The elements of Γ are given by where in the last equality and from now on, we assume that r m is circularly symmetric. This assumption implies that the quantizer output levels are identical for the real and imaginary parts, and thus we use α m to represent both α m,r and α m,i . The expression in (14) has been defined in the literature as the equivalent gain of a non-linear device [42], [43].
As we will see later on, since the elements of Γ depend only on the signals at one stage of the Σ∆ architecture, they are much easier to compute than the elements of Γ 0 . Moreover, the resulting decomposition is consistent with Fig. 2. Given that no precondition is imposed on the correlation E [r m q * l ] for m = l, the model is compatible with the fact that the quantization noise of one stage appears in subsequent stages.
Plugging (13) into (11) and using some algebraic manipulations, we obtain the following mathematical model for the Σ∆ architecture: Equation (15) is the spatial Σ∆ equivalent to the temporal domain Σ∆ description in (7). Similar to the temporal case, (15) indicates that Γ = I should hold for the spatial Σ∆ array to work as desired, that is, to pass x and q through spatial all-pass and high-pass filters, respectively. If Γ = I, then (15) becomes and the m-th element of y is expressed as which explicitly shows the quantization noise-shaping characteristic of the spatial Σ∆ architecture. The only task remaining to complete our proposed linear model is to calculate the power of the equivalent quantization noise. The condition Γ = I for the adequate operation of the Σ∆ scheme determines the quantization levels that have to be set. Setting (14) equal to 1, we obtain the optimum value of α m : It is worth noting that (18) is different from which leads to the Lloyd-Max one-bit quantizer that minimizes the mean-squared-error (MSE) between the input and the output of the quantizer. However, the Lloyd-Max approach makes the quantization error uncorrelated with the quantizer output, but not with the input. While the expression derived in (18) is useful, it is difficult to analytically evaluate the expectations in closed form, and it is not clear how the output level could be tuned using analog processing in the RF chain (e.g., via an AGC or some other type of calibration). To address this issue, we use the assumption that r m is Gaussian inherent in the Bussgang decomposition to find an approximation for α ⋆ m that is easier to deal with, both for the subsequent mathematical analysis and from the viewpoint of a hardware implementation. The validity of the approximation will be apparent in the numerical examples presented later. If r m is Gaussian, we can write In the discussion below, we show how to express (20) in terms of the statistics of the array output x, which provides an analytical solution and clarifies how the quantizer output levels could be set in a practical setting.

2) Quantization Noise Power:
In this section, we calculate the power of the effective quantization noise and the power of the quantizers' inputs, which is needed to properly set the output levels using (20). With Γ = I, (13) becomes Since r m and q m are uncorrelated, and using (20), we obtain To determine E |r m | 2 , we substitute (21) into (11), so that It can be shown that where 4 Moreover, following the same reasoning as in Appendix A of [5], it can be shown that E [x m ′ q * m ] ≈ 0, ∀m, m ′ ∈ M = {1, · · · , M }. This results in R qx ≈ 0. Therefore, Eq. (26) implies that Substituting (22) into (27) and noting that E |r 1 | 2 = E |x 1 | 2 , we obtain the following recursive equality to calculate E |r m | 2 for m > 1: Let where χ can be any element of the set χ ∈ {r, x, q}. Then, using (22) and (28), we have where (32) Equation (30) shows that the calculation of E |r m | 2 needed in (20) can be formulated in terms of the power of the antenna outputs E |x m | 2 , for which simple expressions exist from (1). This further implies that control of E |x m | 2 via an AGC would allow the quantizer output levels to be set without feedback from the digital baseband. In the following remark, we show that, using the optimal quantizer output settings, the power of the quantization noise does not grow with m despite the fact that is propagated from one antenna to the next.
Remark 1. Eq. (31) implies that, by appropriately selecting the quantizers' output levels, the quantization noise power does not increase without bound. In particular, consider the case where the power of the received signal is constant over the array elements, i.e., p x = p x 1. Then, which shows that, in the limit of a large number of antenna elements, the quantization noise power converges to a constant value of approximately 1.33 times the input power.

3) Quantization Noise Power Density:
In the time domain, it is well-known that sampling a band-limited signal by a rate N times larger than the Nyquist rate and down-sampling after quantization can reduce the in-band quantization noise power by a factor of 1/N and 1/N 3 for standard one-bit and Σ∆ modulation, respectively [44]. In this section, we look for a similar behaviour for quantization across an array in space. More precisely, we want to quantify how spatial oversampling, i.e., decreasing the antenna spacing, d/λ, (or equivalently, increasing the number of antennas for spaceconstrained arrays) can reduce the quantization noise power for the in-band angular spectrum. To do so, we define the quantization noise power density as where R is the covariance matrix of the quantization noise. To differentiate the two cases, we denote the covariance matrix of the quantization noise for standard one-bit quantization as R q 1 , and the covariance of the Σ∆ quantization noise as R q Σ∆ . Expressions for these covariance matrices will be derived later in this subsection. Hence, the normalized received quantization noise power over some angular region, Θ, is given by 5 where δ = sin Θ 2 . Next we find P q for standard one-bit and Σ∆ quantization. a) One-bit Quantization: Unlike [5], for standard one-bit quantization, we choose the quantizer output levels as α m = πE |x m | 2 /2 so that y m = Q (x m ) = x m + q m . This causes no loss of generality for standard one-bit quantization, since the value of the quantizer output has no impact on the performance of the resulting system. Therefore, the covariance matrix of the quantization noise can be written as where the arc-sine law [45], [46] is used to obtain and Note that the arc-sine in (37) is applied separately to each element of the matrix argument, and also separately to the real and imaginary parts of the matrix elements. From [5], we have that diag (R y ) = π 2 diag (R x ). Since the off-diagonal elements of Υ are small, we use the approximation sin −1 (x) ≈ ζx, where ζ > 1, to obtain Moreover, from (1), R x becomes 5 To simplify the calculation of the quantization noise power, we assume without loss of generality that the Σ∆ array is steered to broadside (θ = 0).
where for L ≫ 1, u kℓ can be taken as a random variable uniformly distributed in [−δ, δ]. That is, Therefore, Now we are ready to calculate the standard one-bit quantization noise power, P q1 .
Equation (47) (16), the covariance of the quantization noise for the Σ∆ architecture is R q Σ∆ = U −1 R q U −H . We derive an expression for the normalized quantization noise power of the Σ∆ array, P qΣ∆ , in the next proposition.
Proposition 2. The quantization noise power for spatial Σ∆ quantization is We set φ = 0 due to the assumption of u ∈ [−δ, δ] in the definition of the quantization noise power, and we note that Then In addition, from (23), and the fact that R qx ≈ 0, it can be readily shown that E q m q * m±1 ≈ 0. Hence, for the sake of analysis, we approximate E [q m q * m ′ ] ≈ 0, ∀m = m ′ ∈ M, and therefore R q = diag p q . Consequently, By integrating (52) and using some algebraic manipulation, we arrive at (48 where in (a) we have used sinc (x) ≈ 1 − x 2 6 and 1 M Tr for M ≫ 1 and assuming p x = p x 1. Equation (53) Equation (55) states that, for spatial Σ∆ quantization, increasing the number of antennas for an array with a fixed aperture, d 0 , decreases the quantization noise power proportional to 1/M 2 . Hence, the spatial Σ∆ architecture brings about an oversampling gain of M 3 compared to the standard one-bit architecture.
In the next section, we study the spectral efficiency of a massive MIMO system with spatial Σ∆ processing and discuss the impact of the spatial Σ∆ architecture on the system performance.

IV. SPECTRAL EFFICIENCY
In this section, we study the SE of a massive MIMO system with spatial Σ∆ processing. We consider maximum ratio combining (MRC) and zero-forcing (ZF) receivers. We derive here an approximation for the SE of the system with an MRC receiver, and evaluate the SE for the ZF receiver in the next section, numerically. We first present the case where perfect channel state information (CSI) is assumed to be available at the BS, and then we discuss the impact of imperfect CSI on the system performance at the end of the section .
From (1) and (16), the received signal at a BS with a Σ∆ architecture can be modeled as Denoting the linear receiver by W , we havê and the kth element ofŝ is given bŷ where w k is the kth column of W . We assume the BS treats w H k g k as the desired channel and the other terms of (58) as worst-case Gaussian noise when decoding the signal. Consequently, a lower bound for the ergodic achievable SE at the kth user can be written as [48] where (60)

A. MRC Receiver
For the case of an MRC receiver, W = G. The following proposition presents an approximation for the achievable SE of a massive MIMO system with spatial Σ∆ processing and an MRC receiver.
Proposition 3. For a massive MIMO system employing a spatial Σ∆ architecture and an MRC receiver, the SE of the kth user assuming perfect CSI is given by eq. (61) shown at the top of the next page, where Σ ik [48], an approximation for (59) can be calculated as By setting w k = g k and using Lemma 2 of [49] and Lemma 1 of [50], the expected values of the desired signal, interference, and thermal noise can be readily calculated. For the quantization noise term, note that Therefore, In addition, the kth user channel vector can be written as where h kl is the lth element of h k . Hence, which, after some algebraic manipulation, leads to (61) and the proof is complete.
Remark 6. The noise shaping characteristic of the spatial Σ∆ architecture is explicitly manifested in (61). A similar characteristic is observed in [39] for Σ∆ precoding. It shows the importance of the design parameter φ which should be chosen to minimize G = 1 for all users. By writing the steering angle as φ = 2π d λ sin (θ), we have Eq. (67) indicates that G could be made arbitrarily small by decreasing the relative antenna spacing d/λ (the spatial oversampling gain) or sin (θ) − sin (θ kℓ ) (the angle steering gain). However, physical constraints on the antenna spacing and larger angular spreads, Θ, limit the lower bound on G.
For the case that L ≫ 1, sin (θ kℓ ) = u kℓ can be taken as a random variable uniformly distributed in [δ 1 , δ 2 ] where δ 1 = sin θ 0 − Θ 2 and δ 2 = sin θ 0 + Θ 2 . Hence, In this case, the optimal value of the steering angle that minimizes G can be simply derived as which indicates that the optimal steering angle is dependent on δ 1 , δ 2 , and the relative antenna spacing d/λ.

B. ZF receiver
For the ZF receiver, W = G G H G −1 . After substituting this for W in (62), the SE achieved for the kth user with the Σ∆ architecture and ZF receiver can be written as in (70) at the top of the next page. Although (70) does not provide direct insight into the effect of the shaped quantization noise on the SE, in Section V we numerically evaluate this expression and show the superior performance of the Σ∆ architecture compared with standard one-bit quantization.

V. NUMERICAL RESULTS
In this section, we numerically evaluate the SE performance of Σ∆ massive MIMO systems in various scenarios. We assume static-aware power control in the network [51] so that p k = p 0 /β k . In all of the cases considered, we assume M = 100 antennas, K = 10 users, L = 50 coherent paths per user, and an angular spread of Θ = 40 • centered at θ 0 = 30 • . The DoAs for each user are drawn uniformly from the interval [10 • , 50 • ], which corresponds to u = sin(θ) ∈ [0.17, 0.77], and the steering angle of the Σ∆ array is set to φ = 2π d λ sin (θ 0 ). The SNR is defined to be SNR p0 σ 2 n . Fig. 3 shows the simulated and analytically derived quantization noise power density, i.e., ρ q (u) , u ∈ [−1, 1], for Σ∆ and standard one-bit quantization when the relative antenna spacing is d = λ/4. We see that the quantization noise power for the Σ∆ array is substantially lower over the angles where the users are present, while the effect is the opposite for standard one-bit quantization -the quantization noise is higher for angles where the amplitude of the received signals is larger. We also observe that there is excellent agreement between the simulations and our theoretically derived expressions for both cases. Note that the careful design of the quantizer output levels is a critical aspect for achieving the desired Σ∆ noise shaping characteristic shown here.
The impact of spatial oversampling on the shape of the quantization noise spectrum is illustrated in Fig. 4. We see from the figure that, as discussed in Remarks 2, the quantization noise power for the standard one-bit ADC architecture grows as d/λ decreases. Analogously to temporal Σ∆ modulation where increasing the sampling rate helps to push the quantization noise to higher frequencies and widen the quantization-noise-free band, we can reduce the quantization noise power over wider angular regions by placing the antenna elements of the array closer together. For example, when d = λ/2, the Σ∆ quantization noise power is below that of the standard one-bit quantizer over a beamwidth of 40 • . This beamwidth increases to about 80 • , 150 • , and 180 • for d = λ/4, d = λ/8, and d = λ/16, respectively. Mutual coupling will impact these results as d decreases, but both the standard one-bit and Σ∆ approaches would be expected to degrade.
In Fig. 5, we compare the SE performance of Σ∆ and standard one-bit quantization for the case of an MRC receiver. It is clear that the derived theoretical SE expression very closely matches the simulated value. The one-bit Σ∆ implementation achieves a significantly increased SE compared with standard one-bit quantization, and performs nearly identically to an MRC receiver with infinite resolution ADCs. It should be emphasized that this performance gain is achieved without paying a significant penalty in terms of power consumption (as with mixed-ADC architectures) or complicated processing (as required by non-linear receivers).
In Fig. 6, we numerically evaluate the SE when the ZF receiver is employed. The SE improvement of Σ∆ processing is much greater than for the case of MRC. For example, at SNR = 0 dB, about a 50% improvement in SE can be achieved by the spatial Σ∆ architecture compared with standard one-bit quantization, which confirms its ability to provide high SE with a simple architecture and low power consumption.
The effect of channel estimation error on the performance of the algorithms is also shown in Fig. 6 for the ZF receiver. For these results, we used a least squares (LS) channel estimator for each of the algorithms. Fig. 7 shows the performance of the ZF receiver with and without perfect CSI versus the number of antennas. The presence of imperfect CSI obviously degrades all of the algorithms, but we see that the Σ∆ architecture provides a way to successfully bridge the performance gap between standard one-bit and high-resolution quantization with only a minimal increase in hardware complexity.

VI. CONCLUSION
In this paper, we studied the performance of massive MIMO systems employing spatial one-bit Σ∆ quantization. Using an element-wise Bussgang approach, we derived an equivalent linear model in order to analytically characterize the spectral efficiency of a massive MIMO base station with a Σ∆ array, and we compared the results with the performance achieved by an array that employs standard one-bit quantization. Our results demonstrated that the spatial Σ∆ architecture can scale down the quantization noise power proportional to the square of the spatial oversampling rate. This can be interpreted as scaling down the quantization noise power proportional to the inverse square of the number of antennas at the BS for space-constrained arrays. This result gains more importance by noting that in standard one-bit quantization, the quantization noise power grows proportional to the inverse of the spatial oversampling rate, or equivalently, proportional to the number of antennas at the BS in space-constrained arrays. Furthermore, it was shown how this capability allows the spatial Σ∆ architecture to bridge the SE gap between infinite resolution and standard one-bit quantized systems. For the ZF   receiver, the spatial Σ∆ architecture can outperform standard one-bit quantization by about 50%, and achieve almost the same performance as an infinite resolution system for the MRC receiver. While these results were obtained by assuming the availability of perfect CSI at the BS, we showed that the ability to steer the spatial Σ∆ architecture to different angles allows the system to cope with the impact of channel estimation error and, hence, leads to a design that is robust against imperfect CSI.
The Σ∆ array reduces the quantization error over a given sector of angles, and shapes it to spatial frequencies outside the sector. Thus, performance for users within the sector is improved relative to standard one-bit quantization, while interference outside the sector should be removed prior to signal detection. The concept of cell sectorization is consistent with current cellular implementations that divide the cell into 120 • regions using separate antenna arrays and antenna elements with non-uniform angular responses. The size of the sector of "good performance" for the Σ∆ array is a function of the amount of spatial oversampling in the array; the closer the antennas are placed to each other, the wider the sector. While temporal Σ∆ receivers can be designed to oversample by factors of 10 or more to achieve a very high degree of quantization noise reduction, the physical dimensions of the antennas and the deleterious effect of mutual coupling prevent such large oversampling factors for the Σ∆ array. Fortunately, as shown by the numerical results in this paper, reducing the antenna spacing by only a factor of 2 is sufficient to realize large spectral efficiency gains with the Σ∆ architecture. Oversampling by 2 does not create a sector as large as 120 • , but the ability of the Σ∆ array to electronically control the center point of the angular sector provides important flexibility. For example, multiple sectors could be serviced in parallel with a single antenna array by deploying a bank of Σ∆ receivers tuned to different spatial frequencies, in order to cover a wider angular region.