To decrease the power consumption of massive multipleinput multiple-output (MIMO) systems, architectures with low-resolution analog to digital converters (ADCs) have been studied extensively in the literature [1]- [6]. Although coarse quantization degrades the performance of the system, it has been shown that it can be alleviated by increasing the number of antennas [1] or exploiting more complicated signal processing techniques [7]. Recently, the idea of temporal ΣΔ quantization has been extended to the spatial domain [8]- [11]. It has been shown that, using minimal additional hardware in the analog domain, the resulting spatial one-bit ΣΔ architecture can shape the quantization noise to angles of arrival away from those that correspond to the users of interest. Hence, performance close to that of systems with high-resolution ADCs can be achieved while reducing power consumption and complexity.
One drawback of using one-bit ADCs not addressed in previous work is its susceptibility to strong interference, which can occur in MIMO systems when a jammer is present. Since a one-bit ADC has zero dynamic range, a moderately strong jammer can effectively swamp the relatively weak signals of interest and significantly degrade performance. With one-bit ADCs, jammer mitigation must occur in the RF domain prior to sampling, in order for the coarsely quantized ADCs to reveal the dynamics of the signals of interest.
In this paper, we show that the feedback employed by the spatial ΣΔ architecture can be generalized for this purpose. The genesis of the idea comes from work described in [12], which is based on the use of temporally oversampled ΣΔ ADCs. Instead of simply using feedback of the delayed ADC output as in a standard temporal ΣΔ architecture, the approach of [12] employs an analog feedback beamformer (FBB) designed to temporally null the interference. Unlike [12], in the method presented here we take a different approach that does not employ temporal oversampling, but instead uses the spatial ΣΔ architecture. In particular, the feedback between adjacent antennas is generalized to include an FBB signal that also serves to spatially null the interference. We generalize the approach of [11] to develop an equivalent linear model for the ΣΔ array that includes the FBB signal and sets the output level of the one-bit quantizers. Simulations show that while the ordinary spatial ΣΔ architecture is not effective in adequately alleviating the impact of the jammer, our proposed spatial FBB ΣΔ quantizer can provide performance that is close to that of an unquantized system.
In the next section we outline the basic system model. In Section III, some background on temporal FBB ΣΔ modulation is provided and the spatial FBB ΣΔ architecture is proposed. Then we adopt the equivalent linear model developed in [11] to analyze the FBB ΣΔ array. Simulation results are presented in Section IV, followed by our conclusions.
Notation: We use boldface letters to denote vectors, and capitals to denote matrices. The symbols (.) * , (.) T , (.) H , and (.) † represent conjugate, transpose, conjugate transpose, and pseudo inverse, respectively. A circularly-symmetric complex Gaussian (CSCG) random vector with zero mean and covariance matrix R is denoted n ∼ CN(0, R). The identity matrix is denoted by I and the expectation operator by E [.]. For a complex value, x = x r + j x i , we define x r = Re [x] and
Consider the uplink of a massive MIMO system consisting of a legitimate, single-antenna user that sends its signal to a base station (BS) equipped with a uniform linear array of M antennas. In addition, a jammer is present that aims to impair the performance of the legitimate user. Accordingly, the M × 1 received signal at the BS is
where p represents the average transmission power from the user, g U = 1/ √ L L =1 υ a (θ ) is the user's channel vector where L denotes the number of signal paths, υ ∼ CN (0, 1) is the complex channel gain for the -th path, and
denotes the array response vector for angle of arrival θ , where d and λ represent the antenna spacing and the carrier wavelength, respectively. The symbol s U ∈ C transmitted from the user satisfies E |s U | 2 = 1, and n ∼ CN 0, σ 2 n I denotes additive CSCG receiver noise at the BS. In addition, q represents the jammer's average power, g J = 1/ √ L L j=1 υ j a θ j is the channel vector between the jammer and the BS, and s J denotes the jammer's signal, where
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 applied separately to the real and imaginary parts as
The output voltage levels of the one-bit quantizers are represented by α m,r and α m,i . While the value of the output level is irrelevant for standard one-bit quantization, in the case of ΣΔ quantization the selection of adequate output levels is of paramount importance and the necessity for this more general approach will become apparent later 1 . Furthermore, 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. Finally, the received baseband signal at the BS is given by
In this subsection, we elaborate on the temporal FBB ΣΔ modulation approach of [12] to clarify the noise shaping characteristics of this technique. Fig. 1(a) shows a block diagram representing the Nth-order temporal FBB ΣΔ modulator with feedback weights w = [w 0 , • • • , w N -1 ] T . To shape the quantization noise, the weighted output signals are fed back and subtracted from the input (Δ-stage), and then this error is integrated (Σ-stage). 1 While the one-bit ADC output levels will be optimized, this is a one-time optimization and the values do not change as a function of the user scenario or channel realization. Thus the ADCs are still truly "one-bit." 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 FBB ΣΔ quantizer can then be written as
where
Unlike ordinary ΣΔ modulation that passes the signal through an all-pass filter and the quantization noise through a highpass filter, in (6) we see that X(z) is passed through A(z) and Q(z) through B(z) for FBB ΣΔ modulation. Hence, this approach not only provides a tool for shaping the quantization noise, but proper design of the feedback weights w allows for temporal filtering that passes the desired signal while eliminating undesirable contributions from other sources such as a jammer.
The idea underlying temporal FBB ΣΔ modulation can be adapted to the angle domain, in order to spatially shape the quantization noise in a desired way and remove interference. Instead of forming the Δ component using delayed samples of the quantized input as in the temporal case, we use the quantized signals from adjacent antennas. A direct translation of the temporal ΣΔ concept to the angle domain pushes the quantization noise to higher spatial frequencies, which correspond to angles away from the array broadside (|θ| 0 • ), while oversampling (i.e., reduced d/λ) pushes the signals of interest near broadside closer to zero spatial frequency. However, by phase-shifting the quantization error in the feedback loop prior to the Σ stage, the quantization error can be shaped away from a band of spatial frequencies that is not centered at zero. This bandpass approach has been proposed for spatial ΣΔ architectures in [9], [11]. It is worthwhile to note that although ordinary ΣΔ modulation provides a noise shaping characteristic, FBB ΣΔ not only shapes the quantization noise, but also suppresses the extra quantization noise caused by the jammer with appropriate feedback beamforming.
Fig. 2 shows the architecture of the angle-steered FBB ΣΔ array. Using Fig. 2, we can formulate a compact input-output description of the spatial FBB ΣΔ array by defining
V = e -jφ Z -1 UW,
wherefoot_1
and expressing the input to the quantizers as
The output of the angle-steered one-bit FBB ΣΔ array is then defined by
To analyze the performance of spatial ΣΔ processing, we will represent the one-bit quantization operation in ( 14) with an equivalent linear model [11]
where Γ = diag (γ 1 , . . . , γ M ) with
and q denotes the effective quantization noise. In (16), it is assumed that r m is circularly symmetric. This assumption implies that identical values should be selected for the output levels of the real and imaginary quantizers, and thus we will let α m represent both α m,r and α m,i . Following the same reasoning as in [11], we will set Γ = I by choosing an appropriate value for each α m . Therefore, we obtain the following mathematical model for the FBB ΣΔ architecture:
where B = I + V . Equation ( 17) is the spatial equivalent of the temporal FBB ΣΔ description in (6), with the following equivalences:
Note that the condition Γ = I and assuming that r m is approximately Gaussian leads to the following choice for the output levels [11]:
Next, we calculate the power of the equivalent quantization noise q, which is needed both to analytically assess the system performance and to compute α m . In the discussion below, we show how to express (20) in terms of the statistics of the array input x, which illustrates how the quantizer output levels can be analytically chosen in a practical setting. Moreover, we will show how spatial FBB ΣΔ impacts the power of the quantization noise and elaborate on how it differs from the ordinary spatial ΣΔ approach.
With Γ = I, (15) becomes
Since r m and q m are uncorrelated, and using (20), we obtain
To determine E |r m | 2 , we substitute (21) into (13), so that
Let us denote Ψ = B -1 U and Υ = B -1 V . It is clear that Ψ is a lower triangular matrix and Υ is a lower triangular matrix with zeros along the main diagonal. In addition, following the same reasoning as in Appendix A of [1], it can be shown that
Eq. ( 24) implies that
To approximate E |r m | 2 , and for the sake of analysis, we assume R q is diagonal. Since Υ is a lower triangular matrix with zeros along the main diagonal, we only need the first m-1 diagonal elements of R q to specify E |r m | 2 . Hence, we can recursively calculate E |r m | 2 for m > 1 using ( 22) and (25).
In the next section, we show that the diagonal elements of R q are much smaller than those for the ordinary ΣΔ architecture. This is because of the appropriate design of the feedback weights that lead to the elimination of strong interference before the one-bit quantization.
This section describes the results of several Monte Carlo simulations in order to illustrate the performance of the FBB ΣΔ quantizer. In our simulations, we assume L = 20 multipath arrivals for both the legitimate user and the jammer with angles of arrival randomly uniformly distributed in θ l ∈ [θ 0 -δ, θ 0 +δ], where the center angle θ 0 is different for the user and the jammer. We set θ 0 u = -20 • and θ 0 j = 60 • for the desired user and jammer, respectively, with δ = 5 • . We further assume d/λ = 1/4, 8-PSK symbols, and 10 5 trials. The steering angle of the FBB ΣΔ array is set to φ = 2π d λ sin θ 0 u . We let σ 2 n = 1, so that p and q denote the SNR of the user and the jammer, respectively. In all simulations, we consider a strong interference setting with q = 20 dB. We also assume that θ 0 u and θ 0 j are known at the BS 3 . Hence, following the same reasoning as in [12], the feedback weights are estimated as
where
and β is a constant. In the simulations that follow, we selected β = √ 10. Then, the solution in ( 26) is followed by iterative refinement (see Section III-C in [12]) to find the desired feedback weights. Fig. 3 shows the simulated and analytically derived quantization noise power density which is defined as
where R is the covariance matrix of the quantization noise for each approach (standard one-bit, ΣΔ, or FBB ΣΔ). In this figure, we set the order of the FBB ΣΔ filter at N = 50.
We see that the quantization noise power for the FBB ΣΔ array is substantially lower over the angles where the user is 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. In addition, we see that even the ordinary ΣΔ array suffers from large quantization noise in the presence of strong interference. We also observe that there is excellent agreement between the simulations and our theoretically derived expressions for both cases. Note that careful design of the quantizer output levels is a critical component in achieving the desired ΣΔ noise shaping characteristic shown here. In Fig. 4, we compare the symbol error rate of the FBB ΣΔ array with that of a system with high-resolution ADCs and a system with high-resolution ADCs and no strong interference, i.e., x p = √ pg U s U + n, as a benchmark. The methods that do not allow FBB in the RF domain must attempt to cancel the interference digitally, after the quantization. Consequently, for the systems implemented with high-resolution ADCs, standard one-bit ADCs, and the original spatial ΣΔ architecture, we project the sampled signal onto the subspace orthogonal to the interference in the digital domain. Denoting the signals received by the standard one-bit and ΣΔ architectures by y 1 and y ΣΔ , respectively, the signals after the projection for the three methods are given by B -1 U x, B -1 U y 1 , and B -1 U y ΣΔ . We assume perfect channel state information (CSI) is available and use the maximum ratio combiner (MRC) at the BS to decode the 8-PSK symbols. Fig. 4 shows the superior performance of the one-bit FBB ΣΔ architecture which achieves performance equivalent to that of a system with only high resolution ADCs. This performance is achieved with only minimal additional hardware in the analog domain, and thus has significantly reduced complexity and energy consumption compared with a system employing high-resolution ADCs.
We presented a new spatial one-bit FBB ΣΔ architecture for mitigating strong interference in massive MIMO systems with one-bit quantization. We showed that this simple architecture can effectively compensate for the vulnerability of one-bit ADCs against strong interference. The critical challenges in designing this architecture are to find the appropriate output levels for the one-bit quantizers and the values for the feedback weights. A recursive algorithm was proposed to specify the quantizers' output levels. The feedback weights were designed by adopting an algorithm used previously for a temporal FBB ΣΔ implementation. However, the behaviour of the feedback weights indicates that they amount to a spatial beamformer pointing in the direction(s) of the interference, and hence could be designed by a less complicated approach. Interesting directions for future work include studying the impact of angle estimation errors on the performance of the FBB ΣΔ architecture, or using the approach for combined quantization noise shaping and transmit beampattern design for the downlink with low-resolution digital-to-analog converters.
This work was supported by the U.S. National Science Foundation under Grants CCF-1703635 and ECCS-1824565 and by the Spanish Ministry of Science and Universities Grant PRX18/00638.
Note that Z -1 is the spatial domain equivalent of the delay operator z -1 for the z-transform in the time domain.
Note that, although here we consider a single jammer in a known location, the feedback weights can in general be designed to reduce the impact of signals arriving from multiple sectors in which the jammers are known to lie, without precise knowledge of the actual jammer locations.