Towards a Low-SWaP 1024-beam Digital Array: A 32-beam Sub-System at 5.8 GHz

The efficient formation of far-field antenna patterns simultaneously across a multitude of directions is crucially important for wireless communications, radio astronomy, imaging, radar, and electronic warfare. Multibeam beamforming has been usually achieved in the microwave domain using analog techniques (e.g., Rotman lenses [37] and Butler/Nolan matrices [31, 17]). Emerging mmW systems are considering hybrid multibeam beamforming due to its power efficiency and excellent performance for a reasonably small number of antenna elements and user streams [45, 44]. Although digital beamforming requires the control of each individual antenna element in an antenna array, it is promising for the future due to its many inherent advantages, which include [18]: i) maximum flexibility/reconfigurabilty; ii) easy system updates and support for new beamforming algorithms as they emerge; iii) precise control of both the gain and phase of individual antenna elements thus giving better control of the beams; iv) maximum degrees of freedom from a given array; and v) reduced maintenance and calibration requirements.

Element-wise digital beamforming requires a dedicated receiver (or transmitter, in transmit mode) for each antenna element, which is usually a uniformly spaced linear or rectangular array of antennas. Multibeams can be generated by expanding the concept of a phased-array to multiple simultaneous directions by using the fact that each direction of propagation of a carrier wave is associated with two spatial frequencies $\left(\omega_{x},\omega_{y}\right)\in\mathbb{R}^{2}$ across the two orthogonal coordinate axes of a rectangular array aperture. Multiple beam digital beamforming is desired at the lowest possible energy consumption for a given bandwidth, supply voltage, and technology node, which leads to domain-specific architectures that are optimized for low complexity and power consumption.

Here we propose approximate computing-based algorithms and computing architectures that achieve quasi-orthogonal RF beams without using any digital multiplier circuits. The multiplierless nature of the digital computing architectures allow low chip area/size, weight, and power consumption (SWaP) and avoid the need for digital multipliers that have high circuit complexity (transistor count) and power consumption. This is likely to become more critical as wireless systems move to sub-terahertz frequencies and much wider channel bandwidths than used currently [35]. Algorithms that are multiplierless thus lead to substantially reduced SWaP in real-time digital silicon implementations [7, 35].

Multibeam beamforming on linear/rectangular apertures is important for exploiting multi-directional channels in massive-MIMO systems, for example in 5G mm-wave wireless networks. Such systems rely on the combination of beamforming with MIMO theory [21, 36, 12, 45, 35, 44] and as frequencies move to THz ranges, the need for providing thousands of simultaneous beams will emerge due to the small size of the wavelength and physical antenna aperture. Recent work has described different phased array architectures targeting 5G applications [46, 27, 25, 26, 11, 39, 29]. However, most of the literature has been concentrated either on hybrid beamforming systems or fully-analog architectures due to the prohibitive processing complexity of fully-digital beamforming. Other important applications for microwave and mm-wave digital multibeam beamformers include emerging defense applications such as space-based low earth orbit communications, mesh networks between micro satellites, space-based Internet distribution to densely populated areas, and multi-domain mosaic warfare where reliable high-speed wireless connectivity is needed across multiple platforms (air, space, land, and sea), as well as for emerging terahertz imaging systems [35]. The demands of high-capacity wireless networks for such applications can be significantly more difficult to meet than commercial 5G standards since modern military platforms can travel at hypersonic speeds across hundreds of kilometers. Such demanding wireless channel conditions necessitate beamforming gain across wide bandwidths and narrow angles of propagation (i.e., sharp beams) to both thwart detection and also benefit from beamforming gain. Furthermore, 5G networks will eventually require digital beamforming to reduce the overhead associated with the current 3GPP beam search time in the 5G game structure—great reduction in beam pointing can be obtained by simultaneously searching the environment for the best pointing angle, but this is not yet supported in the 5G 3GPP standard [3].

Some recent work has focused on achieving element-wise fully-digital beamforming. The paper in [24] presents a low-power 8-element digital beamforming prototype based on bit-stream processing. The design uses a low-resolution $\Delta\Sigma$ architecture that replaces multipliers with multiplexers. This multiplexer-based architecture achieves lower power and smaller area than conventional digital beamformers, but the design is limited to a 20 MHz bandwidth with only two simultaneous output beams. Another recent paper [23] reports a 16-element 4-beam digital beamformer targeting large scale MIMO for 5G communications systems. It uses a similar multiplexer-based approach as in [24] with an interleaved architecture to support a 100 MHz bandwidth. The work in [32, 49] also report experimental verification of fully digital multibeam beamforming schemes targeting MIMO-based 5G implementations.

The paper [30] presents a spatial DFT-based digital multibeam beamforming implementation scheme for satellite communications. The earlier work of authors in [4] describes a low-complexity algorithm using the spatial DFT based approach for generating 16 simultaneous beams using a ULA. The work in [4] uses a 16-point DFT approximation to generate the simultaneous beams and presents the measured beams of its fully digital implementation targetting 5G MIMO applications. This paper describes a novel low-complexity algorithm for realizing a massive number of simultaneous sharp beams, which are vitally important in coping with the rapid increases in path loss expected in future mmW/sub-THz wireless systems. In particular, we propose a low-SWaP approach to generating 1024 beams using a $32\times 32$ aperture and ultra-low-complexity digital VLSI hardware. We propose a 32-beam subsystem based on a novel 32-point DFT approximation as the building block of such a $32\times 32$ system. The proposed 32-beam subsystem has been implemented at 5.8 GHz and the digital beams have measured and compared with those from exact-DFT-based beams. The measured beams have been used to derive the beam patterns of the corresponding $32\times 32$ rectangular aperture by assuming identical element patterns in all directions.

The paper is structured in as follows. Section I provides a introduction to the paper Section II, followed by the introduction, describes the role of the DFT for spatial filtering in multibeam transceivers. Section III describes a 32-point approximate DFT algorithm with zero multiplicative complexity. Section IV describes an experimental realization of a 32-element receive array that can implement the proposed approximate DFT algorithm for digital beamforming. Experimental results from this array are presented in Section V. Finally, Section VI concludes the paper.

Multibeam beamforming on an $N\times M$ ($N,\leavevmode\nobreak\ M\in\mathbb{Z^{+}}$) linearly spaced rectangular array can be achieved by uniformly sampling the spatial frequency domains to define a set of far-field plane-waves having spatial frequencies determined by setting $\left(\omega_{x},\omega_{y}\right)=\left(\frac{2\pi}{N}k_{1},\frac{2\pi}{M}k_{2}\right)$ where $k_{1}=0,1,\ldots,M-1$ and $k_{2}=0,1,\ldots,N-1$. For this analysis we consider $M=N$ so that the same proposed $N$-point transform can be used row-wise and column-wise in the rectangular aperture for generating two-dimensional (2D) beams. The spatial frequency points $\left(\frac{2\pi}{N}k_{1},\frac{2\pi}{N}k_{2}\right)$ correspond to beams pointing at unique angle pairs indexed by $\left(k_{1},k_{2}\right)\in\mathbb{Z}^{2}$. The corresponding spatially-sampled time-continuous plane waves at the terminals of the array elements can be expressed in a Fourier basis as $E(n_{x},n_{y},t)=E_{0}\sum_{k_{1}=0}^{N-1}\sum_{k_{2}=0}^{N-1}x_{m}(t)e^{j(n_{x}\omega_{x}+n_{y}\omega_{y}+2\pi f_{c}t)}$ where $f_{c}$ is the unmodulated carrier frequency, $E_{0}$ is a constant that sets the signal power, and $x_{m}(t)$ is the complex modulated information component of the signal. Note that we assume that the bandwidth of $x_{m}(t)$ is much smaller than $f_{c}$, since our analysis is only valid for narrowband signals for which the so-called spatial-wideband effect is negligible [28, 47].

In receiver mode, the plane-waves present at the antennas are sampled in the spatial domain, amplified and filtered, down-converted to baseband (or an intermediate frequency (IF), and finally digitized by an analog-to-digital converter (ADC) present at each array location. The digitized signals at each location is complex, i.e., has in-phase ($I$) and quadrature ($Q$) components. For example, down-conversion using a quadrature mixer (which is modeled as multiplication by $e^{-j2\pi f_{c}t}$) leaves the spatial frequency components $E_{BB}(n_{x},n_{y},t)=x_{m,k}(t)e^{j(n_{x}\omega_{x}+n_{y}\omega_{y})}$ intact. As a result, the spatial spectrum of the wave remains localized at $\left(\omega_{x},\omega_{y}\right)$. The creation of a sharp RF beam for extracting directional information for a particular plane-wave therefore involves the application of a 2D spatial bandpass filter having the sharpest possible selectivity centered on a particular frequency pair in the spatial frequency domain. From filter theory, it is known that the discrete Fourier transform (DFT) realizes a filterbank of finite impulse response (FIR) filters with sharp bandpass responses that take the well-known $\operatorname{sinc}(\omega)$ response shape; the peak stopband magnitude for this shape has an asymptote of to $-13.25$ dB for increasing filter order $N$. Therefore, to simultaneously receive an $N\times N$ array of signals, the multibeam beamformer must compute the 2D DFT spatially across the $n_{x}$-$n_{y}$ dimensions of the array.

For transmit applications, the waves to be transmitted at simultaneous multiple directions are applied to the inputs of the 2D inverse DFT (IDFT), with the corresponding IDFT outputs being converted to analog using digital-to-analog converters (DACs), filtered, up-converted to the desired carrier frequency, and amplified before being applied to the input terminals of the transmit array. Thus, the computation of the 2D spatial DFT/IDFT for each new sample of the digital baseband signal is a straightforward technique for achieving a large number of simultaneous RF beams in both receive and transmit modes.

Fig. 1(a) shows the digital beamforming architecture for an $N\times N$ uniform rectangular aperture (URA) that generates $N^{2}$ beams. The block diagram of an $N$-element uniform linear array (ULA) subsystem that acts as a building block for the $N^{2}$ URA is shown in Fig. 1(b). The direct computation of the DFT of an $N$-point vector of input values is defined as the matrix-vector multiplication $\mathbf{X}=\mathbf{F}_{N}\cdot\mathbf{x}$ where $\mathbf{x}$ and $\mathbf{X}$ are both $N$-point complex column vectors and $\mathbf{F}_{N}$ is the $N\times N$ complex matrix, known as the DFT matrix [6]. The direct matrix-vector multiplication requires a number of complex arithmetic operation in $\mathcal{O}(N^{2})$, where $\mathcal{O}(\cdot)$ is O-notation [20, p. 429]. However, because of the symmetries of the DFT matrix, it is possible to compute the matrix-vector product $\mathbf{X}=\mathbf{F}_{N}\cdot\mathbf{x}$ with less than $N^{2}$ complex arithmetic operations. Algorithms that exploit the DFT matrix structure are called fast Fourier transforms (FFTs). FFTs are a famous and well-explored class of fast algorithms based on sparse matrix factorizations, which ultimately reduce the arithmetic complexity to compute $\mathbf{X}=\mathbf{F}_{N}\cdot\mathbf{x}$ to be of $\mathcal{O}(N\>\log N)$. The complexity reduction from $\mathcal{O}(N^{2})$ to $\mathcal{O}(N\>\log N)$ is substantial as $N$ grows large, which explains the importance of the use of FFTs in place of the DFT.

If we consider a single-beam conventional digital beamforming scenario, that would need 3$N$ real multiplications and 5$N$ + 2($N$-1) = 7$N$-2 addition operations per beam. Considering this complexity, if $N$ arbitrary beams are needed, then such system would demand for 3$N^{2}$ real multiplications and $7N^{2}-2N$ real additions. The use of FFTs brings down this complexity to the order of $N\log N$. Using the proposed approach mentioned in section 3, we can obtain FFT-like 32 one-dimensional (1D) beams at no multiplications and with only addition operations. We believe that these kind of large number of beams are required for applications like millimeter wave 5G communications as described in the introduction of the paper. This method can be an overkill for applications that require only few number of beams. But for applications that do need multiple simultaneous beams, the proposed method provides an attractive solution with a much lower power consumption and area in VLSI implementations.