Spatial Sigma-Delta Modulation for Coarsely Quantized Massive MIMO Downlink: Flexible Designs by Convex Optimization

• Published in: IEEE open journal of signal processing Vol. 5; pp. 1 - 20
• Main Authors: Keung, Wai-Yiu, Ma, Wing-Kin
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: <inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX"> Sigma \Delta</tex-math> </inline-formula> modulation; <named-content xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" content-type="math" xlink:type="simple"> <tex-math notation="LaTeX">$\Sigma \Delta$</tex-math> </named-content> modulation; Antenna arrays; Antennas; Bit error rate; coarsely quantized MIMO; convex optimization; Convexity; Delta modulation; Digital to analog converters; Downlink; Downlinking; Errors; Massive MIMO; Massive MIMO downlink; MIMO communication; Modulation; Modulators; Optimization; Precoding; Quantization (signal); Signal design; Vectors
• ISSN: 2644-1322; 2644-1322
• DOI: 10.1109/OJSP.2024.3375653

This paper considers the context of multiuser massive MIMO downlink precoding with low-resolution digital-to-analog converters (DACs) at the transmitter. This subject is motivated by the consideration that it is expensive to employ high-resolution DACs for practical massive MIMO implementations. The challenge with using low-resolution DACs is to overcome the detrimental quantization error effects. Recently, spatial Sigma-Delta (<inline-formula><tex-math notation="LaTeX">\Sigma \Delta</tex-math></inline-formula>) modulation has arisen as a viable way to put quantization errors under control. This approach takes insight from temporal <inline-formula><tex-math notation="LaTeX">\Sigma \Delta</tex-math></inline-formula> modulation in classical DAC studies. Assuming a 1D uniform linear transmit antenna array, the principle is to shape the quantization errors in space such that the shaped quantization errors are pushed away from the user-serving angle sector. In the previous studies, spatial <inline-formula><tex-math notation="LaTeX">\Sigma \Delta</tex-math></inline-formula> modulation was performed by direct application of the basic first- and second-order modulators from the <inline-formula><tex-math notation="LaTeX">\Sigma \Delta</tex-math></inline-formula> literature. In this paper, we develop a general <inline-formula><tex-math notation="LaTeX">\Sigma \Delta</tex-math></inline-formula> modulator design framework for any given order, for any given number of quantization levels, and for any given angle sector. We formulate our design as a problem of maximizing the signal-to-quantization-and-noise ratios (SQNRs) experienced by the users. The formulated problem is convex and can be efficiently solved by available solvers. Our proposed framework offers the alternative option of focused quantization error suppression in accordance with channel state information. Our framework can also be extended to 2D planar transmit antenna arrays. We perform numerical study under different operating conditions, and the numerical results suggest that, given a moderate number of quantization levels, say, 5 to 7 levels, our optimization-based <inline-formula><tex-math notation="LaTeX">\Sigma \Delta</tex-math></inline-formula> modulation schemes can lead to bit error rate performance close to that of the unquantized counterpart.