A Fast True Time-Delay Wideband Multi-Beam Beamforming Algorithm Based on a 16-Beam Approximate-DVM

True-time-delay (TTD) beamformers can generate wideband squint-free beams in both analog and digital signal domains. The delay Vandermonde matrix (DVM) was introduced as a mathematical model that represents a TTD-based multi-beam beamformer while reducing the delays from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(N</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>) to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(N</i> log <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>), where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> = 2<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">^r</i>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i>s ≥ 1) represents the number of beams. In this paper, we propose to reduce the complexity of delays from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(N</i> log <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>) to nearly <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(N)</i> for a small number of beams. More preciously, we present a recursive algorithm to compute the DVM-vector product with a complexity reduction of at least 21% to at most 52% compared to our most recent work, and at least 39% to at most 98% compared to the brute-force DVM-vector calculation. This enhancement was achieved by utilizing 16-beam approximated-DVM (ADVM) building blocks executing recursively with the DVM algorithm. The reduced complexity DVM algorithm achieves nearly linear complexity for smaller input sizes, specifically when <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ≤ 1024. This modification results in a complexity reduction when compared to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(N</i> log <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>) complexity of the DVM algorithm, spanning from 8 to 1024 beams. For example, by computing the DVM-vector product for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> = 8 to 1024 elements antenna arrays, we can obtain wideband RF beams while reducing the required chip area and power consumption by at least 21% at 1024 beams to at most 52% at 16 beams compared to radix-2 DVM algorithm, and also at least 39% at 8 beams to at most 98% at 1024 beams compared to the brute-force DVM-vector product computation. With this reduction, we show that the proposed DVM algorithm is better suited for end-to-end RF-IC design that includes multiple wideband channels. At the end, a signal flow graph, simulated beam patterns at 150 MHz, 300 MHz, 600 MHz, and 1 GHz frequencies based on the proposed ADVM algorithm, and a digital overview are provided to demonstrate the simplicity, efficiency, and accuracy of the proposed TTD multibeam beamformers for RF-IC design.