RF Analog Beamforming Fan Filters Using CMOS All-Pass Time Delay Approximations

Abstract

A continuous-time (CT) radio frequency (RF) antenna array beamformer and analog circuit based on a discrete-space-continuous-time (DSCT) 2-D fan-filter having transfer function H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F,A</sub> (z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sub> ,s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ct</sub> ) is derived. The proposed transfer function is based on a 2-D FIR discrete domain fan filter. The discrete domain prototype is converted to the proposed mixed-domain DSCT analog filter by replacing unit sampled delays with CT analog first-order all-pass networks corresponding to the bilinear transform. First-order all-pass network Φ(s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> ) is a poor approximation to a CT delay exp(- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sT</i> ) . To address this, a novel broadband pre-warping method is proposed to exactly compensate for such “bilinear warping”. A 65 nm CMOS VLSI circuit for Φ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> ) is proposed and an example fan filter with axis oriented at θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> =35 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> , half-fan-angle ε = 5 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> and maximum frequency <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> = 2.6 GHz is simulated employing closed-form expressions, an ABCD parameter based model and 65 nm CMOS simulations in Cadence. A stop-band interference rejection of 38 dB is verified by BSIM4 based simulations. The proposed circuit for Φ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> ) operates at 3.7 mA from a 1.2 V supply. The beamfomer is shown to operate correctly in the presence of PVT variations of Φ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> ).