Bounds on the Minimum Number of Beamformers for Integrated Sensing and Communications

Consider a downlink integrated sensing and communications system where a base station employs linear beam-forming to estimate an unknown vector of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L$</tex> real parameters, while communicating with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex> users. What is the minimum number of beamformers needed to simultaneously perform both tasks? This paper first shows that the minimum number of beamforming vectors is bounded by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K+\sqrt{L(L+1)}/2$</tex>, when the sensing performance is measured in terms of the Cramér-Rao bound involving an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L$</tex> x <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L$</tex> Fisher information matrix, and the communications performance is measured in terms of signal-to-noise-and-interference ratios. This bound can be tightened and also generalized by recognizing that when the sensing metric is a function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex> quadratic terms involving the beamformers, the minimum number of beamformers is at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K+\sqrt{M}$</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex> can be less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L(L+1)/2$</tex>. In particular, for the task of estimating the complex path loss and the angle-of-arrival (AoA) of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> targets (with a total of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L=3N$</tex> parameters), due to the interdependencies in estimating these parameters through the beamformers, we show that the total number of beamformers needed is at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K+\sqrt{3.5N^{2}+0.5N}$</tex>. For the sensing-only scenario with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex> = 0, the minimum number of beamformers needed is asymptotically bounded by 1.871N.