How Many Simultaneous Beamformers are Needed for Integrated Sensing and Communications?

Consider a downlink integrated sensing and communications (ISAC) system in which a base station employs linear beamforming to communicate to $K$ users, while simultaneously uses sensing beams to perform a sensing task of estimating $L$ real parameters. How many beamformers are needed to achieve the best performance for both sensing and communications? This paper establishes bounds on the minimum number of downlink beamformers, in which sensing performance is measured in terms of the Cram\'{e}r-Rao bound for parameter estimation and communications performance is measured in terms of the signal-to-interference-and-noise ratios. We show that an ISAC system requires at most $K + \sqrt{\frac{L(L+1)}{2}}$ beamformers if the remote users have the ability to cancel the interference caused by the sensing beams. If cancelling interference due to the sensing beams is not possible, the bound becomes $\sqrt{K^2 + \frac{L(L+1)}{2}}$. Interestingly, in the latter case, the bound on the number of beamformers is less than the sum of the bounds for each task individually. These results can be extended to sensing tasks for which the performance is measured as a function of $d$ quadratic terms in the beamformers. In this case, the bound becomes $K + \sqrt{d}$ and $\sqrt{K^2 + d}$, respectively. Specifically, for estimating complex path losses and angles-of-arrival of $N_\text{tr}$ targets while communicating to $K$ users, the bound on the minimum number of beamformers scales linearly in $K$ and in $N_\text{tr}$, assuming interference from sensing can be cancelled. When interference cancellation is not possible, the following exact characterization for the case of $N_\text{tr} = 1$ can be obtained: when $K=0$ or $1$, two beamformers should be used; when $K \ge 2$, exactly $K$ beamformers should be used, i.e., communication beamformers alone are already sufficient.