Sphere Packing Analysis for Performance Trade-Off in Joint Communications and Sensing-Part II: Fourier Analysis of Volume

The technology of joint communications and sensing (JCS) is expected to prevail in 6G wireless networks. There exists a performance tradeoff between the functions of communications and sensing in JCS. One effective approach to analyze the tradeoff in JCS is to consider the level sets of a given performance metric of sensing as the signaling space of communication codewords. Then, performance bounds can be obtained for communications using the approach of sphere packing. The principle for generic sensing performance metric has been studied in the first part of this paper. In the second part of this paper, the concrete cases of sensing performance metrics, namely the signal-to-noise ratio (SNR) and integrated sidelobe level (ISL), are studied. The problems are turned into the volume evaluation for the intersection of a (elliptic) sphere (the quadratic approximation of the level set) and a hyperplane (the constraint on the total transmit power). They are solved by using the theory of Fourier-transform-based volume evaluation of convex sets. It is found that the optimal waveform is not unique, thus providing free lunch (although not plenty of) for communications. Another finding is that the communication data rate increases logarithmically with respect to the sensing performance metric degradation.