Analytical construction of a multidimensional carrierless amplitude-phase filter basis

Carrierless amplitude-phase (CAP) modulation is a widely used format in optical transmission, distinguished by its ability to extend the signal constellation beyond the conventional two-dimensional case. However, closed-form analytical expressions for CAP pulse-shaping filters are available only for the conventional two-dimensional CAP. At present, the only known approach for constructing orthonormal CAP transceiver filters for N > 2 relies on solving a minimax optimization problem. In this paper, we demonstrate that this approach may converge to degenerate solutions in the form of time-multiplexed pulse amplitude modulation (PAM) waveforms, undermining the purpose of multidimensionality. As an alternative, we propose an analytical low-numerical-complexity construction of a multidimensional CAP basis, formed as a superposition of frequency-overlapping filters that individually satisfy the conditions for interference-free transmission. For N that are powers of two, the resulting filters exhibit ideal characteristics, whereas for the remaining N, they show a limited, a few dB amplitude non-uniformity. The proposed formulation is more immune to synchronization errors than conventional CAP. In addition, a method for DC suppression is presented.