  The polar receiver architecture is a receiver design that captures the
envelope and phase information of the signal rather than its in-phase and
quadrature components. Several studies have demonstrated the robustness of
polar receivers to phase noise and other nonlinearities. Yet, the
information-theoretic limits of polar receivers with finite-precision
quantizers have not been investigated in the literature. The main contribution
of this work is to identify the optimal signaling strategy for the additive
white Gaussian noise (AWGN) channel with polar quantization at the output. More
precisely, we show that the capacity-achieving modulation scheme has an
amplitude phase shift keying (APSK) structure. Using this result, the capacity
of the AWGN channel with polar quantization at the output is established by
numerically optimizing the probability mass function of the amplitude. The
capacity of the polar-quantized AWGN channel with $b_1$-bit phase quantizer and
optimized single-bit magnitude quantizer is also presented. Our numerical
findings suggest the existence of signal-to-noise ratio (SNR) thresholds, above
which the number of amplitude levels of the optimal APSK scheme and their
respective probabilities change abruptly. Moreover, the manner in which the
capacity-achieving input evolves with increasing SNR depends on the number of
phase quantization bits.
