The Degrees of Freedom of the Oversampled Non-Coherent Channel

The degrees of freedom of a class of discrete-time non-coherent channels with oversampling, termed the Oversampled Non-Coherent (ONC) channel, are shown. The ONC channel is obtained from the classic continuous-time AWGN channel by considering the scenario in which the channel output is also corrupted by phase noise (PN) and processed by a multi-sample receiver. The continuous-time PN process has high variability which results in receiver output samples affected by a discrete-time PN process iid uniformly distributed over the unit circle. The ONC channel models the non-coherent detection scenario in which oversampling is employed for phase recovery but the PN process has such a high variance that its samples appear independent and uniformly distributed despite the oversampling. As such, the assumption of independent and uniformly distributed discrete PN is a limiting assumption that, generally speaking, well approximates the scenario in which the PN coherence time is much smaller than the oversampling time. In this paper, we obtain the generalized degrees of freedom for the case in which the oversampling factor L grows with the transmit power P as $P^{\alpha}$. Perhaps surprisingly, we show that no degree of freedom for reliable information transfer is available for $L \gt P^{2}$. We conjecture that the same capacity asymptotic holds for other PN channels with oversampling, such as the oversampled Wiener PN channel, when the noise variance grows to infinity faster than the sampling rate.