The Cost of Phase Noise in the Waveform Channel: Bandwidth Exponents for Classes of Pulses and Capacity Bounds

In this paper, we determine the receive bandwidth at which phase noise leaves the degrees of freedom of the continuous-time (waveform) channel intact, and we bound its capacity at practical signal-to-noise ratios (SNRs). A Wiener phase of fixed linewidth multiplies the waveform, white noise is added on the whole line, and the receiver arbitrarily processes one band, whose width grows like the SNR to the power b. We show that the noise must not be cut to the transmission window before filtering, since the band then plays no role. Our results are unified, since they hold for classes of pulses defined by checkable conditions. To the best of our knowledge, no pre-log above 3/4 and no converse below 1 were known for this channel. Every bounded pulse confined to one symbol period has pre-log at most 3/4 under any processing. In contrast, a class of overlapping pulses of bounded support and a class of band-limited pulses have pre-log at least 3b for b < 1/3 and the full pre-log 1 for b ≥ 1/3. Conversely, for b < 1/3, the pre-log is below 1 for every piecewise-smooth pulse of bounded support and every band-limited pulse with a regular spectrum. Hence, for every root-raised-cosine and raised-cosine pulse and every Tukey pulse of roll-off below 1, the pre-log is 1 if and only if b ≥ 1/3. For the Tukey pulses, it equals 3b for 1/4 ≤ b < 1/3. The rectangle attains 3/4 for b ≥ 1/2, so this value is not an artifact of discrete-time models, whereas the band lowers it to (1 + b)/2 for 1/3 ≤ b ≤ 1/2. At b = 1, we bracket the capacities of the Tukey and truncated root-raised-cosine pulses of roll-off 1/2 at 20, 30 and 40 dB and normalized linewidths 10⁻³ and 10⁻², with the first upper bounds below the coherent capacity for this channel. For the Tukey pulse at 30 dB and normalized linewidth 10⁻³, our bounds determine the capacity to within 1.37 bits.