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Ewan D. Barr added minres.tex
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\section{Maximum Temporal Resolution}
Appendix~\ref{app:fluct} displays fluctuation power spectra derived from the long-term average profiles of the Parkes Pulsar Timing Array (PPTA) pulsars observed at 50\,cm \cite{Dai_2015}. In these plots, the fluctuation power of each pulsar drops exponentially as a function of spin harmonic, eventually hitting a white noise floor. These plots indicate that 1024 phase bins is sufficient to resolve all of the structure in the mean pulse profiles of most currently observed MSPs. PSR\,J0437$-$4715 and PSR\,J2241$-$5236 are the worst case examples in which fluctuation power approaches the Nyquist limit where spin harmonics will be aliased.
Greater telescope sensitivity increases the signal-to-noise ratio, causing more fluctuation power to rise out of the white noise floor; however, these plots indicate that the number of spin harmonics required to resolve all significant power is proportional only to the logarithm of the S/N; i.e. to first order, the maximum harmonic,
\begin{equation}
H_\mathrm{max} \sim (1 - \log S/N) / k,
\end{equation}
where $k$ is the slope of the line fit to the spectrum of the pulsar, $S/N$ is the signal-to-noise ratio at the y-intercept (approximately the $S/N$ in the first harmonic) and it is assumed that harmonics past S/N~1 are not important. By eye, increasing the S/N by two orders of magnitude will add around 100 to 300 harmonics for 0437 and 2241, so 2048 phase bins should suffice for observing these pulsars with SKA Phase I.
Now, a sub-millisecond pulsar with a spin period around 500 microseconds and a pulse profile that requires 2048 bins to resolve all of the structure in its mean pulse profile would require 200\,ns time resolution, or 5\,MHz wide bands. In the current design, the SKA1.Mid beam-former will supply 10 MHz bands, so we are prepared for this hand-waving worst case scenario.
Having said this, it is currently not feasible to perform phase-coherent dispersion removal across a 10 MHz band at the low end of Band 1 (e.g. centre frequency around 355 MHz) out to the maximum DM of 3000, where the minimum required FFT length is 64 megasamples. This transform length is possible, but the computational burden and GPU RAM required to span the desired bandwidth (i.e. N x 10MHz sub-bands) will likely exceed what can be done in real time with the current design. I don't have a benchmark handy to support this, but I can work on something to explore the tradeoffs; e.g. by reducing the number of pulsars observed simultaneously and dividing the band over multiple processing nodes, greater compute resources can be focused on the extreme systems with both high DM and high temporal resolution requirements.
