Tukey single taper

Recall that a suggested value for $M$ is $2\sqrt{N}$, which for $N=196$ gives $M\approx 28$. This value is compared along with a range of others ( $M=5,10,15,28$) when using Tukey tapering with no spatial smoothing of the autocorrelation estimate. The results are shown in figure 13 (TR=3 secs) and figure 17(a) (TR=1.5 secs).

The first thing to note from figure 13 is that when no autocorrelation estimation is made and the residuals are assumed to be white (i.e. $S=I$), then the boxcar design deviates far more from the theoretical distribution than the single-event design. This is because the single-event design has power at frequencies across the full range and is therefore less effected by not correcting for the coloured noise in the data which is concentrated at low frequency, whereas the boxcar design's power is mostly at its fundamental frequency, which is within the range of low frequency noise. This makes the obvious point that designs which concentrate their power as far away as possible from the low frequency end will suffer less from the low frequency noise in the data. However, the amount of bias for a randomised ISI when no autocorrelation estimation is made is still quite considerable, and so there is still the requirement for the estimation and correction of the autocorrelation.

The values $M=5$ and $M=10$ perform about the same in figure 13(a) and 17(a) and $M=5$ performs slightly better than $M=10$ in figure 13(b). The key point is that lower values of $M$, i.e. those that smooth the spectral density more than is normally recommended, perform better.