The bootstrap implementation of the EKH algorithm is a simple and intuitive
way of understanding what actually occurs but with the obvious limitation that
it is exact only when the amount of the frequency switch is a whole number of
channels. The generalization of the bootstrap to non-whole
numbers of channels follows trivially after expressing its actions in terms
of a convolution in the manner of the original discussion of EKH. As a
function of channel number the convolving function
for a switch of s channels is
where , and the div operation signifies integer division discarding any remainder. If the convolution is implemented with an FFT, the spectrum should be well padded with zeroes at either end.
Figure 6 (click here) shows convolution functions for the cases of 34 and channels. The delta functions of the whole number case at top have a clearer ringing behaviour at bottom; the sinc function is used to interpolate band-limited, critically-sampled functions and the action of convolution with a single off-center sinc is simply to translate by a non-integer number of channels. The phase of the comb in Fig. 6 (click here) has been chosen so that the recovered signal will appear in the same channels as are occupied by the signal phase of the switching cycle. If both the signal and reference phases are symmetrically offset and the recovered signal should appear between them, the term -ms in Eq. (4) can be replaced by -(m+1/2)s.
Note that the simple existence of a fractional channel switching interval does not by itself mandate the use of the bootstrap. The ensuing sacrifice in noise level (a factor of at least if the simple shift and subtract technique can be used) causes larger statistical uncertainties in the estimated line properties, quite possibly exceeding the error introduced by use of a slightly inaccurate switching interval. When features are well-resolved, whole-channel shifts introduce at most only small errors. The convolution implementation of the bootstrap runs much more more slowly than the whole-channel algorithm expressed in Eq. (3).