A "square" pulse, positive or negative, is sent along a uniform transmission line from\ell = 0and is regenerated at\ell = L, with some probability of error. The pulse is possibly regenerated at an intermediate point. In all the most reliable channels and in several others that we have examined, the best intermediate point for regeneration is the midpoint,\ell = L/2, but it is sometimes better still not to regenerate at any intermediate point. Some of the theory might be suggestive for explaining why the "nodes of Ranvier" are approximately equally spaced in neural axons. For some of the less reliable channels the midpoint is locally worst or even absolutely worst, and the optimal intermediate regeneration point is elsewhere, at say\ell = \ell_{1}(or equally good atL - ell_{1}, by symmetry).

The purpose of this note is to show as clearly as possible the mathematical relationship between the two basic fast methods used for the calculation of discrete Fourier transforms and to generalize one of the methods a little further. This method applies to all those linear transformations whose matrices are expressible as direct products.

Let a1, a2, …, aN; b1, b2, …, bN be 2N numbers, each of at most ν binary digits. We wish to calculate the 2M + 1 “lagged products” [equation: see PDF] for some M ⩽ N − 1. We suppose that N is large, and M not too small, and that ν is considerably smaller than the “word-length” of a computer. A comparison is made of three methods of organizing the calculation, in each of which several numbers are packed into a single word.