22X Page 75

(F.T.) of F is defind as the function F^{*} where

(X2) |

where Θ**.**Φ is the scalar product of Θ and Φ when they are considered as vectors with 0 for dot and 1 for cross. For example (U.N.) = 1**.**0+1**.**0+1**.**1+0**.**0= 1. It can easily be shown that F^{**} = F, i.e. that F is the F.T. of F^{*}, so the relation between F and F^{*} is symmetrical.

The "Faltung", F, of two functions F_{1} and F_{2} is defined by the equation

(X3) |

which is clearly also equal to .

It is easy to see that if F is the Faltung of F_{1} and F_{2} then F^{*} = F_{1}^{*}.F_{2}^{*}. In other words the F.T. of a Faltung is times the product of the F.T.'s. Therefore, by equation (X1)

.P.B.^{*}(ΔD) = P.B.^{*}(ΔP) P.B.^{*}(ΔΨ') |
(X4) | |

(see foot-note). |

where P.B.^{*} means the Fourier Transform of the Proportional Bulge.

This gives P.B.^{*}(ΔP) in terms of P.B.^{*}(ΔD) and P.B.^{*}(ΔΨ') and hence P.B.(ΔP) in terms of P.B.(ΔD) and P.B.(ΔΨ'). The process is not as laborious as it sounds in virtue of the rather simple interpretation of an F.T. For example if Θ is the T.P. letter J or vector (1, 1, 0, 1, 0) and if F is a P.B. function, taken as P.B.(ΔD) for definiteness, we have

since F is assumed to be a P.B. function.

Thus P.B.^{*}(J) = P.B.(ΔD_{1+2+4} = .) |
(X5) |

so we see that the F.T. of a P.B.is times the P.B. of the so-called "32-combination count" (R3 p. 49; R5 p. 55), for which the lower half of the Colossus switchboard is well adapted. The equation (X4) is now seen to express the well known and elementary property of the multiplication of P.B.'s.

Observe that P.B.(ΔP) is not quite determinate since

and the expected values of both numerator and denominator of this are zero, if ab=½. The same applies to the arguments 4, 9, 3, T.

Note: P.B.(ΔP) is a function of Θ and should strictly be written as P.B. (ΔP = Θ).

Back to General Report on Tunny. Contents.