| ### Appendix C - Jitter Calculations
Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks
Russell D. Reed and Robert J. Marks II
Copyright © 1999 Massachusetts Institute of Technology

## C.2 Jitter: CDF-PDF Convolution in n Dimensions

The following shows that the convolution of an n-dimensional spherical Gaussian probability density function (PDF) and a Gaussian cumulative distribution function (CDF) results in another Gaussian CDF.

Let f1 (x) be a spherical Gaussian PDF in n-dimensions

(C.3) |

and let F2(x) be a Gaussian CDF of the form

(C.4) |

This can be written as

(C.5) |

where ŵ = w/|w| and σ2 = 1/|w|.

The convolution of F2 and f1 is the n-dimensional integral

(C.6) |

Separate x and α into components parallel and orthogonal to ŵ

where l and k are scalars and y and β are n-dimensional vectors orthogonal to ŵ.

Then

(C.7) |

and

Thus F2(x) * f1(x) reduces to a one-dimensional convolution of a Gaussian CDF with standard deviation σ2 = 1/|w| and a Gaussian PDF with standard deviation σ1. It can be shown (see section C.3) that this is a Gaussian CDF with variance σ23 = σ21 + σ22.

Letting Za denote the Gaussian CDF function with standard deviation a,

(C.8) |

Thus, the convolution of a Gaussian CDF and a Gaussian PDF can be computed by a simple scaling of the original CDF.

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