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Use of Fourier transform in pre-processing

As we saw earlier, Hartley transform based fixed pre-processing is inferior to that based on Fourier transform. An explanation was given on the basis of symmetries and shift invariance in the section gif. Therefore we expect improved performances from Fourier transform even when the pre-processing is adaptive. However a training procedure which preserves the symmetries of weight distributions must be used. Main argument of the use of Hartley transform is to avoid the complex weights. But as seen from fig.gif, even Fourier transform can be implemented as a neural network containing real weights, but with a slightly modified network structure than the usual MLP. We can easily derive the equations which give the forward and backward pass.

Forward pass is given by,


where N denotes the window length, and tex2html_wrap_inline3284 .

If we use the notation


and error is denoted by J, then we can find tex2html_wrap_inline3290 simply by using the chain rule,


We assume that tex2html_wrap_inline3292 is known and tex2html_wrap_inline3294 can simply be found by differentiating eqn.2.1 wrt tex2html_wrap_inline3296 . Thus we get,


Eqns.2.2 and 2.3 define the backward pass. Note that tex2html_wrap_inline3296 can be further back propagated as usual.

Narada Warakagoda
Fri May 10 20:35:10 MET DST 1996

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