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The Fourier transform provides us with a more efficient way to compute convolutions that only uses proportional n log(n) operations. 3) That is, convolution in original space is element-wise multiplication in Fourier space. n-1] // transform data: fft(x[], n, +1) fft(y[], n, +1) // convolution in transformed domain: for i:=0 to n-1 { y[i] := y[i] * x[i] } // transform back: fft(y[], n, -1) // normalize: for i:=0 to n-1 { y[i] := y[i] / n } } It is assumed that the procedure fft() does no normalization.

4. Transpose the matrix. Note the elegance! A variant of the MFA is called four step FFT in [28]. A trivial modification is obtained if the steps are executed in reversed order. The transposed matrix Fourier algorithm (TMFA) for the FFT: 1. Transpose the matrix. 2. Apply a (length C) FFT on each row of the transposed matrix. 3. Multiply each matrix element (index r, c) by exp(σ 2 π i r c/n). 4. Apply a (length R) FFT on each column of the transposed matrix. e. g. in unit strides). In radix 2 (or 2n ) algorithms one even has skips of powers of 2, which is particularly bad on computer systems that use direct mapped cache memory: One piece of cache memory is responsible for caching addresses that lie apart by some power of 2.

Read on for the positive answer. 7 Convolution of real valued data using the MFA Consider the MFA-algorithm for the cyclic convolution as given on page 46 but with real input data: For row 0 which is real after the column FFTs one needs to compute the usual cyclic convolution; for row R/2 which is also purely real after the column FFTs a negacyclic convolution is needed1 , the code for negacyclic convolution is given on page 67.

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