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The transposing back at the end of the routine can be avoided if a backtransform will follow9 , the backtransform must then be called with R and C swapped. The generalization to higher dimensions is straight forward. 10 The matrix Fourier algorithm (MFA) The matrix Fourier algorithm10 (MFA) works for (composite) data lengths n = R C. Consider the input array as a R × C-matrix (R rows, C columns). 7 (matrix Fourier algorithm) The matrix Fourier algorithm (MFA) for the FFT: 1. Apply a (length R) FFT on each column.

E. x with n zeros appended). ω) B {ω} B is the cc. of C {ω2 } C and therefore every B {} B-term is the cc. of the C {} C-term in the same line. Is there a nice and general scheme for real valued convolutions based on the MFA? Read on for the positive answer. 6 s, d lower half plus/minus higher half of x CHAPTER 2. 6 46 Convolution of real valued data using the MFA For row 0 (which is real after the column FFTs) one needs to compute the (usual) cyclic convolution; for row R/2 (also real after the column FFTs) a negacyclic convolution is needed7 , the code for that task is given on page 62.

Let Trc be the operator corresponding to the postprocessing in real_complex_fft_by_fht, and Tcr correspond to the preprocessing in complex_real_fft_by_fht. 21) −1 −1 It should be no surprise that Trc · Tcr = 1, or, equivalently Trc = Tcr and Tcr = Trc . 22) The corresponding code should be obvious. 20. 6 Discrete cosine transform (DCT) by HT The discrete cosine transform wrt. spr] which is its own inverse. 11 (DCT via FHT) Pseudo code for the computation of the DCT via FHT: CHAPTER 3. n-1] // workspace unzip_rev(x, y, n) fht(y[],ldn) cos_rot(y[], x[], n) } (cf.