30-3 Multidimensional fast Fourier transform
We can generalize the -dimensional discrete Fourier transform defined by equation to dimensions. The input is a -dimensional array whose dimensions are , where . We define the -dimensional discrete Fourier transform by the equation
for .
a. Show that we can compute a -dimensional by computing -dimensional s on each dimension in turn. That is, we first compute separate -dimensional s along dimension . Then, using the result of the s along dimension as the input, we compute separate -dimensional s along dimension . Using this result as the input, we compute separate -dimensional s along dimension , and so on, through dimension .
b. Show that the ordering of dimensions does not matter, so that we can compute a -dimensional by computing the -dimensional s in any order of the dimensions.
c. Show that if we compute each -dimensional by computing the fast Fourier transform, the total time to compute a -dimensional is , independent of .
(Omit!)