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Transforms 75
Φk, 1 ≤ k ≤ log 2n, to a procedure whose cost is only proportional to n. Consequently,
the complete FFT can be performed with a cost proportional to nlog 2n, provided n is
a power of 2. This shows that, for large values of n, the procedure is efficient. Whether
this approach is competitive for small values of n, depends strongly on code implemen-
tation. Performance is also affected by different computer architectures.
A clear description, both intuitive and rigorous, of the steps to be followed to minimize
the operations, is given in brigham(1974), where a flow chart is presented. General-
izations to the case where n is not a power of 2, are discussed e.g. in singleton(1969)
and temperton(1983).
Finally, we touch on the problem of reordering the output data, known as un-
scrambling procedure. Consider for example equation (4.3.9). We associate to any index
0 ≤ m ≤ 7, its 3-digit binary representation, i.e.
0 000 0
1 001 4
2 010 2
3 011 6
4 100 1
5 101 5
6 110 3
7 111 7
The last column shows the indices obtained by reading the binary numbers from right to
left. These exactly match the indices of the left-hand side vector in (4.3.9). Reordering
this last set results in the reordering of the output data. The rough outline above can
be easily generalized. Recent variants of the FFT algorithm (self-sorting FFT) do not
require unscrambling of the data after applying the transform. We refer the reader to
specialized publications for further information.