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8.3: Modifications to the Basic Cooley-Tukey FFT

Soon after the paper by Cooley and Tukey, there were improvements and extensions made. One very important discovery was the improvement in efficiency by using a larger radix of 4, 8 or even 16. For example, just as for the radix-2 butterfly, there are no multiplications required for a length-4 DFT, and therefore, a radix-4 FFT would have only twiddle factor multiplications. Because there are half as many stages in a radix-4 FFT, there would be half as many multiplications as in a radix-2 FFT. In practice, because some of the multiplications are by unity, the improvement is not by a factor of two, but it is significant. A radix-4 FFT is easily developed from the basic radix-2 structure by replacing the length-2 butterfly by a length-4 butterfly and making a few other modifications. Programs can be found in and operation counts will be given in Evaluation of the Cooley-Tukey FFT Algorithms.

Increasing the radix to 8 gives some improvement but not as much as from 2 to 4. Increasing it to 16 is theoretically promising but the small decrease in multiplications is somewhat offset by an increase in additions and the program becomes rather long. Other radices are not attractive because they generally require a substantial number of multiplications and additions in the butterflies.

The second method of reducing arithmetic is to remove the unnecessary TF multiplications by plus or minus unity or by plus or minus the square root of minus one. This occurs when the exponent of \(W_N\) is zero or a multiple of \(N/4\). A reduction of additions as well as multiplications is achieved by removing these extraneous complex multiplications since a complex multiplication requires at least two real additions. In a program, this reduction is usually achieved by having special butterflies for the cases where the TF is one or \(j\). As many as four special butterflies may be necessary to remove all unnecessary arithmetic, but in many cases there will be no practical improvement above two or three.

In addition to removing multiplications by one or \(j\), there can be a reduction in multiplications by using a special butterfly for TFs with \(W_{N/8}\), which have equal real and imaginary parts. Also, for computers or hardware with multiplication considerably slower than addition, it is desirable to use an algorithm for complex multiplication that requires three multiplications and three additions rather than the conventional four multiplications and two additions. Note that this gives no reduction in the total number of arithmetic operations, but does give a trade of multiplications for additions. This is one reason not to use complex data types in programs but to explicitly program complex arithmetic.

A time-consuming and unnecessary part of the execution of a FFT program is the calculation of the sine and cosine terms which are the real and imaginary parts of the TFs. There are basically three approaches to obtaining the sine and cosine values. They can be calculated as needed which is what is done in the sample program above. One value per stage can be calculated and the others recursively calculated from those. That method is fast but suffers from accumulated round-off errors. The fastest method is to fetch precalculated values from a stored table. This has the disadvantage of requiring considerable memory space.

If all the N DFT values are not needed, special forms of the FFT can be developed using a process called pruning which removes the operations concerned with the unneeded outputs.

Special algorithms are possible for cases with real data or with symmetric data. The decimation-in-time algorithm can be easily modified to transform real data and save half the arithmetic required for complex data. There are numerous other modifications to deal with special hardware considerations such as an array processor or a special microprocessor such as the Texas Instruments TMS320.

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