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Next we take advantage of the symmetries of the sine and cosine as functions of the frequency index $k$ . Using these symmetries on [link] gives
for $k=0,1,2,\cdots ,N/2-1$ . This again reduces the number of operations by a factor of two, this time because it calculates two outputvalues at a time. The first reduction by a factor of two is always available. The second is possible only if both DFT values are needed. Itis not available if you are calculating only one DFT value. The above development has not dealt with the details that arise with the differencebetween an even and an odd length. That is straightforward.
If the length of the sequence to be transformed is even, there are further symmetries that can be exploited. There will be four data values that areall multiplied by plus or minus the same sine or cosine value. This means a more complicated pre-addition process which is a generalization of thesimple calculation of the even and odd parts in [link] and [link] will reduce the size of the order ${N}^{2}$ part of the algorithm by still another factor of two or four. It the length is divisible by 4, theprocess can be repeated. Indeed, it the length is a power of 2, one can show this process is equivalent to calculating the DFT in terms ofdiscrete cosine and sine transforms [link] , [link] with a resulting arithmetic complexity of order $N\phantom{\rule{0.166667em}{0ex}}log\left(N\right)$ and with a structure that is well suited to real data calculations and pruning.
If the flow-graph of the Cooley-Tukey FFT is compared to the flow-graph of the QFT, one notices both similarities and differences. Both progress instages as the length is continually divided by two. The Cooley-Tukey algorithm uses the periodic properties of the sine and cosine to give thefamiliar horizontal tree of butterflies. The parallel diagonal lines in this graph represent the parallel stepping through the data in synchronismwith the periodic basis functions. The QFT has diagonal lines that connect the first data point with the last, then the second with the nextto last, and so on to give a “star" like picture. This is interesting in that one can look at the flow graph of an algorithm developed by somecompletely different strategy and often find section with the parallel structures and other parts with the star structure. These must be usingsome underlying periodic and symmetric properties of the basis functions.
A careful analysis of the QFT shows that $2N$ additions are necessary to compute the even and odd parts of the input data. This is followed by thelength $N/2$ inner product that requires $4{(N/2)}^{2}={N}^{2}$ real multiplications and an equal number of additions. This is followed by thecalculations necessary for the simultaneous calculations of the first half and last half of $C\left(k\right)$ which requires $4(N/2)=2N$ real additions. This means the total QFT algorithm requires ${M}^{2}$ real multiplications and ${N}^{2}+4N$ real additions. These numbers along with those for the Goertzel algorithm [link] , [link] , [link] and the direct calculation of the DFT are included in the following table. Of the various order- ${N}^{2}$ DFT algorithms, the QFT seems to be the most efficient general method for anarbitrary length $N$ .
Algorithm | Real Mults. | Real Adds | Trig Eval. |
Direct DFT | $4\phantom{\rule{0.166667em}{0ex}}{N}^{2}$ | $4\phantom{\rule{0.166667em}{0ex}}{N}^{2}$ | $2\phantom{\rule{0.166667em}{0ex}}{N}^{2}$ |
Mod. 2nd Order Goertzel | ${N}^{2}+N$ | $2\phantom{\rule{0.166667em}{0ex}}{N}^{2}+N$ | $N$ |
QFT | ${N}^{2}$ | ${N}^{2}+4N$ | $2N$ |
Timings of the algorithms on a PC in milliseconds are given in the following table.
Algorithm | $N=125$ | $N=256$ |
Direct DFT | 4.90 | 19.83 |
Mod. 2O. Goertzel | 1.32 | 5.55 |
QFT | 1.09 | 4.50 |
Chirp + FFT | 1.70 | 3.52 |
These timings track the floating point operation counts fairly well.
The QFT is a straight-forward DFT algorithm that uses all of the possible symmetries of the DFT basis function with no requirements on the lengthbeing composite. These ideas have been proposed before, but have not been published or clearly developed by [link] , [link] , [link] , [link] . It seems that the basic QFT is practical and useful as a general algorithmfor lengths up to a hundred or so. Above that, the chirp z-transform [link] or other filter based methods will be superior. For special cases and shorter lengths, methods based on Winograd's theories willalways be superior. Nevertheless, the QFT has a definite place in the array of DFT algorithms and is not well known. A Fortran program isincluded in the appendix.
It is possible, but unlikely, that further arithmetic reduction could be achieved using the fact that ${W}_{N}$ has unity magnitude as was done in second-order Goertzel algorithm. It is also possible that some way ofcombining the Goertzel and QFT algorithm would have some advantages. A development of a complete QFT decomposition of a DFT of length- ${2}^{M}$ shows interesting structure [link] , [link] and arithmetic complexity comparable to average Cooley-Tukey FFTs. It does seem better suited toreal data calculations with pruning.
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