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2.3: In-Place Calculation of the DFT and Scrambling

Because use of both the type-one and two index maps uncouples the calculations of the rows and columns of the data array, the results of each short length \(N_i\) DFT can be written back over the data as it will not be needed again after that particular row or column is transformed. This is easily seen from Figures in section 2.2 where the DFT of the first row of \(x(n_1,n_2)\) can be put back over the data rather written into a new array. After all the calculations are finished, the total DFT is in the array of the original data. This gives a significant memory savings over using a separate array for the output.

Unfortunately, the use of in-place calculations results in the order of the DFT values being permuted or scrambled. This is because the data is indexed according to the input map equation and the results are put into the same locations rather than the locations dictated by the output map equation. For example with a length-8 radix-2 FFT, the input index map is

\[n=4n_{1}+2n_{2}+n_{3}\]

which to satisfy the equation requires an output map of

\[k=k_{1}+2k_{2}+4k_{3}\]

The in-place calculations will place the DFT results in the locations of the input map and these should be reordered or unscrambled into the locations given by the output map. Examination of these two maps shows the scrambled output to be in a “bit reversed" order.

For certain applications, this scrambled output order is not important, but for many applications, the order must be unscrambled before the DFT can be considered complete. Because the radix of the radix-2 FFT is the same as the base of the binary number representation, the correct address for any term is found by reversing the binary bits of the address. The part of most FFT programs that does this reordering is called a bit-reversed counter. Examples of various unscramblers are found in the appendices.

The development here uses the input map and the resulting algorithm is called “decimation-in-frequency". If the output rather than the input map is used to derive the FFT algorithm so the correct output order is obtained, the input order must be scrambled so that its values are in locations specified by the output map rather than the input map. This algorithm is called “decimation-in-time". The scrambling is the same bit-reverse counting as before, but it precedes the FFT algorithm in this case. The same process of a post-unscrambler or pre-scrambler occurs for the in-place calculations with the type-one maps. It is possible to do the unscrambling while calculating the FFT and to avoid a separate unscrambler. This is done for the Cooley-Tukey FFT and for the PFA.

If a radix-2 FFT is used, the unscrambler is a bit-reversed counter. If a radix-4 FFT is used, the unscrambler is a base-4 reversed counter, and similarly for radix-8 and others. However, if for the radix-4 FFT, the short length-4 DFTs (butterflies) have their outputs in bit-revered order, the output of the total radix-4 FFT will be in bit-reversed order, not base-4 reversed order. This means any radix- \(2^n\) FFT can use the same radix-2 bit-reversed counter as an unscrambler if the proper butterflies are used.

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