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Reference [link] should be consulted for the details of these conditions and examples. Two classes of index maps are definedfrom these conditions.
The map of [link] is called a type-one map when integers $a$ and $b$ exist such that
The map of [link] is called a type-two map when when integers $a$ and $b$ exist such that
The type-one can be used only if the factors of $N$ are relatively prime, but the type-two can be used whether they are relatively prime ornot. Good [link] , Thomas, and Winograd [link] all used the type-one map in their DFT algorithms. Cooley and Tukey [link] used the type-two in their algorithms, both for a fixed radix $(N={R}^{M})$ and a mixed radix [link] .
The frequency index is defined by a map similar to [link] as
where the same conditions, [link] and [link] , are used for determining the uniqueness of this map in terms of the integers ${K}_{3}$ and ${K}_{4}$ .
Two-dimensional arrays for the input data and its DFT are defined using these index maps to give
In some of the following equations, the residue reduction notation will be omitted for clarity. These changes of variablesapplied to the definition of the DFT given in [link] give
where all of the exponents are evaluated modulo $N$ .
The amount of arithmetic required to calculate [link] is the same asin the direct calculation of [link] . However, because of the special nature of the DFT, the integer constants ${K}_{i}$ can be chosen in such a way that the calculations are “uncoupled" andthe arithmetic is reduced. The requirements for this are
When this condition and those for uniqueness in [link] are applied, it is found that the ${K}_{i}$ may always be chosen such that one of the terms in [link] is zero. If the ${N}_{i}$ are relatively prime, it is always possible to make both terms zero. If the ${N}_{i}$ are not relatively prime, only one of the terms can be set to zero. When they are relatively prime, there is a choice, itis possible to either set one or both to zero. This in turn causes one or both of the center two $W$ terms in [link] to become unity.
An example of the Cooley-Tukey radix-4 FFT for a length-16 DFT uses the type-two map with ${K}_{1}=4$ , ${K}_{2}=1$ , ${K}_{3}=1$ , ${K}_{4}=4$ giving
The residue reduction in [link] is not needed here since $n$ does not exceed $N$ as ${n}_{1}$ and ${n}_{2}$ take on their values. Since, in this example, the factors of $N$ have a common factor, only one of the conditions in [link] can hold and, therefore, [link] becomes
Note the definition of ${W}_{N}$ in [link] allows the simple form of ${W}_{16}^{{K}_{1}{K}_{3}}={W}_{4}$
This has the form of a two-dimensional DFT with an extra term ${W}_{16}$ , called a “twiddle factor". The inner sum over ${n}_{1}$ represents four length-4 DFTs, the ${W}_{16}$ term represents 16 complex multiplications, and the outer sum over ${n}_{2}$ represents another four length-4 DFTs. This choice of the ${K}_{i}$ “uncouples" the calculations since the first sum over ${n}_{1}$ for ${n}_{2}=0$ calculates the DFT of the first row of the data array $\widehat{x}({n}_{1},{n}_{2})$ , and those data values are never needed in the succeeding row calculations. The row calculations are independent,and examination of the outer sum shows that the column calculations are likewise independent. This is illustrated in [link] .
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