5.2: The FFT from Factoring the DFT Operator

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The definition of the DFT in Multidimensional Index Mapping can written as a matrix-vector operation by $C=WX \; \; \text{where}\; \; N=8$

\[\begin{bmatrix} C(0)\\ C(1)\\ C(2)\\ C(3)\\ C(4)\\ C(5)\\ C(6)\\ C(7) \end{bmatrix}=\begin{bmatrix} W^{0} & W^{0} & W^{0} & W^{0} & W^{0} & W^{0} & W^{0} & W^{0}\\ W^{0} & W^{1} & W^{2} & W^{3} & W^{4} & W^{5} & W^{6} & W^{7}\\ W^{0} & W^{2} & W^{4} & W^{6} & W^{8} & W^{10} & W^{12} & W^{14}\\ W^{0} & W^{3} & W^{6} & W^{9} & W^{12} & W^{15} & W^{18} & W^{21}\\ W^{0} & W^{4} & W^{8} & W^{12} & W^{16} & W^{20} & W^{24} & W^{28}\\ W^{0} & W^{5} & W^{10} & W^{15} & W^{20} & W^{25} & W^{30} & W^{35}\\ W^{0} & W^{6} & W^{12} & W^{18} & W^{24} & W^{30} & W^{36} & W^{42}\\ W^{0} & W^{7} & W^{14} & W^{21} & W^{28} & W^{35} & W^{42} & W^{49} \end{bmatrix}\begin{bmatrix} x(0)\\ x(1)\\ x(2)\\ x(3)\\ x(4)\\ x(5)\\ x(6)\\ x(7) \end{bmatrix} \nonumber \]

which clearly requires $N^2=64$ complex multiplications and $N(N-1)$ additions. A factorization of the DFT operator, $W$, gives $W=F_{1}F_{2}F_{3}\; \; and\; \; C=F_{1}F_{2}F_{3}X$.

Expanding on that gives

\[\begin{bmatrix} C(0)\\ C(4)\\ C(2)\\ C(6)\\ C(1)\\ C(5)\\ C(3)\\ C(7) \end{bmatrix}=\begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0\\ W^{0} & 0 & -W^{2} & 0 & 0 & 0 & 0 & 0\\ 0 & W^{0} & 0 & -W^{2} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & W^{0} & 0 & -W^{0} & 0\\ 0 & 0 & 0 & 0 & 0 & W^{2} & 0 & -W^{2} \end{bmatrix} \nonumber \]

\[\begin{bmatrix} 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ W_{0} & 0 & 0 & 0 & -W_{0} & 0 & 0 & 0\\ 0 & W_{1} & 0 & 0 & 0 & -W_{1} & 0 & 0\\ 0 & 0 & W_{2} & 0 & 0 & 0 & -W_{2} & 0\\ 0 & 0 & 0 & W_{3} & 0 & 0 & 0 & -W_{3} \end{bmatrix}\begin{bmatrix} x(0)\\ x(1)\\ x(2)\\ x(3)\\ x(4)\\ x(5)\\ x(6)\\ x(7) \end{bmatrix} \nonumber \]

where the $F_i$ matrices are sparse. Note that each has $16\; (\text{or}\; 2N)$ non-zero terms and $F_2$ and $F_3$ have $8\; (\text{or}\; N)$ non-unity terms. If $N=2^M$, then the number of factors is $\log (N)=M$. In another form with the twiddle factors separated so as to count the complex multiplications we have

\[\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & W^{0} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & W^{2} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0& 0 & 0 & W^{0} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & W^{2} \end{bmatrix}\begin{bmatrix} 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 \end{bmatrix} \nonumber \]

\[\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & W^{0} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & W^{1} & 0 & 0\\ 0 & 0 & 0 & 0& 0 & 0 & W^{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & W^{3} \end{bmatrix}\begin{bmatrix} 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 \end{bmatrix}\begin{bmatrix} x(0)\\ x(1)\\ x(2)\\ x(3)\\ x(4)\\ x(5)\\ x(6)\\ x(7) \end{bmatrix} \nonumber \]

which is in the form

$C = A 1 M 1 A 2 M 2 A 3 X " role="presentation" style="position:relative;" tabindex="0">$

described by the index map. $A_1$, $A_2$, and $A_3$ each represents $8$ additions, or, in general, $N$ additions. $M_1$ and $M_2$ each represent $4$ (or $N/2$) multiplications.

This is a very interesting result showing that implementing the DFT using the factored form requires considerably less arithmetic than the single factor definition. Indeed, the form of the formula that Cooley and Tukey derived showing that the amount of arithmetic required by the FFT is on the order of $N\log N$ can be seen from the factored operator formulation.

Much of the theory of the FFT can be developed using operator factoring and it has some advantages for implementation of parallel and vector computer architectures. The eigenspace approach is somewhat of the same type.

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