7.3: Algebraic Derivation of the Cooley-Tukey FFT

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Knowing the polynomial algebra underlying the DFT enables us to derive the Cooley-Tukey FFT algebraically. This means that instead of manipulating the DFT definition, we manipulate the polynomial algebra $C [s] / (s N - 1) " role="presentation" style="position:relative;" tabindex="0">$

\[L_{M}^{N}:i\mapsto \text{iM}\: \text{mod}\: N-1,\: \text{for}\: 0\leq i< N-1,\: N-1\mapsto N-1 \nonumber \]

For example (·means 0),

\[L_{2}^{6}=\begin{bmatrix} 1 & . & . & . & . & .\\ . & . & 1 & . & . & .\\ . & . & . & . & 1 & .\\ . & 1 & . & . & . & .\\ . & . & . & 1 & . & .\\ . & . & . & . & . & 1 \end{bmatrix} \nonumber \]

$L_{N/2}^{N}$ is sometimes called the perfect shuffle.

Further, we use matrix operators; namely the direct sum

\[A\oplus B=\begin{bmatrix} A & \\ & B \end{bmatrix} \nonumber \]

and the Kronecker or tensor product

\[A\otimes B=\left [ a_{k,l}B \right ]_{k,l},\: \: \text{for}\: A=[a_{k,l}] \nonumber \]

In particular,

\[I_n\otimes A=A\oplus ...\oplus A=\begin{bmatrix} A & & & & \\ & . & & & \\ & & . & & \\ & & & . & \\ & & & & A \end{bmatrix} \nonumber \]

is block-diagonal.

We may also construct a larger matrix as a matrix of matrices, e.g.,

\[\begin{bmatrix} A & B\\ B & A \end{bmatrix} \nonumber \]

If an algorithm for a transform is given as a product of sparse matrices built from the constructs above, then an algorithm for the transpose or inverse of the transform can be readily derived using mathematical properties including

\[(AB)^T=B^TA^T,\; \; \; (AB)^{-1}=B^{-1}A^{-1} \nonumber \]

\[(A\oplus B)^T=A^T\oplus B^T,\; \; \; (A\oplus B)^{-1}=A^{-1}\oplus B^{-1} \nonumber \]

\[(A\otimes B)^T=A^T\otimes B^T,\; \; \; (A\otimes B)^{-1}=A^{-1}\otimes B^{-1} \nonumber \]

Permutation matrices are orthogonal, i.e., $P^T=P^{-1}$. The transposition or inversion of diagonal matrices is obvious.

Radix-2 FFT

\[s^{2M}-1=(s^M-1)(s^M+1) \nonumber \]

factors and we can apply the CRT in the following steps:

\[\mathbb{C}[s]/(s^N-1) \nonumber \]

\[\rightarrow \mathbb{C}[s]/(s^M-1)\oplus \mathbb{C}[s]/(s^M+1) \nonumber \]

\[\rightarrow \underset{0\leq i< M}{\oplus }\mathbb{C}[s]/(x-W_{N}^{2i})\oplus \underset{0\leq i< M}{\oplus }\mathbb{C}[s]/(x-W_{M}^{2i+1}) \nonumber \]

\[\rightarrow \underset{0\leq i< N}{\oplus }\mathbb{C}[s]/(x-W_{N}^{i}) \nonumber \]

\[B=\begin{bmatrix} I_M & \ast \\ I_M & \ast \end{bmatrix} \nonumber \]

where the entries $\ast$ are determined next. For the base elements $s^{M+n},\: 0\leq n< M$, we have

\[s^{M+n}=s^n\: \text{mod}\: (s^M-1) \nonumber \]

\[s^{M+n}=-s^n\: \text{mod}\: (s^M+1) \nonumber \]

which yields the final result

\[B=\begin{bmatrix} I_M & I_M\\ I_M & -I_M \end{bmatrix}=\text{DFT}_2\otimes I_M \nonumber \]

The final algorithm obtained is

\[\text{DFT}_{2M}=L_{M}^{N}(\text{DFT}_{M}\oplus \text{DFT}-3_{M})(\text{DFT}_{2}\otimes I_M) \nonumber \]

\[\text{DFT}_{2M}=L_{M}^{N}(I_2\otimes \text{DFT}_{M})(I_M\oplus D_M)(\text{DFT}_{2}\otimes I_M) \nonumber \]

The last expression is the radix-$2$ decimation-in-frequency Cooley-Tukey FFT. The corresponding decimation-in-time version is obtained by transposition using the equation and the symmetry of the DFT:

\[\text{DFT}_{2M}=(\text{DFT}_{2}\otimes I_M))(I_M\oplus D_M)(I_2\otimes \text{DFT}_{M})L_{2}^{N} \nonumber \]

The first decomposition step requires us to compute $s^n\: \text{mod}\: \left ( s^M-W_{K}^{i} \right ),\: 0\leq n< N$. To do so, we decompose the index $n$ as $n=lM+m$ and compute

\[s^n=s^{lM+m}=(s^M)^ls^m\equiv W_{k}^{lm}s^m\: \text{mod}\: (s^M-W_{K}^{i}) \nonumber \]

This shows that the matrix for is given by: $\text{DFT}_K\otimes I_M$

In step equation, each $\mathbb{C}[s]/(s^M-W_{K}^{i})$ is completely decomposed by its polynomial transform

\[\text{DFT}_M(i,K)=\text{DFT}_{M}.\text{diag}_{0\leq i< M}(W_{N}^{ij}) \nonumber \]

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