0.7 Dft and fft: an algebraic view  (Page 4/5)

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Matrix notation

We denote the $N×N$ identity matrix with ${I}_{N}$ , and diagonal matrices with

${diag}_{0\le k

The $N×N$ stride permutation matrix is defined for $N=KM$ by the permutation

${L}_{M}^{N}:\phantom{\rule{4pt}{0ex}}iK+j↦jM+i$

for $0\le i . This definition shows that ${L}_{M}^{N}$ transposes a $K×M$ matrix stored in row-major order. Alternatively, we can write

${L}_{M}^{N}\phantom{\rule{3pt}{0ex}}\text{:}\phantom{\rule{3pt}{0ex}}i\phantom{\rule{3pt}{0ex}}↦\phantom{\rule{3pt}{0ex}}\mathrm{iM}\phantom{\rule{3pt}{0ex}}\text{mod}\phantom{\rule{3pt}{0ex}}N-1,\phantom{\rule{5pt}{0ex}}\text{for}\phantom{\rule{3pt}{0ex}}0\le i

For example ( $·$ means 0),

${L}_{2}^{6}=\left[\begin{array}{cccccc}1& ·& ·& ·& ·& ·\\ ·& ·& 1& ·& ·& ·\\ ·& ·& ·& ·& 1& ·\\ ·& 1& ·& ·& ·& ·\\ ·& ·& ·& 1& ·& ·\\ ·& ·& ·& ·& ·& 1\end{array}\right].$

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

Further, we use matrix operators; namely the direct sum

$A\oplus B=\left[\begin{array}{c}A\\ & B\end{array}\right]$

and the Kronecker or tensor product

$A\otimes B={\left[{a}_{k,\ell }B\right]}_{k,\ell },\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}A=\left[{a}_{k,\ell }\right].$

In particular,

${I}_{n}\otimes A=A\oplus \cdots \oplus A=\left[\begin{array}{c}A\\ & \ddots \\ & & A\end{array}\right]$

is block-diagonal.

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

$\left[\begin{array}{cc}A& B\\ B& A\end{array}\right].$

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 orinverse of the transform can be readily derived using mathematical properties including

$\begin{array}{cc}{\left(AB\right)}^{T}={B}^{T}{A}^{T},\hfill & {\left(AB\right)}^{-1}={B}^{-1}{A}^{-1},\hfill \\ {\left(A\oplus B\right)}^{T}={A}^{T}\oplus {B}^{T},\hfill & {\left(A\oplus B\right)}^{-1}={A}^{-1}\oplus {B}^{-1},\hfill \\ {\left(A\otimes B\right)}^{T}={A}^{T}\otimes {B}^{T},\hfill & {\left(A\otimes B\right)}^{-1}={A}^{-1}\otimes {B}^{-1}.\hfill \end{array}$

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

The DFT decomposes $\mathcal{A}=\mathbb{C}\left[s\right]/\left({s}^{N}-1\right)$ with basis $b=\left(1,s,\cdots ,{s}^{N-1}\right)$ as shown in [link] . We assume $N=2M$ . Then

${s}^{2M}-1=\left({s}^{M}-1\right)\left({s}^{M}+1\right)$

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

$\begin{array}{ccc}& & \mathbb{C}\left[s\right]/\left({s}^{N}-1\right)\hfill \\ & \to & \mathbb{C}\left[s\right]/\left({s}^{M}-1\right)\oplus \mathbb{C}\left[s\right]/\left({s}^{M}+1\right)\hfill \end{array}$
$\begin{array}{ccc}& \to & \underset{0\le i
$\begin{array}{ccc}& \to & \underset{0\le i

As bases in the smaller algebras $\mathbb{C}\left[s\right]/\left({s}^{M}-1\right)$ and $\mathbb{C}\left[s\right]/\left({s}^{M}+1\right)$ , we choose $c=d=\left(1,s,\cdots ,{s}^{M-1}\right)$ . The derivation of an algorithm for ${DFT}_{N}$ based on [link] - [link] is now completely mechanical by reading off the matrix for each of the threedecomposition steps. The product of these matrices is equal to the ${DFT}_{N}$ .

First, we derive the base change matrix $B$ corresponding to [link] . To do so, we have to express the base elements ${s}^{n}\in b$ in the basis $c\cup d$ ; the coordinate vectors are the columns of $B$ . For $0\le n , ${s}^{n}$ is actually contained in $c$ and $d$ , so the first $M$ columns of $B$ are

$B=\left[\begin{array}{cc}{I}_{M}& *\\ {I}_{M}& *\end{array}\right],$

where the entries $*$ are determined next. For the base elements ${s}^{M+n}$ , $0\le n , we have

$\begin{array}{ccc}\hfill {s}^{M+n}& \equiv & {s}^{n}\phantom{\rule{4.pt}{0ex}}\text{mod}\phantom{\rule{4.pt}{0ex}}\left({s}^{M}-1\right),\hfill \\ \hfill {s}^{M+n}& \equiv & -{s}^{n}\phantom{\rule{4.pt}{0ex}}\text{mod}\phantom{\rule{4.pt}{0ex}}\left({s}^{M}+1\right),\hfill \end{array}$

which yields the final result

$B=\left[\begin{array}{cc}{I}_{M}& \phantom{-}{I}_{M}\\ {I}_{M}& -{I}_{M}\end{array}\right]={DFT}_{2}\otimes {I}_{M}.$

Next, we consider step [link] . $\mathbb{C}\left[s\right]/\left({s}^{M}-1\right)$ is decomposed by ${DFT}_{M}$ and $\mathbb{C}\left[s\right]/\left({s}^{M}+1\right)$ by ${DFT\text{-3}}_{M}$ in [link] .

Finally, the permutation in step [link] is the perfect shuffle ${L}_{M}^{N}$ , which interleaves the even and odd spectral components (even and odd exponents of ${W}_{N}$ ).

The final algorithm obtained is

${DFT}_{2M}={L}_{M}^{N}\left({DFT}_{M}\oplus {DFT\text{-3}}_{M}\right)\left({DFT}_{2}\otimes {I}_{M}\right).$

To obtain a better known form, we use ${DFT\text{-3}}_{M}={DFT}_{M}{D}_{M}$ , with ${D}_{M}={diag}_{0\le i , which is evident from [link] . It yields

$\begin{array}{ccc}\hfill {DFT}_{2M}& =& {L}_{M}^{N}\left({DFT}_{M}\oplus {DFT}_{M}{D}_{M}\right)\left({DFT}_{2}\otimes {I}_{M}\right)\hfill \\ & =& {L}_{M}^{N}\left({I}_{2}\otimes {DFT}_{M}\right)\left({I}_{M}\oplus {D}_{M}\right)\left({DFT}_{2}\otimes {I}_{M}\right).\hfill \end{array}$

The last expression is the radix-2 decimation-in-frequency Cooley-Tukey FFT. The corresponding decimation-in-time version isobtained by transposition using [link] and the symmetry of the DFT:

${DFT}_{2M}=\left({DFT}_{2}\otimes {I}_{M}\right)\left({I}_{M}\oplus {D}_{M}\right)\left({I}_{2}\otimes {DFT}_{M}\right){L}_{2}^{N}.$

The entries of the diagonal matrix ${I}_{M}\oplus {D}_{M}$ are commonly called twiddle factors .

The above method for deriving DFT algorithms is used extensively in [link] .

To algebraically derive the general-radix FFT, we use the decomposition property of ${s}^{N}-1$ . Namely, if $N=KM$ then

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