# 0.2 Bilinear forms for circular convolution  (Page 2/2)

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Similarly, we can write a bilinear form for cyclotomic convolution.Let $d$ be any positive integer and let $X\left(s\right)$ and $H\left(s\right)$ be polynomials of degree $\phi \left(d\right)-1$ where $\phi \left(·\right)$ is the Euler totient function.If $A$ , $B$ and $C$ are matrices satisfying ${\left({C}_{{\Phi }_{d}}\right)}^{k}=\mathrm{Cdiag}\left(B{e}_{k}\right)A$ for $0\le k\le \phi \left(d\right)-1$ , then the coefficients of $Y\left(s\right)={⟨X\left(s\right)H\left(s\right)⟩}_{{\Phi }_{d}\left(s\right)}$ are given by $y=C\left\{Bh*Ax\right\}$ . As above, if $y=C\left\{Bh*Ax\right\}$ computes the $d$ -cyclotomic convolution, then we say “ $\left(A,B,C\right)$ describes a bilinear form for ${\Phi }_{d}\left(s\right)$ convolution."

But since ${⟨X\left(s\right)H\left(s\right)⟩}_{{\Phi }_{d}\left(s\right)}$ can be found by computing the product of $X\left(s\right)$ and $H\left(s\right)$ and reducing the result, a cyclotomic convolution algorithmcan always be derived by following a linear convolution algorithm by the appropriate reductionoperation: If $G$ is the appropriate reduction matrix and if $\left(A,B,F\right)$ describes a bilinear form for a $\phi \left(d\right)$ point linear convolution, then $\left(A,B,GF\right)$ describes a bilinear form for ${\Phi }_{d}\left(s\right)$ convolution. That is, $y=GF\left\{Bh*Ax\right\}$ computes the coefficients of ${⟨X\left(s\right)H\left(s\right)⟩}_{{\Phi }_{d}\left(s\right)}$ .

## Circular convolution

By using the Chinese Remainder Theorem for polynomials, circular convolution can be decomposed into disjointcyclotomic convolutions. Let $p$ be a prime and consider $p$ point circular convolution.Above we found that

${S}_{p}={R}_{p}^{-1}\left[\begin{array}{cc}1& \\ & {C}_{{\Phi }_{p}}\end{array}\right]{R}_{p}$

and therefore

${S}_{p}^{k}={R}_{p}^{-1}\left[\begin{array}{cc}1& \\ & {C}_{{\Phi }_{p}}^{k}\end{array}\right]{R}_{p}.$

If $\left({A}_{p},{B}_{p},{C}_{p}\right)$ describes a bilinear form for ${\Phi }_{p}\left(s\right)$ convolution, then

${S}_{p}^{k}={R}_{p}^{-1}\left[\begin{array}{cc}1& \\ & {C}_{p}\end{array}\right]\text{diag}\left(\left[\begin{array}{cc}1& \\ & {B}_{p}\end{array}\right],{R}_{p},{e}_{k}\right)\left[\begin{array}{cc}1& \\ & {A}_{p}\end{array}\right]{R}_{p}$

and consequently the circular convolution of $h$ and $x$ can be computed by

$y={R}_{p}^{-1}C\left\{B{R}_{p}h*A{R}_{p}x\right\}$

where $A=1\oplus {A}_{p}$ , $B=1\oplus {B}_{p}$ and $C=1\oplus {C}_{p}$ . We say $\left(A,B,C\right)$ describes a bilinear form for $p$ point circular convolution. Note that if $\left(D,E,F\right)$ describes a $\left(p-1\right)$ point linear convolution then ${A}_{p}$ , ${B}_{p}$ and ${C}_{p}$ can be taken to be ${A}_{p}=D$ , ${B}_{p}=E$ and ${C}_{p}={G}_{p}F$ where ${G}_{p}$ represents the appropriate reduction operations.Specifically, ${G}_{p}$ is given by Equation 42 from Preliminaries .

Next we consider ${p}^{e}$ point circular convolution.Recall that ${S}_{{p}^{e}}={R}_{{p}^{e}}^{-1}\left({\oplus }_{i=0}^{e},{C}_{{\Phi }_{{p}^{i}}}\right){R}_{{p}^{e}}$ as in Equation 27 from Preliminaries so that the circular convolution is decomposed into a set of $e+1$ disjoint ${\Phi }_{{p}^{i}}\left(s\right)$ convolutions. If $\left({A}_{{p}^{i}},{B}_{{p}^{i}},{C}_{{p}^{i}}\right)$ describes a bilinear form for ${\Phi }_{{p}^{i}}\left(s\right)$ convolution and if

$\begin{array}{ccc}\hfill A& =& 1\oplus {A}_{p}\oplus \cdots \oplus {A}_{{p}^{e}}\hfill \\ \hfill B& =& 1\oplus {B}_{p}\oplus \cdots \oplus {B}_{{p}^{e}}\hfill \\ \hfill C& =& 1\oplus {C}_{p}\oplus \cdots \oplus {C}_{{p}^{e}}\hfill \end{array}$

then $\left(A{R}_{{p}^{e}},B{R}_{{p}^{e}},{R}_{{p}^{e}}^{-1}C\right)$ describes a bilinear form for ${p}^{e}$ point circular convolution. In particular, if $\left({D}_{d},{E}_{d},{F}_{d}\right)$ describes a bilinear form for $d$ point linear convolution, then ${A}_{{p}^{i}}$ , ${B}_{{p}^{i}}$ and ${C}_{{p}^{i}}$ can be taken to be

$\begin{array}{ccc}\hfill {A}_{{p}^{i}}& =& {D}_{\phi \left({p}^{i}\right)}\hfill \\ \hfill {B}_{{p}^{i}}& =& {E}_{\phi \left({p}^{i}\right)}\hfill \\ \hfill {C}_{{p}^{i}}& =& {G}_{{p}^{i}}{F}_{\phi \left({p}^{i}\right)}\hfill \end{array}$

where ${G}_{{p}^{i}}$ represents the appropriate reduction operation and $\phi \left(·\right)$ is the Euler totient function. Specifically, ${G}_{{p}^{i}}$ has the following form

${G}_{{p}^{i}}=\left[\begin{array}{ccc}{I}_{\left(p-1\right){p}^{i-1}}& -{1}_{\mathrm{-p}-1}\otimes {I}_{{p}^{i-1}}& \left[\begin{array}{c}{I}_{\left(p-2\right){p}^{i-1}-1}\\ {0}_{{p}^{i-1}+1,\left(p-2\right){p}^{i-1}-1}\end{array}\right]\end{array}\right]$

if $p\ge 3$ , while

${G}_{{2}^{i}}=\left[\begin{array}{cc}{I}_{{2}^{i-1}}& \left[\begin{array}{c}-{I}_{{2}^{i-1}-1}\\ {0}_{1,{2}^{i-1}-1}\end{array}\right]\end{array}\right].$

Note that the matrix ${R}_{{p}^{e}}$ block diagonalizes ${S}_{{p}^{e}}$ and each diagonal block represents a cyclotomic convolution.Correspondingly, the matrices $A$ , $B$ and $C$ of the bilinear form also have a block diagonal structure.

## The split nesting algorithm

We now describe the split-nesting algorithm for general length circular convolution [link] . Let $n={p}_{1}^{{e}_{1}}\cdots {p}_{k}^{{e}_{k}}$ where ${p}_{i}$ are distinct primes. We have seen that

${S}_{n}={P}^{t}{R}^{-1}\left(\underset{d|n}{\oplus },\Psi ,\left(d\right)\right)RP$

where $P$ is the prime factor permutation $P={P}_{{p}_{1}^{{e}_{1}},\cdots ,{p}_{k}^{{e}_{k}}}$ and $R$ represents the reduction operations. For example, see Equation 46 in Preliminaries . $RP$ block diagonalizes ${S}_{n}$ and each diagonal block represents a multi-dimensional cyclotomicconvolution. To obtain a bilinear form for a multi-dimensional convolution,we can combine bilinear forms for one-dimensional convolutions.If $\left({A}_{{p}_{j}^{i}},{B}_{{p}_{j}^{i}},{C}_{{p}_{j}^{i}}\right)$ describes a bilinear form for ${\Phi }_{{p}_{j}^{i}}\left(s\right)$ convolution and if

$\begin{array}{ccc}\hfill A& =& {\oplus }_{d|n}{A}_{d}\hfill \\ \hfill B& =& {\oplus }_{d|n}{B}_{d}\hfill \\ \hfill C& =& {\oplus }_{d|n}{C}_{d}\hfill \end{array}$

with

$\begin{array}{ccc}\hfill {A}_{d}& =& {\otimes }_{p|d,p\in \mathcal{P}}{A}_{{H}_{d}\left(p\right)}\hfill \\ \hfill {B}_{d}& =& {\otimes }_{p|d,p\in \mathcal{P}}{B}_{{H}_{d}\left(p\right)}\hfill \\ \hfill {C}_{d}& =& {\otimes }_{p|d,p\in \mathcal{P}}{C}_{{H}_{d}\left(p\right)}\hfill \end{array}$

where ${H}_{d}\left(p\right)$ is the highest power of $p$ dividing $d$ , and $\mathcal{P}$ is the set of primes, then $\left(ARP,BRP,{P}^{t}{R}^{-1}C\right)$ describes a bilinear form for $n$ point circular convolution. That is

$y={P}^{t}{R}^{-1}C\left\{B,R,P,h,*,A,R,P,x\right\}$

computes the circular convolution of $h$ and $x$ .

As above $\left({A}_{{p}_{j}^{i}},{B}_{{p}_{j}^{i}},{C}_{{p}_{j}^{i}}\right)$ can be taken to be $\left({D}_{\phi \left({p}_{j}^{i}\right)},{E}_{\phi \left({p}_{j}^{i}\right)},{G}_{{p}_{j}^{i}}{F}_{\phi \left({p}_{j}^{i}\right)}\right)$ where $\left({D}_{d},{E}_{d},{F}_{d}\right)$ describes a bilinear form for $d$ point linear convolution. This is one particular choice for $\left({A}_{{p}_{j}^{i}},{B}_{{p}_{j}^{i}},{C}_{{p}_{j}^{i}}\right)$ - other bilinear forms for cyclotomic convolution that are not derived from linear convolution algorithms exist.

A 45 point circular convolution algorithm:

$y={P}^{t}{R}^{-1}C\left\{B,R,P,h,*,A,R,P,x\right\}$

where

$\begin{array}{ccc}\hfill P& =& {P}_{9,5}\hfill \\ \hfill R& =& {R}_{9,5}\hfill \\ \hfill A& =& 1\oplus {A}_{3}\oplus {A}_{9}\oplus {A}_{5}\oplus \left({A}_{3}\otimes {A}_{5}\right)\oplus \left({A}_{9}\otimes {A}_{5}\right)\hfill \\ \hfill B& =& 1\oplus {B}_{3}\oplus {B}_{9}\oplus {B}_{5}\oplus \left({B}_{3}\otimes {B}_{5}\right)\oplus \left({B}_{9}\otimes {B}_{5}\right)\hfill \\ \hfill C& =& 1\oplus {C}_{3}\oplus {C}_{9}\oplus {C}_{5}\oplus \left({C}_{3}\otimes {C}_{5}\right)\oplus \left({C}_{9}\otimes {C}_{5}\right)\hfill \end{array}$

and where $\left({A}_{{p}_{j}^{i}},{B}_{{p}_{j}^{i}},{C}_{{p}_{j}^{i}}\right)$ describes a bilinear form for ${\Phi }_{{p}_{j}^{i}}\left(s\right)$ convolution.

## The matrix exchange property

The matrix exchange property is a useful technique that, under certain circumstances,allows one to save computation in carrying out the action of bilinear forms [link] . Suppose

$y=C\left\{A,x,*,B,h\right\}$

as in [link] . When $h$ is known and fixed, $Bh$ can be pre-computed so that $y$ can be found using only the operations represented by $C$ and $A$ and the point by point multiplications denoted by $*$ . The operation of $B$ is absorbed into the multiplicative constants.Note that in [link] , the matrix corresponding to $C$ is more complicated than is $B$ . It is therefore advantageous to absorb the workof $C$ instead of $B$ into the multiplicative constants if possible. This can be done when $y$ is the circular convolution of $x$ and $h$ by using the matrix exchange property.

To explain the matrix exchange property we draw from [link] . Note that $y=Cdiag\left(Ax\right)Bh$ , so that $\mathrm{Cdiag}\left(Ax\right)B$ must be the corresponding circulant matrix,

$\mathrm{Cdiag}\left(Ax\right)B=\left[\begin{array}{cccc}{x}_{0}& {x}_{n-1}& \cdots & {x}_{1}\\ {x}_{1}& {x}_{0}& & {x}_{2}\\ ⋮& & & \\ {x}_{n-1}& {x}_{n-2}& & {x}_{0}\end{array}\right].$

Since $\mathrm{Cdiag}\left(Ax\right)B=J{\left(\mathrm{Cdiag},\left(,A,x,\right),B\right)}^{t}J$ where $J$ is the reversal matrix, one gets

$\begin{array}{ccc}\hfill y& =& C\left\{A,x,*,B,h\right\}\hfill \\ & =& \mathrm{Cdiag}\left(Ax\right)Bh\hfill \\ & =& J{\left(\mathrm{Cdiag},\left(,A,x,\right),B\right)}^{t}Jh\hfill \\ & =& J{B}^{t}diag\left(Ax\right){C}^{t}Jh\hfill \\ & =& J{B}^{t}\left\{A,x,*,{C}^{t},J,h\right\}\hfill \end{array}$

As noted in [link] , the matrix exchange property can be used whenever $y=T\left(x\right)h$ where $T\left(x\right)$ satisfies $T\left(x\right)={J}_{1}T{\left(x\right)}^{t}{J}_{2}$ for some matrices ${J}_{1}$ and ${J}_{2}$ . In that case one gets $y={J}_{1}{B}^{t}\left\{A,x,*,{C}^{t},{J}_{2},h\right\}$ .

Applying the matrix exchange property to [link] one gets

$y=J{P}^{t}{R}^{t}{B}^{t}\left\{{C}^{t},{R}^{-t},P,J,h,*,A,R,P,x\right\}.$

A 45 point circular convolution algorithm:

$y=J{P}^{t}{R}^{t}{B}^{t}\left\{u,*,A,R,P,x\right\}$

where $u={C}^{t}{R}^{-t}PJh$ and

$\begin{array}{ccc}\hfill P& =& {P}_{9,5}\hfill \\ \hfill R& =& {R}_{9,5}\hfill \\ \hfill A& =& 1\oplus {A}_{3}\oplus {A}_{9}\oplus {A}_{5}\oplus \left({A}_{3}\otimes {A}_{5}\right)\oplus \left({A}_{9}\otimes {A}_{5}\right)\hfill \\ \hfill {B}^{t}& =& 1\oplus {B}_{3}^{t}\oplus {B}_{9}^{t}\oplus {B}_{5}^{t}\oplus \left({B}_{3}^{t}\otimes {B}_{5}^{t}\right)\oplus \left({B}_{9}^{t}\otimes {B}_{5}^{t}\right)\hfill \\ \hfill {C}^{t}& =& 1\oplus {C}_{3}^{t}\oplus {C}_{9}^{t}\oplus {C}_{5}^{t}\oplus \left({C}_{3}^{t}\otimes {C}_{5}^{t}\right)\oplus \left({C}_{9}^{t}\otimes {C}_{5}^{t}\right)\hfill \end{array}$

and where $\left({A}_{{p}_{j}^{i}},{B}_{{p}_{j}^{i}},{C}_{{p}_{j}^{i}}\right)$ describes a bilinear form for ${\Phi }_{{p}_{j}^{i}}\left(s\right)$ convolution.

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