# 0.8 Appendix: bilinear forms for linear convolution

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## Appendix: bilinear forms for linear convolution

The following is a collection of bilinear forms for linear convolution.In each case $\left({D}_{n},{D}_{n},{F}_{n}\right)$ describes a bilinear form for $n$ point linear convolution. That is

$y={F}_{n}\left\{{D}_{n},h,*,{D}_{n},x\right\}$

computes the linear convolution of the $n$ point sequences $h$ and $x$ .

For each ${D}_{n}$ we give Matlab programs that compute ${D}_{n}x$ and ${D}_{n}^{t}x$ , and for each ${F}_{n}$ we give a Matlab program that computes ${F}_{n}^{t}x$ . When the matrix exchange algorithm is employed in the design of circular convolutionalgorithms, these are the relevant operations.

## 2 point linear convolution

${D}_{2}$ can be implemented with 1 addition, ${D}_{2}^{t}$ with two additions.

${D}_{2}=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 1& 1\end{array}\right]$
${F}_{2}=\left[\begin{array}{ccc}1& 0& 0\\ -1& -1& 1\\ 0& 1& 0\end{array}\right]$
function y = D2(x) y = zeros(3,1);y(1) = x(1); y(2) = x(2);y(3) = x(1) + x(2); function y = D2t(x) y = zeros(2,1);y(1) = x(1)+x(3); y(2) = x(2)+x(3); function y = F2t(x) y = zeros(3,1);y(1) = x(1)-x(2); y(2) = -x(2)+x(3);y(3) = x(2);

## 3 point linear convolution

${D}_{3}$ can be implemented in 7 additions, ${D}_{3}^{t}$ in 9 additions.

${D}_{3}=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 1\\ 1& -1& 1\\ 1& 2& 4\\ 0& 0& 1\end{array}\right]$
${F}_{3}=\frac{1}{6}\left[\begin{array}{ccccc}6& 0& 0& 0& 0\\ -3& 6& -2& -1& 12\\ -6& 3& 3& 0& -6\\ 3& -3& -1& 1& -12\\ 0& 0& 0& 0& 6\end{array}\right]$
function y = D3(x) y = zeros(5,1);a = x(2)+x(3); b = x(3)-x(2);y(1) = x(1); y(2) = x(1)+a;y(3) = x(1)+b; y(4) = a+a+b+y(2);y(5) = x(3); function y = D3t(x) y = zeros(3,1);y(1) = x(2)+x(3)+x(4); a = x(4)+x(4);y(2) = x(2)-x(3)+a; y(3) = y(1)+x(4)+a;y(1) = y(1)+x(1); y(3) = y(3)+x(5); function y = F3t(x) y = zeros(5,1);y(1) = 6*x(1)-3*x(2)-6*x(3)+3*x(4); y(2) = 6*x(2)+3*x(3)-3*x(4);y(3) = -2*x(2)+3*x(3)-x(4); y(4) = -x(2)+x(4);y(5) = 12*x(2)-6*x(3)-12*x(4)+6*x(5); y = y/6;

## 4 point linear convolution

${D}_{4}={D}_{2}\otimes {D}_{2}$
${F}_{4}=\left[\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& -1& 1& 0& 0& 0& 0& 0& 0\\ -1& 1& 0& -1& 0& 0& 1& 0& 0\\ 1& 1& -1& 1& 1& -1& -1& -1& 1\\ 0& -1& 0& 1& -1& 0& 0& 1& 0\\ 0& 0& 0& -1& -1& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\end{array}\right]$
function y = F4t(x) y = zeros(7,1);y(1) = x(1)-x(2)-x(3)+x(4); y(2) = -x(2)+x(3)+x(4)-x(5);y(3) = x(2)-x(4); y(4) = -x(3)+x(4)+x(5)-x(6);y(5) = x(4)-x(5)-x(6)+x(7); y(6) = -x(4)+x(6);y(7) = x(3)-x(4); y(8) = -x(4)+x(5);y(9) = x(4);

## 6 point linear convolution

${D}_{6}={D}_{2}\otimes {D}_{3}$
${F}_{6}=\frac{1}{6}\left[\begin{array}{ccccccccccccccc}6& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -3& 6& -2& -1& 12& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -6& 3& 3& 0& -6& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -3& -3& -1& 1& -12& -6& 0& 0& 0& 0& 6& 0& 0& 0& 0\\ 3& -6& 2& 1& -6& 3& -6& 2& 1& -12& -3& 6& -2& -1& 12\\ 6& -3& -3& 0& 6& 6& -3& -3& 0& 6& -6& 3& 3& 0& -6\\ -3& 3& 1& -1& 12& 3& 3& 1& -1& 12& 3& -3& -1& 1& -12\\ 0& 0& 0& 0& -6& -3& 6& -2& -1& 6& 0& 0& 0& 0& 6\\ 0& 0& 0& 0& 0& -6& 3& 3& 0& -6& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 3& -3& -1& 1& -12& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 6& 0& 0& 0& 0& 0\end{array}\right]$
function y = F6t(x)y = zeros(15,1); y(1) = 6*x(1)-3*x(2)-6*x(3)-3*x(4)+3*x(5)+6*x(6)-3*x(7);y(2) = 6*x(2)+3*x(3)-3*x(4)-6*x(5)-3*x(6)+3*x(7); y(3) = -2*x(2)+3*x(3)-x(4)+2*x(5)-3*x(6)+x(7);y(4) = -x(2)+x(4)+x(5)-x(7); y(5) = 12*x(2)-6*x(3)-12*x(4)-6*x(5)+6*x(6)+12*x(7)-6*x(8);y(6) = -6*x(4)+3*x(5)+6*x(6)+3*x(7)-3*x(8)-6*x(9)+3*x(10); y(7) = -6*x(5)-3*x(6)+3*x(7)+6*x(8)+3*x(9)-3*x(10);y(8) = 2*x(5)-3*x(6)+x(7)-2*x(8)+3*x(9)-x(10); y(9) = x(5)-x(7)-x(8)+x(10);y(10) = -12*x(5)+6*x(6)+12*x(7)+6*x(8)-6*x(9)-12*x(10)+6*x(11); y(11) = 6*x(4)-3*x(5)-6*x(6)+3*x(7);y(12) = 6*x(5)+3*x(6)-3*x(7); y(13) = -2*x(5)+3*x(6)-x(7);y(14) = -x(5)+x(7); y(15) = 12*x(5)-6*x(6)-12*x(7)+6*x(8);y = y/6;

## 8 point linear convolution

${D}_{8}={D}_{2}\otimes {D}_{2}\otimes {D}_{2}$
${F}_{8}=\left[\begin{array}{ccccccccccccccccccccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& 1& 0& -1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 1& -1& 1& 1& -1& -1& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& -1& 0& 1& -1& 0& 0& 1& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 1& -1& -1& -1& 1& 0& 0& 0& 1& 1& -1& 0& 0& 0& 0& 0& 0& -1& -1& 1& 0& 0& 0& 0& 0& 0\\ 1& -1& 0& 1& 1& 0& -1& 0& 0& 1& -1& 0& 1& 0& 0& -1& 0& 0& -1& 1& 0& -1& 0& 0& 1& 0& 0\\ -1& -1& 1& -1& -1& 1& 1& 1& -1& -1& -1& 1& -1& -1& 1& 1& 1& -1& 1& 1& -1& 1& 1& -1& -1& -1& 1\\ 0& 1& 0& -1& 1& 0& 0& -1& 0& 1& 1& 0& -1& 1& 0& 0& -1& 0& 0& -1& 0& 1& -1& 0& 0& 1& 0\\ 0& 0& 0& 1& 1& -1& 0& 0& 0& -1& -1& 1& 1& 1& -1& 0& 0& 0& 0& 0& 0& -1& -1& 1& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 0& 0& -1& 1& 0& -1& -1& 0& 1& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& -1& 1& 1& -1& -1& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 1& -1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$
function y = F8t(x) y = zeros(27,1);y(1) = x(1)-x(2)-x(3)+x(4)-x(5)+x(6)+x(7)-x(8); y(2) = -x(2)+x(3)+x(4)-x(5)+x(6)-x(7)-x(8)+x(9);y(3) = x(2)-x(4)-x(6)+x(8); y(4) = -x(3)+x(4)+x(5)-x(6)+x(7)-x(8)-x(9)+x(10);y(5) = x(4)-x(5)-x(6)+x(7)-x(8)+x(9)+x(10)-x(11); y(6) = -x(4)+x(6)+x(8)-x(10);y(7) = x(3)-x(4)-x(7)+x(8); y(8) = -x(4)+x(5)+x(8)-x(9);y(9) = x(4)-x(8); y(10) = -x(5)+x(6)+x(7)-x(8)+x(9)-x(10)-x(11)+x(12);y(11) = x(6)-x(7)-x(8)+x(9)-x(10)+x(11)+x(12)-x(13); y(12) = -x(6)+x(8)+x(10)-x(12);y(13) = x(7)-x(8)-x(9)+x(10)-x(11)+x(12)+x(13)-x(14); y(14) = -x(8)+x(9)+x(10)-x(11)+x(12)-x(13)-x(14)+x(15);y(15) = x(8)-x(10)-x(12)+x(14); y(16) = -x(7)+x(8)+x(11)-x(12);y(17) = x(8)-x(9)-x(12)+x(13); y(18) = -x(8)+x(12);y(19) = x(5)-x(6)-x(7)+x(8); y(20) = -x(6)+x(7)+x(8)-x(9);y(21) = x(6)-x(8); y(22) = -x(7)+x(8)+x(9)-x(10);y(23) = x(8)-x(9)-x(10)+x(11); y(24) = -x(8)+x(10);y(25) = x(7)-x(8); y(26) = -x(8)+x(9);y(27) = x(8);

## 18 point linear convolution

${D}_{8}={D}_{2}\otimes {D}_{3}\otimes {D}_{3}$

${F}_{18}$ and the program F18t are too big to print.

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