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Appendix: Bilinear Forms for Linear Convolution

Module by: Ivan Selesnick, C. Sidney Burrus. E-mail the authors

Appendix: Bilinear Forms for Linear Convolution

The following is a collection of bilinear forms for linear convolution. In each case (Dn,Dn,Fn)(Dn,Dn,Fn) describes a bilinear form for nn point linear convolution. That is

y = F n D n h * D n x y = F n D n h * D n x
(1)

computes the linear convolution of the nn point sequences hh and xx.

For each DnDn we give Matlab programs that compute DnxDnx and DntxDntx, and for each FnFn we give a Matlab program that computes FntxFntx. When the matrix exchange algorithm is employed in the design of circular convolution algorithms, these are the relevant operations.

2 point linear convolution

D2D2 can be implemented with 1 addition, D2tD2t with two additions.

D 2 = [ 1 0 0 1 1 1 ] D 2 =[ 1 0 0 1 1 1 ]
(2)
F 2 = [ 1 0 0 -1 -1 1 0 1 0 ] F 2 =[ 1 0 0 -1 -1 1 0 1 0 ]
(3)

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

D3D3 can be implemented in 7 additions, D3tD3t in 9 additions.

D 3 = [ 1 0 0 1 1 1 1 -1 1 1 2 4 0 0 1 ] D 3 =[ 1 0 0 1 1 1 1 -1 1 1 2 4 0 0 1 ]
(4)
F 3 = 1 6 [ 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 ] F 3 = 1 6 [ 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 ]
(5)

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 ⊗ D 2 D 4 = D 2 ⊗ D 2
(6)
F 4 = [ 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 ] F 4 =[ 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 ]
(7)

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 ⊗ D 3 D 6 = D 2 ⊗ D 3
(8)
F 6 = 1 6 [ 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 ] F 6 = 1 6 [ 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 ]
(9)

        
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 ⊗ D 2 ⊗ D 2 D 8 = D 2 ⊗ D 2 ⊗ D 2
(10)

F 8 = [ 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 ] F 8 =[ 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 ]
(11)

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 ⊗ D 3 ⊗ D 3 D 8 = D 2 ⊗ D 3 ⊗ D 3
(12)

F18F18 and the program F18t are too big to print.

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