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Appendix: Matlab Functions For Circular Convolution and Prime Length FFTs

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

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Programs for Reduction Operations

The reduction matrix of Equation 44 in Preliminaries is implemented by KRED which calls RED . Its transpose and inverse transpose are implemented by tRED , tRED , itKRED and itRED .

function x = KRED(P,E,K,x)
% x = KRED(P,E,K,x);
% P : P = [P(1),...,P(K)];
% E : E = [E(K),...,E(K)];
for i = 1:K
   a = prod(P(1:i-1).^E(1:i-1));
   c = prod(P(i+1:K).^E(i+1:K));
   p = P(i);
   e = E(i);
   for j = e-1:-1:0
      x(1:a*c*(p^(j+1))) = RED(p,a,c*(p^j),x(1:a*c*(p^(j+1))));
   end
end

function y = RED(p,a,c,x)
% y = RED(p,a,c,x);
y = zeros(a*c*p,1);
for i = 0:c:(a-1)*c
   for j = 0:c-1
      y(i+j+1) = x(i*p+j+1);
      for k = 0:c:c*(p-2)
         y(i+j+1) = y(i+j+1) + x(i*p+j+k+c+1);
         y(i*(p-1)+j+k+a*c+1) = x(i*p+j+k+1) - x(i*p+j+c*(p-1)+1);
      end
   end
end

function x = tKRED(P,E,K,x)
% x = tKRED(P,E,K,x);
% (transpose)
% P : P = [P(1),...,P(K)];
% E : E = [E(K),...,E(K)];
for i = K:-1:1
   a = prod(P(1:i-1).^E(1:i-1));
   c = prod(P(i+1:K).^E(i+1:K));
   p = P(i);
   e = E(i);
   for j = 0:e-1
      x(1:a*c*(p^(j+1))) = tRED(p,a,c*(p^j),x(1:a*c*(p^(j+1))));
   end
end

function y = tRED(p,a,c,x)
% y = tRED(p,a,c,x);
% (transpose)
y = zeros(a*c*p,1);
for i = 0:c:(a-1)*c
   for j = 0:c-1
      y(i*p+j+c*(p-1)+1) = x(i+j+1);
      for k = 0:c:c*(p-2)
         y(i*p+j+k+1) = x(i+j+1) + x(i*(p-1)+j+k+a*c+1);
         y(i*p+j+c*(p-1)+1) = y(i*p+j+c*(p-1)+1) - x(i*(p-1)+j+k+a*c+1);
      end
   end
end

Programs for I ⊗ Dk ⊗ I

The operations of Im⊗D2⊗InIm⊗D2⊗In and Im⊗D3⊗InIm⊗D3⊗In are carried out by ID2I and ID3I . Their transposes by ID2tI and ID3tI . The functions D2 and D3 are listed in the appendix, `Bilinear Forms for Linear Convolution.' Two versions of ID2I are listed here. One of them calls D2 in a loop, while the other version puts the D2 code in the loop instead of using a function call. There are several ways to implement the form I⊗D2⊗II⊗D2⊗I. But this is a simple and straightforward method. It is modeled after IAI in the text.

function y = ID2I(m,n,x)
y = zeros(m*n*3,1);
v = 0:n:2*n;
u = 0:n:n;
for i = 0:m-1
   for j = 0:n-1
      y(v+i*3*n+j+1) = D2(x(u+i*2*n+j+1));
   end
end

function y = ID2I(m,n,x)
y = zeros(m*n*3,1);
for i = 0:n:n*(m-1)
   i2 = 2*i;
   i3 = 3*i;
   for j = 1:n
      j2 = i2 + j;
      j3 = i3 + j;
      y(j3) = x(j2);
      y(n+j3) = x(n+j2);
      y(2*n+j3) = x(j2) + x(n+j2);
   end
end

function y = ID2tI(m,n,x)
y = zeros(m*n*2,1);
v = 0:n:n;
u = 0:n:2*n;
for i = 0:m-1
   for j = 0:n-1
      y(v+i*2*n+j+1) = D2t(x(u+i*3*n+j+1));
   end
end

function y = ID3I(m,n,x)
y = zeros(m*n*5,1);
v = 0:n:4*n;
u = 0:n:2*n;
for i = 0:m-1
   for j = 0:n-1
      y(v+i*5*n+j+1) = D3(x(u+i*3*n+j+1));
   end
end

function y = ID3tI(m,n,x)
y = zeros(m*n*3,1);
v = 0:n:2*n;
u = 0:n:4*n;
for i = 0:m-1
   for j = 0:n-1
      y(v+i*3*n+j+1) = D3t(x(u+i*5*n+j+1));
   end
end

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