Convolution in frequency domain (NOT CONVOLUTION IN TIME DOMAIN)

I know there are two theorem:

1. convolution in time domain equals multiplication in frequency domain;
2. multiplication in time domain equals convolution in frequency domain;

I am confused with the implementation with the 2nd in MATLAB. Here is my codes:

    a = 1:5;
    b = a+3;
    A = fft(a);
    B = fft(b);
    c = a .* b
    ans =
         4    10    18    28    40
    D = conv(A, B);
    d = real(ifft(D))
    d =
      Columns 1 through 5
           11.111       50.544       26.081       58.919       39.694
      Columns 6 through 9
           73.642       52.666       95.071        42.27

so my quesition is why the "a.*b" is not the same with "real(ifft(D))", is somewhere wrong? I know in the 1st theorem, the Nfft should be length(A)+length(B)-1, however, it seems not work here.

Any help will be thanks!

Answer 1

[Matt J]

Cyclic convolution is the dual of multiplication when dealing with discrete fourier transforms. You are doing linear convolution.

[chen xy]

Thanks for answer, and what's the difference between the cyclic and linear convolution in implementation ? For the above codes, how should I rectify ?

[Matt J]

One way to implement cyclic convolution is as a matrix multiplication, using this

http://www.mathworks.com/matlabcentral/fileexchange/26292-regular-control-point-interpolation-matrix-with-boundary-conditions

For you, it would be

N=length(A);
CirculantMatrix = interpMatrix(A,1,N,1,'circ');
C=CirculantMatrix*B(:); %cyclic convolution
c=ifft(C,'symmetric').'/N

c =

      4.0000   10.0000   18.0000   28.0000   40.0000

[chen xy]

It works well, thx a lot!

[[e-mail]]

Hi guys

How can I do the same for two matrices of the same size? say:

x = [3 5 4; 7 6 1; -1 2 0];

y = [2 7 1; 2 -3 2; 5 6 9];

how can I get the same results as x.*y using convolution of the fft2 of these two matrices?

Thanks in advance