MATLAB: Convolution in frequency domain (NOT CONVOLUTION IN TIME DOMAIN)

Question

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

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