Iterative phase determination and information content in Fourier transformation

 

^{Iterative phase determination and information content in Fourier transformation}

Information content basis for iterative phase determination of Fourier transformation

Consider a linear array of N complex values

a(j), j = 0 to N-1.

a(j) contains 2N independent real values. It’s discrete Fourier transformation is an array of N complex values

Fa(2pi*j/N), j = -int(N/2) to N-int(N/2)-1.

If a(j) is real, it contains only N independent real values, then Fa(j) = [Fa(-j)]’, so Fa(j) only contains N independent real values.

If a(j) is real and >=0 for all j, then it contains N independent non-negative values. Then, the N real values in the Fourier transformation Fa(j) are no longer independent.

In an expanded support, A(j) = a(j), for j = 0 to N-1, A(j) = 0 for j=N to 2N-1, the Fourier transformation FA( 2pi*j/(2N)), j = -N to N-1.

If a(j) is real and >=0, then FA(j) = [FA(-j)]’, and |FA(j)| might be the (N+1) independent non-negative values, and the phase factors are not independent and can be determined iteratively?