pmtm (Multitaper) calculates PSD from Fourier coefficients as abs(Xx)²; shouldn’t that be Xx.conj(Xx)?

In MATLAB 7.11.2(R2010b) Service Pack 2, in the Signal Processing Toolbox file pmtm.m (Multitaper), line 232, the current code uses abs(Xx).², where XX is the set of raw Fourier coefficients, to eventually calculate the PSD. Shouldn’t that be Xx.conj(Xx) since power is defined as the square of the norm of the Fourier coefficients?

% Compute DFT using FFT or Goertzel
[Xx,w] = computeDFT(E(:,1:k).*x(:,ones(1,k)),nfft,Fs);
%Sk = abs(Xx).^2; % OLD CODE HERE
Sk = Xx.*conj(Xx); % AK CHANGE HERE

ANSWER

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There is no difference between

abs(Z).^2

And

Z.*conj(Z)

for a complex vector. They are equivalent. In fact, for a complex number the modulus (norm) is defined to be the square root of the product of Z and its conjugate.

Just convince yourself for a single complex number

Z = 1+1i;
abs(Z)^2
Z*conj(Z)

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