For a vector A of length N
B = fft(A)
The Parseva'sl theorem tells
norm(A) = norm(B) / sqrt(N)
So actually the unit of
B / sqrt(N)
is unit of A.
Now the problem is to know what is the unit of sqrt(N). N depends on your sampling. In principle it's unitless since
N = T / Fs
where T is the acquisition period and Fs is the sampling frequency. But you observe that if you change Fs while keeping T constant, N changed inversonally to Fs. So it's up to you to "interpret" the unit of N (which is sometime not clear).
May be it's better to compute a more natural normalization
C = fft(A)/sqrt(length(A))
Then the unit of C is less ambiguous and it's the unit of A (the energies - l2 norm - of A and C are identical).
For PWELCH it depends on the windows size (which plays the same role as N in FFT) and the window-function that PWELCH uses. The proper normalization is to divide the ouput by the l2-norm of the window-function computed on the sliding window. You can have those information with little reverse engineering and the information provided by the document page. But it's not straignforward.
