Greg, I think that's almost true. If you define the PSD as |F(w)|^2 then certainly the phase information gets lost in the modulus operation and the autocorrelation is unrecoverable.
More precisely however, the PSD is defined on infinite stationary processes as lim(T->inf) (1/T) * E(|F(w)|^2) where E is the expectation operator. This expands to:
lim(T->inf) (1/T) * E(F(w)*F'(w))
where F' is the complex conjugate of F.
Now provided the original process was real, F(w) = integral(f(t)*e^(-2*pi*i*w*t), w.r.t. t) and F'(w) = integral(f(s)*e^(2*pi*i*w*s), w.r.t. s)
So F(w)*F'(w) is just the double integral of f(t)*f(s)*e^(-2*pi*i*w*(t-s))
If you're really careful with the limits and the double integral, you can change variables to tau and another arbitrary variable, evaluate the arbitrary variable integral and arrive at
F(w)*F'(w) = integral( (T-tau) * f(t)*f(t-tau)*e^(-2*pi*i*w*(tau), w.r.t. tau)
If you apply the expectation operator, it only applies to the f(t)*f(t-tau) term and if you multiply by (1/T) you get:
(1/T) * E(F(w)*F'(w)) = integral( (1-tau/T) * E(f(t)*f(t-tau))*e^(-2*pi*i*w*(tau), w.r.t. tau)
Of course if we now take the limit as T->inf the tau/T term disappears the right hand side is just the Fourier transform of E(f(t)*f(t-tau)).
If we do a bit more handwaving, E(f(t)*f(t-tau)) is the autocorrelation of f(t).
So if all these conditions are true and you squint hard enough, the PSD of f(t) is equal to the Fourier transform of the autocorrelation of f(t). In fact, this is sometimes how it is defined. Theoretically then, the inverse fourier transform of the PSD gives you back the autocorrelation.
But I think by the time you convert to discrete, limit to a finite signal, and incorporate the fancy estimation stuff in MATLAB's pwelch function, this tenuous link to the autocorrelation might be a bit hard to recover!
PS. I'm sorry the math notation is going to be very hard to follow. I tried to make it as clean as possible. It is possible to render LaTeX or something instead?
