Quantum uncertainty and wave-particle duality

I have always felt that Heisenberg's Uncertainty Principle was a manifestation of wave-particle duality, rather than a simple consequence of the quantum randomness.  The principle is literally a theorem in Fourier Analysis rather than a theorem in probability theory.  Recent experimental results reported by Tim Wogan in Physics World seem to provide evidence for this interpretation.  The relationship between uncertainty and wave-particle duality is not as simple as I would have thought, but nonetheless seems to be a better way to interpret it than to merely attribute it to the inherently random nature of quantum phenomena.