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There is a problem with that approach, in that the "asymetric squarewave" actually reverses polarity at its lower point. In fact, it will try to discharge any smoothing capacitor placed across the zener.
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Yes, but it doesn't matter much, it just adds to the ripple slightly. You do not need another diode - well so long as the capacitor can stand the worst case start-up situation of being unlucky with the first cycle - something to check. I think the reverse aspect will be somewhat compensated by the fact that it is a square wave.