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Extra info for Advances in Neural Information Processing Systems 2
2. MINIMUM COST OF ENTROPY One question still unanswered is, What would be the minimum cost of entropy required for the demon to operate the trapdoor? Let the arrival molecules at the trapdoor at an instant be one, two, or more molecules. Then the computer is required to provide the demon with a "yes" or a "no" information. For example, if a single molecule is approaching the trapdoor (say, a high-velocity one), the demon will open the trapdoor to allow the molecule to go through. Otherwise, he will stand still.
However, in certain signal detections, an increase in the signal-to-noise ratio does not necessarily guarantee minimizing the detection error. Nevertheless, minimizing the detection error can always be obtained by using a decision process. Let us consider the detection of binary signals. 98) One can write P(a = Q/b) = P(a = Q)P(b/a = 0) P(0 = l/b) P(a = l)p(b/a = 1)' " J where P(0) is the a priori probability of 0; that is, a = I corresponds to the signal presented, and 0 = 0 corresponds to no signal.
For his transformation. 130) in which the WDF has been normalized to unity for simplicity. One of the interesting features of the WDF is the inversion. Apart from the association with a constant phase factor, the transformation is unique, as can be shown, W(t, v)e~i4nvt dt = U(2v)U*(Q), (r, v)ei4nvt dv = w(2t)M*(0). 133) W(T, V) eXp[/47lv(T - Tmax)] dv = U(2l - Tma>*(Tmax). 134) 40 1. Entropy Information and Optics (a) Fig. 14. Wigner distributions, (a) For a single tone signal, (b) For a pulse signal.