Optimal Entropy - Constrained Scalar Quantization of aUniform
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چکیده
Optimal scalar quantization subject to an entropy constraint is studied for a wide class of diierence distortion measures including rth power distortions with r > 0. It is proved that if the source is uniformly distributed over an interval, then for any entropy constraint R (in nats), an optimal quantizer has N = e R interval cells such that N ? 1 cells have equal length d and one cell has length c d. The cell lengths are uniquely determined by the requirement that the entropy constraint is satissed with equality. Based on this result, a parametric representation of the minimum achievable distortion D h (R) as a function of the entropy constraint R is obtained for a uniform source. The D h (R) curve turns out to be nonconvex in general. Moreover, for the squared error distortion it is shown that D h (R) is a piecewise concave function, and that a scalar quantizer achieving the lower convex hull of D h (R) exists only at rates R = log N, where N is a positive integer.
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