Bounds on performance of optimum quantizers

A quantizer Q divides the range 0, 1 of a random variable x into K quantizing intervals the i th such interval having length xᵢ. We define the quantization error for a particular value of x (unusually) as the length of the quantizing interval in which x finds itself, and measure quantizer performance (unusually) by the r th mean value of the quantizing interval lengths Mᵣ (Q) = x^r^{1/r}, averaging with respect to the distribution function F of the random variable x. Q₁ is defined to be an optimum quantizer if Mᵣ (Q₁) Mᵣ (Q) for all Q. The unusual definitions restrict the results to bounded random variables, but lead to general and precise results. We define a class Q^ of quasi-optimum quantizers; Q₂ is in Q^ if the different intervals xᵢ make equal contributions to the mean r th power of the interval size so that Pr \ xᵢ \ x₈^ₑ is constant for all i. Theorems 1, 2, 3, and 4 prove that Q₂ Q^ exists and is unique for given F, K, and r: that 1 KMᵣ (Q₂) KMᵣ (Q₁) Iᵣ, where Iᵣ = \₀^{1 f (x) ᵖ dx\}^ 1/q, f is the density of the absolutely continuous part of the distribution function F of x, p = 1/ (1+ r), and q = r / (1 + r): that lim KMᵣ (Q₂) = Iᵣ as K ; and that if KMᵣ (Q) = Iᵣ for finite K, then Q=Q^.