Bounds on performance of optimum quantizers
Peter Eliaš · IEEE Transactions on Information Theory · 1970
A quantizerQdivides the range [0, 1] of a random variablexintoKquantizing intervals theith such interval having length\Delta x_i. We define the quantization error for a particular value ofx(unusually) as the length of the quantizing interval in whichxfinds itself, and measure quantizer performance (unusually) by therth mean value of the quantizing interval lengthsM_r (Q) = \overline{\Delta x^{r^{1/r}}}, averaging with respect to the distribution functionFof the random variablex.Q_1is defined to be an optimum quantizer ifM_r (Q_1) \leq M_r (Q)for allQ. The unusual definitions restrict the results to bounded random variables, but lead to general and precise results. We define a classQ^{\ast}of quasi-optimum quantizers;Q_2is inQ^{\ast}if the different intervals\Delta x_imake equal contributions to the meanrth power of the interval size so thatPr \{ \Delta x_i \} \Delta x_{i^{r}}is constant for alli. Theorems 1, 2, 3, and 4 prove thatQ_2 \in Q^{\ast}exists and is unique for givenF, K, andr: that1 \geq KM_r (Q_2) \geq KM_r (Q_1) \geq I_r, whereI_r = \{\int_0^{1} f (x)^p dx\}^ {1/q}, fis the density of the absolutely continuous part of the distribution functionFofx, p = 1/(1+ r), andq = r /(1 + r): thatlim KM_r (Q_2) = I_rasK \rightarrow \infty; and that ifKM_r (Q) = I_rfor finiteK, thenQ=Q^{\ast}.