Bennett HighResolution Analysis of Quantizer Distortion For fixedrate memoryless VQ there - PDF document

Bennett-1 Bennett's IntegralA formula for the mean-squared error distortion of a "high-resolution"Zador's formulaAn approximation to the OPTA function -0.04 -0.02 0.040.06 33.5 -0.04 -0.02 0.040.06 0.4 0.6 0.8 1.4 1.6 Bennett-3 x1x2 1x2 Bennett-5 ¥Dimension k¥Size M¥The distribution or density of points/cells over ¥Some way of characterizing the shapes of the cells as a function of Bennett-7 1M 1( x) ( x) d xwhere( x. It is also called the "point" or "cell density" function.The formula above is called "Bennett's integral". It was first derived by Bennett. W. Bennett, "Spectra of quantized signals," Bell Syst. Tech. J., vol. 27, pp. 446-472, July 1948.2A. Gersho, "Asymptotically optimal block quantizers," IEEE Trans. Inform. Thy., vol. 25, pp. 373-380, 1979. Bennett-8 ( x) x) x) d x = 1 x) d x ( x) |A|  AM ( ( x) M( x) Therefore, # .( x) |A| = |A| ( x) Bennett-9 Bennett-11 D= X)|| Si i ( x) d xk f ( wiSi i x because f X( x) ( wi Sif X( x) ( wi Si ( f ( wi) 1k , wi) w) = w|| x = "moment of inertia" (mi) of S about w w) of the shape of S from its size. w) = w|| xk S 1 d x w) w) = w|| xk Bennett-13 w) is not affected by a scaling nor a translation. Thus, it is determined z = a x : x w) = w|| xk S w|| zk z = x, a z = d xS a2 w|| zk (ak ) |S| w|| zk v: S  v = { z = x+ v : x v)= v|| xk S w|| zk z = x- v, d z = d x ( wik M, ) derived earlier f ( wi, by the definition of nmi f ( ) |S i oM ( wi) |M ( wi) |Therefore,M , f ( wi( wi) | M , wo f x( ( x) x = Bennett's integralwhere the last " Bennett-15Special case: Scalar quantization (k=1)12 Then - fX(x)2 (a)the partition divides [a,b] into M cells of width (c)the codepoints (levels) are in the centers of the cells (i.e. they are b M  b-a X ab fX(x) 2 ab fX2 Bennett-17 :high-resolution conditions X can be approximated byD 1M m( x)( x) ( x) d +Most cells are small enough that the prob. density can be approximated as con-+Neighboring cells have similar sizes and shapes, i.e. cell size & shape change +The quantization density is approximately +The ( x) x = m(Si, wi x) 3S. Na and D. Neuhoff, "Bennett's integral for vector quantizers," IEEE Trans. Inform. Thy., vol. 41, pp. 886-900, July 1995. Bennett-18 D= X)|| Si i ( x) d xk f ( wiSi i f ( wi) 1k , ) MI of S f ( ) m(S , recall: wi, wi) k | M f ( wi) m( wi)( wi) | ,recall: |S wi) i, wi)1M m( x)( x) ( x) d xby the definition of an integral Bennett-19 ¥Bennett's integral identifies point density and inertial profile as key¥When M is large, both left and righthand sides of the Bennett integral relation m( x)( x) f X( d x ¥Bennett's integral shows that distortion decreases as , x and the codevector of the cell in which ¥Equivalently, SNR increases 6 dB for each one bit increase of rate. = 2 x)( x) ( x) d x 2 D = 10 log10 22R 2  ( f X( d x 2  ( f X( d x ¥Usually, we don't employ a quantization density or inertial profile to describe a¥Usually, ¥Quantization density and inertial profile are, generally, idealizations or models. ¥We don't use the following as definitions because if we did, a quantizer would x) = x) = m(S wi ¥Sketch of why Property 5 on p. 10 implies Property 2 x) d x =  SiA ( x) d M  ) AM ) ¥When, as usual, ¥When, as usual, m¥We see from Bennett's integral that to make D small, we want larger quantization ( Bennett-23 w is w) = S w|| (S) ¥NMI is the same for cubes of all dimensions. x : a x : -1/2   w = (0, m(S)= 1 i k "i313 | k !"i12 k "i323 |12 k "i333 |12 12 k 112 12 ¥The NMI of various cell shapescell shapedimensionNMI2.104 cubeany 2#$3 circle2 sphere3 spherek 2%e 1 1 s2krectanglek1 1k &k)2'k)21/k ¥Shapes that tend to make NMI smaller+Spheroidal rather than oblong s. +More finely faceted ¥A spheres has the lowest NMI of any cell of a given dimension.¥NMI of a sphere decreases with dimension to the limit 1/2 Bennett-27 % k % ( )!k! %"ne nN %"ne nn12n k#k "2%ek k*+k2  ) "1#k "2%ek k/2-2/k 2%e k 2%e 12%e . Bennett-29 : m( x)( x) ( x) d x+M( x) x) in probability as M  M( x) = 1M ( x) x) as M x) = , y +{M ( +diam(cell of S X) ( +Bennett's integral is finite S. Na and D. Neuhoff, "Bennett's integral for vector quantizers," IEEE Trans. Inform. Thy., vol. 41, pp. 886-900, July 1995.