analyzed a minimum-MSE quantizer for a pdf in which fulu) = f over an interval of |size L, fulu) = f over an interval of size L, and fu(u) = 0 elsewhere. Let M be the total number of representation points to be used, with M in the first interval and M = M - M in the second. Assume (from symmetry) that the quantization intervals are of equal size A = L/M in interval 1 and of equal size A = L/M in interval 2. Assume that Mis very large, so that we can approximately minimize the MSE over M, M without an integer constraint on M, M (that is, assume that M, M can be arbitrary real numbers). (a) Show that the MSE is minimized if Af1/ = 2/3, -1/3 i.e. the quantization interval sizes are inversely proportional to the cube root of the density. [Hint. Use a Lagrange multiplier to perform the minimization. That is, to minimize a function MSE(A1, A2) subject to a constraint M = f(A,A2), first minimize MSE(A, A) + AfA1, A) without the constraint, and, second, choose A so that the solution meets the constraint.] (b) Show that the minimum MSE under the above assumption is given by 3 (L 1/ +42/) 22 12M MSE = (c) Assume that the Lloyd-Max algorithm is started with 0 < M < M representation points in the first interval and M = M - M points in the second interval. Explain where the Lloyd- Max algorithm converges for this starting point. Assume from here on that the distance between the two intervals is very large. (d) Redo part (c) under the assumption that the Lloyd-Max algorithm is started with 0 < M < M - 2 representation points in the first interval, one point between the two intervals, and the remaining points in the second interval. (e) Express the exact minimum MSE as a minimum over M-1 possibilities, with one term for each choice of 0 < M < M. (Assume there are no representation points between the two intervals.) (f) Now consider an arbitrary choice of A and A (with no constraint on M). Show that the entropy of the set of quantization points is given by H(V)=-fL, log(fA) -fL log(f4). (g) Show that if the MSE is minimized subject to a constraint on this entropy (ignoring the integer constraint on quantization levels), then A = A. * 2 1 1 1 1 S 1 LC