analyzed a minimum-MSE quantizer for a pdf in which fulu) = f over an interval of...
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analyzed a minimum-MSE quantizer for a pdf in which fulu) = f₁ over an interval of |size L₁, fulu) = f₂ over an interval of size L₂, and fu(u) = 0 elsewhere. Let M be the total number of representation points to be used, with M₁ in the first interval and M₂ = M - M₁ in the second. Assume (from symmetry) that the quantization intervals are of equal size A₁ = L₁/M₁ in interval 1 and of equal size A₂ = L₂/M₂ in interval 2. Assume that Mis very large, so that we can approximately minimize the MSE over M₁, M₂ without an integer constraint on M₁, M₂ (that is, assume that M₁, M₂ can be arbitrary real numbers). (a) Show that the MSE is minimized if A₁f1/³ = ₂2/3, -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(A₁,A2), first minimize MSE(A₁, A₂) + AfA1, A₂) 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 (L₁ƒ 1/³ +4₂₁ƒ2/³)³ 2√2² 12M² MSE = (c) Assume that the Lloyd-Max algorithm is started with 0 < M₁ < M representation points in the first interval and M₂ = M - M₁ 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 < M₁ < 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 < M₁ < M. (Assume there are no representation points between the two intervals.) (f) Now consider an arbitrary choice of A₁ and A₂ (with no constraint on M). Show that the entropy of the set of quantization points is given by H(V)=-f₁L, log(f₁A₁) -f₂L₂ log(f₂4₂). (g) Show that if the MSE is minimized subject to a constraint on this entropy (ignoring the integer constraint on quantization levels), then A₁ = A₂. * 2 1 1 1 1 S 1 LC analyzed a minimum-MSE quantizer for a pdf in which fulu) = f₁ over an interval of |size L₁, fulu) = f₂ over an interval of size L₂, and fu(u) = 0 elsewhere. Let M be the total number of representation points to be used, with M₁ in the first interval and M₂ = M - M₁ in the second. Assume (from symmetry) that the quantization intervals are of equal size A₁ = L₁/M₁ in interval 1 and of equal size A₂ = L₂/M₂ in interval 2. Assume that Mis very large, so that we can approximately minimize the MSE over M₁, M₂ without an integer constraint on M₁, M₂ (that is, assume that M₁, M₂ can be arbitrary real numbers). (a) Show that the MSE is minimized if A₁f1/³ = ₂2/3, -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(A₁,A2), first minimize MSE(A₁, A₂) + AfA1, A₂) 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 (L₁ƒ 1/³ +4₂₁ƒ2/³)³ 2√2² 12M² MSE = (c) Assume that the Lloyd-Max algorithm is started with 0 < M₁ < M representation points in the first interval and M₂ = M - M₁ 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 < M₁ < 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 < M₁ < M. (Assume there are no representation points between the two intervals.) (f) Now consider an arbitrary choice of A₁ and A₂ (with no constraint on M). Show that the entropy of the set of quantization points is given by H(V)=-f₁L, log(f₁A₁) -f₂L₂ log(f₂4₂). (g) Show that if the MSE is minimized subject to a constraint on this entropy (ignoring the integer constraint on quantization levels), then A₁ = A₂. * 2 1 1 1 1 S 1 LC
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Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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