Optimal clipping for a Laplace-distributed tensor

ProofAdvanced⏱ 1 hourQuantizationclippingmseoptimisationlaplace

Problem

Let $X$ have the Laplace density $f(x) = \frac{1}{2\lambda}e^{-|x|/\lambda}$ , quantized with $b$ bits over $[-A,A]$ .

Compute the clipping term $2\int_A^\infty (x-A)^2 f(x)\,dx$ in closed form.
Write the total MSE, treating $\P(|X|\le A)\approx1$ , and derive the stationarity condition for the optimal $A$ .
Solve numerically for $b = 4$ and $b = 8$ with $\lambda = 1$ , and comment on how the optimal loading factor moves with $b$ .

Write a complete argument. Every inequality you use should be one you could state precisely.

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