solns6 - ECE 1502 Information Theory Problem Set 6...

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ECE 1502 — Information Theory Problem Set 6 solutions 1 April 11, 2006 13.1 One bit quantization of a Gaussian random variable. Let X ∼ N (0 , σ 2 ) and let the distortion measure be squared error. With one bit quantization, the obvious reconstruction regions are the positive and negative real axes. The reconstruction point is the centroid of each region. For example, for the positive real line, the centroid a is a = 0 x 2 2 πσ 2 e - x 2 2 σ 2 dx (1) = 0 σ 2 π e - y dy (2) = σ 2 π , (3) using the substitution y = x 2 / 2 σ 2 . The expected distortion for one bit quantization is D = 0 -∞ x + σ 2 π 2 1 2 πσ 2 e - x 2 2 σ 2 dx (4) + 0 x - σ 2 π 2 1 2 πσ 2 e - x 2 2 σ 2 dx (5) = 2 -∞ x 2 + σ 2 2 π 1 2 πσ 2 e - x 2 2 σ 2 dx (6) - 2 0 - 2 2 π 1 2 πσ 2 e - x 2 2 σ 2 dx (7) = σ 2 + 2 π σ 2 - 4 1 2 π σ 2 2 π (8) = σ 2 π - 2 π . (9) 13.2 Rate Distortion . We wish to evaluate the rate distortion function R ( D ) = min p x | x ): ( x, ˆ x ) p ( x ) p x | x ) d ( x, ˆ x ) D I ( X ; ˆ X ) . (10) Since d (0 , 1) = , we must have p (0 , 1) = 0 for a finite distortion. Thus, the distortion D = p (1 , 0), and hence we have the following joint distribution for ( X, ˆ X ) (assuming D 1 2 ). p ( x, ˆ x ) = 1 2 0 D 1 2 - D (11) The mutual information for this joint distribution is R ( D ) = I ( X ; ˆ X ) = H ( X ) - H ( X | ˆ X ) (12) 1 Solutions to problems from the text are supplied courtesy of Joy A. Thomas.
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