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

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Problem Set 6 solutions 1 April 11, 2006 13.1 One bit quantization of a Gaussian random variable. Let X ∼ N (0 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 = Z 0 x 2 2 πσ 2 e - x 2 2 σ 2 dx (1) = Z 0 σ r 2 π e - y dy (2) = σ r 2 π , (3) using the substitution y = x 2 / 2 σ 2 . The expected distortion for one bit quantization is D = Z 0 -∞ ± x + σ r 2 π ! 2 1 2 πσ 2 e - x 2 2 σ 2 dx (4) + Z 0 ± x - σ r 2 π ! 2 1 2 πσ 2 e - x 2 2 σ 2 dx (5) = 2 Z -∞ ² x 2 + σ 2 2 π ³ 1 2 πσ 2 e - x 2 2 σ 2 dx (6) - 2 Z 0 ± - 2 r 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 r 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 µ (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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This note was uploaded on 11/19/2010 for the course ANTH 122 taught by Professor 323 during the Spring '10 term at Centennial College.

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

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