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Unformatted text preview: ECE 534: Elements of Information Theory, Fall 2010 Homework 10 SOLUTIONS all by Davide Basilio Bartolini Ex. 10.1 Text One-bit quantization of a single Gaussian random variable . Let X N (0 , 2 ) and let the distortion measure be squared error. Here we do not allow block descriptions. Show that the optimum reproduction points for 1-bit quantization are radicalBig 2 and that the expected distortion for 1-bit quantization is - 2 2 . Compare this with the distortion rate bound D = 2 2- 2 R for R = 1. Solution If we are using a 1-bit quantization, we are allowed to use up to two reproduction points (say, p- and p + ) for which two reconstruction regions are to be defined. It is quite straightforward that the optimal reconstruction regions in this case are the positive and negative axes; the reconstruction points may be found as the expectation for each over the given Gaussian distribution. For the positive axis: p + = integraldisplay x 2 2 2 e- x 2 2 dx = integraldisplay 2 2 2 2 e- y dy parenleftbigg substituting y = x 2 2 2 dy = 2 x 2 2 dx = x 2 dx parenrightbigg = integraldisplay radicalbigg 2 e- y dy = bracketleftBigg radicalbigg 2 e- y bracketrightBigg = radicalbigg 2 1 and, by symmetry, p- = radicalBig 2 . Now we can compute the expected distortion D as the expected squared error: D = E f ( x ) [( X X ) 2 ] = integraldisplay- parenleftBigg x + radicalbigg 2 2 parenrightBigg 2 1 2 2 e- x 2 2 2 dx + integraldisplay parenleftBigg x radicalbigg 2 2 parenrightBigg 2 1 2 2 e- x 2 2 2 dx = 2 integraldisplay -...
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This note was uploaded on 01/19/2012 for the course ECE 534 taught by Professor Natashadevroye during the Fall '10 term at Ill. Chicago.
- Fall '10