{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW10s[1]

HW10s[1] - ECE 534 Elements of Information Theory Fall 2010...

This preview shows pages 1–3. Sign up to view the full content.

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 , σ 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 π σ and that the expected distortion for 1-bit quantization is π - 2 π σ 2 . Compare this with the distortion rate bound D = σ 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 0 x 2 2 πσ 2 e - x 2 σ 2 dx = integraldisplay 0 σ 2 2 2 πσ 2 e - y dy parenleftbigg substituting y = x 2 2 σ 2 dy = 2 x 2 σ 2 dx = x σ 2 dx parenrightbigg = integraldisplay 0 σ radicalbigg 2 π e - y dy = bracketleftBigg σ radicalbigg 2 π e - y bracketrightBigg 0 = σ radicalbigg 2 π 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
and, by symmetry, p - = σ radicalBig 2 π . Now we can compute the expected distortion D as the expected squared error: D = E f ( x ) [( X ˆ X ) 2 ] = integraldisplay 0 -∞ parenleftBigg x + radicalbigg 2 σ 2 π parenrightBigg 2 1 2 πσ 2 e - x 2 2 σ 2 dx + integraldisplay 0 parenleftBigg x radicalbigg 2 σ 2 π parenrightBigg 2 1 2 πσ 2 e - x 2 2 σ 2 dx = 2 integraldisplay -∞ parenleftbigg x 2 + 2 σ 2 π parenrightbigg 1 2 πσ 2 e - x 2 2 σ
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 5

HW10s[1] - ECE 534 Elements of Information Theory Fall 2010...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online