# Soln01 - ECE 342 Communication Theory Fall 2005, Solutions...

This preview shows pages 1–2. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ECE 342 Communication Theory Fall 2005, Solutions to Homework 1 Prof. Tiffany J Li www: http://www.eecs.lehigh.edu/ ∼ jingli/teach email: [email protected] Problem 2.1 1) 2 = Z ∞-∞ x ( t )- N X i =1 α i φ i ( t ) 2 dt = Z ∞-∞ x ( t )- N X i =1 α i φ i ( t ) ! x * ( t )- N X j =1 α * j φ * j ( t ) dt = Z ∞-∞ | x ( t ) | 2 dt- N X i =1 α i Z ∞-∞ φ i ( t ) x * ( t ) dt- N X j =1 α * j Z ∞-∞ φ * j ( t ) x ( t ) dt + N X i =1 N X j =1 α i α * j Z ∞-∞ φ i ( t ) φ * j dt = Z ∞-∞ | x ( t ) | 2 dt + N X i =1 | α i | 2- N X i =1 α i Z ∞-∞ φ i ( t ) x * ( t ) dt- N X j =1 α * j Z ∞-∞ φ * j ( t ) x ( t ) dt Completing the square in terms of α i we obtain 2 = Z ∞-∞ | x ( t ) | 2 dt- N X i =1 Z ∞-∞ φ * i ( t ) x ( t ) dt 2 + N X i =1 α i- Z ∞-∞ φ * i ( t ) x ( t ) dt 2 The first two terms are independent of α ’s and the last term is always positive. Therefore the minimum is achieved for α i = Z ∞-∞ φ * i ( t ) x ( t ) dt which causes the last term to vanish.which causes the last term to vanish....
View Full Document

## This note was uploaded on 08/06/2008 for the course ECE 342 taught by Professor Li during the Fall '05 term at Lehigh University .

### Page1 / 4

Soln01 - ECE 342 Communication Theory Fall 2005, Solutions...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online