lect2 - Kevin Buckley - 2007 1 ECE 8770 Topics in Digital...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Kevin Buckley - 2007 1 ECE 8770 Topics in Digital Comm. - Sp2007 Lecture 2 1 Digital Communications System, and Signal & Chan- nel Representation 1.4 Signal Space Representation This Section of the course notes corresponds to Section 4.2 of the course text. The objective here is to develop a generally applicable framework for studying a dig- itally modulated communication signal. In this Section we will introduce this frame- work, termed signal space representation, and in Section [1.5] we will apply it to several common digital communication modulation schemes. This signal space representation of digital communication symbols will be based on: the equivalent lowpass representation of symbols; a basis expansion of the set of symbols employed in the modulation scheme; and a Euclidean measure of the distance between symbols (i.e. a geometric represen- tation). Later, when we discuss the channel and demodulator, we will combine this signal space representation of a modulation scheme with the equivalent lowpass representation of a digital communication system. Kevin Buckley - 2007 2 1.4.1 Linear Space Concepts Consider n-dimensional complex vector v k : v k = [ v 1 ,k , v 2 ,k , ...v n,k ] T . (1) The inner product of two such vectors v k and v j is defined as v k v j = v H j v k = n summationdisplay i =1 v i,k v i,j . (2) Two vectors, v k and v j , are orthogonal is v k v j = 0 . (3) The norm of a vector v k is || v k || = parenleftBig v H k v k parenrightBig 1 2 (4) A vector v k has unit norm if || v k || = 1. Given a set of m n vectors { v 1 , v 2 , , v m } , they are linearly independent if no one vector can be written as linear combination of the m- 1 others. (Then the column rank of the matrix V = [ v 1 , v 2 , ..., v m ] is m , i.e. V is full-rank.) If n n-dimensional vectors { v 1 , v 2 , , v n } form an orthonormal set , i.e. if v k v j = ( k- j ); all k, j , (5) then this set forms an orthonormal basis for the n-dimensional complex vector space where these vectors reside. Orthonormal bases facilitate representation of vectors in the n-dimensional space. Let { v 1 , v 2 , , v n } be an orthonormal basis. Then any n-dimensional complex vector v can be expanded (and represented) as v = n summationdisplay k =1 s k v k = V s (6) where V = [ v 1 , v 2 , ..., v n ], s = [ s 1 , s 2 , ..., s n ] T , and s k = v H k v . A set of m n-dimensional vectors is low rank if its span has dimension less than n . Kevin Buckley - 2007 3 Consider an arbitrary vector v , a set of m < n orthonormal vectors { v 1 , v 2 , , v m } , and the matrix V = [ v 1 , v 2 , , v m ]. In general, v can not be represented as a linear combination of the m orthonormal vectors. Even so, consider the rank- m (low-rank) representation of v : v = m summationdisplay k =1 s k v k = V s (7) with, as before, s k = v H k v . The error vector is e = v- v = v- V s . (8) The s used above, ( s = V H v ), minimizes the Euclidean norm of the error,...
View Full Document

Page1 / 23

lect2 - Kevin Buckley - 2007 1 ECE 8770 Topics in Digital...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online