This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Kevin Buckley  2007 1 ECE 8770 Topics in Digital Comm.  Sp’2007 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 ndimensional 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 fullrank.) If n ndimensional vectors { v 1 , v 2 , ··· , v n } form an orthonormal set , i.e. if v k · v j = δ ( k j ); all k, j , (5) then this set forms an orthonormal basis for the ndimensional complex vector space where these vectors reside. Orthonormal bases facilitate representation of vectors in the ndimensional space. Let { v 1 , v 2 , ··· , v n } be an orthonormal basis. Then any ndimensional 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 ndimensional 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 (lowrank) representation of v : ˆ 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 ˆ v = v V s . (8) The s used above, ( s = V H v ), minimizes the Euclidean norm of the error,...
View
Full
Document
This document was uploaded on 10/12/2009.
 Spring '09

Click to edit the document details