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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 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 = ( 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 : 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,...
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 Spring '09

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