Lecture 22 - ECE 614: Principles of Digital Communications,...

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Unformatted text preview: ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins Spri Spring 2010 ng 2010 Lecture 22: April 13, 2010 Outline: System Analysis Based on Signal Space Models Linear Block Codes 1 ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins Spri Spring 2010 ng 2010 Abstract Linear Space Let V be a collection of objects. Define a + operation between two objects in V . Closure : for all x , y V , we have x + y V Associative : x +( y + z )=( x + y )+ z Commutative : x + y = y + x There exists a zero element satisfying x + = x for all x V. For each x V , there exists an additive inverse w V , such that x + w = . Let F be a field. Define a * operation between a member of F and an object of V , ch that such that Closure : for all x V , a F , a* x V Distributative : for all x , y V, a F , a* ( x + y )=( a* x )+( a* y ) r all a b * =( * ( * for all x V, a , b F , ( a+b ) x ( a x )+( b x ) Then V is an abstract linear space . alar addition space member 2 scalar addition addition ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins Spri Spring 2010 ng 2010 Projecting Signals onto A Signal Space t r t t r t e t y t r t r t e t t r t t r 3 t x ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins Spri Spring 2010 ng 2010 The Gram-Schmidt Orthonormalization Procedure , , , set waveform a Start with 2 1 t s t s t s M le whi to 1 For t e M m m end rest he renumber t and waveform this discard t e t e t m m m end where i m i i mi m m s ,s t s t s t s t e 1 1 4 m m i mi ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins Spri Spring 2010 ng 2010 An Orthogonal Basis? 2 s third step e 1 s 1 21 2 s s 2 2 e fourth step 1 2 21 , s s second step 2 1 rst step 5 first step ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins Spri Spring 2010 ng 2010 Transformation Between ON Basis space. signal same the of basis ON two be ~ , , ~ , ~ , , , , Let 2 1 2 1 t t t t t t M M...
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This note was uploaded on 03/14/2011 for the course STAT 614 taught by Professor Marymeyer during the Spring '10 term at Colorado State.

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Lecture 22 - ECE 614: Principles of Digital Communications,...

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