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lecture07 - Lecture Notes 7 Lecture 7 Goals Signals as...

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Lecture Notes 7 Lecture 7: Goals Signals as Vectors, Noise as Vectors Optimum Detection in AWGN EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 1 / 93

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Lecture Notes 7 Composition of Signals A set of functions ϕ i ( t ) is said to be orthonormal if integraldisplay ϕ i ( t ) ϕ * j ( t ) dt = braceleftbigg 1 i = j 0 i negationslash = j . Given a set of orthogonal signals { ϕ i ( t ) , i = 0 , 1 , ..., N 1 } and a set of M vectors s m = ( s m , 0 , ..., s m , N - 1 ) , m = 0 , 1 , ..., M 1 we can construct a set of M signals as s m ( t ) = N - 1 summationdisplay i - 0 s m , n φ n ( t ) EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 2 / 93
Lecture Notes 7 Decomposition of Signals (Gram-Schmidt) Given a set of signals s 0 ( t ) , ..., s M - 1 ( t ) there exists a set of orthonormal signals ϕ 0 ( t ) , ϕ 1 ( t ) , ..., ϕ N - 1 ( t ) with N M such that s i ( t ) = N - 1 summationdisplay m = 0 s i , m ϕ m ( t ) . The coefficients are determined as s i , l = integraldisplay s i ( t ) ϕ * l ( t ) dt . EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 3 / 93

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Lecture Notes 7 Gram-Schmidt: Step 1, Step 2 u 0 ( t ) = s 0 ( t ) φ 0 ( t ) = u 0 ( t ) / || u 0 ( t ) || s 1 , 0 = ( s 1 ( t ) , φ 0 ( t )) u 1 ( t ) = s 1 ( t ) s 1 , 0 φ 0 ( t ) φ 1 ( t ) = u 1 ( t ) / || u 1 ( t ) || The waveform s 1 , 0 φ 0 ( t ) is the component of s 1 ( t ) in di- rection of φ 0 ( t ) -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 s 0 s 1 φ 0 ( t ) s 1 , 0 φ 0 ( t ) u 1 ( t ) φ 1 ( t ) EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 4 / 93
Lecture Notes 7 Gram-Schmidt: Step 3 s 2 , 1 = ( s 2 ( t ) , φ 1 ( t )) s 2 , 0 = ( s 2 ( t ) , φ 0 ( t )) u 2 ( t ) = s 2 ( t ) s 2 , 1 φ 1 ( t ) s 2 , 0 φ 2 ( t ) φ 2 ( t ) = u 2 ( t ) / || u 2 ( t ) || EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 5 / 93

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Lecture Notes 7 Properties 1 integraltext s i ( t ) s * j ( t ) dt = N - 1 l = 0 s i , l s * j , l 2 integraltext | s i ( t ) | 2 dt = N - 1 l = 0 | s i , l | 2 3 integraltext | s i ( t ) s j ( t ) | 2 dt = N - 1 l = 0 | s i , l s j , l | 2 EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 6 / 93
Lecture Notes 7 Proof of 1 integraldisplay s i ( t ) s * j ( t ) dt = integraldisplay N - 1 summationdisplay l = 0 s i , l ϕ l ( t ) N - 1 summationdisplay m = 0 s * j , m ϕ * m ( t ) dt = N - 1 summationdisplay l = 0 s i , l N - 1 summationdisplay m = 0 s * j , m integraldisplay ϕ l ( t ) ϕ * m ( t ) dt = N - 1 summationdisplay l = 0 s i , l s * j , l EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 7 / 93

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Lecture Notes 7 From a Vector to a Signal (Signal Composition) s i Serial to Parallel s i , 0 s i , 1 s i , N - 1 ϕ 0 ( t ) × ϕ 1 ( t ) × ϕ N - 1 ( t ) × + s i ( t ) EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 8 / 93
Lecture Notes 7 From a Signal to a Vector (Signal Decomposition) s i ( t ) ϕ * 0 ( t ) × ϕ * 1 ( t ) × ϕ * N - 1 ( t ) × integraltext s i ( t ) ϕ * 0 ( t ) dt integraltext s i ( t ) ϕ * 1 ( t ) dt integraltext s i ( t ) ϕ * N - 1 ( t ) dt s i , 0 s i , 1 s i , N - 1 EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 9 / 93

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Lecture Notes 7 Example 1 Consider the following set of four signals. T 2 T 3 T 4 T 5 T 6 T 7 T A s 0 ( t ) t T 2 T 3 T 4 T 5 T 6 T 7 T A s 1 ( t ) t T 2 T 3 T 4 T 5 T 6 T 7 T A s 2 ( t ) t T 2 T 3 T 4 T 5 T 6 T 7 T A s 3 ( t ) t EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 10 / 93
Lecture Notes 7 Example 1: Orthogonal Basis (not orthonormal) T 2 T 3 T 4 T 5 T 6 T 7 T u 0 ( t ) t -3/4 1/4 1/4 1/4 1/2 T 2 T 3 T 4 T 5 T 6 T 7 T u 1 ( t ) t T 2 T 3 T 4 T 5 T 6 T 7 T 1/7 -5/7 2/7 2/7 1/7 1 u 2 ( t ) t T 2 T 3 T 4 T 5 T 6 T 7 T 1/6 1/6 -4/6 2/6 1/6 1/6 1 u 3 ( t ) t EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 11 / 93

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Lecture Notes 7 Example 1: Orthonormal Basis φ 0 ( t ) , φ 1 ( t ) , φ 2 ( t ) , φ 3 ( t ) t T 2 T 3 T 4 T
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