Of michigan 2 t e 2 2t sin2 fc t pt t cos2 fc t

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Unformatted text preview: ), φ1 (t ), ..., φN −1 (t ) with N ≤ M such that si (t ) = N −1 si ,n φn (t ), n =0 si ,n = si (t )φi (t )dt , vector → waveform (modulation) waveform → vector (demodulation.). Thus we can always map a set of M arbitrary signals into a set of M vectors of length N ≤ M and a set of N orthogonal waveforms. EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 118 / 130 Lecture Notes 5 Signal Space Concepts Example: s0 (t ) = ApT (t ) s1 (t ) = −ApT (t ) In this case we can take φ0 (t ) = 1 √ pT (t ). T Then √ s0 (t ) = A T φ0 (t ) √ = A2 T φ0 (t ) √ = E φ0 (t ) √ s1 (t ) = − E φ0 (t ) EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 119 / 130 Lecture Notes 5 Signal Space Concepts Example: s0 (t ) = s1 (t ) √ E = s2 (t ) s3 (t ) s4 (t ) s5 (t ) s6 (t ) s7 (t ) 2/T cos(2π fc t )pT (t ) E /2 2/T cos(2π fc t )pT (t ) + √ E 2/T sin(2π fc t )pT (t ) = E /2 2/T sin(2π fc t )pT (t ) =− E /2 2/T sin(2π fc t )pT (t ) E /2 2/T cos(2π fc t )pT (t ) − √ = − E 2/T sin(2π fc t )pT (t ) E /2 2/T sin(2π fc t )pT (t ) E /2 2/T cos(2π fc t )pT (t ) + √ = − E 2/T cos(2π fc t )pT (t ) =− = E /2 2/T cos(2π fc t )pT (t ) − In this case φ0 (t ) = EECS 455 (Univ. of Michigan) 2 T E /2 2/T sin(2π fc t )pT (t ) cos(2π fc t )pT (t ) and φ1 (t ) = Fall 2012 2 T sin(2π fc t )pT (t ). September 6, 2012 120 / 130 Lecture Notes 5 Signal Space Concepts Signals as Vectors With these orthogonal basis functions we can write the signals as √ s0 (t ) = E φ0 (t ) s1 (t ) = s2 (t ) = √ s3 (t ) = − E /2φ0 (t ) + E φ1 (t ) √ E /2φ0 (t ) + s4 (t ) = − E φ0 (t ) s5 (t ) = − √ E /2φ0 (t ) − s6 (t ) = − E φ1 (t ) s7 (t ) = EECS 455 (Univ. of Michigan) E /2φ1 (t ) E /2φ0 (t ) − Fall 2012 E /2φ1 (t ) E /2φ1 (t ) E /2φ1 (t ) September 6, 2012 121 / 130 Lecture Notes 5 Signal Space Concepts Signals as Vectors We can represent the signals as vectors as long as an orthogonal basis function is understood. √ s0 = ( E , 0) s1 = ( E /2, √ s2 = (0, E ) s3 = (− √ E /2, s4 = (− E , 0) E /2, − √ = (0, − E ) s5 = (− s6 s7 = ( E /2, − EECS 455 (Univ. of Michigan) E /2) Fall 2012 E /2) E /2) E /2) September 6, 2012 122 / 130 Lecture Notes 5 Signal Space Concepts 8-PSK φ1 (t ) s3 s2 √ E s0 √ φ0 (t ) E s4 √ −E s5 EECS 455 (Univ. of Michigan) s1 √ s7 −E s6 Fall 2012 September 6, 2012 123 / 130 Lecture Notes 5 Signal Space Concepts 16-QAM φ1 (t ) s0 s1 s2 s3 s4 s5 s6 s7 φ0 (t ) s8 s10 s11 s12 EECS 455 (Univ. of Michigan) s9 s13 s14 s15 Fall 2012 September 6, 2012 124 / 130 Lecture Notes 5 Signal Space Concepts Correlation and Distance Consider the representation of signals in terms of orthonormal waveforms. si (t ) = N −1 si ,l φl (t ) l =0 The correlation of two signals is si (t )sj∗ (t )dt = N −1 si ,l φl (t ) l =0 = N −1 N −1 N −1 sj∗,m φ∗ (t )dt m m =0 si ,l sj∗,m φl (t )φ∗ (t )dt m l =0 m =0 = N −1 si ,l sj∗,l = (si , sj ) l =0 EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 125 / 130 Lecture Notes 5 Signal Space Concepts Correlation and Distance If i = j then |si (t )|2 dt = EECS 455 (Univ. of Michigan) Fall 2012 N −1 l =0 |si ,l |2 . September 6, 2012 126 / 130 Lecture Notes 5 Signal Space Concepts Correlation and Distance The distance between two signals is 2 dE (si , sj ) = = = |si (t ) − sj (t )|2 dt |si (t )|2 − si (t )sj∗ (t ) − si∗ (t )sj (t ) + |sj (t )|2 dt N −1 l =0 = N −1 l =0 = N 1 l =0 EECS 455 (Univ. of Michigan) |si ,l |2 − N −1 l =0 si ,l sj∗,l − N −1 si∗,l sj ,l + N −1 l =0 l =0 |sj ,l |2 |si ,l |2 − si ,l sj∗,l − si∗,l sj ,l + |sj ,l |2 |si ,l − sj ,l |2 = ||si − sj ||2 Fall 2012 September 6, 2012 127 / 130 Lecture Notes 5 Signal Space Concepts Noise as Vectors Let n(t ) be a (real) white noise process with power spectral density N0 /2. Let φ0 (t ), φ1 (t ), ... be a complete orthonormal set of functions on the interval [0, T ] (e.g. Fourier series). Then the random process can be represented in terms of a sequence of random variables and the orthonormal functions. n(t ) = ∞ ni φi (t ). n =0 The representation is accurate in the mean-square sense. That is N lim E [(n(t ) − n→∞ EECS 455 (Univ. of Michigan) ni φi (t ))2 ] = 0. i =0 Fall 2012 September 6, 2012 128 / 130 Lecture Notes 5 Signal Space Concepts Noise as Vectors The random variables can be determined as ni = n(t )φi (t )dt . Note that E [ni ] = 0 and Var[ni ] = E [|ni |2 ] = E[ n(t )φi (t )dt n∗ (s)φ∗ (s)ds] i = E [n(t )n∗ (s)]φi (t )φ∗ (s)dtds i = N0 δ (t − s)φi (t )φ∗ (s)dtds i 2 N0 φi (t )φ∗ (t )dt i 2 = = EECS 455 (Univ. of Michigan) N0 2 Fall 2012 September 6, 2012 129 / 130 Lecture Notes 5 Signal Space Concepts Noise as Vectors Using a similar technique it can be shown that the random variables ni and nj are uncorrelated. Since they are also Gaussian they are also independent. EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 130 / 130...
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This note was uploaded on 02/12/2014 for the course EECS 455 taught by Professor Stark during the Fall '08 term at University of Michigan.

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