Of michigan 16 16 46 26 16 16 t t fall 2012 1 2t

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Unformatted text preview: ) 1/7 -5/7 2/7 2/7 1/7 T T 1 2T 3T 4T 5T 6T 7T EECS 455 (Univ. of Michigan) 1/6 1/6 -4/6 2/6 1/6 1/6 t T Fall 2012 1 2T 3T 4T 5T 6T 7T t October 3, 2012 11 / 93 Lecture Notes 7 Example 1: Orthonormal Basis φ0 (t ), φ1 (t ), φ2 (t ), φ3 (t ) T 2T 3T 4T 5T 6T 7T t s0 (t ) = 2.00φ0 (t ) + 0.00φ1 (t ) + 0.00φ2 (t ) + 0.00φ3 (t ) s1 (t ) = 1.50φ0 (t ) + 1.32φ1 (t ) + 0.00φ2 (t ) + 0.00φ3 (t ) s2 (t ) = 1.00φ0 (t ) + 1.13φ1 (t ) + 1.31φ2 (t ) + 0.00φ3 (t ) s3 (t ) = 0.50φ0 (t ) + 0.94φ1 (t ) + 1.09φ2 (t ) + 1.29φ3 (t ) EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 12 / 93 Lecture Notes 7 Correlation vs. Filtering Consider the computation of r (t )φ∗ (t )dt . i A filter with input r (t ) and impulse response h(t ) = φ∗ (T − t ) i sampled at time T has output = h(T − t )r (t )dt = = ri φ∗ (T − (T − t ))r (t )dt i φ∗ (t )r (t )dt i So either a correlator whereby the received signal is correlated with the orthonormal signal can be used to obtain ri OR a matched filter with impulse reponse h(t ) = φ∗ (T − t ) which is i sampled at time t = T can be used to obtain ri . EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 13 / 93 Lecture Notes 7 Correlation vs. Filtering t=T r (t ) ri h(t ) = φ∗ (T − t ) i t=T r (t ) h(t ) = pT (T − t ) ri φi (t ) EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 14 / 93 Lecture Notes 7 Example 1: Time orthogonal φ0 (t ) = φ1 (t ) = φ2 (t ) = 1 p (t ) TT 1 p (t − T ) TT 1 p (t − 2T ) TT φ0 (t ), φ1 (t ), φ2 (t ) T EECS 455 (Univ. of Michigan) 2T Fall 2012 3T t October 3, 2012 15 / 93 Lecture Notes 7 Example 2: Phase Orthogonal 2 cos(2π fc t )pT (t ) T 2 φ1 (t ) = − sin(2π fc t )pT (t ) T φ0 (t ) = φ0 (t ), φ1 (t ) T EECS 455 (Univ. of Michigan) Fall 2012 t October 3, 2012 16 / 93 Lecture Notes 7 Example 3: Square-Root Raised Cosine Pulses Let x (t ) = sin(π (1 − α)t /T ) + 4αt /T cos(π (1 + α)t /T ) . π [1 − (4αt /T )2 ]t /T φ0 (t ) = x (t ) φ1 (t ) = x (t − T ) φ2 (t ) = x (t − 2T ) φ3 (t ) = x (t − 3T ) EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 17 / 93 Lecture Notes 7 Example 3: Square-Root Raised Cosine Pulses φ0 (t ),φ1 (t ),φ2 (t ),φ3 (t ) T EECS 455 (Univ. of Michigan) t 2T 3T 4T Fall 2012 October 3, 2012 18 / 93 Lecture Notes 7 Example 4: Time, Phase Orthogonal 2 cos(2π fc t )pT (t ) T 2 φ1 (t ) = − sin(2π fc t )pT (t ) T 2 φ2 (t ) = cos(2π fc t )pT (t − T ) T 2 φ3 (t ) = − sin(2π fc t )pT (t − T ) T φ0 (t ) = EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 19 / 93 Lecture Notes 7 Example 4: Time, Phase Orthogonal φ0 (t ),φ1 (t ),φ2 (t ),φ3 (t ) T EECS 455 (Univ. of Michigan) Fall 2012 2T October 3, 2012 t 20 / 93 Lecture Notes 7 Example 5: Frequency Orthogonal φ0 (t ) = φ1 (t ) = φ2 (t ) = (fi − fj ) = 2 cos(2π f0 t )pT (t ) T 2 cos(2π f1 t )pT (t ) T 2 cos(2π f2 t )pT (t ) T n 2T φ0 (t ),φ1 (t ),φ2 (t ) T EECS 455 (Univ. of Michigan) Fall 2012 t October 3, 2012 21 / 93 Lecture Notes 7 Example 5: Frequency Orthogonal φ0 (t ) T t T t T t φ1 (t ) φ2 (t ) EECS 455 (Univ. of Michigan) Fall 2012 October 3, 2012 22 / 93 Lecture Notes 7 Example 5: Frequency Orthogonal √ √ √ s0 = (+ E , + E , + E ) √ √ √ s1 = (+...
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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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