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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: SquareRoot 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: SquareRoot 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.
 Fall '08
 Stark

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