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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 8PSK
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 16QAM
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 meansquare 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.
 Fall '08
 Stark

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