Maximum Likelihood Sequence Detection · SPSC Maximum Likelihood Sequence Detection 25 The Viterbi...
Transcript of Maximum Likelihood Sequence Detection · SPSC Maximum Likelihood Sequence Detection 25 The Viterbi...
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Maximum Likelihood Sequence Detection 1SPSC
Maximum Likelihood Sequence Detection
ChannelML DetectionError Probability
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Maximum Likelihood Sequence Detection 2SPSC
Channel
Delay Spreadtime dispersionintersymbol interference (ISI).frequency selective fading
Channel Modelpassband PAMbaseband PAM
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Maximum Likelihood Sequence Detection 3SPSC
Channel Model
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Maximum Likelihood Sequence Detection 4SPSC
Discrete-Time Equivalent Channel Model for PAM
2 2 2( )j TE
mP e G j m B j m F j m
T T Tω π π πω ω ω
∞
=−∞
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦∑
( )( )2
202( )j TZ T
m
NS e F j mT
πω ω∞
=−∞
= +∑
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Maximum Likelihood Sequence Detection 5SPSC
Matched Filter as Receiver Front End (1)
matched filter as receive filterdiscrete-time equivalent channel model
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Maximum Likelihood Sequence Detection 6SPSC
Matched Filter as Receiver Front End (2)
autocorrelation of baseband receive pulse shape h(t)
*( ) ( ) ( )h k h t h t kT dtρ∞
−∞
= −∫
( )( )2
21( ) ( )j T j kTh h T
k m
S e k e H j mT
πω ωρ ω∞ ∞
=−∞ =−∞
= = +∑ ∑folded spectrum
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Maximum Likelihood Sequence Detection 7SPSC
Whitening of the Matched Filter (1)
matched filter colored noise
0( ) 2 ( )j T j TZ hS e N S eω ω=
spectral factorization2( ) ( ) (1/ )hS z M z M zγ ∗ ∗=
minimal phase and allpass system
min( ) ( ) ( )h apS z H z H z=
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Maximum Likelihood Sequence Detection 8SPSC
Whitening of the Matched Filter (2)
equalization with inverse minimal phase filter
noise process has white power spectrum
02( )NNS zK
=
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Maximum Likelihood Sequence Detection 9SPSC
Detection
ML Detection of a Single SymbolML Detection of a Signal VectorML Detection with IntersymbolInterferenceSequence Detection
Markov ChainsMarkov Chain Signal GeneratorThe Viterbi Algorithm
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Maximum Likelihood Sequence Detection 10SPSC
Detection
Estimation transmitted signal is contiuous-valuedDetection transmitted signal is discrete-valuedModel for detection:
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Maximum Likelihood Sequence Detection 11SPSC
ML Detection of a Single Symbol
Special case of MAP detector if
ML choosesTo maximize likelihoodMeasure of the quality
Aâε Ω| ( | )Y Ap y â
( )Ap â const=
||
( | ) ( )( | )
( )Y A A
AYY
p y â p âp â y
p y=
Pr[ ] Pr[ ]error â a= ≠
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Maximum Likelihood Sequence Detection 12SPSC
ML Detection of a Signal Vector
Vector of SymbolsMaximize Equivalent to maximize
Equivalent to minimizing
| |ˆ ˆ ˆ ˆ( | ) ( | ) ( )f f f= − = −Y S N S Ny s y s s y s
2/ 2 2
1 1ˆ ˆ( ) exp(2 ) 2M Mfπ σ σ
⎛ ⎞− = − −⎜ ⎟⎝ ⎠
N y s y s
ˆ−y s
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Maximum Likelihood Sequence Detection 13SPSC
ML Detection With IntersymbolInterference (1)
,0kh k M≤ ≤
A= +Y h N
Generator is LTI filterInput single data symbol AModel
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Maximum Likelihood Sequence Detection 14SPSC
ML Detection With IntersymbolInterference (2)
ML minimizes distance âh and observation y
Equivalent to maximizing
2 2 2 22 ,â â â− = − +y h y y h h
2 22 , â â−y h h
[ ] 0, *m m k k k
my h y h− =
= =∑y h
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Maximum Likelihood Sequence Detection 15SPSC
ML Detection With IntersymbolInterference (3)
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Maximum Likelihood Sequence Detection 16SPSC
ML Detection With IntersymbolInterference (4)
Exponential complexity Message of K M-ary symbolsMK matched filtersMK comparisons
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Maximum Likelihood Sequence Detection 17SPSC
Sequence Detection
Markov ChainsMarkov Chain Signal GeneratorThe Viterbi Algorithm
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Maximum Likelihood Sequence Detection 18SPSC
Markov Chains (1)
Independent of past samples
Homogenous if independent of kState transition diagram
( ) ( )1 1 1| , ,... |k k k k kp p+ − +Ψ Ψ Ψ = Ψ Ψ
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Maximum Likelihood Sequence Detection 19SPSC
Markov Chains (2)
Trellis diagram
NodeBranchPath
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Maximum Likelihood Sequence Detection 20SPSC
Markov Chain Signal Generator (1)
Sequence of homogenous Markov chain states
State transitions
Observation function
State of shift-register
kΨ
1( , )k k kS g ψ ψ +=
1 10
( , )M
k k i ki
g h Aψ ψ + −=
= ∑
[ ]1 2, ,...,k k k k MX X X− − −Ψ =
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Maximum Likelihood Sequence Detection 21SPSC
ISI Model
Shift-register process
Markov Chain Signal Generator (2)
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Maximum Likelihood Sequence Detection 22SPSC
Markov Chain Signal Generator Example
10.5k k kh δ δ −= +
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Maximum Likelihood Sequence Detection 23SPSC
The Viterbi Algorithm (1)
A. Viterbi of UCLA in 1967Homogenous Markov chainLinear complexity growing with message length KApplication for maximization problems
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Maximum Likelihood Sequence Detection 24SPSC
The Viterbi Algorithm (2)
Sequence of inputs = path through the trellis Assign Path metric = Σ branch metricsChoose lowest path metric =minimize
2k kbranch metric y s= −
ˆ−y s
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Maximum Likelihood Sequence Detection 25SPSC
The Viterbi Algorithm (3)
Survivor path of k-1 = smallest path metric to node k-1Only hold survivor pathFor node k choose smallest branch metric + survivor path of k+1
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Maximum Likelihood Sequence Detection 26SPSC
The Viterbi Algorithm Example (1)
Observation sequence 0.2, 0.6, 0.9, 0.1Impulse response of channel
AWGNState transitions
10.5k k kh δ δ −= +
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Maximum Likelihood Sequence Detection 27SPSC
The Viterbi Algorithm Example (2)
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Maximum Likelihood Sequence Detection 28SPSC
The Viterbi Algorithm Example (3)
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Maximum Likelihood Sequence Detection 29SPSC
Error Probability Calculation
Error EventDetection ErrorUpper Bound of Detection ErrorLower Bound of Detection ErrorSymbol Error Probability
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Maximum Likelihood Sequence Detection 30SPSC
Error Event
(a) length 1, metric from real sequence(b) length 2, metric from real sequence
1.25
3.5
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Maximum Likelihood Sequence Detection 31SPSC
Detection Error
w(e) … total number of detection errors in error event ePr[e] depends on real path and chosen path estimate
Pr[detection error] Pr[ ] ( )e E
e w eε
= ∑[ ] ˆPr[ ] Pr Pre ⎡ ⎤= Ψ Ψ Ψ⎣ ⎦
ˆPr ⎡ ⎤Ψ Ψ⎣ ⎦
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Maximum Likelihood Sequence Detection 32SPSC
Upper Bound of Detection Error (1)
Q … cumulative probability distributiond …Euclidian distance of real an chosen path
( )ˆ ˆPr | ( , ) / 2Q d σ⎡ ⎤Ψ Ψ ≤ Ψ Ψ⎣ ⎦
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Maximum Likelihood Sequence Detection 33SPSC
Upper Bound of Detection Error (2)
Only terms of minimal distanceOthers decay exponentially Approaches
( )minPr[detection error] ( ) Pr[ ] / 2 other termse E
w e Q dε
σ≤ Ψ +∑
min( / 2 )RQ d σ
( ) Pr[ ]e B
R w eε
= Ψ∑
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Maximum Likelihood Sequence Detection 34SPSC
Lower Bound of Detection Error (1)
Pr[detection error] Pr[ ] Pr[an error event]e E
eε
≥ =∑
minPr[an error event | ] ( ( ) / 2 )Q d σΨ ≥ Ψ
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Maximum Likelihood Sequence Detection 35SPSC
Lower Bound of Detection Error (2)
Using total probability
Only minimal distance error events
minPr[detection error] Pr[ ] ( ( ) / 2 )Q d σΨ
≥ Ψ Ψ∑
minPr[detection error] ( ( ) / 2 )PQ d σ≥ Ψ
Pr[ ]A
PεΨ
= Ψ∑
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Maximum Likelihood Sequence Detection 36SPSC
Symbol Error Probability (1)
Upper and lower bound together
Consider C between P and R
min min( / 2 ) Pr[detection error] ( ( ) / 2 )PQ d RQ dσ σ≤ ≤ Ψ
minPr[detection error] ( / 2 )CQ d σ≈
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Maximum Likelihood Sequence Detection 37SPSC
Symbol Error Probability (2)
One detection error, one ore more bit errorsOne input Xk by n source bits
1 Pr[detection error] Pr[bit error] Pr[detection error]n
≤ ≤
Pr[detection error] Pr[bit error]≈