Maximum Likelihood Sequence Detection - spsc

Maximum Likelihood Sequence
Detection
• Channel
• ML Detection
• Error Probability
SPSC
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Channel
• Delay Spread
• time dispersion
• intersymbol interference (ISI).
• frequency selective fading
• Channel Model
• passband PAM
• baseband PAM
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Channel Model
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Discrete-Time Equivalent
Channel Model for PAM
∞
jωT
⎡ ⎛ 2π ⎞⎤ ⎡ ⎛ 2π ⎞⎤ ⎡ ⎛ 2π
⎞⎤
P( e ) = ∑ G⎢ j⎜ω− m⎟ BE
j⎜ω− m⎟ F j⎜ω−
m⎟
m=−∞
T
⎥ ⎢
T
⎥ ⎢
T
⎥
⎣ ⎝ ⎠⎦ ⎣ ⎝ ⎠⎦ ⎣ ⎝ ⎠⎦
2N
0
( j T
∞
ω
SZ
e ) = ∑ F j + m
T m=−∞
( ( 2 π
ω )) 2
T
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Matched Filter as
Receiver Front End (1)
• matched filter as receive filter
• discrete-time equivalent channel model
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Matched Filter as
Receiver Front End (2)
• autocorrelation of baseband receive
pulse shape h(t)
ρ
∞
*
h( ) = ∫ ( ) ( − )
−∞
k h t h t kT dt
• folded spectrum
∞
∞
jωT
jωkT
1
S ( e ) = ∑ ρ ( k)
e = ∑
T
H j + m
( ( 2 π
ω )) 2
h h T
k=−∞
m=−∞
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Whitening of the
Matched Filter (1)
• matched filter colored noise
jωT
S ( e ) = 2 N S ( e )
• spectral factorization
Z
2
0
h
jωT
Sh ( z ) = γ M ( z ) M (1/ z )
• minimal phase and allpass system
S ( z) = H ( z) H ( z)
h
min
∗
ap
∗
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Whitening of the
Matched Filter (2)
• equalization with inverse minimal phase filter
• noise process has white power spectrum
S ( ) N
z
=
2N
K
0
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Detection
• ML Detection of a Single Symbol
• ML Detection of a Signal Vector
• ML Detection with Intersymbol
Interference
• Sequence Detection
• Markov Chains
• Markov Chain Signal Generator
• The Viterbi Algorithm
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Detection
• Estimation transmitted signal is contiuousvalued
• Detection transmitted signal is discretevalued
• Model for detection:
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ML Detection of a Single Symbol
• Special case of MAP detector if
p ( ) A
â = const
p
AY |
( â| y)
=
p ( y| â) p ( â)
YA |
p
Y
( y)
A
• ML chooses
âε Ω A
• To maximize likelihood
• Measure of the quality
p
|
( y| â)
YA
Pr[ error] = Pr[ â ≠ a]
SPSC
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ML Detection of a Signal Vector
• Vector of Symbols
• Maximize f ˆ ˆ ˆ ˆ
YS |
( y | s) = fNS |
( y − s| s) = fN( y −s)
• Equivalent to maximize
1 1
f ( y ˆ
⎛
− s ) exp ˆ
N
=
M /2 M ⎜− −
2
(2 π) σ ⎝ 2σ
y s
2
⎞
⎟
⎠
• Equivalent to minimizing
y
− sˆ
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ML Detection With Intersymbol
Interference (1)
• Generator is LTI filter
• Input single data symbol A
• Model
h ,0≤
k ≤ M
k
Y= hA
+ N
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ML Detection With Intersymbol
Interference (2)
• ML minimizes distance âh and
observation y
2 2 2 2
y− hâ = y − 2 y,
h â+
h â
• Equivalent to maximizing
2 2
2 yh , â−
h â
[ ]
0
yh , = ∑ yh
m m
= yk * h − k k =
m
SPSC
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ML Detection With Intersymbol
Interference (3)
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ML Detection With Intersymbol
Interference (4)
• Exponential complexity
• Message of K M-ary symbols
M K matched filters
M K comparisons
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Sequence Detection
• Markov Chains
• Markov Chain Signal Generator
• The Viterbi Algorithm
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Markov Chains (1)
• Independent of past samples
p
Ψ | Ψ , Ψ ,... = p Ψ | Ψ
( ) ( )
k+ 1 k k− 1 k+
1 k
• Homogenous if independent of k
• State transition diagram
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Markov Chains (2)
• Trellis diagram
• Node
• Branch
• Path
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Markov Chain
Signal Generator (1)
• Sequence of homogenous Markov
chain states
• State transitions
Ψ k
• Observation function
• State of shift-register
Ψ =
S = g ψ ψ
k k k + 1
( , )
M
g( ψ , ψ +
) = ∑ hA −
[ X , X ,..., X ]
k k−1 k−2
k−M
k k 1 i k 1
i=
0
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• ISI Model
Markov Chain
Signal Generator (2)
• Shift-register process
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Markov Chain
Signal Generator Example
h = δ + 0.5δ k k k −
1
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The Viterbi Algorithm (1)
• A. Viterbi of UCLA in 1967
• Homogenous Markov chain
• Linear complexity growing with
message length K
• Application for maximization problems
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The Viterbi Algorithm (2)
• Sequence of inputs = path through the
trellis
• Assign
branch metric = y −s
• Path metric = Σ branch metrics
• Choose lowest path metric =
minimize y− sˆ
k
k
2
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The Viterbi Algorithm (3)
• Survivor path of k-1 =
smallest path metric to node k-1
• Only hold survivor path
• For node k choose smallest
branch metric + survivor path of k+1
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The Viterbi Algorithm
Example (1)
• Observation sequence
{0.2, 0.6, 0.9, 0.1}
• Impulse response of channel
• AWGN
• State transitions
h = δ + 0.5δ k k k −
1
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The Viterbi Algorithm
Example (2)
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The Viterbi Algorithm
Example (3)
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Error Probability Calculation
• Error Event
• Detection Error
• Upper Bound of Detection Error
• Lower Bound of Detection Error
• Symbol Error Probability
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Error Event
• (a) length 1,
metric from real sequence
• (b) length 2,
metric from real sequence
1.25
3.5
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Detection Error
Pr[detection error] = ∑ Pr[ ewe ] ( )
[ ] ˆ
Pr[ e] = Pr Ψ Pr ⎡Ψ Ψ⎤
⎣ ⎦
eε
E
• w(e) … total number of detection errors
in error event e
• Pr[e] depends on real path and chosen
path estimate ˆ ⎤
Pr ⎡
Ψ Ψ
⎣
⎦
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Upper Bound of
Detection Error (1)
( Ψˆ
Ψ σ )
Pr ⎡ ˆ | ⎤
⎣
Ψ Ψ
⎦
≤Q d( , ) / 2
• Q … cumulative probability distribution
• d …Euclidian distance of real an chosen
path
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Upper Bound of
Detection Error (2)
• Only terms of minimal distance
• Others decay exponentially
• Approaches
( σ )
≤∑
we Ψ Q dmin
+
eε
E
Pr[detection error] ( ) Pr[ ] / 2 other terms
RQ( dmin
/2 σ )
R
= ∑ w()Pr[ e Ψ]
eε
B
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Lower Bound of
Detection Error (1)
Pr[detection error] ≥ ∑ Pr[ e] = Pr[an error event]
eε
E
Pr[an error event | Ψ] ≥Qd
( ( Ψ)/2
σ )
min
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Lower Bound of
Detection Error (2)
• Using total probability
≥∑
Ψ Qd Ψ
min
Ψ
Pr[detection error] Pr[ ] ( ( ) / 2 σ )
• Only minimal distance error events
Pr[detection error] ≥ PQ( d ( Ψ) / 2 σ )
min
P
= ∑ Pr[ Ψ]
Ψε A
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Symbol Error Probability (1)
• Upper and lower bound together
PQ( d / 2 σ ) ≤ Pr[detection error] ≤ RQ( d ( Ψ) / 2 σ )
min
min
• Consider C between P and R
Pr[detection error] ≈ CQ( dmin
/ 2 σ )
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Symbol Error Probability (2)
• One detection error, one ore more bit
errors
• One input X k by n source bits
1 Pr[detection error] ≤ Pr[bit error] ≤ Pr[detection error]
n
Pr[detection error]
≈
Pr[bit error]
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