markovian error models based on jeffrey s. slack finite state markov models for error bursts on the...
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Markovian Error Models
Based onJeffrey S. Slack
FINITE STATE MARKOV MODELS FOR ERROR BURSTS ON
THE LAND MOBILE SATELLITE CHANNEL
Bernoulli model
• Independent bit errors• Bit Error Rate (BER)=P• Frame Error Rate (FER)=F• N: bits per frame
F=1-(1-P)n ¼ n P
• Example P=1E-9, n=1500*8, F=1E-5• Example P=1E-3, n=500, F=0,4
Signal strength and bursty errors
Gilbert Model
No errors in state (good) 1
Bernoulli model in state (bad) 2
(BER=1-h)
Determining Model Parameters
• Match average BER
• Match Error Gap Distribution
U(n)=P(00..0) (at least n good bits in row)
• Match Block Error Probability
P(m,n)=probability of m errors in block of n bits
Mapping Transition Probabilities to u(n) and P(m,n)
P11,P12,P21,P22 ! u(n),P(m,n)
P*11,P*12,P*21,P*22 Ã u*(n),P*(m,n)
Matching Error Gaps
Matching Block Error Probabilities
Elliot Model
Bernoulli model in state 1(BER=1-k)Bernoulli model in state 2(BER=1-h)
BEP for the Elliot ModelAssumed: 1-h >> 1-k
h and transition probabilities determined as for the Gilbert model
K determined from BEP
Matching Error Gaps
Matching BEP
The McCullough model
Random error state
Bursty error state
State change allways on error
BEP for the McCullough model
Estimation
Results
Best k-value
The Fritchman model
Transition between error free state prohibited
(for tractability)
Error Gap Probabilities
PBA PB
• B-states are now attractive
• Probabilities for staying in A-states are the same for the two transition matrices
PAB
PAB=0
PB=I
PA
Error Gap Probabilities
Error Gap/Cluster Probabilities
Measured Error Cluster Probabilities
Straight line -> geometric -> only one dominating eigenvalue -> only one errorstate
Block Error Probabilities
Estimation
Matching Error Gap
Matching Block Errors