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