he time and condition hold at time = n-1. An input k therefore source channel matrix determines a stationary sequence of input- corresponds to the placing of a ball labelled k in the unoccupied slot, output pairs (X Y ), for n = 1,2, An input, output and joint and the opening of the trapdoor then determines the corresponding output and the next state. If at time t = n the channel is in state 10' and an input 1 is applied, then one ball labelled 0 and one ball labelled 1 rest over the trapdoors. With a probability of 1/2 the ball 1 falls through, leaving the ball () in the occupied slot. The channel then moves to state input = 0 and emits an output of 1. With probability 1/2 the 0 ball falls through, sending the channel into the state $01 and an output 0 is emitted. The behaviour of this channel may thus be described by two matrices, M and M called the channel matrices, and whose components are the state transition probabilities under the input 0 and 1 respectively, and a function that associates an output with each input-state pair, Figure 4.5. Another model that is successful in characterising error bursts and impulsive noise in channels is the SO called Gilbert model, where the channel is modelled as a discrete memoryless binary symmetric channel for which the probability of error is a time varying par- input = 1 ameter. The changes in probability of error are modelled as a Markov process as shown in Figure 4.6. The error generating mechanism in the channel occupies one of the three states. When the channel is in state 2, the probability of error during a bit interval is 10-2 and the channel stays in this state during the succeeding bit interval with a probability of .998. The channel may make a transi- tion to state 1, which has a bit error probability of 0.5, and since the Figure 4.5 Channel matrices for the trapdoor channel Continuous channels 4/11 0.9999998 0.998 5.10-2 5.10-3 State Probability of error Figure 4.6 A three state Gilbert model uncertainty may therefore be defined respectively as in Equations 