Random Signals for Engineers using MATLAB and Mathcad Copyright 1999 Springer Verlag NY Example 5.3 Poisson Distribution and Random Processes A random process can be generated that is a sum of unit steps that occur at times, t i, that are Poisson distributed. The Random Process S becomes n s (t ) u t t i i 1 This random process can be generated by recognizing that ti is the exponentially distributed inter event times and s(t) is just the index of each ti . N=20; lam=2; z=-1/lam*log(rand(20,1)); t(1)=0; for i=1:20 t(i+1)=t(i)+z(i); end i=1:20; stairs(t(1:20),i) 20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 The next process generated is the so-called Random telegrapher wave. It is similar to the random binary transmission process of Example 5.2 where sequences of +1, -1 are generated. However, the possible transitions are not periodic at time T as the binary waveform. For this Random process we will generate 3 sample processes from the ensemble. As above, we first generate three sets of event times by generating the exponential inter-event times and summing these to obtain the random event time sequences. The three exponential inter-event times are generated in the loop are z 1 log rand ( N ,1) lam The cumsum function is used to sum the variable for each i. The Telegraph waveform is formed using a vector set of alternating +1 and -1 sequence with each event. Finally this waveform can be plotted in a subgraph for each instance of the ensemble. Since the random number generator does not repeat each of the three plots are members of the ensemble of random processes generated. N=70; for j=1:3 subplot(3,1,j) z=-1/lam*log(rand(N,1)); zc=cumsum(z); y(1)=-1; for i=1:(N-1) if y(i)==-1 y(i+1)=1; else y(i+1)=-1; end end stairs(zc,y) axis([0 35 -1.5 1.5]) end 1 0 -1 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 1 0 -1 1 0 -1