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Program 2: c18 cdmahmm2.m in Software Display GTIN-13 in Software Program 2: c18 cdmahmm2.m

Program 2: c18 cdmahmm2.m generate, create gtin-13 none on software projects iPad In the second program, an er Software EAN-13 ror vector is generated by processing a sequence of symbols through the channel represented by the semi-Markov model generated by the preceding program. In addition, the probability of error is generated using three techniques. The rst bit error probability is generated by the original CDMA simulation.

The second bit error probability is the BER predicted by the semiMarkov model (raising A to a high power as described in 15). The third bit error probability is determined by counting the errors resulting from passing a large number of symbols (25,000 in this case) through the semi-Markov channel model. The MATLAB program follows: % File: c18_cdmahmm2.

m load cdmadata1 NN = 25000 [out] = c18_errvector(A_matrix,NN); % % load data from c18_cdmahmm1 % number of points to be used % generate error vector. i i i i i TranterBook 2003/11/ Software GS1-13 18 16:12 page 731 #749. Section 18.1. A Code-Division Multiple Access System % Compute and display three error probabilities. % pe2 = A_matrix^100; pe2 = pe2(1,3); pe3 = sum(out/NN); a = [ The predicted error probabilities for the CDMA system: ]; b = [ From the original simulation PE = ,num2str(BER), . ]; c = [ Predicted from the semi-Markov model PE = ,num2str(pe2), .

]; d = [ From the reconstructed error vector ,num2str(pe3), . ]; % disp(a) disp(b) % display PE from simulation disp(c) % display PE predicted from semi-Markov model disp(d) % display PE from reconstructed error vecor save cdmadata2 out % End of script file. The error vector based on the HMM is generated by the function c18 errvector, which is essentially identical to c15 errvector, which was originally discussed in 15.

The program c18 errvector is given in Appendix C. The only differences between c18 errvector and c15 errvector is that c18 errvector is a function rather than a script and that it generates the error vector for a speci c state transition matrix. Executing the second program, c18 cdmahmm2, provides the following results: >> c18_cdmahmm2 NN = 25000 The predicted error probabilities for the CDMA system: From the original simulation PE = 0.

03185. Predicted from the semi-Markov model PE = 0.032054.

From the reconstructed error vector 0.03204. We see that the three error probabilities agree closely.

It should also be noted that the error probabilities agree with the point given on Figure 18.5 for K = 1 (middle curve) and Eb /N0 = 5 dB..

Program 3: c18 cdmahmm3 The third program allows com parison of the original error sequence generated by the CDMA simulation and the error sequence generated by the semi-Markov model. We use two comparisons. The rst of these is to plot Pr {0m .

1} for both error sequences, EAN 13 for None as was done in 15. The second method of comparison is the histogram of the error-free runs for both sequences. The MATLAB code for accomplishing this follows:.

i i i i i TranterBook 2003/11/ Software EAN 13 18 16:12 page 732 #750. 732 % File: c18_cdmahmm3.m l oad cdmadata1 load cdmadata2 runcode2 = c15_seglength(out); c15_intervals2(runcode1,runcode2) % % Build histograms. % aa1 = runcode1(1,:); efd1 = aa1(1:2:length(aa1)); aa2 = runcode2(1,:); efd2 = aa2(1:2:length(aa2)); figure subplot(2,1,1) [N,x] = hist(efd1,20); %hist(efd1,x) bar(x,N,1) xlabel( Histogram Bin ) ylabel( Number of Samples ) subplot(2,1,2) hist(efd2,x); xlabel( Histogram Bin ) ylabel( Number of Samples ) % End of script file.

. Two Example Simulations 18 . % load data from c18_cdmahmm1 % load data from c18_cdmahmm2 % display intervals Executing the program yields the interval display illustrated in Figure 18.6, and the histograms of the error-free run lengths illustrated in Figure 18.7.

In both of these gures, the results from the CDMA system simulation are shown in the top frame, and the results from the semi-Markov model are shown in the bottom frame. Both gures show reasonably good agreement. In Figure 18.

7 note the di erence in the scaling for the bar heights. This di erence in scaling results because the CDMA simulation was performed for 100,000 symbols, and in testing the semi-Markov model, 25,000 symbols were used. Thus, the scale di erence is four.

There are several reasons for the di erences in the top and bottom frames in Figures 18.6 and 18.7.

First, the HMM must be an accurate representation of the original channel. This requires that the data sequence upon which the model is derived be su ciently long, and that a su cient number of iterations in deriving the model be performed, in order to ensure that convergence is reached. Also, an appropriate number of states must be used.

In addition, once the model is derived it must be tested by processing a su cient number of symbols in order to obtain statistically reliable results. These concerns are typically addressed by performing experiments such as those discussed here..

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