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i i i generate, create ean13 none for software projects pdf417 2d barcode i i TranterBook 2003 ean13+2 for None /11/18 16:12 page 615 #633. Section 15.6. Two Examples Irrespective of the algo rithms used, the Markov model for a discrete channel must be computed for di erent values of the parameters of the underlying physical channel. If the underlying physical channel is changed in any way, the discrete channel model must be reestimated. For example, if a discrete channel model is developed for a given channel, then the Markov model must be developed from measurements or waveform-level simulations for di erent values of the parameters of the channel including S/N.

Thus a parametrized set of Markov models for the underlying channel can be developed and used for the design and analysis of error control coders, interleavers, etc.. Two Examples We conclude this chapter with two examples that demonstrate the determination of a Markov channel and the determination of a semi-Markov model. The ow of these examples is as follows: 1. In the rst example (Example 15.

4), the error vector generated in Example 15.3 is used as input to the Baum-Welch algorithm. The result is an estimated channel model.

The error probability and the run-length distribution resulting from this model are determined and are then compared to the error probability and run-length distribution of the original sequence generated in the rst example. 2. The second example (Example 15.

5) is similar to the rst example. The essential di erence is that a block equivalent semi-Markov model is used rather than a Markov model. The motivation for using a semi-Markov model is that the semi-Markov model leads to a substantial reduction in the time required to derive the model from measured or simulated data.

Example 15.4. In this example we generate an error sequence of length N = 20,000 with a known two-state model de ned by A= and B= 0.

95 0.50 0.05 0.

50 (15.84) 0.95 0.

05 0.10 0.90 (15.

83). The Baum-Welch algorithm ean13+5 for None is then used to estimate the model using the generated error sequence. Note that we assume a three-state model when the Baum-Welch model is applied. This may at rst seem strange but, given simulation or measurement data, the channel model that produced the data is not known.

The MATLAB dialog, with some of the nonessential dialog eliminated, is as follows:. i i i i i TranterBook 2003 Software EAN13 /11/18 16:12 page 616 #634. Discrete Channel Models 15 . c15 errvector Accept def ault values Enter y for yes or n for no > n Enter N, the number of points to be generated > 20000 Enter A, the state transition matrix > [0.95 0.05; 0.

1 0.9] Enter B, the error distribution matrx > [0.95 0.

5; 0.05 0.5] out1 = out; c15 bwa(20,3,out) Enter the initial state transition matrix P > [0.

9 0.05 0.05; 0.

1 0.8 0.1; 0.

1 0.2 0.7] Enter the initial state probability vector pye > [0.

3 0.3 0.4] Enter the initial output symbol probability matrix.

.. B > [0.

9 0.8.7; 0.

1 0.2 0.3] After one iteration we have4 0.

9051 0.0469 0.0481 A1 = 0.

0991 0.7931 0.1078 0.

0895 0.1856 0.7249.

B1 =. 0.9206 0.7462 0.

5848 0.0 794 0.2538 0.

4152. Also, after 10 iteration GTIN-13 for None s, the estimated state transition matrix is 0.9415 0.0300 0.

0285 = 0.1157 0.7504 0.

1340 0.0669 0.1382 0.

7949 0.9547 0.6745 0.

4257 0.0453 0.3255 0.

5743. B10 =. At the 20th iteration, w GS1-13 for None hich is the termination point, the estimated state transition matrix is 0.9437 0.0299 0.

0263 0.9522 0.6637 0.

4313 B20 = A20 = 0.1173 0.7425 0.

1402 0.0478 0.3363 0.

5687 0.0663 0.1324 0.

8013 The log likelihood function is illustrated in Figure 15.13. It can be seen that the log likelihood function converges in about 10 iterations.

Note that this conclusion is consistent with the preceding computations of Ak and Bk for k = 10 and k = 20. Note also that, as discussed previously, the likelihood numbers are very small. We now generate a second sequence, out2, using the model estimated by the Baum-Welch algorithm.

This gives, with the dialog dealing with default values suppressed:. 4 It should be mentioned that rerunning this program may result in estimates for A and B that are di erent from the results given here, even though the same parameters and initial conditions are used. The reason for this lies in the fact that the error vector, upon which the estimates of b A and B are based, is a sample function of a random process. In addition, the rows of A and b b b the columns of B may not sum to one when the elements of A and B are represented with nite precision.

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