**Project Choice 1: Convolutional Encoder and Viterbi Decoder with Finite Traceback**

A popular convolutional encoder is the 64-state, rate-1 /2 encoder described by the following generator matrix:

G(D) = [D^{6} + D^{5} + D^{3} +D^{2} +1 D^{6} +D^{3} +D^{2} + D +1].

1. Determine the Hamming d_{free}, L_{D} 4, the number of neighbors N , and N_{b} for this code.

2. Using the built-in functions in Matlab Communications toolbox, write a program to convert the generator matrix to trellis (poly2trellis), encode the random binary information bits (convene), add bit errors to the channel, and finally hard decode the noisy sequence (i.e. Hamming distance) using the continuous Viterbi algorithm (vitdec) with various tracebeck depths as discussed below.

Your code should allow you to compute the simulated bit error rate by comparing the random bits transmitted to the received bits. The interesting region to simulate should for probabilities of error between 10^{-1} and 10^{-2 }when using a traceback depth equal to L_{D}. Simulate the code with traceback depths of L_{D}/2, L_{D}, 2L_{D}, and 5 times the constraint length (35 in this case), where the constraint length is the impulse response length of the feedforward implementation of the code (7 in this case). Note that a common rule of thumb is to use a traceback depth equal to 5 times the constraint length.

Simulate at least the following probabilities of channel bit error:

10^{-0.75}, 10^{-1}, 10^{-1.25}, 10^{-1.5}, 10^{-1.75}, 10^{-2}. You can check other channel bit error rates if you have time. Run each simulation until you have recorded at least 100 bit errors. The more bit errors you collect, the smother your curves will be. Turn in your source code and your results.

3. Evaluate and plot the Van de Meeberg transfer function bound (slide 45 of Lecture 10) on decoded information bit error rate for this case for numerous channel bit error probabilities including the six points mentioned above. Turn in your source code and your results. Be sure to plot enough points to get a good characterization of the pole.

4. Prepare a plot showing all of your results in one figure. Plot BER (1og10 scale) vs, channel bit error probability (also log 10 scale). Also write a two-page report discussing your project. This brief report will give you some technical writing experience, which is something you need as part of your graduate training. Write the two pages as if you were explaining to your manager at a company how you completed the project include the key results, discuss any key plots (which you should include in the body of the report, but they don't count in computing the two pages) as well as a discussion of what parts were most interesting (and why) or what parts were most difficult (and why).

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