Table 14.8

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CHANNEL CODING TECHNIQUES

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Listing of Reed Solomon (7, 3) encoding and decoding % % % % % % % % % % % Information Number of bits per symbol Word lengths for code Number of information symbols Galois array Specify RS generation polynomial Encode the information symbols Transmit and receive Error eraser Error eraser Decoding

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msg1= [3 5 0]; m = 3; n = 2^m-1; k= 3; msg=gf([msg1],m) gen=rsgenpoly(n,k); code = rsenc(msg,n,k,gen) rxcode = code; rxcode(1)= 2; rxcode(7) = 2; decod = rsdec(code,n,k,gen)

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is generated. The received rxcode contains two errors. The decoded symbols are stored in decod. All these arrays are listed in Table 14.7. Procedures of the experiment are listed as follows: 1. Start MATLAB and change the working directory to ..\experiments\exp14.4.1_RS_codec. 2. Run the experiment by typing rs_codec in the MATLAB command window. 3. Modify the error locations and the number of errors in a frame to check decoder output decod. The maximum correctable errors in a frame are two symbols. 4. Modify the code to use RS(15, 11). Repeat the same tests. 5. Write oating-point C implementation of the RS(7, 3) and compare the result with the MATLAB output. 6. Convert the oating-point C functions to C55x assembly functions, and compare the experiment results against the MATLAB results.

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14.4.2 Reed Solomon Coding Using Simulink

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Using the same generation polynomial given in the previous experiment and the same RS(7, 3) coding, we can use a Simulink model to simulate a wireless system. As shown in Figure 14.14, we have signal source Bernoulli binary generator, RS(7, 3) encoder and decoder modules, modulation and demodulation modules, parallel to serial and serial to parallel conversions, AWGN channel, and bit-error monitors. The parameters such as codeword length, bit-error rate (BER), n and k of RS(n, k) can be con gured. For example, double click on the AWGN module to change signal-to-noise ratio (SNR) which will affect the BER. Procedures of the experiment are listed as follows: 1. Start MATLAB and change the working directory to ..\experiments\exp 14.4.2_RS_ simulink. 2. Run the experiment by typing Reed_Solomon in the MATLAB command window. This starts the Simulink and creates a window as shown in Figure 14.14.

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EXPERIMENTS AND PROGRAM EXAMPLES

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Simulink model for Reed Solomon code

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3. Modify SNR of WGN channel, and then check the BER. 4. Modify the code to use RS(15, 11) encoder and decoder. Repeat the same tests. 5. Compare the BER before and after correction. As shown in Figure 14.14, there are 4678 and 14 bit errors corresponding to BER of 0.005948 and 0.00004153, respectively, before and after correction. Note that the bit rate before correction is 7/3 times higher than that of after correction.

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14.4.3 Veri cation of RS(255, 239) Generation Polynomial

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As discussed in Example 14.2, we used the generation polynomial {g0 , g1 , . . . , g16 } = {0x4f, 0x2c, 0x51, 0x64, 0x31, 0xb7, 0x38, 0x11, 0xe8, 0xbb, 0x7e, 0x68, 0x1f, 0x67, 0x34, 0x76, 0x01}, stored in gpoly_alpha, and the primitive polynomial p(x) = 1 + x 2 + x 3 + x 4 + x 8 . All i , i = 1, . . . , 16, are the roots of this generation polynomial de ned in Equation (14.5). With the given primitive polynomial, we can compose a table for GF(256). The elements are populated as follows:

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0 7 = 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80 8 = 4 + 3 + 2 + 1 = 0x1d 9 = 8 1 = ( 4 + 3 + 2 + 1 ) 1 = 5 + 4 + 3 + 1 = 0x3a

Using the same method, all data for i , i = 0, . . . , 255 over GF(256) can be calculated and this data is generated in the table alpha_tabl[ ]. Using a similar method, we can also convert the generation polynomial to index-based table as int gpoly_index[17] = {0x88, 0xf0, 0xd0, 0xc3, 0xb5, 0x9e, 0xc9, 0x64,0x0b, 0x53, 0xa7, 0x6b, 0x71, 0x6e, 0x6a, 0x79, 0x00}. With these tables, we can verify that i , i = 1, . . . , 16, are all the roots of g(x). Table 14.9 lists the C code to verify if g( i ) = 0 for i = 1, . . . , 16. The les used for this experiment are listed in Table 14.10.

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