Simulations of QAM Modulation over AWGN and Rayleigh Flat Fading
C.2 Theoretical Performances of QAM Modulation over Rayleigh Flat Fading Channel
C.2.1 Bit Error Probabilities from Symbol Error Probabilities un- un-der Rayleigh Flat Fading Channel
Traditionally, we can use Pb = logPs
2M to derive the bit error rate from already known symbol error rate. But after we scrutinize the records about the number of error bits in the error symbols, we find it is a slight distortion in this method. Table C.6 to Table C.10 record the simulation results and the theoretical values under the Rayleigh flat fading channel. Fig. C.5 shows the bit error rate probability of traditionally method and our simulations under the Rayleigh flat fading channel. Finally, we can get such a conclusion about the relationship between symbol error rate and bit error rate. With the higher of the Eb/N0, the inferential lines of bit error rate will get closer to the real simulations.
1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1
5 10 15 20 25 30 35 40 45 50 55 60 65
symbol-error-rate
Eb/No
simulation Qpsk simulation 16Qam simulation 64Qam simulation 256Qam simulation 1024Qam Qpsk 16Qam 64Qam 256Qam 1024Qam
Figure C.3: Symbol error probability versus Eb/N0 for QAM modulations under the Rayleigh flat fading channel.
1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1
5 10 15 20 25 30 35 40 45 50 55 60 65
symbol-error-rate
Eb/No
Qpsk Gaussian Qpsk Rayleigh 16Qam Gaussian 16Qam Rayleigh 64Qam Gaussian 64Qam Rayleigh 256Qam Gaussian 256Qam Rayleigh 1024Qam Gaussian 1024Qam Rayleigh
Figure C.4: Symbol error probability versus Eb/N0 for QAM modulations under the AWGN and Rayleigh flat fading channels.
1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1
5 10 15 20 25 30 35 40 45 50 55 60 65
bit-error-rate
Eb/No
simulation Qpsk simulation 16Qam simulation 64Qam simulation 256Qam simulation 1024Qam Qpsk Ps/2 16Qam Ps/4 64Qam Ps/6 256Qam Ps/8 1024Qam Ps/10
Figure C.5: Bit error probability versus Eb/N0 for QAM modulations under the Rayleigh flat fading channel.
Table C.4: Symbol error rates of theory and simulation are sistent, and the bit error rate of simulation result also con-forms to the distributions of the bit error numbers in the er-ror symbols. Formula Pb=(1*Perror1bit +2*Perror2bit +3*Perror3bit +4*Perror4bit +5*Perror5bit +6*Perror6bit +7*Perror7bit +8*Perror8bit )*Ps/8
256QAM under AWGN channel
T heory Simul. Simul. 1bit 2bit
Eb/N0 Ps Ps Pb Perror Perror
5 0.796408 0.796288 0.159249 0.524451866 0.363660629 10 0.525592 0.525096 0.078498 0.807792762 0.188496301 13 0.294254 0.293576 0.039895 0.912897653 0.087065499 16 0.0967384 0.096488 0.012367 0.974652612 0.025347388 17 0.0552126 0.054976 0.006969 0.985916955 0.014083045 18 0.0275839 0.027488 0.00346 0.992863877 0.007136123 19 0.0116892 0.011624 0.001457 0.996845038 0.003154962 20 0.00403837 0.004016 0.000502 0.999085616 0.000914384
21 0.00108161 0.00108 0.000135 1 0
22 0.000210678 0.000208 0.000026 1 0
23 2.75209E-05 0.00002744 0.00000343 1 0
3bit 4bit 5bit 6bit F ormula.
Perror Perror Perror Perror Pb
0.09990847 0.011483004 0.000489566 6.46566E-06 0.159249356
0.00369191 1.90269E-05 0 0 0.07849773
3.68475E-05 0 0 0 0.039894747
0 0 0 0 0.012366715
0 0 0 0 0.006968779
0 0 0 0 0.00346052
0 0 0 0 0.001457584
0 0 0 0 0.000502459
0 0 0 0 0.000135
0 0 0 0 0.000026
0 0 0 0 0.00000343
Table C.5: Symbol error rates of theory and simulation are sistent, and the bit error rate of simulation result also con-forms to the distributions of the bit error numbers in the er-ror symbols. Formula Pb=(1*Perror1bit +2*Perror2bit +3*Perror3bit +4*Perror4bit +5*Perror5bit +6*Perror6bit +7*Perror7bit +8*Perror8bit +9*Perror9bit +10*Perror10bit)*Ps/10
1024QAM under AWGN channel
T heory Simul. Simul. 1bit 2bit
Eb/N0 Ps Ps Pb Perror Perror
5 0.930807 0.93078 0.21471 0.2518 0.3526 10 0.814896 0.81463 0.1333 0.5022 0.3738 15 0.544463 0.54379 0.065679 0.7965 0.1991 17 0.389009 0.38832 0.043626 0.8768 0.1229 20 0.161122 0.16062 0.016765 0.9579 0.0421 23 0.0299324 0.02982 0.003004 0.9924 0.0076 25 0.00449938 0.00442 0.000443 0.9986 0.0014 26 0.00122714 0.00123 0.000123 0.9997 0.0003 27 0.000244515 0.00024 0.000024 1 0 28 3.28644E-05 0.000032 0.0000032 1 0
3bit 4bit 5bit 6bit 7bit 8bit F ormula.
Perror Perror Perror Perror Perror Perror Pb
0.2612 0.108 0.0239 0.0023 2.94E-05 9.0E-08 0.214710
0.1097 0.0136 0.00065 1.1E-05 0 0 0.1333
0.0043 2.43E-05 0 0 0 0 0.065679
0.00027 0 0 0 0 0 0.04363
5.29E-07 0 0 0 0 0 0.01674
0 0 0 0 0 0 0.003004
0 0 0 0 0 0 0.0004426
0 0 0 0 0 0 0.000123
0 0 0 0 0 0 0.000024
0 0 0 0 0 0 0.0000032
Table C.6: Symbol error rates of theory and simulation are consis-tent, and the bit error rate of simulation result also conforms to the distributions of the bit error numbers in the error symbols. Formula Pb=(1*Perror1bit +2*Perror2bit )*Ps/2
Qpsk under Rayleigh flat fading channel
T heory Simul. Simul. 1bit 2bit F ormula.
Eb/N0 Ps Ps Pb Perror Perror Pb
5 0.115273 0.115478 0.064282 0.88663 0.11336 0.064284675 6 0.0954077 0.095371 0.052986 0.88878 0.11121 0.052988928 7 0.0784119 0.078294 0.043447 0.89024 0.10975 0.043443673 8 0.0640557 0.063994 0.03543 0.88993 0.11006 0.035518684 10 0.0421319 0.042121 0.02326 0.89514 0.10485 0.023268879 13 0.0219135 0.021927 0.01207 0.89884 0.10115 0.012072564 15 0.0140236 0.014034 0.00772 0.89988 0.10011 0.007719519 18 0.00711514 0.007116 0.003917 0.89914 0.10085 0.003916836 20 0.00451 0.004498 0.002476 0.89830 0.10169 0.002477711 23 0.00226926 0.002243 0.001234 0.89935 0.10064 0.001234376 25 0.0014339 0.001433 0.000788 0.90022 0.09977 0.000787991 30 0.000454217 0.000451 0.000248 0.90251 0.09748 0.000247482 35 0.000143714 0.000145 0.00008 0.90138 0.09861 7.96496E-05 40 4.54541E-05 0.000047 0.000026 0.90298 0.09701 2.57797E-05 45 1.43746E-05 0.0000144 0.0000073 1 0 0.0000072 50 0.000004545 0.00000454 0.0000023 1 0 0.00000227
Table C.7: Symbol error rates of theory and simulation are consis-tent, and the bit error rate of simulation result also conforms to the distributions of the bit error numbers in the error symbols. Formula Pb=(1*Perror1bit +2*Perror2bit +3*Perror3bit +4*Perror4bit ) *Ps/4
16QAM under Rayleigh flat fading channel
T heory Simul. Simul. 1bit 2bit 3bit 4bit F ormula.
Eb/N0 Ps Ps Pb Perror Perror Perror Perror Pb
5 0.3138 0.3138 0.103 0.743 0.207 0.044 0.0067 0.103 10 0.1346 0.13456 0.042 0.785 0.176 0.034 0.0051 0.042 15 0.04810 0.0481 0.0149 0.801 0.164 0.030 0.0045 0.0149 20 0.015870 0.01587 0.00488 0.807 0.159 0.029 0.0043 0.00488 25 0.005088 0.00509 0.00156 0.808 0.159 0.029 0.0042 0.001563 30 0.001616 0.001615 0.000496 0.808 0.158 0.029 0.0045 0.0004963 35 0.000512 0.000514 0.000158 0.809 0.159 0.028 0.0041 0.000158 40 0.000162 0.000162 0.00005 0.809 0.156 0.029 0.0054 4.99E-05 45 5.12E-05 0.0000513 0.000015 0.808 0.157 0.031 0.0047 1.52E-05 50 1.62E-05 0.0000161 0.000005 0.810 0.157 0.028 0.0054 4.97E-06
Table C.8: Symbol error rates of theory and simulation are consis-tent, and the bit error rate of simulation result also conforms to the distributions of the bit error numbers in the error symbols. Formula Pb=(1*Perror1bit +2*Perror2bit +3*Perror3bit +4*Perror4bit +5*Perror5bit +6*Perror6bit )*Ps/6
64QAM under Rayleigh flat fading channel
T heory Simul. Simul. 1bit 2bit
Eb/N0 Ps Ps Pb Perror Perror
5 0.586 0.585 0.1534 0.5988 0.2729 10 0.327 0.3276 0.0767 0.7013 0.2183 15 0.1376 0.1378 0.0307 0.7485 0.1885 20 0.0487 0.04868 0.0106 0.7655 0.1777 25 0.01599 0.015968 0.0035 0.7713 0.1744 30 0.00512 0.005117 0.0011 0.7728 0.1727 35 0.00163 0.001626 0.00035 0.7729 0.1730 40 0.000515 0.000515 0.00011 0.7733 0.1729 45 0.000163 0.000161 0.000035 0.7761 0.1689 50 5.15E-05 0.0000515 0.000011 0.7738 0.1731 55 1.63E-05 0.0000163 0.0000037 0.7738 0.1739
3bit 4bit 5bit 6bit F ormula.
Perror Perror Perror Perror Pb
0.09227 0.0290 0.00617 0.000647 0.15337 0.05885 0.0173 0.00366 0.000385 0.0767 0.04621 0.0135 0.00279 0.000291 0.03066 0.04198 0.0120 0.00251 0.000211 0.01062 0.03996 0.0116 0.00230 0.000253 0.003479 0.04008 0.0116 0.00235 0.000264 0.001107 0.03978 0.0115 0.00242 0.000251 0.000351 0.03990 0.0112 0.00235 0.000214 0.0001112 0.04037 0.0119 0.00225 0.000341 3.478E-05 0.03892 0.0109 0.00295 0.000210 1.11E-05 0.03582 0.0142 0.00208 0 3.68E-06
Table C.9: Symbol error rates of theory and simulation are sistent, and the bit error rate of simulation result also con-forms to the distributions of the bit error numbers in the er-ror symbols. Formula Pb=(1*Perror1bit +2*Perror2bit +3*Perror3bit +4*Perror4bit +5*Perror5bit +6*Perror6bit +7*Perror7bit +8*Perror8bit )*Ps/8
256QAM under Rayleigh flat fading channel
T heory Simul. Simul. 1bit 2bit
Eb/N0 Ps Ps Pb Perror Perror
5 0.8147 0.8142 0.20457 0.40587 0.3224 10 0.6039 0.6038 0.124213 0.57388 0.2760 15 0.3357 0.335514 0.060935 0.68268 0.2230 20 0.1403 0.140212 0.02409 0.73135 0.1947 25 0.04948 0.049482 0.008322 0.74988 0.1833 30 0.0162 0.01617 0.00271 0.75316 0.1813 35 0.005197 0.005208 0.000868 0.75689 0.1798 40 0.0016499 0.00165 0.000275 0.75557 0.1802 45 0.000522 0.000521 0.000087 0.76042 0.1757 50 0.00016525 0.000163 0.000027 0.75598 0.1772 55 0.0000522 0.000052 0.00000864 0.76371 0.1697
3bit 4bit 5bit 6bit 7bit 8bit F ormula.
Perror Perror Perror Perror Perror Perror Pb
0.1678 0.07210 0.024518 0.006103 0.001036 9.577E-05 0.20456 0.0987 0.03668 0.011366 0.002756 0.000454 4.718E-05 0.12421 0.0635 0.02206 0.006755 0.001563 0.000265 2.197E-05 0.06093 0.0502 0.01709 0.005086 0.001254 0.000197 2.207E-05 0.02408 0.0454 0.01547 0.004635 0.001050 0.000159 1.298E-05 0.00832 0.0450 0.01474 0.004607 0.000998 0.000151 2.269E-05 0.00270 0.0430 0.01472 0.004203 0.001094 0.000150 1.765E-05 0.00086 0.0440 0.01433 0.004239 0.001132 0.000388 3.236E-05 0.000276 0.0440 0.01437 0.004132 0.001086 0.000191 0 8.661E-05 0.0456 0.01570 0.004591 0.000865 0 0 2.726E-05 0.0482 0.01324 0.004006 0.000890 0.000222 0 8.6397E-06
Table C.10: Symbol error rates of theory and simulation are consistent, and the bit error rate of simulation result also con-forms to the distributions of the bit error numbers in the er-ror symbols. Formula Pb=(1*Perror1bit +2*Perror2bit +3*Perror3bit +4*Perror4bit +5*Perror5bit +6*Perror6bit +7*Perror7bit +8*Perror8bit +9*Perror9bit +10*Perror10bit)*Ps/10
1024QAM under Rayleigh flat fading channel
T heory Simul. Simul. 1bit 2bit 3bit 4bit
Eb/N0 Ps Ps Pb Perror Perror Perror Perror
5 0.93409 0.93452 0.249104 0.21634 0.29135 0.24568 0.14653 10 0.830626 0.830518 0.174076 0.38368 0.31989 0.17357 0.07949 15 0.63 0.62 0.104901 0.55868 0.27850 0.10275 0.04025 20 0.347162 0.347162 0.051435 0.75743 0.19155 0.04178 0.00785 25 0.1457 0.14557 0.02031 0.72314 0.19776 0.05169 0.01874 30 0.05148 0.051408 0.00701 0.74331 0.18509 0.04710 0.01665 35 0.0169 0.01685 0.002285 0.74860 0.18076 0.04694 0.01645 40 0.00541 0.00544 0.00074 0.74693 0.18337 0.04618 0.01622 45 0.001718 0.001725 0.000234 0.74911 0.17974 0.04710 0.01488 50 0.000544 0.000546 0.000074 0.75246 0.17900 0.04723 0.01525 55 0.000173 0.000175 0.000024 0.75143 0.18122 0.04318 0.01567 60 0.0000544 0.000054 0.0000073 0.75136 0.18865 0.03816 0.01672
5bit 6bit 7bit 8bit 9bit 10bit F ormula.
Perror Perror Perror Perror Perror Perror Pb
0.06771 0.02408 0.00675 0.001305 0.000217 1.189E-05 0.24905 0.03040 0.00973 0.00258 0.000550 8.0586E-05 4.78E-06 0.17406 0.01413 0.00432 0.00108 0.000223 3.119E-05 2.5E-06 0.10485 0.00120 0.00015 1.59E-05 1.827E-06 1.073E-07 0 0.0518 0.00618 0.00184 0.000512 0.000104861 1.39E-05 0 0.02031 0.00567 0.00162 0.000445 8.416E-05 9.35E-06 0 0.00700 0.00522 0.00147 0.000457 4.428E-05 2.95E-05 0 0.00228
0.00566 0.00135 0.000218 4.361E-05 0 0 0.00073
0.00707 0.00171 0.000244 0.000122 0 0 0.00023
0.00438 0.00131 0.000302 5.035E-05 0 0 7.346E-05
0.00591 0.00175 0.000799 0 0 0 2.365E-05
0.00327 0.00109 0.000727 0 0 0 7.225E-06
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