Concluding Remarks

Declaration 2 Publications

6. Conclusion

6.3 Concluding Remarks

In this dissertation, a trellis code-aided bandwidth efficiency improvement technique that improves the bandwidth efficiency of space-time block coded wireless communication systems without compromising the link reliability (error performance) has been presented. The presented technique has been investigated for differentially transmitted Alamouti STBC in the form of TC- DSTBC, and for USTLD in the form of E-USTLD. The significance of the trellis code-aided bandwidth efficiency improvement technique for space-time block coded systems is that it addresses the high speed data demands of modern digital communication systems. However, the bandwidth efficiency improvement has been achieved at the cost of high detection complexity, hence, an OP based LC near-optimal detection scheme for E-USTLD has been presented.

Moreover, the work presented in this dissertation has opened other avenues for future research.

53

Appendix A

The E-USTLD signals received prior to each decoding segment 𝑖 are expressed as:

𝒚_1,𝑖= √^𝜌₂ 𝒉_1,𝑖¹ 𝑥_𝑞¹_𝑖+ √^𝜌₂ 𝒉_1,𝑖² 𝑥_𝑟²_𝑖e^{j𝜃𝑘𝑖}+ 𝜼_1,𝑖 (A.1)

𝒚_2,𝑖 = √^𝜌₂ 𝒉¹_2,𝑖𝑥̅_𝑟²_𝑖+ √^𝜌₂ 𝒉_2,𝑖² 𝑥̅_𝑞¹_𝑖e^{j𝜃𝑘𝑖}+ 𝜼_2,𝑖 (A.2) Assuming that all symbol pairs are detected correctly, while the encoding trellis path is detected with errors, the PEP conditioned on 𝑯₁ and 𝑯₂ can be expressed as:

𝑃(𝑡_𝑘→𝑔→ 𝑡̂_𝑘→𝑔|𝑯₁𝑯₂) = 𝑃 (∑ {‖𝒚_1,𝑖− √^𝜌₂ 𝒉_1,𝑖¹ 𝑥_𝑞¹_𝑖− √^𝜌₂ 𝒉_2,𝑖¹ 𝑥_𝑟²_𝑖e^{j𝜃𝑘̂𝑖}‖

𝐹 2 𝑧 +

𝑖=1

‖𝒚_2,𝑖− √^𝜌₂ 𝒉¹_2,𝑖𝑥̅_𝑟²_𝑖− √^𝜌₂ 𝒉_2,𝑖² 𝑥̅_𝑞¹_𝑖e^{j𝜃𝑘̂𝑖} ‖

𝐹 2

} < ∑ {‖𝒚_1,𝑖− √^𝜌₂ 𝒉_1,𝑖¹ 𝑥_𝑞¹_𝑖− √^𝜌₂ 𝒉_2,𝑖¹ 𝑥²_𝑟𝑖e^{j𝜃𝑘𝑖}‖

𝐹 2 𝑧 +

𝑖=1

‖𝒚_2,𝑖− √^𝜌₂ 𝒉_2,𝑖¹ 𝑥̅_𝑟²_𝑖− √^𝜌₂ 𝒉_2,𝑖² 𝑥̅_𝑞¹_𝑖e^{j𝜃𝑘𝑖} ‖

𝐹 2

}) (A.3)

where 𝑧 is the length of the shortest error event path. Substituting (A.1) and (A.2) into (A.3) yields:

𝑃(𝑡_𝑘→𝑔 → 𝑡̂_𝑘→𝑔|𝑯₁𝑯₂) = 𝑃 (∑ {‖√^𝜌₂ 𝒉¹_2,𝑖𝑥_𝑟²_𝑖(e^{j𝜃𝑘𝑖}− e^{j𝜃𝑘̂𝑖}) + 𝜼_1,𝑖‖

𝐹 2 𝑧 +

𝑖=1

‖√^𝜌₂ 𝒉_2,𝑖² 𝑥̅_𝑞¹_𝑖(e^{j𝜃𝑘𝑖}− e^{j𝜃𝑘̂𝑖}) + 𝜼_2,𝑖‖

𝐹 2

} < ∑^𝑧_𝑖=1{‖𝜼_1,𝑖‖_𝐹²+ ‖𝜼_2,𝑖‖_𝐹²}) (A.4)

Let 𝑨_𝑖 = √^𝜌₂ 𝒉_2,𝑖¹ 𝑥_𝑟²_𝑖(e^{j𝜃𝑘𝑖}− e^{j𝜃𝑘̂𝑖}) and 𝑩_𝑖 = √^𝜌₂ 𝒉_2,𝑖² 𝑥̅_𝑞¹_𝑖(e^{j𝜃𝑘𝑖}− e^{j𝜃𝑘̂𝑖}). Applying the triangle inequality as in Naidoo et al [28] yields:

𝑃(𝑡_𝑘→𝑔→ 𝑡̂_𝑘→𝑔|𝑯₁𝑯₂)

= 𝑃 (∑ {‖𝑨_𝑖‖_𝐹²− ‖𝜼_1,𝑖‖

𝐹

2+ ‖𝑩_𝑖‖_𝐹²− ‖𝜼_2,𝑖‖

𝐹 2}

𝑧

𝑖=1

< ∑ {‖𝜼_1,𝑖‖

𝐹

2+ ‖𝜼_2,𝑖‖

𝐹 2}

𝑧

𝑖=1

)

(A.5) Considering that 𝜼_1,𝑖 and 𝜼_2,𝑖 are random Gaussian vectors with independent entries, the sum

∑^𝑧_𝑖=1{‖𝜼_1,𝑖‖_𝐹²+ ‖𝜼_2,𝑖‖_𝐹²} can be written as ‖∑^𝑧_𝑖=1(𝜼_1,𝑖+ 𝜼_2,𝑖)‖_𝐹². Therefore, (A.5) can be further simplified as:

54 𝑃(𝑡_𝑘→𝑔 → 𝑡̂_𝑘→𝑔|𝑯₁𝑯₂) = 𝑃 (‖∑(𝜼_1,𝑖+ 𝜼_2,𝑖)

𝑧

𝑖=1

‖

𝐹 2

> ∑{‖𝑨_𝑖‖_𝐹²+ ‖𝑩_𝑖‖_𝐹²}

𝑧

𝑖=1

)

= 𝑃 (‖∑^𝑧_𝑖=1(𝜼_1,𝑖+ 𝜼_2,𝑖)‖_𝐹 > √¹

2∑^𝑧_𝑖=1(‖𝑨_𝑖‖_𝐹²+ ‖𝑩_𝑖‖_𝐹²)) (A.6) Let 𝜼_T, with entries that are Gaussian RVs distributed as 𝐶𝑁(0,1) be defined as:

𝜼_T=^{∑𝑧𝑖=1(𝜼1,𝑖+𝜼2,𝑖)}

√2𝑧 (A.7) where 2𝑧 is the variance of each entry of the sum ∑^𝑧_𝑖=1(𝜼_1,𝑖+ 𝜼_2,𝑖). Substituting (A.7) into (A.6) yields:

𝑃(𝑡_𝑘→𝑔 → 𝑡̂_𝑘→𝑔|𝑯₁𝑯₂) = 𝑃 (‖ 𝜼_T‖_𝐹 > √1

4𝑧∑(‖𝑨_𝑖‖_𝐹²+ ‖𝑩_𝑖‖_𝐹²)

𝑧

𝑖=1

)

= 𝑄 (√1

4𝑧∑(‖𝑨_𝑖‖_𝐹²+ ‖𝑩_𝑖‖_𝐹²)

𝑧

𝑖=1

)

= 𝑄 (√∑ ^𝜌

8𝑧‖𝒉_2,𝑖¹ ‖

𝐹

2|𝑥_𝑟²_𝑖|²𝑑_𝑖+ ^𝜌

8𝑧‖𝒉_2,𝑖² ‖

𝐹

2|𝑥̅_𝑞¹_𝑖|²𝑑_𝑖

𝑧𝑖=1 ) (A.8)

where 𝑑_𝑖 = |e^{j𝜃𝑘𝑖}− e^{j𝜃𝑘̂𝑖}|², 𝑖 ∈ [1: 𝑧].

Let 𝜅_1,𝑖=_8𝑧^𝜌 ‖𝒉_2,𝑖¹ ‖

𝐹

2|𝑥_𝑟²_𝑖|²𝑑_𝑖 and 𝜅_2,𝑖 =_8𝑧^𝜌‖𝒉_2,𝑖² ‖

𝐹

2|𝑥̅_𝑞¹_𝑖|²𝑑_𝑖, therefore

𝑃(𝑡_𝑘→𝑔→ 𝑡̂_𝑘→𝑔|𝑯₁𝑯₂) = 𝑄 (√∑( 𝜅_1,𝑖+ 𝜅_2,𝑖)

𝑧

𝑖=1

)

(A.9) Note that employing trellis code-aided mapping of additional bits, sets the squared distances as 𝑑₁= 4, 𝑑₂= 2 and 𝑑₃= 4 for a scheme with one additional bit and z = 3, while for a scheme with two additional bits, 𝑑_𝑖 = 4; 𝑖 ∈ [1: 𝑧] and 𝑧 = 2.

55

Appendix B

Figure A-1: 64-QAM Gray-coded labeling map 𝜔^𝐺 [11]

Figure A-2: 64-QAM optimized labeling map 𝜔^𝑂 [11]

56 Figure A-3: 16-QAM Gray-coded labeling map 𝜔^𝐺[11]

Figure A-4: 16-QAM optimized labeling map 𝜔^𝑂 [11]

57

References

[1] S. K. Roy and N. Jain, “BER performance analysi of Alamouti coding technique in Rayleigh fading channel,” International Journal of Science and Research, vol. 3, no. 12, pp. 480- 483, Dec. 2014.

[2] S. M. Alamouti, “A Simple transmit diversity technique for wireless communications,”

IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1451-1458, Oct.

1998.

[3] C. Xu, S. X. N. Li Wang and L. Hanzo, “Multiple-symbol differential sphere detection- aided differential space-time block codes using QAM constellations,” IEEE Signal Processing Letters, vol. 18, no. 9, pp. 497-500, Sep. 2011.

[4] V. Tarokh and H. Jafarkhani, “A differential detection scheme for transmit diversity,” IEEE Journal on Selected Areas in Communications, vol. 18, no. 7, pp. 1169-1174, July 2000.

[5] D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Transactions on Communications, vol. 38, no. 3, pp. 300-308, Mar. 1990.

[6] L. Li, Z. Fang, Y. Zhu and Z. Wang, “Generalized differential transmission for STBC systems,” in IEEE Global Telecommunications (GLOBECOM) Conference, New Orleans, pp. 1-5, Dec. 2008.

[7] E. B. Slimane, S. Jarboui, I. Lamouchi and A. Bouallegue, “Concatenated differential space- time block codes and four dimensional 8-PSK trellis coded modulation for wireless MIMO system with Rayleigh fading noise,” International Journal of Computer Science and Information Technologies, vol. 3, no. 2, pp. 3456-3459, Apr. 2012.

[8] H. Lee, M. Siti, W. Zhu and M. Fitz, “Super-orthogonal space-time block code using a unitary expansion,” in IEEE 60th Annual Conference on Vehicular Technology, Los Angeles, vol. 4, pp. 2513-2517, Sep. 2004.

[9] W. T. Chen, “Spectral efficiency analysis for LTE networks,” in IEEE Fourth International Conference on Consumer Electronics, Berlin, pp. 93-95, Sep. 2014.

[10] E. Basar , U. Aygolu, E. Panayirci and H. V. Poor, “Space-time block coded spatial modulation,” IEEE Transactions on Communications, vol. 59, no. 3, pp. 823-832, Mar.

2011.

58 [11] K. Govindasamy, H. Xu and N. Pillay, “Space-time block coded spatial modulation with

labeling diversity,” International Journal of Communication Systems, vol. e3395.

https://doi.org/10.1002/dac.3395, 2017.

[12] Q. Ling and T. T. Li, “Efficiency Improvement for Alamouti Codes,” in IEEE 40th Annual Conference on Information Sciences and Systems, Princeton, pp. 569-572, Mar. 2006.

[13] Z. A. Baloch, M. U. Baloch and N. Hussain, “Efficiency improvement of space time block codes,” International Journal of Communications, Network and System Sciences, vol. 3, no.

6, pp. 507-510, June 2010.

[14] J. Liu, “Wireless multipath fading channels modeling and simulation based on sum-of- sinsoids,” in First IEEE International Conference on Computer Communication and the Internet, Wuhan, pp. 165-168, Oct. 2016.

[15] F. E. Mahmood, F. Y. Abdullah and H. A. Al-Tayyar, “Analysis, simulation and modeling of mobile Rayleigh fading channel using Labview,” in IEEE Antenna and Propagation Conference, Loughborough, pp. 1-6, Nov. 2011.

[16] C. S. Hwang, S. H. Nam, J. Chung and V. Tarokh, “Differential space-time block codes using nonconstant modulus constellations,” IEEE Transactions on Signal Processing, vol.

51, no. 11, pp. 2955-2964, Nov. 2003.

[17] H. Xu, K. Govindasamy and N. Pillay, “Uncoded space-time labeling diversity,” IEEE Communication Letters, vol. 20, no. 8, pp. 1511-1514, June 2016.

[18] X. Li, A. Chindapol and J. A. Ritcey, “Bit-interleaved coded modulation with iterative decoding and 8 PSK signaling,” IEEE Transactions on Communications, vol. 50, no. 8, pp.

1250-1257, Aug. 2002.

[19] Y. Huang and J. A. Ritcey, “Optimal constellation labeling for iteratively decoded bit- interleaved space-time coded modulation,” IEEE Transactions in Information Theory, vol.

51, no. 5, pp. 1865-1871, May 2005.

[20] M. Krasicki and P. Szulakiewicz, “Boosted space-time diversity scheme for wireless communications,” Electronics Letters, vol. 45, no. 16, pp. 843-845, July 2009.

[21] E. J. Kaminsky, J. Ayo and K. V. Cartwright, “TCM without constellation expansion,”

Journal of Communications and Networks, vol. 4, no. 2, pp. 90-96, June 2002.

59 [22] G. Ungerboeck, “Trellis-coded modulation with redundant signal sets,” IEEE

Communication Magazine, vol. 25, no. 2, pp. 5-21, Feb. 1987.

[23] S. Benedetto, M. Mondin and G. Montorsi, “Performance evaluation of trellis coded modulation schemes,” Proceedings of the IEEE, vol. 82, no. 6, pp. 833-855, June 1994.

[24] S. Das, N. Al-Dhahir and R. Calderbank, “Novel full-diversity high-rate STBC for 2 and 4 transmit antennas,” IEEE Communications Letters, vol. 10, no. 3, pp. 171-173, Mar. 2006.

[25] H. Xu and N. Pillay, “Simple near-maximum-likelihood low-complexity detection scheme for Alamouti space-time block coded spatial modulation,” IET Communications, vol. 8, no.

15, pp. 2611-2618, May 2014.

[26] S. Siwamogsatham and M. F. Fitz, “Improved high rate space-time codes via orthogonality and set partitioning,” in IEEE Wireless Communication and Networking Conference, Orlando, vol. 1, pp. 264-270, Mar. 2002.

[27] S. S. H. Bidaki, S. Talebi and M. Shahabinejad, “A full-rate full-diversity 2\times2 space- time block code with linear complexity for the maximum likelihood receiver,” IEEE Communications Letters, vol. 15, no. 8, pp. 842-844, Aug. 2011.

[28] N. R. Naidoo, H. Xu and T. Quazi, “Spatial modulation: optimal detector asymptotic performance and multiple-stage detection,” IET Communications, vol. 5, no. 10, pp. 1368- 1376, July 2011.

[29] Z. Paruk and H. Xu, “Performance analysis and simplified detection for two-dimensional signal space diversity with MRC reception,” SAIEE Africa Research Journal, vol. 104, no.

3, pp. 97-106, Sep. 2013.

[30] R. A. Abdallah, S. J. Lee, M. Goel and N. R. Shanbhag, “Low-power pre-coding based Viterbi decoder for tail-biting convolutional codes,” in Signal Processing Systems, Tampere, pp. 185-190, Oct. 2009.

[31] A. Hedayat, H. Shah and A. Nosratinia, “Analysis of Space–time coding in correlated fading channels,” IEEE Transactions on Wireless Communications, vol. 4, no. 6, pp. 2882-2891, Nov. 2005.

[32] E. Basar and I. Altunbas, “Space-time channel modulation,” IEEE Transactions on Vehicular Technology, vol. 66, no. 8, pp. 7609-7614, Aug. 2017.