可攜式多媒體應用和移動通訊已成為我們生活中不可或缺的一部分。眾所周知，使用多個天線來創建多輸入多輸出（MIMO）無線通道可以顯著提高數據速率和可靠性以增強資料的傳輸品質。對於給定數量的發射和接收天線，Zheng和Tse提出在分集和復用增益之間存在連續的線性權衡(Diversity–Multiplexing Gain Tradeoff)，並在高訊號雜訊比的條件下利用合適的空時編碼量化了可實現的最大分集和復用增益。
近年來，空間時間碼被證明了在多根天線下，在多條衰落通道中利用此編碼技術可以提高系統的資料傳輸率和並增加更正錯誤的能力已實現巨大的性能提升。在空時編碼中，利用在發射機處採用多根傳送天線，對傳輸資料符元進行在空間和時間相互交錯的智能編碼，以便於多工增益和分集多樣性從中獲得提升。
在本論文中，我們計算了在頻率非選擇性衰退通道下線性調變之空間時間編碼設計準則的分集復用權衡。此編碼設計準則基於特殊的時間空間碼架構模型，由一個迴旋編碼器及一個空間編碼器所組成。此空間時間碼利用除法代數提供簡單的數學模式來建造線性調變之時間空間碼，以達到分集復用權衡。而在相同的天線條件下，相較於一般的空時碼，迴旋編碼器將提供更高的多重性增益，以確保更可靠的傳輸品質。而解碼器是利用維特比解碼器，用於執行最大似然順序解碼。最後在相同頻寬效率下不同時間空間編碼做模擬圖之比較與驗證理論推導的正確性。

In this thesis, we derived a simple diversity-multiplexing tradeoff of a space-time coding scheme for linear modulation over frequency-nonselective fading channels. With an optimal convolutional temporal encoder and the Golden spatial code, which achieves the full transmit spatial diversity, the maximum overall diversity gain is Lr*Lt*ECL over fast fading channels, where Lr is the number of receive antennas, Lt the number of transmit antennas, ECL the effective code length of the convolutional encoder. With the Viterbi decoder, simulation results demonstrate the theoretical analysis and justify the effectiveness of our coding scheme.

1 Introduction 4
2 A Scheme for Space-Time Coding with Linear Modulation 7
2.1 A Space-Time Coding Scheme . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Space-Time Code design Criteria for Linear Modulation . . . . . . . . . . 9
3 A Spatial Encoder for Full Rank Matrix Design 13
3.1 Algebraic Properties of Spatial Encoder . . . . . . . . . . . . . . . . . . 13
3.1.1 Division Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.2 Embedding Field Extension into Mnn(F) . . . . . . . . . . . . . 14
3.2 Golden Code: A Space-Time Code with Full Rate and Full Diversity . . 16
3.2.1 Construction of Golden Code . . . . . . . . . . . . . . . . . . . 16
3.2.2 Cubic Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.3 Minimum Determinant of the Golden Code . . . . . . . . . . . . 19
4 The Diversity Gain and Multiplexing Gain of a Space-Time Coding
Scheme 23
4.1 Diversity-Multiplexing Tradeo . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Design Criteria of a Coding Scheme over Fast Fading Channel . . . . . . 24
4.2.1 The DMT of a Space Time Coding Scheme with the Golden Code
over Fast Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Design Criteria of a Coding Scheme over Slow Fading Channel . . . . . . 33
4.3.1 The DMT of a Space Time Coding Scheme with the Golden Code
over Slow Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 MLSD Viterbi Decoder and Simulation Results 37
5.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 The Viterbi Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Simulation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3.1 Parameter Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3.2 The Diversity-Multiplexing Tradeo of a Space Time Coding Scheme 41
5.3.3 Performance Comparison over Fast Fading Channels . . . . . . . 42
6 Conclusion 45

[1] J.-C. Belore, G. Rekaya, and E. Viterbo, "The golden code: A 22 full-rate space-
time code with non-vanishing determinants", IEEE Transactions on Information The-
ory, vol. 51, no. 4, April 2005.
[2] L. Zheng and D. Tse, "Diversity and multiplexing: A fundamental tradeo in multiple
antenna channels", EEE Trans. Inform. Theory, 49(5), pp. 10731096, May2003.
[3] G. J. Foschini and M. J. Gans, "On limits of wireless communications in a fading
environment when using multiple antennas," Wireless Personal Commun., vol. 6, pp.
311-335, Mar. 1998.
[4] F. Oggier, J.-C. Belore, and E. Viterbo, "Cyclic division algebras: A tool for space-
time coding", Foundations and Trends in Commun. andInf. Theory, vol. 4, no. 1.
[5] J.-C. Belore and G. Rekaya, "Quaternionic lattices for space-time coding," in
Proceedings of ITW 2003, Paris, April 2003.
[6] P. Elia, K.R. Kumar, S.A. Pawar, P.V. Kumar and Hsiao-feng Lu, "Explicit space-
time codes that achieve the diversity-multiplexing gain tradeo",International Sym-
posium on Information Theory, ISIT 2005, p. 896900, Adelaide, Australia, 4-9th Sept.
2005.
[7] S. M. Alamouti, "A simple transmit diversity technique for wireless communications,"
IEEE J. Select. Areas Commun., vol. 16, pp. 1451-1458, Oct. 1998.
[8] I. E. Teletar, "Capacity of multi-antenna Gaussian channels," Europ. Trans. Telecom-
mun., vol. 10, pp. 585-595, Nov./Dec. 1999.
[9] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time codes for high data rate
wireless communication: performance criterion and code construction," IEEE Trans.
Inform. Theory, vol. 44, no. 2, pp. 744-765, Mar. 1998.
[10] H. Yao and G. W. Wornell, "Achieving the full MIMO diversity-multiplexing fron-
tier with rotation-based space-time codes", in Proc. AllertonConf. Communication,
Control, and Computing, Monticello, IL, Oct.2003.
[11] M.-C. Shih, "Space-time code design for QPSK modulation over frequency- nonse-
lective fading channels," Master's thesis, National Tsing Hua University, Tai- wan,
2007.
[12] H.-Y. Cheng, "Space-time code design over frequency- nonselective fading channels
by division algebras," Master's thesis, National Tsing Hua University, Taiwan, 2008.
[13] C.-M. Li, "Space-Time Code Design for Linear Modulation over Frequency-Selective
Fading Channels," Master's thesis, National Tsing Hua University, Taiwan, 2017.
[14] C.-C. Lu and J.-P. Chen, An optimal test of Zq-convolutional coded CPFSK for
communications over Rayleigh fading channels," to be submitted for publication.
[15] I. N. Herstein, Abstract Algebra, 3rd edn. January 30, 1996.
[16] R. J. McEliece, Finite Fields for Computer Scientists and Engineers. Kluwer, 1989.
[17] J. G. Proakis, Digital Communications, 4th edn. McGraw-Hill, 2001.