臺灣博碩士論文加值系統

廣義分頻多工為一極富可能性之調變方式，藉由使用原型濾波器，其具有低頻帶外輻射之特性。然而，當使用文獻中常見的一些原型濾波器時，廣義分頻多工系統通常是非正交的，因而招致相較於正交分頻多工，在接收機均方誤差與符元錯誤率上的頻帶內效能退化。

本論文提出一種新的基於矩陣來將廣義分頻多工傳送機矩陣參數化之方式，而非如傳統上利用原型濾波器之基於向量的參數化方式。此新的參數化方式使我們很容易推導廣義分頻多工（傳送機）矩陣的性質，包括使廣義分頻多工矩陣為非奇異與么正各自的條件。

藉由使用此新的參數化方式，我們推導出使最小均方誤差接收機之一種低複雜度實現方式存在的充分且必要條件。若選用么正的廣義分頻多工傳送機矩陣，此實現方式在多路徑通道的情況下存在。在此實現方式不存在的情況下，我們提出一種低複雜度次最佳最小均方誤差接收機，其效能近似最小均方誤差接收機之效能。

此新的參數化方式也使我們能推導出可最小化接收機均方誤差的最佳原型濾波器。在許多情況下，這些最佳原型濾波器對應於使用么正的廣義分頻多工矩陣。在廣義分頻多工系統使用這些最佳原型濾波器不會導致雜訊增強的問題，使得系統展現與正交分頻多工相同的均方誤差效能。

此外，基於提出的矩陣式參數化方式，我們為廣義分頻多工發展出一個能最小化頻帶外輻射並保持優良頻帶內效能的原型濾波器最佳化演算法。透過將特徵矩陣作為最佳化變數，該濾波器設計問題被表示為一個非凸問題。在一些轉換之後，我們提出一個迭代解決兩個凸問題的演算法來處理原問題。模擬結果顯示在相同頻譜效率的情況下，相較於正交分頻多工與文獻中的原型濾波器，最佳化所得的濾波器在頻帶外輻射與符元錯誤率上皆表現最好。

Generalized frequency division multiplexing (GFDM) is a promising modulation scheme featuring low out-of-band (OOB) radiation, which is achieved through the use of prototype filters. However, GFDM systems are usually non-orthogonal with prototype filters commonly used in the literature, incurring in-band performance degradation in receiver mean square error (MSE) and symbol error rate (SER) compared to that achieved through orthogonal frequency division multiplexing (OFDM).

In this thesis, a new matrix-based characterization of GFDM transmitter matrices is proposed, as opposed to traditional vector-based characterization with prototype filters. The new characterization facilitates deriving properties of GFDM (transmitter) matrices, including conditions for GFDM matrices being nonsingular and unitary, respectively.

Using the new characterization, the necessary and sufficient conditions for the existence of a form of low-complexity implementation for a minimum mean square error (MMSE) receiver are derived. Such an implementation exists under multipath channels if the GFDM transmitter matrix is selected to be unitary. For cases where this implementation does not exist, a low-complexity suboptimal MMSE receiver is proposed, with its performance approximating that of an MMSE receiver.

The new characterization also enables derivations of optimal prototype filters in terms of minimizing receiver MSE. They are found to correspond to the use of unitary GFDM matrices under many scenarios. The use of such optimal filters in GFDM systems does not cause the problem of noise enhancement, thereby demonstrating the same MSE performance as OFDM.

In addition, based on the proposed matrix characterization, a filter optimization algorithm that minimizes OOB radiation while maintaining good in-band performance is developed for GFDM. Through the characteristic matrix as the optimizing variable, the filter design problem is formulated as a nonconvex problem. After some transformations, an algorithm in which two convex problems are solved iteratively is proposed to tackle the original problem. Simulation results show that under the same spectral efficiency, optimized filters perform the best in terms of both OOB radiation and SER performance, compared to OFDM and prototype filters existing in the literature.

誌謝 i
摘要 iii
Abstract v
Contents vii
List of Figures xi
List of Tables xv
1 Introduction 1
2 Characterization of GFDM Systems 5
2.1 Characterization of GFDM Matrices: Basic Definitions .......... 6
2.2 GFDM Transmitter Implementations .................... 8
2.2.1 Direct implementation ....................... 9
2.2.2 Frequency-domain implementation ................ 9
2.2.3 Characteristic-matrix-domain implementation ........... 9
2.3 Unitary and Invertible GFDM Matrices .................. 11
3 GFDM Receiver Implementations 13
3.1 Low-Complexity ZF Receivers ....................... 14
3.2 Low-Complexity MMSE Receivers .................... 15
3.3 Low-Complexity Approximated MMSE Receivers ............ 19
3.3.1 Simulation Results ......................... 19
3.4 Remarks on Soft-Output Demodulation .................. 21
4 Complexity Analysis 25
4.1 Additional Complexity Comparison Results ................ 30
5 Power Spectral Density and OOB Leakage 33
6 Optimal Prototype Filters that Minimize MSE 37
6.1 Optimization Results for ZF Receivers ................... 38
6.2 Optimization Results for MMSE Receivers ................ 40
6.3 Comparison of Prototype Filter Candidates ................ 42
6.4 Simulation Results ............................. 45
6.4.1 MSE and SER Performance .................... 45
6.4.2 PAPR ................................ 51
6.4.3 OOB Leakage ............................ 52
6.4.4 Additional Simulation Results for SER Performance ....... 53
7 Optimal Prototype Filters that Minimize MSE and OOB Radiation 57
7.1 Problem Formulation ............................ 57
7.2 Proposed Algorithm ............................. 58
7.3 Simulation Results ............................. 61
7.3.1 Parameter Settings ......................... 61
7.3.2 Simulation Results for the Case of η = 1 ............. 62
7.3.3 Influence of the Weight w ..................... 65
7.3.4 Simulation Results for the Case of η > 1 ............. 66
7.4 Future Work ................................. 67
8 Multiple Access with Optimized Prototype Filters 69
9 Conclusions 73
Bibliography 74
A Proof of Theorem 5 81
B Proof of Theorem 6 83
C Proof of Corollaries 1 and 2 85

[1] G. Fettweis, M. Krondorf, and S. Bittner, “GFDM - generalized frequency division multiplexing,” in Veh. Technol. Conf., 2009. VTC Spring 2009. IEEE 69th, Apr. 2009, pp. 1–4.
[2] N. Michailow, M. Matthé, I. Gaspar, A. Caldevilla, L. Mendes, A. Festag, and G. Fettweis, “Generalized frequency division multiplexing for 5th generation cellular networks,” IEEE Trans. Commun., vol. 62, no. 9, pp. 3045–3061, Sep. 2014.
[3] M. Matthé, N. Michailow, I. Gaspar, and G. Fettweis, “Influence of pulse shaping on bit error rate performance and out of band radiation of generalized frequency division multiplexing,” in Proc. IEEE ICC Workshop, 2014, pp. 43–48.
[4] J. Bingham, “Multicarrier modulation for data transmission: an idea whose time has come,” IEEE Commun. Mag., vol. 28, no. 5, pp. 5–14, May 1990.
[5] M. Matthé, L. Mendes, and G. Fettweis, “Generalized frequency division multiplexing in a Gabor transform setting,” IEEE Commun. Lett., vol. 18, no. 8, pp. 1379–1382, Aug. 2014.
[6] R. Datta, N. Michailow, M. Lentmaier, and G. Fettweis, “GFDM interference cancellation for flexible cognitive radio PHY design,” in Veh. Technol. Conf. (VTC Fall), 2012 IEEE, Sep. 2012, pp. 1–5.
[7] R. Datta, G. Fettweis, Z. Kollár, and P. Horváth, “FBMC and GFDM interference cancellation schemes for flexible digital radio PHY design,” in Digital System Design (DSD), 2011 14th Euromicro Conf. on, Aug. 2011, pp. 335–339.
[8] M. Matthé, L. Mendes, I. Gaspar, N. Michailow, D. Zhang, and G. Fettweis, “Precoded GFDM transceiver with low complexity time domain processing,” EURASIP J. on Wireless Commun. and Networking, vol. 2016, no. 1, p. 138, 2016.
[9] A. Farhang, N. Marchetti, and L. E. Doyle, “Low-complexity modem design for GFDM,” IEEE Trans. Signal Process., vol. 64, no. 6, pp. 1507–1518, Mar. 2016.
[10] M. Matthé, I. Gaspar, D. Zhang, and G. Fettweis, “Reduced complexity calculation of LMMSE filter coefficients for GFDM,” in Veh. Technol. Conf. (VTC Fall), 2015 IEEE 82nd, Sept 2015, pp. 1–2.
[11] M. Matthé, D. Zhang, and G. Fettweis, “Iterative detection using MMSE-PIC demapping for MIMO-GFDM systems,” in Eur. Wireless 2016; 22th Eur. Wireless Conf., May 2016, pp. 1–7.
[12] N. Michailow, I. Gaspar, S. Krone, M. Lentmaier, and G. Fettweis, “Generalized frequency division multiplexing: Analysis of an alternative multi-carrier technique for next generation cellular systems,” in 2012 Int. Syst. on Wireless Commun. Syst. (ISWCS), Aug 2012, pp. 171–175.
[13] N. Michailow, S. Krone, M. Lentmaier, and G. Fettweis, “Bit error rate performance of generalized frequency division multiplexing,” in Veh. Technol. Conf. (VTC Fall), 2012 IEEE, Sep. 2012, pp. 1–5.
[14] S. K. Bandari, A. Drosopoulos, and V. V. Mani, “Exact SER expressions of GFDM in Nakagami-m and Rician fading channels,” in Eur. Wireless 2015; 21th Eur. Wireless Conf.; Proc. of, May 2015, pp. 1–6.
[15] B. M. Alves, L. L. Mendes, D. A. Guimaraes, and I. S. Gaspar, “Performance of GFDM over frequency-selective channels,” in Proc. Int. Workshop on Telecommun., 2013.
[16] S. Tiwari, S. S. Das, and K. K. Bandyopadhyay, “Precoded generalised frequency division multiplexing system to combat inter-carrier interference: performance analysis,” IET Commun., vol. 9, no. 15, pp. 1829–1841, 2015.
[17] A. RezazadehReyhani, A. Farhang, and B. Farhang-Boroujeny, “Circularly pulse-shaped waveforms for 5G: Options and comparisons,” in 2015 IEEE Global Commun. Conf. (GLOBECOM), Dec. 2015, pp. 1–7.
[18] M. Matthé, L. L. Mendes, and G. Fettweis, “Asynchronous multi-user uplink transmission with generalized frequency division multiplexing,” in 2015 IEEE Int. Conf. on Commun. Workshop (ICCW), Jun. 2015, pp. 2269–2275.
[19] M. Matthé, I. Gaspar, D. Zhang, and G. Fettweis, “Near-ML detection for MIMO-GFDM,” in Veh. Technol. Conf. (VTC Fall), 2015 IEEE 82nd, Sep. 2015, pp. 1–2.
[20] M. Matthé, L. L. Mendes, N. Michailow, D. Zhang, and G. Fettweis, “Widely linear estimation for space-time-coded GFDM in low-latency applications,” IEEE Trans. Commun., vol. 63, no. 11, pp. 4501–4509, Nov. 2015.
[21] Z. Sharifian, M. J. Omidi, A. Farhang, and H. Saeedi-Sourck, “Polynomial-based compressing and iterative expanding for PAPR reduction in GFDM,” in 2015 23rd Iranian Conf. on Elect. Eng., May 2015, pp. 518–523.
[22] S. Li, Y. Zhao, L. He, Z. Wu, and Y. Li, “Design and performance analysis of a GFDM-DCSK communication system,” in 2016 13th IEEE Annu. Consumer Commun. Networking Conf. (CCNC), Jan. 2016, pp. 802–803.
[23] S. Han, Y. Sung, and Y. H. Lee, “Filter design for generalized frequency-division multiplexing,” IEEE Trans. Signal Process., submitted for publication. [Online]. Available: https://arxiv.org/abs/1607.04004v2
[24] I. Gaspar and G. Fettweis, “An embedded midamble synchronization approach for generalized frequency division multiplexing,” in 2015 IEEE Global Commun. Conf. (GLOBECOM), Dec. 2015, pp. 1–5.
[25] L. Chang, G. Y. Li, J. Li, and R. Li, “Blind parameter estimation of GFDM signals over frequency-selective fading channels,” IEEE Trans. Commun., vol. 64, no. 3, pp. 1120–1131, Mar. 2016.
[26] P. S. Wang and D. W. Lin, “Maximum-likelihood blind synchronization for GFDM systems,” IEEE Signal Process. Lett., vol. 23, no. 6, pp. 790–794, Jun. 2016.
[27] I. Gaspar, L. Mendes, M. Matthé, N. Michailow, A. Festag, and G. Fettweis, “LTE-compatible 5G PHY based on generalized frequency division multiplexing,” in 2014 11th Int. Symp. on Wireless Commun. Syst. (ISWCS), Aug. 2014, pp. 209–213.
[28] I. Gaspar, A. Festag, and G. Fettweis, “Synchronization using a pseudo-circular preamble for generalized frequency division multiplexing in vehicular communication,” in Veh. Technol. Conf. (VTC Fall), 2015 IEEE 82nd, Sep. 2015, pp. 1–5.
[29] X.-G. Xia, “A family of pulse-shaping filters with ISI-free matched and unmatched filter properties,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1157–1158, Oct. 1997.
[30] P.-C. Chen, B. Su, and Y. Huang, “Matrix characterization for GFDM systems: Low-complexity MMSE receivers and optimal prototype filters,” IEEE Trans. Signal Process., to appear. [Online]. Available: http://arxiv.org/abs/1611.04429v4
[31] P.-C. Chen and B. Su, “Filter optimization of out-of-band radiation with performance constraints for GFDM systems,” accepted to the 18th IEEE Int. Workshop on Signal Process. Advances in Wireless Commun. (SPAWC), 2017.
[32] H. Bolcskei and F. Hlawatsch, “Discrete Zak transforms, polyphase transforms, and applications,” IEEE Trans. Signal Process., vol. 45, no. 4, pp. 851–866, Apr. 1997.
[33] A. J. Laub, Matrix Analysis for Scientists and Engineers. SIAM: Society for Industrial and Applied Mathematics, 2004.
[34] D. W. Lin and P. S. Wang, “On the configuration-dependent singularity of GFDM pulse-shaping filter banks,” IEEE Commun. Lett., vol. 20, no. 10, pp. 1975–1978, Oct. 2016.
[35] M. Zibulski and Y. Y. Zeevi, “Oversampling in the Gabor scheme,” IEEE Trans. Signal Process., vol. 41, no. 8, pp. 2679–2687, Aug. 1993.
[36] Y.-P. Lin, S.-M. Phoong, and P. P. Vaidyanathan, Filter Bank Transceivers for OFDM and DMT Systems. New York, NY, USA: Cambridge University Press, 2010.
[37] G. Matsaglia and G. P. H. Styan, “Equalities and inequalities for ranks of matrices,” Linear and multilinear Algebra, vol. 2, no. 3, pp. 269–292, 1974.
[38] C. Eckart and G. Young, “The approximation of one matrix by another of lower rank,” Psychometrika, vol. 1, no. 3, pp. 211–218, 1936.
[39] M. Brand, “Fast low-rank modifications of the thin singular value decomposition,” Linear Algebra and Its Appl., vol. 415, no. 1, pp. 20–30, May 2006.
[40] P. Duhamel and M. Vetterli, “Fast Fourier transforms: A tutorial review and a state of the art,” Signal Process., vol. 19, no. 4, pp. 259–299, 1990.
[41] G. Strang, Introduction to linear algebra, 4th ed. Wellesley Cambridge Press, 2009.
[42] S. Haykin, Communication Systems, 4th ed. Wiley Publishing, 2001.
[43] S. Sesia, M. Baker, and I. Toufik, LTE-the UMTS long term evolution: from theory to practice. John Wiley & Sons, 2011.
[44] Z. Sharifian, M. J. Omidi, H. Saeedi-Sourck, and A. Farhang, “Linear precoding for PAPR reduction of GFDMA,” IEEE Wireless Commun. Lett., vol. 5, no. 5, pp. 520–523, Oct. 2016.
[45] S. H. Han and J. H. Lee, “An overview of peak-to-average power ratio reduction techniques for multicarrier transmission,” IEEE Wireless Commun., vol. 12, no. 2, pp. 56–65, Apr. 2005.
[46] Y.-P. Lin and S.-M. Phoong, “BER minimized OFDM systems with channel independent precoders,” IEEE Trans. Signal Process., vol. 51, no. 9, pp. 2369–2380, Sep. 2003.
[47] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge University Press, 2009.
[48] J. Dattorro, Convex optimization & Euclidean distance geometry. Meboo Publishing, 2016.
[49] M. Grant and S. Boyd,“CVX: Matlab software for disciplined convex programming, version 2.1,” http://cvxr.com/cvx, Mar. 2014.