壓縮感知這項技術至少已存在五十多年，而較常被應用在許多領域的訊號處理技術。壓縮感知的原理，是從相對較少的資訊進而還原出數量較多欲得知的訊號，藉此達成資源的節省。在無線通訊領域上，由於多重路徑通道的稀疏特性，我們可以藉由以壓縮感知為基礎的方法來處理在正交分頻多工中的稀疏通道估測問題。
　　藉由最小化測量矩陣的最大相關性質的方法估算出稀疏通道而不是使用熟知的有限等距性質，其中在領航信號的樣式設計上，有別於一般的設計使用純粹隨機選取領航信號的位置，本文提出新的方法選取領航信號，我們發現選取某些領航信號的位置會得出不好的結果，因此我們提出新的選取方法縮減我們所要隨機選取的基底範圍，根據通道的長度和多重路徑通道的數量而在領航信號的相對距離上使用固定的倍數，以減少多重路徑搜尋的範圍，縮小其在多重路徑裡位置的選取，進而達到更好的最小化測量矩陣。模擬結果顯示，相對於被廣泛採用的準則，由本文提出的設計準則所產生的領航信號，可以測量到更小的最小化測量矩陣的最大相關性質的值，因此在稀疏通道估測上可以有更低以及更穩定的均方誤差表現。

Compressed-sensing (CS) is a technique that has invented at least fifteen years. It usually applied the signal processing technique in lots of field. For the principle of Compressed-sensing, we would like to recover the estimated signal. A linear transformation will transform the estimated signal to another domain which has smaller dimension. Therefore, the information of received signal will smaller than the estimated signal. We apply some compressed sensing algorithm to recover the estimated signal. Since the character of the sparse channel of multiple accesses in the wireless communication, we can estimate the sparse channel in orthogonal frequency division multiplexing (OFDM) system by Compressed-sensing based.
　　Instead of the well-known restricted isometry property, we estimate the sparse channel by calculating the minimum of the maximum relation of the measurement matrix. On the design of the pilot pattern, the way of choosing pilot pattern normally use purely random search. In this work, differ from the usual base of purely random search, we propose a new approach to design the pilot pattern. We find that some subcarriers for choosing as pilot subcarriers are bad choose. As a result, we propose a way to reduce the searching base in order to generate a better pilot pattern. According to the number of channel taps and the number of subcarriers of OFDM system, we design the pilot pattern in the reduced base which is fixed the multiple of the relative distance of pilot subcarriers. We use different multiple of the relative distance by different criterion. The simulation results display that the proposed method of designing pilot pattern is superior to the normal way of designed pilot pattern. We compare the different algorithm of designing pilot pattern in a way of minimum of the maximum relation of the measurement matrix. As a result, it can display a smaller and more stable mean square error in the sparse channel estimation.

CONTENTS
口試委員會審定書 #
中文摘要 i
ABSTRACT ii
誌謝 iv
CONTENTS v
LIST OF FIGURES vii
LIST OF TABLES ix
Chapter 1 Introduction 1
1.1 Background 1
1.2 Related Works 4
Chapter 2 System Model 7
2.1 An Overview of OFDM system and pilot design based on mutual coherence 7
2.2 Sparse Channel Estimation via Compressed Sensing 10
2.3 The Objective function Based on Mutual Coherence 15
Chapter 3 Proposed Method 16
3.1 The Proposed Criterion Description 16
3.2 The Lower Bound of the Mutual Coherence 21
3.3 The Character of The Difference Sets 24
Chapter 4 Simulation Results 26
4.1 The Different Size of a Set 27
4.2 The Welch Bound Constraints with Reduction-Based Pilot Allocation 28
4.3 The Proposed Method Simulate in Different Scenario 30
Chapter 5 Conclusion 44
References 45

[1] Y. Li, L. J. Cimini, and N. R. Sollenberger, “Robust channel estimation for OFDM systems with rapid dispersive fading channels,” IEEE Trans. Commun., vol. 46, pp. 902–915, Jul. 1998.
[2] Z. Tang, R. C. Cannizzaro, G. Leus, and P. Banelli, “Pilot-assisted time-varying channel estimation for OFDM systems,” IEEE T. Signal. Proces., vol. 55, no. 5, pp. 2226–2238, 2007.
[3] D. Needell, “Topics in Compressed Sensing,” PhD Dissertation, Mathematics, Univ. of California, Davis, May 2009.
[4] S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l-1 regularized least squares,” IEEE J. Sel. Topics Signal Process., vol. 1, no. 4, pp. 606–617, Dec. 2007.
[5] J. Tropp,A. Gilbert “Signal recovery from random measurement via orthogonal matching pursuit”IEEE Transactions on Information Theory,53(12)(2007),pp.4655-4666.
[6] D. L. Donoho, A. Maleki, and A. Montanari, “Message passing algorithms for compressed sensing,” Proc. Nat. Acad. Sci., vol. 106, no. 45, pp. 18914–18919, Nov. 2009.
[7] W. Dai and O. Milenkovic. Subspace pursuit for compressive sensing signal reconstruction. IEEE Trans. Inform. Theory, 55(5):2230–2249, 2009.
[8] E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE vol. 52, no. 12, pp. 5406–5425, Dec. 2006.Trans. Inf. Theory,
[9] Jian Wang and ByonghyoShim , “A Simple Proof of the Mutual Incoherence Condition for Orthogonal Matching Pursuit”, arXiv 1105.4408v1 (cs.It), 23 May (2011).
[10] M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, New York, 2010) (Cited on p. 27.)
[11] L. Najjar, “Pilot allocation by Genetic Algorithms for sparse channel estimation in OFDM systems,” in Proc. of European Signal Process. Conf. (EUSIPCO), pp. 1–5, Sep. 2013
[12] C. Qi, G. Yue, L. Wu, and A. Nallanathan, “Pilot Design for Sparse Channel Estimation in OFDM-Based Cognitive Radio Systems,” IEEE Trans. Veh. Technol., vol. 63, no. 2, pp. 982–987, Feb. 2014.
[13] C. Qi, G. Yue, L. Wu, Y. Huang, and A. Nallanathan, “Pilot design schemes for sparse channel estimation in OFDM systems,” IEEE Trans. Veh. Tech., vol. 64, no. 4, pp. 1493–1505, Apr. 2015.
[14] C. Qi and L. Wu, “A study of deterministic pilot allocation for sparse channel estimation in OFDM systems,” IEEE Commun. Lett., vol. 16, no. 5, pp. 742–744, May 2012.
[15] O. Taheri, Sergiy A. Vorobyov, “Sparse channel estimation with Lp-norm and reweighted L1-norm penalized least mean squares,” in Proc. IEEE ICASSP, Prague, Czech Republic, 22-27 May 2011.
[16] G. Davis, S. Mallat, and M. Avellaneda, “Greedy adaptive approximation,”　Constr. Approx., vol. 13, pp. 57–98, 1997.
[17] C. Qi and L. Wu, “A study of deterministic pilot allocation for sparse channel estimation in OFDM systems,” IEEE Commmun. Lett., vol. 16, no. 5, pp. 742–744, May 2012.
[18] M. Khosravi and S. Mashhadi, “Joint pilot power & pattern design for compressive OFDM channel estimation,” IEEE Communications Letters, vol. 19, no. 1, pp. 50-53, January 2015.
[19] P. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun, vol. 42, pp. 2908–2914, Oct. 1994.
[20] La Jolla Cyclic Difference Set Repository. [Online]. Available: http://
www.ccrwest.org/diffsets/ds_list.pdf.