在未來的通訊系統中，通道容量需要大幅度的成長，才能夠滿足未來對於通訊系統的服務品質的要求，這是一個被廣為討論的課題。傳統的傳送方式需要一串訓練序列對通道進行估測，如果我們能夠縮短訓練序列的長度，甚至是不需要的話，那麼頻寬擁擠的問題將會獲得改善。
由於先前的通訊系統結構，無論是接收端或是傳送端，本身所能擁有的天線數量很有限，但透過近年不斷的發展並更新通訊系統的技術。在未來的通訊系統中，很有可能會引用大規模多輸入多輸出系統的概念，應用於未來的通訊系統中，那麼我們要進行訊號到達角度的估測的問題，就會變成一個稀疏訊號重建的問題，因此我們能夠利用已被探討過的壓縮性感知演算法來進行角度的估測。
近年來，感知無線電及大規模多輸入多輸出的概念已被提出，藉由將這兩種概念與通道訊號方向估測進行結合，或許可以視為解決頻寬擁擠問題的一個方法，因為在波長太短時，傳統的角度估測方法會沒辦法進行估測，因此本篇論文中主要討論如何將壓縮感測技術應用在通道訊號方向的估測。

In the future, the channel requirement increase rapidly. How to improve the communication systems performance has become an important issue. The conventional method to transmit signals need pilot sequence. The pilot sequence also need frequency band to transmit. If we do the channel estimation, we can transmit signals with less pilot sequence or without the pilot sequence. The problem of the channel requirement will be improved.
In the previous communication systems, the number of antennas in transmitter and receiver is not large. By the developing of the communication systems in the recent years, the concept of the massive MIMO will be applied in the future. When the number of the DOA is much smaller than the number of antennas in the receiver, the DOA estimation problem can be regard as the famous sparse signal reconstruction problem.
Recently, the concept of cognitive radio and massive Multi-input Multi-output (MIMO) system model has been proposed and it is possible to improve the communication systems performance by combining the concept of the channel estimation that the channel estimation based on direction of arrival (DOA) can be regarded as a solution to this problem. Since the resolution of the channel estimation based on DOA with compressive sampling is higher than the resolution of the conventional DOA method Multiple Signal Classification (MUSIC), we focus on the channel estimation based on DOA with compressive sampling in this thesis.

Chinese Abstract i
English Abstract ii
Contents iii
1 Introduction 1
2 System Model 3
2.1 Cognitive Radio Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 System Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Sparse Signal Representation 12
3.1 Compressive Sensing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Sparse signal representation in spatial domain for DOA estimation . . . . . . 15
3.3 Analysis of DOA Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.1 Multiple Signal Classication [1] . . . . . . . . . . . . . . . . . . . . . 19
3.3.2 Compressed Sampling for DOA [2] . . . . . . . . . . . . . . . . . . . 22
4 Compressive Sensing Theory Algorithms for DOA estimation 25
4.1 Orthogonal Matching Pursuit [3] . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Compressive Sampling Matching Pursuit [4] . . . . . . . . . . . . . . . . . . 28
4.3 Subspace Pursuit [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 DOA Estimation Based on Existing Compressive Sensing Algorithms . . . . 32
5 Simulations 33
5.1 Comparison of Resolution with Dierent Number of antennas . . . . . . . . 34
5.2 Comparison of Resolution with Existing Compressive Sensing Algorithms and
MUSIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Comparison of the Performance with Dierent SNR . . . . . . . . . . . . . . 40
5.4 Comparison of the Performance with Dierent Number of the Measurement . 42
5.5 Relationship between the Number of the Measurement and SNR with Target
Detection Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Conclusion 46
Bibliography 47

[1] B. Friedlander, "A sensitivity analysis of the music algorithm," Acoustics, Speech and Signal Processing, IEEE Transactions on, vol. 38, pp. 1740-1751, Oct 1990.
[2] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. New York, NY, USA: Cambridge University Press, 2005.
[3] J. Tropp and A. Gilbert, "Signal recovery from random measurements via orthogonal matching pursuit," Information Theory, IEEE Transactions on, vol. 53, pp. 4655-4666, Dec 2007.
[4] D. Needell and J. Tropp, "Cosamp: Iterative signal recovery from incomplete and inaccurate samples," Applied and Computational Harmonic Analysis, vol. 26, no. 3, pp. 301 -321, 2009.
[5] W. Dai and O. Milenkovic, "Subspace pursuit for compressive sensing signal reconstruction,"
Information Theory, IEEE Transactions on, vol. 55, pp. 2230-2249, May 2009.
[6] G. Scutari, D. Palomar, and S. Barbarossa, "Cognitive mimo radio," Signal Processing Magazine, IEEE, vol. 25, pp. 46-59, November 2008.
[7] S. Haykin, "Cognitive radio: brain-empowered wireless communications," Selected Areas in Communications, IEEE Journal on, vol. 23, pp. 201-220, Feb 2005.
[8] D. Donoho, "Compressed sensing," Information Theory, IEEE Transactions on, vol. 52, pp. 1289-1306, April 2006.
[9] E. Candes and M.Wakin, "An introduction to compressive sampling," Signal Processing Magazine, IEEE, vol. 25, pp. 21-30, March 2008.
[10] A. Gurbuz, J. McClellan, and V. Cevher, "A compressive beamforming method," in Acoustics, Speech and Signal Processing, 2008. ICASSP 2008. IEEE International Conference on, pp. 2617-2620, March 2008.
[11] V. Cevher, A. Gurbuz, J. McClellan, and R. Chellappa, "Compressive wireless arrays for bearing estimation," in Acoustics, Speech and Signal Processing, 2008. ICASSP 2008. IEEE International Conference on, pp. 2497-2500, March 2008.
[12] R. Ertel, P. Cardieri, K. Sowerby, T. Rappaport, and J. Reed, "Overview of spatial channel models for antenna array communication systems," Personal Communications,
IEEE, vol. 5, pp. 10-22, Feb 1998.
[13] J. Kermoal, L. Schumacher, K. Pedersen, P. Mogensen, and F. Frederiksen, "A stochastic mimo radio channel model with experimental validation," Selected Areas in Communications, IEEE Journal on, vol. 20, pp. 1211-1226, Aug 2002.
[14] S. D. Pan, Shuo and M. E. Bialkowski., "Mimo capacity for spatial channel model scenarios.," Proc. of Aust. Comm. Thr. Wkshp, pp. 25-29, 2007.
[15] D. Malioutov, M. Cetin, and A. Willsky, "A sparse signal reconstruction perspective for source localization with sensor arrays," Signal Processing, IEEE Transactions on, vol. 53, pp. 3010-3022, Aug 2005.
[16] S. Mallat and Z. Zhang, "Matching pursuits with time-frequency dictionaries," Signal Processing, IEEE Transactions on, vol. 41, pp. 3397-3415, Dec 1993.
[17] R. Schmidt, "Multiple emitter location and signal parameter estimation," Antennas and Propagation, IEEE Transactions on, vol. 34, pp. 276-280, Mar 1986.
[18] A. Ferreol, P. Larzabal, and M. Viberg, "Statistical analysis of the music algorithm in the presence of modeling errors, taking into account the resolution probability," Signal Processing, IEEE Transactions on, vol. 58, pp. 4156-4166, Aug 2010. 49