在多天線系統(MIMO)中，利用空間上多路複用(Multiplexing)之傳送天線波束賦形技術(Beamforming)是一種用來分離不同天線所傳來串流的技術。在有限的回饋頻寬下，天線波束賦形的矩陣必須被量化並回饋到傳送端。一般常見的方法，是利用奇異值分解(SVD)來設計出預先編碼器跟解碼器，然後基於Householder反射處理後，再回饋向量層面經過量化的天線波束賦形矩陣到傳送端。
在先前的論文中，已有人使用LU分解(LUD)來求得天線波束賦形矩陣。利用L下三角矩陣的特性，使得LU分解可以藉由迭代的方式來產生，因此在實作方面也更容易被實現。但是先前的論文並沒有討論要如何將天線波束賦形矩陣回饋到傳送端。為了可以解決此問題，在這篇論文中，我們將提出一種不同於以往的方法，我們推導出一個可以使L矩陣行的範數(Norm)為1的LU分解。藉此就可以在LU分解後，將向量層面經過量化的天線波束賦形矩陣回饋到傳送端。我們也會證實，在傳送端及接收端天線個數不多於10的情形下，LU分解在計算複雜度上會低於奇異值分解。而且利用LU分解的向量層面量化比起利用奇異值分解的向量層面量化，在回饋效率及編碼本(Codebook)搜尋時間都有較好的效能。最後，藉由模擬的結果，可以顯示出LU分解的系統比起奇異值分解的系統，會有較佳的誤碼率(BER)效能。而在信雜噪比(SINR)方面，使用LU分解之系統也有較佳的表現。

Spatial multiplexing with transmit beamforming is a technique for multi-input multi-output (MIMO) systems to separate transmit streams that are sent from different antennas. Under the constraint of limited feedback bandwidth, the beamforming matrix must be quantized and then fed back to transmitter. The conventional method utilizes singular value decomposition (SVD) to design the precoder and the decoder, and then feedback the beamforming matrix to transmitter with vector-wise quantization based on Householder reflection. In this thesis, we propose a different approach using LU decomposition (LUD) to obtain the beamforming matrix. We show that LUD has lower computational complexity than SVD, and vector-wise quantization with LUD has better feedback efficiency than that with SVD. Finally the simulation results show LUD system has better bit error rate (BER) performance than SVD system.

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