具週期性前導序列之正交分頻多工/正交分頻多工存取系統中最大似然載波頻率偏移估計

Abstract

正交分頻多工（orthogonal frequency division multiplexing ; OFDM）是一多載波調變技術，以其高頻譜效率而廣為人知，目前此技術已被使用於許多無線通訊系統中。OFDM之所以有高的頻譜使用效率是因為將頻譜切成多個子載波，且子載波間相互重疊且正交。然而當載波頻率偏移(carrier frequency offset; CFO)存在時，正交性會遭到破壞，造成子載波間的相互干擾而降低通訊的品質。因此在真正的數據傳輸前，我們必須先估計並補償CFO。本論文旨在研究OFDM相關系統之CFO最大似然(Maximum likelihood; ML)估計，我們的系統假設在接收端可以接收到未知的週期性訊號。本論文分別就OFDM、正交多頻分工存取(orthogonal frequency division multiple access; OFDMA)上行(uplink)和合作式放大傳輸(amplify-and-forward; AF) OFDM系統，提出其對應之ML解。 在傳統的正交分頻多工系統中，ML解需計算相關矩陣(correlation matrix)的反矩陣，但隨著子載波數的增加，其複雜度將大到無法實現。為避免直接計算該反矩陣，本論文提出一個新的ML法來直接求得CFO的閉合解(closed-form solution)，此法的優點是計算複雜度極低，該ML法亦可延伸至同時估計CFO和符元時間偏移(symbol timing offset)。為評估在CFO的估計效能，我們導出該系統之CFO的理論下限也就是CRB (Cram r rao bound)。本論文接著探討交錯型(interleaved) OFDMA系統上行中的CFO估計問題。由於系統特性使然，此系統接收的訊號一定是週期性的，因此無需額外的訓練訊號。跟OFDM系統一樣我們需計算一相關矩陣的反矩陣，但OFDMA系統因牽涉到多使用者的CFO估計，所以此反矩陣很難計算。我們因此提出使用級數展開法(series expansion) 將此反矩陣展開，並保留適當的低階項後，我們可以得到一個閉合解，然後以解根的方式解得CFO。最後我們推導出相對應的CRB，實驗發現無論是在傳統的OFDM系統或OFDMA系統中，本論文所提的ML解皆可逼近CRB。 最後，本論文探討AF-OFDM系統中的CFO估計和功率分配問題，在此系統中雜訊為有色雜訊(color noise)，然而此特性並未在之前的研究中被討論，因此現有的ML解皆非最佳，現有的CRB也不適用。因為CFO的ML解變的相當複雜無法導出閉合解，本論文提出使用梯度下降法(gradient descent method)來求得CFO的ML解。此外，有色雜訊使得此系統中的CRB過於複雜以致於無法得到一簡單的閉合解。本論文做了一些近似來推導出CRB的閉合解。實驗發現此近似的CRB是準確的且我們所提出的ML解可以逼近此CRB。我們接者提出兩個功率分配法將CRB最小化，並利用梯度下降法求得其解。模擬發現，本論文提出的功率分配法不但可以大幅改善CFO估計的準確性，也可提升了系統的訊雜比(signal-to-noise ratio)。

Orthogonal frequency division multiplexing (OFDM), being a multicarrier modulation technique, is well known for its high spectral efficiency and has been adopted in many wireless systems. The high efficiency of OFDM comes from the fact that the spectrums of its subcarriers are overlapped and orthogonal each other. However, when the carrier frequency offset (CFO) is present, the orthogonality between the subcarriers is lost and inter carrier interference (ICI) is induced causing performance degradation. As a result, CFO must be estimated and compensated before the actual transmission can be conducted. In this dissertation, we study the maximum-likelihood (ML) methods for CFO estimation in OFDM-based systems, assuming that unknown periodic received sequences are available at the receivers. Specifically, we solve the ML CFO estimation problems in conventional OFDM, orthogonal frequency division multiple access (OFDMA) uplink, and cooperative amplify-and-forward (AF) OFDM systems.

In conventional OFDM systems, the ML CFO estimator requires the inversion of an correlation matrix. When the number of subcarriers is large, the computational complexity can become prohibitively high. We then develop a new ML method that can yield a closed-form solution without the inversion. The advantage of the proposed method is that the required computational complexity is low. The proposed method is further extended to a joint CFO and symbol timing offset (STO) estimation. Theoretical Cram$\acute{e}$r-Rao lower bounds (CRBs) are also derived to verify the optimality of the proposed approaches. We then investigate the CFO estimation problem in interleaved OFDMA uplink systems. Since the periodicity is inherent in OFDMA systems, no training sequences are required. As previously, there is an correlation matrix in the likelihood function to be inverted. Since multi-users are involved, the CFOs in the likelihood function become intractable after the matrix inversion. We propose using a series expansion method to express the inverted matrix. By properly truncating the expansion, we can obtain a closed-form expression, solve the optimum CFOs with a root-finding method, and derive the corresponding CRB. Simulations show that the performance of the proposed method can approach the CRB.

We finally consider the ML CFO estimation and the power allocation problem in cooperative AF-OFDM systems. In this scenario, the noise at the destination becomes colored. The colored-noise problem has not been considered before. Thus, the existing ML methods are not optimal and the existing CRBs are not valid. Since the likelihood function is a complicated function of the CFO, we then propose a gradient-descent method to solve the problem. The expression for the CRB for the CFO estimation in AF-OFDM systems is even more complicated and a simple solution cannot be obtained. We then propose an approximation method such that a closed-form solution can also be derived. Simulations show that the approximated CRB is accurate and the performance of the proposed gradient-descent method can approach the CRB. Minimizing the approximated CRB, we further propose two power allocation algorithms (PAA), implemented with constrained gradient-based method, for the source and relays. Simulations show that not only the performance of the CFO estimation is greatly enhanced, but also the signal-to-noise ratio (SNR) between source and destination is improved.