正交分頻多工系統之最大可靠度頻率偏移估計法

Abstract

本論文主要在探討如何解決正交分頻多工系統 (Orthogonal Frequency Division Multiplexing, OFDM) 中存在的載波頻率偏移問題。載波頻率偏移的不確定性取決於OFDM 所應用的系統，造成的頻率偏差可能會達到幾十個次載波（Subcarrier）的頻寬和，此時載波頻率偏移通常被分割成整數及小數偏移兩部分處理。本論文提出完整的最佳及次佳的解決方案以補償載波頻率偏移的問題。 基於最大可靠度原則 (Maximum Likelihood Principle) 以及可獲得領航符元 (Pilot Symbols) 的情形下，我們推導了多個高效率的演算法以估計載波頻率偏移的整數及小數部分。藉由使用多個相同的領航符元以推導最大可靠度小數頻率偏移估計法，我們證明了所有先前以相關性原則 (Correlation-based) 所獲得的演算法僅僅使用了一部份的充分統計 (Sufficient Statistic)。 但此最大可靠度估測器必須在可能發生載波頻率偏移的範圍內做全面性搜尋並比較而選出一個具有最大可靠度的載波頻率偏移估計值；我們轉換此種需要全面性搜尋的方法，使其變成僅需搜尋並解出一個頻譜多項式（Spectrum polynomial）的根來估測載波頻率偏移而大大降低全面性搜尋的計算複雜度。同時進一步適當的縮減頻譜多項式的維度，我們可明確地得到此多項式根的數式表示法，因此更加簡化搜尋根的過程。 以上所提出的利用解頻譜多項式的根並搜尋而獲得最佳根的過程是可以被更進一步再簡化而幾乎不會付出任何性能損失的代價；在此我們使用一個事實即可估測出最大可靠度載波頻率偏移估計值的根必定落在此頻譜多項式的一個特定的因式上，而解出此因式的根的過程是幾乎不花費任何計算量的，因此此估測法非常吸引人，因為其具有相當低的複雜度並且即使在低訊號雜音比 (Signal to Noise Ratio, SNR) 時仍有優秀的性能表現。我們也提出詳細的數學分析，分析出這些演算法的平均平方誤差值（Mean-squared Error, MSE）並以數學模擬來驗證我們的分析結果。 一旦小數部分的載波頻率偏移被估計及補償後，整數載波頻率偏移所造成的頻率不確定性問題也是必須被即刻解決的。在此我們提出一個特殊領航符元結構使得小數及整數部分的頻率偏移都可利用此符元估算。根據此符元結構，一個新式的最大可靠度（ML）整數載波估計法被提出。由於此符元結構是一些先前文獻上的提案所設計的符元的延伸，因此我們的推導也提供這些提案一個整體的理論基礎。為了降低計算複雜度，我們以此最大可靠度估計法為基礎而研究並探討其他可能變形的估計法。最後，我們建議使用一個兩階段的方法來進一步降低複雜度而不會付出估測性能損失的代價。

The frequency uncertainty associated with an orthogonal frequency division multiplexing (OFDM) signal, depending on the application, can be as large as many tens subcarrier spacings. This carrier frequency offset (CFO) uncertainty is usually partitioned into an integer part and a fractional part and must be compensated for before other synchronization and demodulation operations take place. This thesis presents complete optimal (in the generalized maximum likelihood sense) and near-optimal solutions for the above CFO compensation problem. Based on the generalized maximum likelihood (ML) principle and assuming the availability of pilot symbols, we derive efficient algorithms for estimating either the fractional part or the integer part of the CFO, assuming a fractional-then-integer frequency synchronization procedure. By deriving the ML fractional CFO estimates based on repetitive identical training symbols, we show that all previous correlation-based algorithms use only a part of the sufficient statistic. We then convert the problem of obtaining the ML solution from searching exhaustively over the entire uncertainty range to that of solving a spectrum polynomial whence greatly reduce the computational load. By properly truncating the spectrum polynomial, we obtain a closed-form expression for the corresponding zeros so that the root-searching procedure is much simplified. The complexity of locating the desired root is further reduced at almost no expense of performance degradation by an alternate algorithm that uses the fact that the solution is related to the root of a special factor of the polynomial. This alternate method is very attractive for its simplicity and excellent performance that, even at low signal-to-noise ratios (SNRs), is very close to the corresponding Cram\'{e}r-Rao lower bound. We also present detailed analysis of the mean-squared error (MSE) performance and validate our analysis by simulations. The frequency ambiguity due to the presence of an integer CFO should be resolved once the fractional part has been estimated. We propose a special pilot symbols structure that can be used in both fractional and integer CFO estimations. A class of new ML integer CFO estimation algorithms is derived and the associated performance is presented. The pilot structure is a generalization of some earlier proposals and our derivation gives these proposal a unified theoretical foundation. To reduce the estimation complexity, we examine some variations of the ML estimate and suggest a two-stage method whose performance is almost as good as that of the optimal estimate.