正交分頻多工 (OFDM，orthogonal frequency-division multiplexing) 技術提供了一種在多重路徑通道傳輸大量、高速資料之方便可行的方法。OFDM具有以下幾個優點：(a) 在實現上可利用FFT (fast Fourier transform)、IFFT (inverse fast Fourier transform) 來減少複雜度與成本；(b) 在對抗 impulse noise 與 multipath fading 方面有良好效果；(c) 由於OFDM子通道頻譜間是相互重疊的，所以具有較高的頻寬使用效率；(d) 因為每一子通道的頻率響應約為定值，所以不需複雜的等化器。基於以上原因，OFDM在高速數位資料通訊與多媒體通訊服務上，已受到相當多的重視。
然而，OFDM技術的一個主要問題，便是OFDM系統較單一載波通訊系統擁有較大的峰值對平均功率比 (PAPR，peak-to-average power ratio)。較大的PAPR會增加類比/數位轉換器與數位/類比轉換器的複雜度，並且降低射頻端功率放大器的效率。在降低PAPR的方法中，selected mapping (SLM) 與partial transmit sequences (PTS) 二方法，因為具有優良的PAPR縮減表現，而且對整個訊號頻譜並沒有產生破壞性的影響，而受人矚目。然而，此類方法的一個缺點便是需要有一組IFFT，以產生可供此類方法選擇的“候選人”訊號，因此SLM與PTS方法所包含的計算複雜度皆很高。在本篇論文中，我們將探討縮減SLM計算複雜度的方法，並且提出一些具有低複雜度的PAPR縮減架構。
首先，我們提出兩個低複雜度不需要乘法的轉換，並利用此轉換替代在SLM方法中的IFFT，以對N點的IFFT來說，我們所提出的轉換只需3N個複數加法，相較於IFFT來說，其所需的計算複雜度明顯的降低許多。我們所提出的轉換的基本想法是我們可以利用在SLM方法中的一個IFFT輸出去產生另外一個IFFT輸出，而不需真正去計算另外一個IFFT的運算。利用這兩個低複雜度的轉換，我們提出了一些新的SLM架構與一個結合SLM與PTS的架構，在這些架構中至少有一半的IFFT轉換被省略。經由電腦的模擬驗證，我們得知這些新的SLM架構，不僅具有較低的計算複雜度，而且在PAPR縮減效能方面與傳統的SLM架構大致相同。
接下來，我們擴展上述的兩種轉換，設計出一類新的具低複雜度轉換，此類轉換同樣不需要複數的乘法運算，並且可用以取代在傳統SLM方法中的IFFT。利用此類轉換，我們提出了兩個新的具有極低複雜度的SLM架構；在第一個架構中我們只需要一個IFFT來產生所需的一組“候選人”訊號，而在另一個架構中，我們只需要二個IFFT。經由電腦的模擬驗證，我們得知第一個新的SLM架構，其在PAPR縮減效能方面較傳統的SLM架構略差一點，而第二個新的SLM架構則具有與傳統的SLM架構幾乎一樣的PAPR縮減效能。雖然這兩個新的SLM架構可以極低的計算複雜度來提供良好的PAPR縮減效能，然而此兩個方法中所使用的低複雜度轉換，卻會對某些子通道上要傳送的資料能量造成一些衰減，進而增加了整個系統的資料錯誤率 (BER，bit error rate)。
為了解決上述問題，我們修改了上述二個新的SLM架構。在修改的SLM架構中，我們其實只是簡單地將Hadamard轉換放於各個原先架構中的IFFT之前。放置Hadamard轉換的目的是將所要傳送的資料打散到各個OFDM子通道中而不是集中在一個子通道上，如此，某一通道中的能量衰減效應並不會完全集中在某一特定的資料上，因而資料錯誤率也不致增加太多。利用此種方法，我們可以有效的解決之前二個SLM架構中的資料錯誤率增加的問題，而其所需增加的計算複雜度卻是極少，因為Hadamard轉換可用極有效率的快速Hadamard轉換加以實現。經由電腦的模擬驗證，我們發現最後所提出的二個修改過的SLM架構不僅資料錯誤率增加很少，而且相較於傳統的SLM架構，具有更優良PAPR縮減效能。

Due to the resistance to multipath channel effects, orthogonal frequency-division multiplexing (OFDM) schemes have been adopted for a number of high-bit-rate transmission systems. One major drawback of OFDM is the high peak-to-average power ratio (PAPR) of the output signal. Selected mapping (SLM) and partial transmit sequences (PTS) methods can provide good performance on PAPR reduction and have no adverse effects on the signal spectrum. However, these methods require a bank of inverse fast Fourier transforms (IFFT’s) to generate a set of candidate signals and thus involve high computational complexity. In this thesis, we focus on reducing the computational complexity of the SLM method and present several new low-complexity schemes for PAPR reduction.
In this thesis, we first propose two low-complexity multiplication-free conversion processes to replace the IFFT’s in the SLM method, where each conversion process for an N-point IFFT involves only 3N complex additions. The basic idea of these proposed conversions is that we can utilize one IFFT output to generate another IFFT output in the SLM method. With these proposed conversions, we then develop several new SLM schemes and a combined SLM & PTS method, in which half of the IFFT blocks are reduced at least. Computer simulation results show that, as compared to the conventional SLM method, these new SLM schemes have approximately the same PAPR reduction performance under the same number of candidate signals for transmission selection.
To further reduce the complexity of the SLM scheme, we extend the first-part results to form a new kind of low-complexity conversions. By using these conversions to replace the IFFT’s in the conventional SLM method, we develop two novel SLM schemes with much lower complexity than the conventional one; the first method uses only one IFFT block to generate the set of candidate signals, while the second one uses two IFFT blocks. Computer simulation results show that, as compared to the conventional SLM scheme, the first proposed approach has slightly worse PAPR reduction performance and the second proposed one reaches almost the same PAPR reduction performance. Although these two proposed SLM schemes can provide pretty good PAPR performance with very low computational complexity, the conversions they use may attenuate the transmitted signal power on some subcarriers, and would degrade the bit-error-rate (BER) performance of the OFDM systems.
To alleviate the above BER degradation problem, we modify the two proposed SLM schemes by inserting a Hadamard transform before the IFFT operation. The purpose of the Hadamard transform used here is to spread the transmitted data to all subcarriers such that the signal space diversity of the transmitted data is increased. The increased computational complexity of the modified SLM schemes is not significant, because the Hadamard transform can be implemented efficiently by a fast algorithm. Computer simulation results show that the modified SLM approaches not only gain BER improvements over the original proposed SLM schemes, but also have better PAPR reduction performance than the conventional SLM scheme.

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