本論文將排列陣列應用在物理層和跨層通訊。在物理層，我們將排列陣列對應到天線，進而傳送位元。在跨層通訊，我們將排列陣列對應到網路層的傳輸路徑，進而傳送資料。
本論文首先分析排列陣列在物裡層快衰變通道使用一根天線的情況。我們首先分析其位元錯誤率，透過分析結果，給出在使用一根天線的情況，最好的傳輸模式為，排列陣列和正交振幅調變的位元分配的平衡。更進一步的分析多樣性在排列調變使用不同參數下的影響，給出了分配正交振幅調變應該要和排列陣列的漢明矩陣裡的最小值相同。接著分析在物理層慢衰變通道使用一根天線的情況。首先分析其位元錯誤率，因為慢衰退通道使得不同的時間點通道有了相關性，使用快通道的方法只能在特別的情況求得解析解。對於一般的情況，我們使用伽馬分佈來近似其結果。最後透過一個類神經網路來學習近似的結果來降低計算位元錯誤率的複雜度。由於我們是使用伽馬分佈來近似，伽馬分布的參數有其物理意義，其中一個參數對應於通訊的多樣性，我們可以得到天線的數量會影響到排列調變在物理層通訊的效能影響。最後針對使用在快通道物理層下使用多根天線的情況，我們透過分析位元錯誤率，給出了一個完整的傳輸建議，針對多樣性的位元的分佈和使用一根天線時一樣，而在使用正交振幅調變最小使用到四八。
在跨層通訊的部分，我們分析了在跨層通訊快通道下使用多條路徑的情形。我們分析其位元錯誤率，由於有兩種不同的動差方程式。分析的結果，需考慮到兩種不同的動差方程式，進而求得最後的結果。由於分析的結果較為複雜，我們使用在物理層得到的結論。結果應證在物理層得到的結果在跨層的也是通用的。
最後我們分析了跨層通訊的下限錯誤率，得到下限錯誤率和傳送正交振幅調變的數量和排列陣列的漢明矩陣的最小值有相關。除此之外我們也提出旋轉排列傳輸可以進一步的改善排列傳輸在物理層及跨層通訊之間的效能。對於排列陣列，我們也提出一個兩階段式的演算法，來產生出不同參數的排列陣列。

The thesis is focus on permutation array applying to physical layer communication and cross-layer communication. In the physical layer communication, permutation array is mapping to the antennas in the transmitter. In cross-layer communication, permutation array is mapping to the paths which are construct from source to destination in the ad-hoc cognitive radio network.
The thesis is first analyze the fast fading physical layer communication which activate only one transmit antenna. Then we analyze the bit error rate. Base on the analysis results, we show that distributing bit balance between QAM and permutation array can get the best performance. Furthermore, we analyze the diversity result of distributing the QAM, we get the better way is to distribute the QAM equal to the minimal value in the hamming distance matrix of permutation array. Next we analyze the slow fading physical layer communication with activating one transmit antenna. Then we analyze the bit error rate. Due to the slow fading channel, the analysis we use in fast fading only work in special case. To general case, we use gamma distribution to approximate. In order to reduce the complexity of evaluating the bit error rate, neural network is used to learn the result of gamma approximation. Due to the gamma approximation, we get some physical meaning from the parameter. One parameter of gamma distribution is diversity, so we find that the number of transmit antenna have a big impact on the performance of permutation modulation in slow fading physical layer communication. For the fast fading physical layer communication which activating more than one transmit antenna, we give that the minimal constellation of QAM should not be less than 4 and 8.
For cross-layer communication which using multiple path, we analyze the bit error rate. Because we have two different moment generating function, the result should base on these two moment generating function. Due to the complex result, we apply the same way use in physical layer communication. The good performance imply the result in physical layer can also be applied to cross-layer. Finally we also analyze the error bound, we get the result that the error bound is related to the number of transmitting QAM and the minimal value in hamming distance array in permutation array.
In addition, we propose a rotated permutation transmission to improve the performance in both physical layer communication and cross-layer communication. For permutation array, we also propose a two level algorithm to generate the permutation array for different parameters.

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