迴旋碼編碼器其非均等錯誤保護特性之研究

Abstract

在很多通訊系統中，我們所傳送的訊息也許有某些部分是較為其它部分來的重要。因此當我們要傳送這個訊息進入通道前，我們希望這訊息中某些重要的部份能有更大的保護，進而在接收端所得到的資料裡重要的部份能夠更可靠。傳送訊息進通道前我們會對訊息使用非均等錯誤率編碼來給予不同程度的保護，傳送進入通道後，再將接收端所收到的向量透過解碼來取得訊息。擁有較大保護的訊息部份將有較高 的錯誤更正能力，使得解碼出來的訊息能夠與原始傳送的訊息更為相近。早期大都是運用線性區塊碼來進行非均等錯誤率保護，漸漸的也開始發展使用迴旋碼來進行。文獻中已有研究指出，在任何的迴旋碼中，都會存在一個最佳編碼器來實行非均等錯誤率保護。很不幸地，並非所有迴旋碼都能有兼具最小延遲元件及最佳非均等錯誤率保護能力的最佳編碼器。因此給定任一迴旋碼，我們希望都能夠找到一個擁 有最少延遲元件的最佳編碼器。利用我們提出的定理結果，可以直接算出實現一個迴旋碼編碼器所需要的最小延遲元件數，並且利用代數的方法來解釋出為什麼在一個(n,k)迴旋碼的多項式編碼器中，所有k×k 子矩陣其行列式之最大的度值不會超過實現此編碼器所需要最少的延遲元件數。最後，我們提出一個簡單的演算法來得到具有最少延遲元件數的最佳編碼器，並且保證此編碼器所產生出來的字碼，經過通道後，將接收端所接收到的向量解碼不會發生無窮項位元錯誤的情形。最後，我們亦證明了某一些迴旋碼皆會存在一個兼具最少延遲元件與最佳非均等錯誤率保護能力的最佳編碼器。

In many communication systems, the transmitted data may have a structure that some parts of the information are more important than that in the other parts. Channel coding with unequal error protection (UEP) is usually employed in such systems so that stronger protection could be applied to the important parts to enhance the quality of communication. At the earliest, block codes were used to perform UEP mostly. Recently, studies of UEP have been expanded to convolutional codes. Previous results showed that there exists at least one UEP-optimal generator matrix with the greatest separation vector for every convolutional code. However, unfortunately, not all convolutional codes can have a UEP-optimal generator matrix which also keeps the minimal complexity for both of encoding and decoding. In this thesis, we show that we can calculate the McMillan degree of a generator matrix directly without decomposing it by using the Smith Algorithm. From this result, we also illustrate why the internal degree of a polynomial generator matrix is not greater than its McMillan degree. Besides, we provide a procedure for searching an optimal polynomial generator matrix with the lowest McMillan degree, and also we show that for some classes of convolutional codes there always exist generator matrices which are both optimal and minimal.