本篇論文探討在馬可夫高斯通道下不同碼率低密度同位檢查碼(LDPC)的表現
。我們是基於有限域的加法組(Additive Group)來建構同位檢查矩陣的，其矩陣為一個準循環的形式。並分別在不同通道環境中模擬。建立在有限域的低密度同位檢查碼，其在可加性高斯白雜訊(AWGN)通道下擁有很好的表現，更重要的是，這種有限域低密度同位檢查碼被證明具有低的錯誤層次(error floor)，此種特性對於通訊或資料儲存系統相當重要。而大部分建立在有限域的低密度同位檢查碼是準循環的，因此能夠利用簡單的位移暫存器[1]進行編碼且具有低的線性複雜度[2]。在對於有線或無線通訊中，脈衝雜訊是一個重要的議題。由於常見的脈衝雜訊通道模型，屬於無記憶性型，其發生雜訊的時機具有隨機性，無法去描述真實通道的特性，故發展出基於馬可夫鏈特性的雜訊碼可夫高斯通道模型(Markov-Gaussian Channel)。
為了解決脈衝雜訊，在本論文中使用了基於有限域的QC¬-LDPC結合了最大後驗機率(MAP)來進行通道估測並且利用和使用積演算法(SPA)進行解碼以降低位元錯誤率。

In this paper, we study the performance of LDPC code which have different Coding rate over Markov-Gaussian channels. We construct parity check matrices base on subgroups of a finite field, and its matrix is a quasi-cyclic form. And simulation in different channel environment. LDPC code which based on finite fields perform well over the binary-input AWGN channel. Most importantly, these finite-field LDPC codes were shown to have low error floors, which is important for communication and data-storage systems. Most of the LDPC code constructions base on finite fields are quasi-cyclic and hence they can be efficiently encoded using simple shift-registers[1] with linear complexity[2]. In the wired and wireless communication systems Impulse noise is a important issue . The common impulse noise models. they are memoryless, which mean that their occurrences are random. However, they can’t describe the characteristics of the real channel. A memory channel such as Markov-Gaussian is introduced to address the characteristics of the real channel.
To address the impulsive noise, QC-LDPC base on finite field is used in this paper to combine the maximum a posteriori (MAP) Estimation for channel estimation and use the SPA algorithm to reduce the bit error rate.

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