使用三維空間填充曲線的類比通道編碼

Abstract

在本篇論文中，我們採用類比通道編碼的方式，將訊號傳經由高斯白雜訊通道。這個方式是建立在香農及科捷利尼科夫的映射方法，就是將一個獨立且同分布的輸入源樣本映射到通道樣本內的空間填充曲線上的一個點。在編碼器上，我們採用三維空間希爾伯特空間填充曲線，且同時包含了不同的分形維度的希爾伯特空間填充曲線，用於一個一比三的頻寬擴展映射。藉由使用不同分形維度的希爾伯特空間填充曲線，我們能找到對於不同的通道訊雜比有個比較好的效能。在解碼器上，我們使用最大似然法，及最小平均方差法。接著我們調查得到曲線長度和系統效能上的關係，和討論在低通道訊雜比的時候的跳線議題、在較高通道雜訊比時的效能趨近線，以及在希爾伯特曲線中的直角議題。最後，我們將三維空間希爾伯特空間填充曲線效能和理論可達到最佳效能做比較，以及將二維空間希爾伯特空間填充曲線的效能和三維空間希爾伯特空間填充曲線的效能做比較。

In this paper, we consider the problem of designing bandwidth expansion with analog channel coding schemes for transmitting sources over AWGN channels. The scheme is based on the Shannon-Kotel’nikov mapping, where the samples of an i.i.d source are mapped into a channel samples using space-filling curves. In the encoder, we use the three-dimension Hilbert space-filling curves with different fractal dimension for 1:3 bandwidth-expansion mapping. By using different fractal dimension Hilbert curves, we can have better performance for the different channel SNR. In the decoder, we use ML and MMSE decoder. And we investigate the relationship between the length of curve and the performance of the system, and discuss the jump-line issue at low channel SNR, the approximate line of the system performance at higher channel SNR and the square-corner issue in the Hilbert curve. Finally, we compare the performance of the 3D space Hilbert curves for 1:3 mapping with OPTA, and compare the 2D space Hilbert curves for 1:2 mapping with 3D space Hilbert curves for 1:3 mapping.