Adaptive Analog Fountain for Wireless Channels Mahyar Shirvanimoghaddam, Yonghui Li and Branka Vucetic School of Electrical and Information Engineering The University of Sydney Sydney, NSW, Australia Emails:{mahyar.shirvanimoghaddam, yonghui.li, branka.vucetic}@sydney.edu.au Abstract—In this paper, we propose an analog rateless code to achieve high spectral-efficient adaptive transmission and increase the system throughput in AWGN channels. In the proposed analog rateless coding scheme, each coded symbol is generated from a number of information bits that are selected uniformly at random and multiplied by some real values obtained randomly from a predetermined probability distribution function, called weight distribution. The analog rateless codes can be described by a weighted bipartite graph. However, unlike the conventional bipartite graph, where the combining coefficients are the binary symbols, the combining coefficients in the weighted bipartite graph of analog rateless codes are real numbers selected from a finite set. As a result, the conventional sum-product decoder cannot be directly applied. We have developed a simple decoding algorithm, called 2-Sum verification decoder, for the proposed analog rateless codes. Its performance is evaluated by using Sum- Or tree analysis. The code degree and weight distributions are optimized to maximize the error recovery probability of the 2- Sum verification decoder. Simulation results shows the proposed code can approach the channel capacity within one bit across a wide range of SNRs. Index Terms—Adaptive Analog Fountain, Analog Rateless Codes, Wireless Channel. I. I NTRODUCTION Transmission of signals over wireless links suffers from se- vere channel impairment, including noise, fading, pathloss and interference. Thus, how to effectively increase the throughput of a wireless transmission in such time-varying channels has been one of the key research focuses in wireless communica- tions. Ideally, to achieve high throughput, the communication protocol needs to adapt well to variations in noise, fading, interference and so on, while performing well in all channel conditions. One solution to this problem is to use a large number of physical layer configurations, such as channel codes with different design rates, signal constellations, and bit-to-symbols mapping strategies. Most existing systems use such an adaptive approach. For example, the high-throughput mode of 802.11n uses an adaptive modulation and coding (AMC) scheme based on LDPC codes, where 16 modulation and coding schemes (MCS) with 4 code rates and 4 different modulation types are used [1]. However, in such adaptive approaches, the transmitter needs to select between these large number of configurations according to the feedback of real-time channel conditions obtained from the receiver. This dramatically in- creases the system complexity. Furthermore, in cases of rapid or unpredictable channel variations, the transmitter cannot precisely follow the channel condition, leading to a significant performance loss in the wireless network. Therefore, how to design the adaptive transmission without the knowledge of channel statistics or conditions at the trans- mitter becomes increasingly important. One possible solution is to use rateless codes between the sender and receiver. Rateless codes were originally designed and optimized for erasure channels [2, 3], but later extended to noisy channels [4], such as binary symmetric channel [5], Gaussian channel [6], and fading channel [7]. The main property of rateless codes that makes them suitable for wireless channels is that the number of generated coded symbols is potentially limitless. As a result, the transmitter can keep transmitting until the desti- nation successfully decodes all information symbols and sends an acknowledgement to the transmitter. In AWGN channels, in order to achieve the capacity-approaching performance, the transmitter needs a priori knowledge of channel SNR and design the degree distributions accordingly. However, the degree distribution functions are very sensitive to channel SNRs, thus the optimized degree distribution in a specific SNR may not work properly in other SNRs [8]. The problem of designing of rateless codes for unknown Gaussian channels was discussed in [8], but only existing degree distributions optimized for erasure channel are used. In [6], a layered approach was proposed to use fixed- rate channel codes, like LDPC, and generate transmission symbols in a rateless fashion. It is shown that by increasing the number of layers, the channel capacity can be achieved by the proposed rateless code. However, the practical deployment of this method requires further research. Another non-linear type of rateless codes based on hash functions was proposed in [1] for Gaussian channels. Despite the good performance, the encoding and decoding complexity of such codes are prohibitively high and thus impractical for real applications. Recently, a seamless rate adaptation strategy was introduced in [9], which uses a regular weighted bipartite graph to generate coded symbols. The main idea behind this scheme is to directly map information bits to the modulated signals, and generate as many signals as required by the receiver. The degree of coded symbols is fixed at 8 and all information bits have the same degree. Despite its good performance, no systematic design strategy has been developed to find the optimum degree distribution of the proposed code. In this paper, we propose a new type of rateless code, referred to as the analog rateless codes. Each analog rateless