# Analog Codes on Graphs

- 2006

#### Abstract

There are many examples of channels for which a combined coding and modulation scheme is needed to approach capacity. In some cases, it is desirable to employ codes which result in a graceful degradation of information rate with decreasing SNR[2, 3, 4]. It is also known that in some applications like Broadcast channels[1], it is impossible to achieve capacity if source and channel coding are separated. In these situations, codes over larger alphabets can be advantageous. In this work, we consider analog codes, whose alphabet is the real line. Traditionally, decoding analog codes has been difficult[5, 6, 7], requiring estimation techniques like Kalman filtering. In this paper, we use capacity-approaching codes defined on graphs[8, 9, 10] to construct analog codes that can be efficiently encoded and decoded in practice. To this end, we introduce a novel superposition strategy that admits infinitely many resolutions. This superposition strategy allows us to present an effective iterative decoder for the analog codes proposed herein, based on the sum product algorithm. The resulting coding scheme performs close to the Shannon capacity of a memoryless, band-limited AWGN channel over a wide range of SNRs. Furthermore, we construct bandwidth efficient codes by truncating analog codes. We find that these perform well in comparison to MPSK cutoff rates.