Sending a Bivariate Gaussian Over a Gaussian MAC

Abstract

We study the power versus distortion trade-off for the distributed transmission of a memoryless bi-variate Gaussian source over a two-to-one average-power limited Gaussian multiple-access channel. In this problem, each of two separate transmitters observes a different component of a memoryless bi-variate Gaussian source. The two transmitters then describe their source component to a common receiver via an average-power constrained Gaussian multiple-access channel. From the output of the multiple-access channel, the receiver wishes to reconstruct each source component with the least possible expected squared-error distortion. Our interest is in characterizing the distortion pairs that are simultaneously achievable on the two source components.We present sufficient conditions and necessary conditions for the achievability of a distortion pair. These conditions are expressed as a function of the channel signal-to-noise ratio (SNR) and of the source correlation. In several cases the necessary conditions and sufficient conditions are shown to agree. In particular, we show that if the channel SNR is below a certain threshold, then an uncoded transmission scheme is optimal. We also derive the precise high-SNR asymptotics of an optimal scheme.

A Unified Approach to Hybrid Coding

“…In Section II, we establish a sufficient condition for lossy communication that recovers and generalizes several existing results on joint source-channel coding over this channel model. In particular, when specialized to the Gaussian communication problem studied in [37], our result recovers the hybrid analog-digital scheme derived by Lapidoth and Tinguely. 2) Relay Networks: Following [39], we propose a channel coding scheme for noisy relay networks based on hybrid coding in Section III.…”

“…Corollary 2 (Berger [52] and Tung [53]): 6] via a hybrid analog-digital scheme that combines uncoded transmission and vector quantization. We remark that the encoders used in the proof of Theorem 1 are the same as the ones described in [37], but the exact operation of the decoder is somewhat different. Lapidoth and Tinguely developed a Gaussian-specific minimum-distance decoding technique to go around the issue of dependency between message and codebook.…”

“…Similar schemes have been proposed [33]- [36] for point-to-point channels with state and source side information at the receiver. In the multiuser setting, Lapidoth and Tinguely [37], [38] showed that hybrid analog-digital coding is asymptotically optimal in the limiting regime of high signal-to-noise ratios for the problem of sending a bivariate Gaussian source over a Gaussian multiple access channel, where each transmitter has access to only one source component, while Tian, Diggavi, and Shamai [24] showed that hybrid analog-digital coding is optimal for the dual problem of sending a bivariate Gaussian source over a Gaussian broadcast channel, where each receiver is interested in only one component of the source.…”

Joint source-channel coding via hybrid coding

“…The above sufficient condition generalized several previous results in the literature including the ones of Berger [14] and Tung [15] for distributed lossy source coding, Lapidoth and Tinguely [4] for lossy communication of a bivariate Gaussian source over a Gaussian multiple access channel, Han and Costa [16] for lossless communication of correlated sources over a broadcast channel, and the result of Tian, Diggavi, and Shamai [5] for lossy communication of a bivariate Gaussian source over a Gaussian broadcast channel, and of Han and Kobayashi [13] for communication of independent messages over the interference channel. Furthermore, the above theorem improves upon the previous result by Liu and Chen [17] on lossy communication over the interference channel.…”

“…Hybrid coding has been previously proposed as an alternative to Shannon's source-channel separation architecture [2], [3], [4], [5]. However, while most existing works in the literature focus on communication of Gaussian sources over additive Gaussian noise channels under squared error distortion measure, we propose a new generalized architecture that can be applied to any lossy communication problem over discrete memoryless channels.…”