On Lossy Multi-Connectivity: Finite Blocklength Performance and Second-Order Asymptotics

We consider the lossy transmission of a single source over parallel additive white Gaussian noise channels with independent quasi-static fading, which we term the lossy multi-connectivity problem. We assume that only the decoder has access to the channel state information. Motivated by ultra-reliable and low latency communication requirements, we are interested in the finite blocklength performance of the problem, i.e., the minimal excess-distortion probability of transmitting $k$ source symbols over $n$ channel uses. By generalizing non-asymptotic bounds by Kostina and Verdú for the lossy joint source-channel coding problem, we derive non-asymptotic achievability and converse bounds for the lossy multi-connectivity problem. Using these non-asymptotic bounds and under mild conditions on the fading distribution, we derive approximations for the finite blocklength performance in the spirit of second-order asymptotics for any discrete memoryless source under an arbitrary bounded distortion measure. Furthermore, in the achievability part, we analyze the performance of a universal coding scheme by modifying the universal joint source-channel coding scheme by Csiszár and using a generalized minimum distance decoder. Our results demonstrate that the asymptotic notions of outage probability and outage capacity are in fact reasonable criteria even in the finite blocklength regime. Finally, we illustrate our results via numerical examples.