Information Velocity of Cascaded Gaussian Channels with Feedback

TL;DR

It is shown that a positive velocity exists as long as the arrival rate is below the individual Gaussian channel capacity, and an explicit positive lower bound on the velocity is provided, for any packet size is given.

Abstract

We consider a line network of nodes, connected by additive white Gaussian noise channels, equipped with local feedback. We study the velocity at which information spreads over this network. For transmission of a data packet, we give an explicit positive lower bound on the velocity, for any packet size. Furthermore, we consider streaming, that is, transmission of data packets generated at a given average arrival rate. We show that a positive velocity exists as long as the arrival rate is below the individual Gaussian channel capacity, and provide an explicit lower bound. Our analysis involves applying pulse-amplitude modulation to the data (successively in the streaming case), and using linear mean-squared error estimation at the network nodes. Due to the analog linear nature of the scheme, the results extend to any additive noise. For general noise, we derive exponential error-probability bounds. Moreover, for (sub-)Gaussian noise we show a doubly-exponential behavior, which reduces to the celebrated Schalkwijk-Kailath scheme when considering a single node. Viewing the constellation as an "analog source", we also provide bounds on the exponential decay of the mean-squared error of source transmission over the network.

Figures (4)

• Figure 1: Block diagram of the system. ${X_{r}\left( t \right)}$, ${Y_{r}\left( t \right)}$, and ${Z_{r}\left( t \right)}$ are the channel input, output, and noise, respectively, at node $r$ at time $t$. Each $T$ time steps, a new packet is generated. At every time step, all the nodes decode all the hitherto arrived packets.
• Figure 2: Data packet streaming: visualization packet estimations at nodes $r = 0, 1, 2$ across time, for time period $T=4$, and four bits per packet $\ell=4$. The packets at the transmitter (node $0$) indicate arrival time to transmitter, while the packets at nodes $1$ and $2$ indicate decoded packets.
• Figure 3: Successively refined source: The achievable MSE exponent. The plot depicts the exponent as a function of velocity, at fixed SNR $P=10$, as a function of the refinement rate $R$. The upper curve is $E_1(v)$, which is $E_S(v)$ for any rate above the capacity $C=\log(11)/2\cong 1.2$ nats. Below one sees the first region of the exponent at rates $1$, $0.5$ and $0.1$ nats (from upper to lower curves); in the second region ($\frac{1-\eta}{\eta}\leq v \leq P$) the exponent equals $E_1(v)$.
• Figure 4: Lower bound on the streaming IV as a function of the rate average rate $R$\ref{['eq:def:R']} for SNRs $P = 0.1, 1, 10, 100$ and the linear limit $P \to 0$. Relative coordinates are used: the lower bound on the streaming IV of Theorem \ref{['thm:main']} is normalized by the lower bound on the single-packet IV $P$ of Theorem \ref{['thm:1source']}, whereas the average rate $R$ is normalized by $C$\ref{['eq:C']}.

Theorems & Definitions (25)

• Remark 1: average rate
• Remark 2: finite horizon
• Remark 3: delayed hops
• Lemma 1
• Lemma 2
• Theorem 1
• Remark 4
• Lemma 3
• Theorem 2
• Lemma 4