The Noisy Feedback Converse for a Gaussian Channel

Introduction

In [1], one of the authors (AC) determined a lower bound on the reliability function of a Gaussian noise channel with Gaussian noisy feedback. The aim of the present paper is to provide an upper bound on the very same reliability function.

Fig. [fig:my_label] shows the events which occur in a noisy feedback communication system with time increasing from left to right.

In Fig. [fig:my_label2], which is for the noiseless or perfect feedback case, the y and f stochastic processes have been equated.

In the figures, the arrows indicate dependencies between the stochastic processes. In Fig. [fig:my_label3], in the "g" sequence, thre are two component "sub-sequences":

1. The "odd" subsequence which is the x-sequence or input sequence, and

2. The "even" subsequence which is the y/f-sequence or output sequence.

In another view, if we want to just look at the inputs, we can simply `stare' at the x-sequence as in the following figure, Fig. [fig:my_label4].

If we relabel g(1) as h(1), g(3) as h(2), and so on, upto g(2n-1) as h(n), then we obtain the next figure, Fig. [fig:my_label5].

Define ${}={N} {E}$. Then ${}$ is a dynamical system.

In an alternate approach, the following figure, Fig. [fig:my_label6], looks at the even subsequence, which is the output.

Next, we let alpha(1) be g(2), alpha(2) be g(4), alpha(3) be g(6) and so on until we let alpha(n) = g(2n), where n is a positive integer. Further, we let ${}^{'} = {E} {N}$. As a result, we have the mapping from alpha(1) to alpha(2), called ${}^{'}$. Thus, alpha(2) = ${}^{'} ((1))$ and in general,

The random dynamical system of [eq:1] will be having statistics which are independent of time because ${N}$ is time-independent and so also ${E}$ has time-independent stats.

Noisy Feedback Case

If we had a noisy feedback case, then $f(1) y(1)$ but rather $f(1) = y(1) + noise$. So the encoder can be treated as adding this noise before using y(1) to generate x(2). So it is a noisy encoder ${E}_{noisy}$. Thus ${}^{"}={E}_{noisy} {N}$ and so

The random dynamical system of ([eq:2]) will also be having statistics which are independent of time.

Lyapunov Exponent Intuition

In this section we present some intuition on the Lyapunov exponent as it relates to our problem setup.

The Lyapunov exponent of our RDS ${}^{"}$ will be high iff the system diverges in phase space iff two closeby initial conditions lead to divergent trajectories of the system.

But, if the two trajectories are divergent, for closeby initial conditions, then resolution of decoding would have increased. That is, you would easily be able to say, given a trajectory, which initial condition it came from, or at least, that it is from a distinct set of possible initial conditions.

Even if the clouds y1 and y1' clouds had some small overlap, a large Lyapunov exponent would lead to a huge gap in a short time between the resultant clouds. Under ML decoding, this would imply a very low probability of error in short time, or a huge error exponent.

Application of Intuition

Consider a channel with noisy feedback, such as the DMC, the Gaussian channel or a Vector Gaussian channel, which models an ephaptic tract. As per the development in the preceding pages, we can model them as random dynamical systems and find the corresponding Lyapunov exponent. The larger the LE, the larger would be the error exponent.

As a procedure, we can first determine the RDS corresponding to the communication system, then perturb this so that we get a `closest' chaotic communication system. That latter system's Lyapunov exponent will be proportional to a tight upper bound on the original random dynamical communication system's error exponent. Here the assumption is that chaotic systems are a different class altogether.

Finding the Lyapunov exponent

Our first step is to find the Lyapunov exponent of the random dynamical system shown in Equation [eq:2]. For this we will need to refer to the multiplicative ergodic theorem [2] and the book [3].

Before we can write the Lyapunov exponent in the most general case, we need the map $^{'}$. We can specialize to the binary case. That is we consider only antipodal signaling over the AWGN channel on the forward link as well as the feedback link. Thus the noise is Gaussian but at each step we have either a `O' (just a label for `-1') or `1'.

The most general non-noisy encoder would be

Here $m$ is the message, which can also belong to a binary set (two possibilities) and $Y_{1}^{t}$ is a length $t$ bit sequence. This thus represents a feedback encoder.

We cam assume initially that there is only a length 1 feedback. However, for illustration purposes (see Figure [fig:my_label8]), let us consider a length 2 feedback at each step. That is, it is a fixed length feedback code instead of a variable length one.

For this feedback code, we can write

We can write the above succinctly as

where ${a,b,c} {prime, non-prime}$ and ${K}_{a,b,c}$ is a coefficient which is either 0 or 1 and indicates if the corresponding product term is present or not.

For equating with a Random Dynamical System, the encoder has to be a fixed mapping applied at every instant. So, our formulation is applicable perhaps only to fixed length codes as opposed to variable length codes. This is fine for starters, but this assumption will have to be relaxed to find the most general upper bound. A new theory, broadening that of Random Dynamical Systems as defined in Arnold's book [3], might have to be developed ^†Note however that if a random dynamical system is described in terms of a triple, then within the choice of maps we can include the variable length codes [4].

Next we look at the channel. Let it be an additive noise channel. The input-output mapping is therefore the identity map, perturbed by noise. Thus,

Thus we can finally write $^{'}$ for this special case as $^{'}= {E} {N}$. This gives us

to which noise would be added. This is the Random Dynamical System corresponding to the output $y$ of the channel. Our next step would be to consider the binary case where distances are Hamming distances and then to find the Lyapunov exponent which would tell us about the divergence rates of trajectories. We will have to find the perturbed RDS of $^{'}$ with the smallest Lyapunov exponent such that the system is still chaotic in order to upper bound the reliability function.