# How Much Information Can Be Sent Reliably from Halifax to Dartmouth Under the Water?

The notion bandwidth efficiency is in the heart of information theory, following from Claude Shannon’s groundbreaking work, that paved the way for the digital age. Claude Shannon, in his famous article in 1948 with the title “A Mathematical Theory of Communication”, proved two important facts that shifted the paradigm of wireless communication from analog to the digital.

First, he showed that we can represent all sorts of information around the world in two digits, 0, and 1, hence the notion of digital communication. He also showed that it is indeed possible to calculate the highest achievable rates on reliable communication. Following from Shannon’s work, calculating the natural limits on channel capacity as a function of Signal-to-Noise Ratio (SNR) has become one of the most important analysis tools for the wireless communication system design.

Here we follow Shannon’s foot steps and to our best knowledge, calculate the highest achievable information rates of a triply-selective shallow, horizontal underwater acoustic channel water channel for the first time with a comparison to the already-established radio channel capacity. We follow a semi-analytic approach for our calculations, the details of which are beyond the scope of this article.

Before proceeding to the results, let us note different interpretations of the channel capacity. In our context, the most sensible way to interpret the notion of the channel capacity is to think of it as the highest achievable bandwidth efficiency. To define the bandwidth efficiency, we can measure our data rate in terms of how many bits we are able to transmit per second, and then divide that data rate by the bandwidth. This way, we can analyze how efficiently we are making use of the available bandwidth. For instance, if our data rate is 10 Kbits/s (10 thousand bits per second) across a channel with 5 KHz bandwidth, then we have a bandwidth efficiency factor of (10Kbits/s / 5KHz)= 2 [bits/s/Hz]. That looks like we are doing great, as we are able to transmit twice as much 5KHz data as our bandwidth, in other words our bandwidth efficiency is 200%. If however, our data rate is measured to be 1 Kbits/s across the same channel with the same bandwidth and SNR, then our bandwidth efficiency is only 20%.

Now our goal is to calculate how much bandwidth efficiency our channel can actually accommodate and how well we are doing compared to the natural limit that our channel can handle. Thereafter we can make an informed decision on how well ROAM is doing and how much room there is for potential improvement.

As seen in Fig. 1, we study the achievable information rates for five different channels. The red dash-dot curve is what Claude Shannon considered in his original article where he offered a universal performance bound for the Additive White Gaussian Noise (AWGN) channel without any limitations on the modulation order. While Shannon’s capacity analysis leads to a readily available formula, we have to acknowledge that with modulation constraints, Shannon limit is far too optimistic and therefore we need tighter, more realistic achievable limits on the highest information rate for reliable communication. In order to do so, we refer to our semi-analytical approach, the results of which are presented for three different LTE channel models as well as for the Halifax Harbor underwater acoustic channel with 2 Km length and less than 20 m depth.

Let us begin describing the results in Fig. 1 with the LTE Extended Pedestrian — A Channel presented with the purple curve. This channel model, as the name implies, considers the communication with hand sets and mobile stations with a dominant Line-of-Sight path. Although this channel presents frequency selectivity, it is not in a severe form (*as compared to the other channels we study herein*) and therefore it has the highest capacity among the the four channels we study, and closest capacity to the universal Shannon limit.

Let us now look at the LTE Extended Vehicular — A and LTE Extended Typical Urban channels, the capacities of which are presented with blue solid and yellow dashed curves, respectively, in Fig. 1. These channels differ from the LTE Extended Pedestrian — A channel in their much more severe frequency-selective nature. Hence their capacity is visibly smaller.

Finally let us turn our attention to the Halifax Harbor Underwater Acoustic channel, the capacity of which, is given with the solid black curve. This is an extreme shallow water environment, and it presents considerably higher variations of frequency-selectivity than the LTE channels above, and it is also highly time-varying. Therefore it is not surprising that it leads to a substantially smaller capacity than the radio channels under the same SNR.

For a quantitative comparison, let us focus on SNR = 2 dB. The Shannon limit at this SNR is approximately 1.37 [bits/s/Hz]. The first LTE channel we consider (*purple *in Fig.1) offers 1.25 [bits/s/Hz] capacity, or equivalently, 125% maximum bandwidth efficiency. The two other LTE channels (*solid blue and dashed yellow *in Fig. 1) present approximately 1.08 [bits/s/Hz] maximum information rate, or equivalently 108% maximum bandwidth efficiency. Halifax Harbor, on the other hand, can only accommodate 0.7 [bits/s/Hz] information rate, or equivalently at most 70% bandwidth efficiency, which is 36% less capacity than the closest radio link at 2 dB SNR. The gap widens at higher SNR.

This is a mathematical evidence to the interference-limited nature of the shallow-water horizontal underwater channels, as even if we improve the SNR significantly, the channel capacity is still limited due to the presence of interference inflicted by the time-variability of the channel.

Let us conclude by marking ROAM’s performance on the graph in Fig. 1. ROAM consistently delivered 0.6 [bits/s/Hz] information rate during our trials on February 17, 2022, which is very close to the maximum achievable limit that the Halifax Harbor could accommodate. This means that ROAM, in fact, operated within (0.6/0.7)= 86% of the natural limits of the channel capacity, as a testimony to its very high performance.