The underwater acoustic channel is characterized by a path loss that depends not only on the transmission distance, but also on the signal frequency. Signals transmitted from one user to another over a distance $l$ are subject to a power loss of $l^{-\alpha}{a(f)}^{-l}$. Although a terrestrial radio channel can be modeled similarly, the underwater acoustic channel has different characteristics. The spreading factor $\alpha$, related to the geometry of propagation, has values in the range $1 \leq \alpha \leq 2$. The absorption coefficient $a(f)$ is a rapidly increasing function of frequency: it is three orders of magnitude greater at 100 kHz than at a few Hz. Existing results for capacity of wireless networks correspond to scenarios for which $a(f) = 1$, or a constant greater than one, and $\alpha \geq 2$. These results cannot be applied to underwater acoustic networks in which the attenuation varies over the system bandwidth. We use a water-filling argument to assess the minimum transmission power and optimum transmission band as functions of the link distance and desired data rate, and study the capacity scaling laws under this model.
Deep Dive into Capacity Scaling Laws for Underwater Networks.
The underwater acoustic channel is characterized by a path loss that depends not only on the transmission distance, but also on the signal frequency. Signals transmitted from one user to another over a distance $l$ are subject to a power loss of $l^{-\alpha}{a(f)}^{-l}$. Although a terrestrial radio channel can be modeled similarly, the underwater acoustic channel has different characteristics. The spreading factor $\alpha$, related to the geometry of propagation, has values in the range $1 \leq \alpha \leq 2$. The absorption coefficient $a(f)$ is a rapidly increasing function of frequency: it is three orders of magnitude greater at 100 kHz than at a few Hz. Existing results for capacity of wireless networks correspond to scenarios for which $a(f) = 1$, or a constant greater than one, and $\alpha \geq 2$. These results cannot be applied to underwater acoustic networks in which the attenuation varies over the system bandwidth. We use a water-filling argument to assess the minimum transm
The seminal work by [1] studied wireless networks, modeled as a set of n nodes that exchange information, with the aim of determining what amount of information the source nodes can send to the destination as the number n grows. The original results obtained for nodes deployed in a disk of unit area motivated the study of capacity scaling laws in different scenarios, ranging from achievability results in random deployments using percolation theory [4] or cooperation between nodes [3], to the impact of node mobility over the capacity of the network, e.g. [2]. Reference [5] provides a good overview of the different assumptions and scaling laws for radio wireless networks.
The underwater acoustic channel is characterized by a path loss that depends not only on the transmission distance, but also on the signal frequency [7]. Signals transmitted over a distance l are subject to a power loss of l -α a(f ) -l . Although a terrestrial radio channel can be modeled similarly, the underwater acoustic channel has different characteristics. The spreading factor α, related to the geometry of propagation, has values in the range 1 ≤ α ≤ 2, where α = 1 corresponds to cylindrical spreading. Also, the absorption coefficient a(f ) is a rapidly increasing function of frequency, e.g. it is three orders of magnitude greater at 100 kHz than at a few Hz [7]. Finally, the power spectral density of the noise underwater is highly dependent on frequency.
Existing capacity scaling laws for wireless radio networks correspond to scenarios for which a(f ) = 1, or a constant greater than one, and α ≥ 2, e.g. [1], [4]. These results cannot be directly applied to underwater acoustic networks in which the attenuation varies over the system bandwidth and α ≤ 2. We study the scaling laws under a model that considers a water-filling argument to assess the minimum transmission power and optimum transmission band as functions of the link distance and desired data rate [8]. In particular, we study the case of arbitrarily deployed networks in a disk of unit area, and follow a similar procedure as in [1] to derive an upper bound on capacity. In this sense, we provide an extension of the work in [1] under a more complicated power loss model.
We show that the amount of information that can be exchanged by each source-destination pair in underwater acoustic networks goes to zero as the number of nodes n goes to infinity. This occurs at least at a rate n -1/α e -W 0 (O(n -1/α )) , where W 0 represents the branch zero of the Lambert function. We illustrate that this throughput per source-destination pair has two different regions. For small n, the throughput decreases very slowly as n increases. For large n, it decreases almost as n -1/α . Thus for large enough n, the throughput decreases more rapidly in underwater networks than in typical radio channels, because of the difference in the path loss exponent α.
The paper is organized as follows. In Section II, we present the underwater channel model. In Section III, we analyze the scaling laws for the case of a network transmitting in an arbitrarily chosen narrow band. In Section IV, we study scaling laws for the low-power/narrow-band case, with optimal bandwidth allocation using a waterfilling argument. In section V, we consider the case in which the nodes can transmit at high power over a wide transmission band. Conclusions are summarized in Section VI.
An underwater acoustic channel is characterized by an attenuation that depends on the distance l and the signal frequency f as
where l ref is a reference distance (typically 1 m).
A common empirical model used for the absorption a(f ) is Thorp’s formula [7] which captures the dependence on the frequency. This absorption a(f ) is an increasing function of f . The spreading factor describes the geometry of propagation and is typically 1 ≤ α ≤ 2, e.g. α = 1 and α = 2 correspond to cylindrical and spherical spreading, respectively. The noise in an acoustic channel can be modeled through four basic sources: turbulence, thermal noise, shipping, and waves. It has a power spectral density (psd) which depends on the frequency, the shipping activity s, and the wind speed w in m/s [7].
The complete model for a colored Gaussian underwater link was presented in [8] where power was allocated through waterfilling. In the absence of multipath and channel fading, the relationship among capacity, transmission power, and optimal transmission band of a point-to-point link is given by [8]
where N (f ) is the psd of the noise, B(l, C) is the optimum band of operation and K(l, C) is a constant. The transmission power associated with a particular choice of (l, C) is given by
where the psd of the signal is S(l, C, f
A distinguishing feature of the underwater acoustic channel is the dependence of the optimal transmission band on the link distance [8]. Fig. 1 illustrates the optimal center frequency f c (l) as a function of distance. The optimal center frequency is define
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