On the Effects of Frequency Scaling Over Capacity Scaling in Underwater Networks—Part II: Dense Network Model

Abstract

This is the second in a two-part series of papers on information-theoretic capacity scaling laws for an underwater acoustic network. Part II focuses on a dense network scenario, where nodes are deployed in a unit area. By deriving a cut-set upper bound on the capacity scaling, we first show that there exists either a bandwidth or power limitation, or both, according to the operating regimes (i.e., path-loss attenuation regimes), thus yielding the upper bound that follows three fundamentally different information transfer arguments. In addition, an achievability result based on the multi-hop (MH) transmission is presented for dense networks. MH is shown to guarantee the order optimality under certain operating regimes. More specifically, it turns out that scaling the carrier frequency faster than or as $n^{1/4}$ is instrumental towards achieving the order optimality of the MH protocol.

Appendix

1.1 Proof of Lemma 1

Upper and lower bounds on $P_L^{(k)}$ in a dense network are derived by basically following the same node indexing and layering techniques as those in Part I. We refer to Appendix A.2 and Figure 4 in [1] for the detailed description (note that the destination nodes are, however, located at positions $\left(\frac{k_x}{\sqrt{n}}, \frac{k_y}{\sqrt{n}}\right)$ in dense networks). Similarly to the extended network case, from (9), the term $P_{L}^{(k)}$ is then given by

$$\begin{aligned} P_{L}^{(k)}= \frac{P}{c_0}\sum _{i_x=1}^{\sqrt{n}/2}\sum _{i_y=1}^{\sqrt{n}}\frac{n^{\alpha /2}}{\left((i_x+k_x-1)^2+(i_y-k_y)^2\right)^{\alpha /2}a(f)^{\sqrt{\left((i_x+k_x-1)^2+(i_y-k_y)^2\right)/n}}}. \end{aligned}$$

First, focus on how to obtain an upper bound for $P_{L}^{(k)}$. Assuming that all the nodes in each layer are moved onto the innermost boundary of the corresponding ring, we then have

$$\begin{aligned} P_{L}^{(k)}&\le \frac{Pn^{\alpha /2}}{c_0}\sum _{i^{\prime }=k_x}^{k_x+\sqrt{n}/2-1} \frac{c_5(i^{\prime }+1)}{i^{\prime \alpha }a(f)^{i^{\prime }\sqrt{n}}} \nonumber \\&\le \frac{2c_5Pn^{\alpha /2}}{c_0}\sum _{i^{\prime }=k_x}^{\sqrt{n}}\frac{1}{i^{\prime \alpha -1}a(f)^{i^{\prime }\sqrt{n}}} \nonumber \\&= c_6Pn^{\alpha /2}\sum _{i^{\prime }=k_x}^{\sqrt{n}}\frac{1}{i^{\prime \alpha -1} \left(1+\epsilon _0\right)^{i^{\prime }n^{\beta -1/2}}} \end{aligned}$$

(20)

for some positive constants $c_0$, $c_5$, and $c_6$ independent of $n$ and an arbitrarily small $\epsilon _0>0$, where the equality comes from the relationship (8) between $a(f)$ and $\beta$. We first consider the case where $k_x=o\left(n^{1/2-\beta +\epsilon }\right)$ for an arbitrarily small $\epsilon >0$. Under this condition, from the fact that the term $i^{\prime \alpha -1}$ in the RHS of (20) is dominant in terms of upper-bounding $P_{L}^{(k)}$ for $i^{\prime }=k_x,\ldots ,\sqrt{n}$, (20) is further bounded by

$$\begin{aligned} P_{L}^{(k)}&\le c_6Pn^{\alpha /2}\sum _{i^{\prime }=k_x}^{\sqrt{n}}\frac{1}{i^{\prime \alpha -1}} \\&\le c_6Pn^{\alpha /2}\left(\frac{1}{k_x^{\alpha -1}}+\int \limits _{k_x}^{\sqrt{n}}\frac{1}{x^{\alpha -1}}dx\right), \end{aligned}$$

which yields $P_{L}^{(k)}=O\left(n^{\alpha /2}(\sqrt{n})^{2-\alpha }\right)=O(n)$ for $1\le \alpha <2$ and $P_{L}^{(k)}=O\left(n\log n\right)$ for $\alpha =2$. When $k_x=\varOmega (n^{1/2-\beta +\epsilon })$, the upper bound (20) for $P_{L}^{(k)}$ is dominated by the term $\left(1+\epsilon _0\right)^{i^{\prime }n^{\beta -1/2}}$, and thus is given by

$$\begin{aligned} P_{L}^{(k)}&\le c_6Pn^{\alpha /2}\sum _{i^{\prime }=k_x}^{\sqrt{n}}\frac{1}{\left(1+\epsilon _0\right)^{i^{\prime }n^{\beta -1/2}}} \nonumber \\&\le c_6Pn^{\alpha /2}\left(\frac{1}{\left(1+\epsilon _0\right)^{k_x n^{\beta -1/2}}}+\int \limits _{k_x}^{\sqrt{n}}\frac{1}{\left(1+\epsilon _0\right)^{x n^{\beta -1/2}}}dx\right) \nonumber \\&= c_6Pn^{\alpha /2}\left(\frac{1}{\left(1+\epsilon _0\right)^{k_x n^{\beta -1/2}}}+\int \limits _{1}^{\sqrt{n}/k_x}\frac{k_x}{\left(1+\epsilon _0\right)^{x k_x n^{\beta -1/2}}}dx\right) \nonumber \\&\le c_6Pn^{\alpha /2}\left(\frac{1}{\left(1+\epsilon _0\right)^{k_x n^{\beta -1/2}}}+\frac{n^{1/2-\beta }}{\left(1+\epsilon _0\right)^{k_x n^{\beta -1/2}}}\right) \nonumber \\&\le \frac{c_7Pn^{\alpha /2}}{\left(1+\epsilon _0\right)^{k_x n^{\beta -1/2}}}\max \left\{ 1,n^{1/2-\beta }\right\} \end{aligned}$$

(21)

for some constant $c_7>0$ independent of $n$, which is the last result in (10).

Next, let us turn to deriving a lower bound for $P_{L}^{(k)}$. Since each layer has at least one node that is onto the innermost boundary of the corresponding ring, the lower bound similarly follows

$$\begin{aligned} P_{L}^{(k)}&\ge \frac{Pn^{\alpha /2}}{c_0}\sum _{i^{\prime }=k_x}^{k_x+\sqrt{n}/2-1} \frac{1}{i^{\prime \alpha }a(f)^{i^{\prime }\sqrt{n}}} \nonumber \\&= c_6Pn^{\alpha /2}\sum _{i^{\prime }=k_x}^{k_x+\sqrt{n}/2-1}\frac{1}{i^{\prime \alpha }\left(1+\epsilon _0\right)^{i^{\prime }n^{\beta -1/2}}}. \end{aligned}$$

(22)

For the condition $k_x=o(n^{1/2-\beta +\epsilon })$, (22) is represented as

$$\begin{aligned} P_{L}^{(k)}&\ge c_6Pn^{\alpha /2}\sum _{i^{\prime }=k_x}^{k_x+\sqrt{n}/2-1}\frac{1}{i^{\prime \alpha }\left(1+\epsilon _0\right)^{i^{\prime }n^{\beta -1/2}}} \\&\ge c_6Pn^{\alpha /2}\sum _{i^{\prime }=k_x}^{2k_x-1}\frac{1}{i^{\prime \alpha }\left(1+\epsilon _0\right)^{i^{\prime }n^{\beta -1/2}}} \\&\ge \frac{c_6Pn^{\alpha /2}}{n^{\epsilon ^{\prime }}}\sum _{i^{\prime }=k_x}^{2k_x-1}\frac{1}{i^{\prime \alpha }} \\&\ge \frac{c_8Pn^{\alpha /2}}{n^{\epsilon ^{\prime }}k_x^{\alpha -1}} \end{aligned}$$

for an arbitrarily small $\epsilon ^{\prime }>0$ and some constant $c_8>0$ independent of $n$, where the third inequality holds due to $\left(1+\epsilon _0\right)^{k_x n^{\beta -1/2}}=O(n^{\epsilon ^{\prime }})$. On the other hand, similarly as in the steps of (21), the condition $k_x=\varOmega (n^{1/2-\beta +\epsilon })$ yields the following lower bound for $P_{L}^{(k)}$:

$$\begin{aligned} P_{L}^{(k)}&\ge c_6P \sum _{i^{\prime }=k_x}^{k_x+\sqrt{n}/2-1}\frac{1}{\left(1+\epsilon _0\right)^{i^{\prime }n^{\beta -1/2}}} \\&\ge c_6P \left(\frac{1}{\left(1+\epsilon _0\right)^{k_x n^{\beta -1/2}}}+\int \limits _{k_x+1}^{k_x+\sqrt{n}/2-1}\frac{1}{\left(1+\epsilon _0\right)^{x n^{\beta -1/2}}}dx\right) \\&\ge c_9P\left(\frac{1}{\left(1+\epsilon _0\right)^{k_x n^{\beta -1/2}}}+\frac{n^{1/2-\beta }}{\left(1+\epsilon _0\right)^{(k_x+1) n^{\beta -1/2}}}\right) \\&\ge \frac{c_9P}{\left(1+\epsilon _0\right)^{k_x n^{\beta -1/2}}}\max \left\{ 1,\frac{n^{1/2-\beta }}{\left(1+\epsilon _0\right)^{n^{\beta -1/2}}}\right\} \end{aligned}$$

some constant $c_9>0$ independent of $n$, which finally complete the proof of the lemma.

1.2 Proof of Lemma 2

The layering technique used in Part I is similarly applied (see Fig. 2 in [1]). From (1), the total interference power $P_I$ at each node from simultaneously transmitting nodes is upper-bounded by

$$\begin{aligned} P_I&= \sum _{k=1}^{\sqrt{n}}(8k)\frac{P\min \left\{ 1,\frac{a(f)^{1/\sqrt{n}}N(f)}{n^{\alpha /2}}\right\} }{c_0(k/\sqrt{n})^\alpha a(f)^{k/\sqrt{n}}} \nonumber \\&= \frac{8Pn^{\alpha /2}\min \left\{ 1,\frac{a(f)^{1/\sqrt{n}}N(f)}{n^{\alpha /2}}\right\} }{c_0}\sum _{k=1}^{\sqrt{n}}\frac{1}{k^{\alpha -1}a(f)^{k/\sqrt{n}}}. \end{aligned}$$

(23)

Using (4) and (8), the upper bound (23) on $P_I$ can be expressed as

$$\begin{aligned} P_I&\le c_{10}Pn^{\alpha /2}\min \left\{ 1,\frac{(1+\epsilon _0)^{n^{\beta -1/2}}}{n^{(\alpha +\beta a_0)/2}}\right\} \sum \limits _{k=1}^{\sqrt{n}}\frac{1}{k^{\alpha -1}(1+\epsilon _0)^{kn^{\beta -1/2}}} \\&\le \left\{ \begin{array}{lll} \frac{c_{10}P}{n^{\beta a_0/2}}\sum \limits _{k=1}^{\sqrt{n}}\frac{1}{k^{\alpha -1}(1+\epsilon _0)^{kn^{\beta -1/2}}}&\quad \mathrm{if}\; 0\le \beta <1/2 \\ \frac{c_{10}P}{n^{\beta a_0/2}}\sum \limits _{k=1}^{\sqrt{n}}\frac{1}{(1+\epsilon _0)^{k}}&\quad \mathrm{if}\; \beta =1/2 \\ c_{10}Pn^{\alpha /2}\sum \limits _{k=1}^{\sqrt{n}}\frac{1}{(1+\epsilon _0)^{kn^{\beta -1/2}}}&\quad \mathrm{if} \;\beta >1/2 \end{array}\right. \\&\le \left\{ \begin{array}{lll} \frac{c_{10}P}{n^{\beta a_0/2}}\sum \limits _{k=1}^{\sqrt{n}}\frac{1}{k^{\alpha -1}(1+\epsilon _0)^{kn^{\beta -1/2}}}&\quad \mathrm{if}\; 0\le \beta <1/2 \\ \frac{c_{10}P}{n^{\beta a_0/2}}\frac{1}{(1+\epsilon _0)-1}&\quad \mathrm{if}\; \beta =1/2 \\ c_{10}Pn^{\alpha /2}\frac{1}{(1+\epsilon _0)^{n^{\beta -1/2}}-1}&\quad \mathrm{if}\; \beta >1/2 \end{array}\right. \\&\le \left\{ \begin{array}{lll} \frac{c_{10}P}{n^{\beta a_0/2}}\sum \limits _{k=1}^{\sqrt{n}}\frac{1}{k^{\alpha -1}(1+\epsilon _0)^{kn^{\beta -1/2}}}&\quad \mathrm{if}\; 0\le \beta <1/2 \\ \frac{c_{11}P}{n^{\beta a_0/2}}&\quad \mathrm{if}\; \beta =1/2 \\ \frac{c_{11}Pn^{\alpha /2}}{(1+\epsilon _0)^{n^{\beta -1/2}}}&\quad \mathrm{if} \;\beta >1/2 \\ \end{array}\right. \end{aligned}$$

for some positive constants $c_{10}$ and $c_{11}$ independent of $n$. Based on the argument in Appendix 6.1, when $0\le \beta <1/2$, it follows that

$$\begin{aligned} \sum _{k=1}^{\sqrt{n}}\frac{1}{k^{\alpha -1}(1+\epsilon _0)^{kn^{\beta -1/2}}}&= \sum _{k=1}^{n^{1/2-\beta }-1}\frac{1}{k^{\alpha -1}(1+\epsilon _0)^{kn^{\beta -1/2}}} + \sum _{k=n^{1/2-\beta }}^{\sqrt{n}}\frac{1}{k^{\alpha -1}(1+\epsilon _0)^{kn^{\beta -1/2}}}\\&\le \sum _{k=1}^{n^{1/2-\beta }-1}\frac{1}{k^{\alpha -1}} + \frac{1}{n^{(1/2-\beta )(\alpha -1)}}\sum _{k=n^{1/2-\beta }}^{\sqrt{n}}\frac{1}{(1+\epsilon _0)^{kn^{\beta -1/2}}} \\&\le \left(1+\int \limits _1^{n^{1/2-\beta }}\frac{1}{x^{\alpha -1}}dx\right) \\&+ \frac{1}{n^{(1/2-\beta )(\alpha -1)}}\left(\frac{1}{1+\epsilon _0}+\int \limits _{n^{1/2-\beta }}^{\sqrt{n}}\frac{1}{(1+\epsilon _0)^{xn^{\beta -1/2}}}dx\right) \\&\le 2\int \limits _1^{n^{1/2-\beta }}\frac{1}{x^{\alpha -1}}dx + \frac{2}{n^{(1/2-\beta )(\alpha -1)}}\int \limits _{1}^{n^\beta }\frac{n^{1/2-\beta }}{(1+\epsilon _0)^{x}}dx \\&\le \left\{ \begin{array}{lll} 4n^{(1/2-\beta )(2-\alpha )}&\quad \mathrm{if}\; 1\le \alpha <2 \\ \log n&\quad \mathrm{if}\; \alpha =2, \end{array}\right. \end{aligned}$$

which results in $P_I=O\left(\frac{\max \left\{ n^{(1/2-\beta )(2-\alpha )},\log n\right\} }{n^{\beta a_0/2}}\right)$. This completes the proof of the lemma.