Adaptive Turbo Coded Modulation for Shallow Underwater Acoustic Communications

Abstract

In this paper the performance of a wireless communications system over shallow underwater acoustic channels is investigated when adaptive modulation and coding techniques with receiver diversity are used. It is assumed that the communication system experiences Ricean shadowed fading. We obtain the analytical figures of the proposed rate-adaptive transmission schemes, emphasizing in the spectral efficiency and the average bit error rate. These analytical expressions are compared to Monte-Carlo simulations corroborating the analytical results.

Appendices

Appendix 1: Cumulative Distribution Function

The definition of the CDF is

$$\begin{aligned} {\mathcal {F}}_{\gamma _t}(\gamma ) = \frac{1}{2\pi i}\oint _{c-i \infty }^{c+i \infty } \mathcal {L}[{\mathcal {F}}_{\gamma _t}(\gamma );s]\,e^{s\gamma }\,ds. \end{aligned}$$

(15)

The property $\mathcal {L}[{\mathcal {F}}_R(r);s]= \mathcal {L}[f_R(r);s]/s$ of the Laplace transform is used to obtain

$$\begin{aligned} {\mathcal {F}}_{\gamma _t}(\gamma )= \frac{1}{2\pi i}\oint _{c-i \infty }^{c+i \infty } \frac{1}{s}\, \mathcal {L}[f_{\gamma _t}(\gamma );s]\,e^{s\gamma }\,ds. \end{aligned}$$

(16)

The Laplace transform of the probability density function (PDF) and the MGF are related through ${\mathcal {M}}_R(-s)=\mathcal {L}[f_R(r);s]$, which gives the sought expression

$$\begin{aligned} {\mathcal {F}}_{\gamma _t}(\gamma )=\frac{1}{2\pi i} \oint _{c-i \infty }^{c+i \infty } \frac{1}{s}\, {\mathcal {M}}_{\gamma _t}(-s)\,e^{s\gamma }\,ds. \end{aligned}$$

(17)

Let’s use the function $\Xi (s)$ in the integrand of (17)

$$\begin{aligned} \Xi (s)=\frac{1}{s}\,{\mathcal {M}}_{\gamma _t}(-s)\,e^{s\gamma }. \end{aligned}$$

(18)

When (4) is used in (18), we have

$$\begin{aligned} \Xi (s)=\frac{1}{s}\left( 1+\frac{s}{x_0} \right) ^{\!\!(m-1)L}\prod _{\ell =1}^L\!\left( 1+\frac{s}{x_\ell } \right) ^{-m}\!\!e^{s\gamma }. \end{aligned}$$

(19)

The function $\Xi (s)$ has a simple pole at $p_0=0$, a zero at $z_0=-x_0$ along with $L$ poles at $\{p_k\}_{k=1}^{L}=-\{x_\ell \}_{\ell =1}^L$ of different orders, according to the integer value $m \ge 0$ and the number $L$ of channels.

We need to solve the integral in (17). The singularities of $\Xi (s)$ and the chosen integral contour are shown in Fig. 11

Fig. 11

Singularities of function $\Xi (s)$ and the integration path

Since the inequality

$$\begin{aligned} \left| \int _{{\mathcal {P}}_\infty }\Xi (s)\,ds\right| \le \frac{c}{R}\;\prod _{\ell =1}^L\left( \frac{R}{x_\ell } \right) ^{-m} \end{aligned}$$

(20)

is satisfied for a value of the radius $R$ (shown in Fig. 11) large enough and an appropriate constant $c$, it is true that

$$\begin{aligned} \lim _{R\rightarrow \infty }\left| \int _{{\mathcal {P}}_\infty }\!\!\Xi (s)\,ds\right| \le \lim _{R\rightarrow \infty } 2\pi R \Bigl |\,\Xi (s)|_{{\mathcal {P}}_\infty }\Bigr | =0. \end{aligned}$$

(21)

Cauchy’s theorem allows the CDF to be written as

$$\begin{aligned} {\mathcal {F}}_{\gamma _t}(\gamma )&= \frac{1}{2\pi i}\oint _{\varepsilon -i \infty }^{\varepsilon +i \infty }\Xi (s)\,ds\nonumber \\&= \frac{1}{2\pi i}\sum _{k=1}^{L}\oint _{{\mathcal {C}}(p_k)}\!\!\Xi (s)\,ds -\frac{1}{2\pi i}\lim _{R\rightarrow \infty }\int _{{\mathcal {P}}_\infty }\!\!\Xi (s)\,ds, \end{aligned}$$

(22)

where $\varepsilon >0$. Eventually, the residue theorem allows the CDF to be calculated as the sum of the residues of the function $\Xi (s)$ at the pole at $p_0=0$ and the $L$ poles at $\{p_k\}_{k=1}^{L}=-\{x_\ell \}_{\ell =1}^L$, that is,

$$\begin{aligned} {\mathcal {F}}_{\gamma _t}(\gamma )=\sum _{r=\,0}^{L} \text{ Res }\,[\,\Xi (s);p_r]. \end{aligned}$$

(23)

We obtain

$$\begin{aligned} {\mathcal {F}}_{\gamma _t}(\gamma )&= (x_0)^{-(m-1)L}\prod \limits _{\ell =1}^L \left( x_{\ell } \right) ^m \left[ (x_0+s)^{(m-1)L}\prod \limits _{\ell =1}^{L}(x_{\ell }+s)^{-m}e^{s\gamma } \Biggl |_{s\,=\,0} \!\! \right. \nonumber \\&\quad + \sum _{k=1}^{L}\frac{1}{(m-1)!} \, \frac{\partial ^{\,m-1}}{\partial s^{\,m-1}}\Biggl |_{s\,=\,-x_k} \!\! \left( \frac{1}{s}(x_0+s)^{(m-1)L}.\left. \!\!\! \prod \limits _{\ell =1; \,\ell \not = k}^{L}(x_{\ell }+s)^{-m}e^{s\gamma }\right) \right] \!.\nonumber \\ \end{aligned}$$

(24)

Appendix 2: Complementary Moment Generating Function

We need to calculate the complementary MGF of Eq. (8) for $\gamma _t$. The Laplace transform and its properties are use to arrive to

$$\begin{aligned} {\mathcal {G}}_{\gamma _t}(s;\zeta )&= \frac{1}{2\pi i}\oint _{c-i \infty }^{c+i \infty } \mathcal {L}[{\mathcal {G}}_{\gamma _t}(s;\zeta );p]\,e^{p\,\zeta }\,dp \nonumber \\&= \frac{1}{2\pi i}\oint _{c-i \infty }^{c+i \infty } \frac{1}{p}\, {\mathcal {M}}_{\gamma _t}(s-p)\,e^{p\,\zeta }\,dp\,. \end{aligned}$$

(25)

Let’s define the new function in the integrand of (25)

$$\begin{aligned} \Xi (p,s)&= \frac{1}{p}\,{\mathcal {M}}_{\gamma _t}(s-p)\,e^{p\,\zeta } \nonumber \\&= \frac{e^{p\,\zeta }}{p}\, \left( 1-\frac{s-p}{x_0} \right) ^{\!(m-1)L} \prod \limits _{\ell =1}^L\left( 1-\frac{s-p}{x_\ell } \right) ^{-m}. \end{aligned}$$

(26)

The new function $\Xi (p,s)$ has, relative to variable $p$, a simple pole at $p_0=0$, a simple or multiple zero at $z_0=-(x_0-s)$, and $L$ poles at $\{p_k\}_{k=1}^{L}=-(\{x_\ell \}_{\ell =1}^{L}-s)$ of different orders, according to the integer value $m \ge 0$ and the number $L$ of channels. Compared to (19), a shift of value $s$ of the zero and multiple poles positions can be observed. Now again, when Cauchy’s and residues theorems are used the function ${\mathcal {G}}_\gamma (s;\zeta )$ is calculated as the sum of the residues of $\Xi (p,s)$ to obtain

$$\begin{aligned}&{\mathcal {G}}_{\gamma _t}(s;\zeta ) = (x_0) ^{-(m-1)L} \prod \limits _{\ell =1}^L\left( x_{\ell } \right) ^m \! \left[ (x_0-s+p)^{(m-1)L} \prod \limits _{\ell =1}^{L}(x_{\ell }-s+p)^{-m}e^{p\,\zeta }\Biggl |_{p\,=\,0} \right. \nonumber \\&\quad +\sum _{k=1}^{L}\!\frac{1}{(m-1)!}\; \frac{\partial ^{\,m-1}}{\partial p^{\,m-1}} \Biggl |_{p\,=\,-(x_k-s)} \left. \!\!\! \left( \frac{1}{p}\,(x_0-s+p)^{(m-1)L} \!\!\!\prod \limits _{\ell =1; \ell \not = k}^{L} \!\!\!\!(x_{\ell }-s+p)^{-m}e^{p\,\zeta }\right) \!\! \right] .\nonumber \\ \end{aligned}$$

(27)