Expected distortion of simultaneous transmission of multiple source symbols with multicarrier frequency hopping modulation

Abstract

In this paper, we consider the problem of transmitting a Gaussian source over a slow-fading channel, when the aim is to adjust system parameters so that the source is reconstructed with minimum expected distortion. In the considered scenario, the spatial distribution of users is modelled by a homogeneous Poisson point process. These users transmit their symbols by means of multicarrier frequency hopping modulation in which the frequency diversity characteristic is implemented by dividing the available bandwidth into several subbands where each subband consists of several subchannels. The number of subbands has a significant effect on the improvement of the system’s performance. We can have as many descriptions as the number of available subbands. In this way, we consider a combination of symmetric multiple description coding as the source coding and direct channel transmission coding as the channel coding so as to code the source signal into several descriptions which have the same number as the subbands. Considering separate source-channel coding strategy, we compute the expected distortion and evaluate the effects of system parameters on it. Furthermore, with the goal of minimizing the expected distortion, the optimum values of some system parameters in general and also for the case where the bandwidth expansion ratio is equal to one, are calculated. Through the optimization process, the optimum number of subbands and the optimum value of source coding rate are determined which lead to a significant improvement in the network performance.

Appendix

The channel coefficients have Rayleigh distribution; therefore, the squared fading coefficients have an exponential distribution, i.e;

$$\begin{aligned} p\left( |\gamma _{o,i}|^{2}> x\right) =e^{-\frac{x}{\acute{\sigma }^{2}}}. \end{aligned}$$

(24)

Thus

$$\begin{aligned} p(SINR_{i}>T)&=p\left[ \frac{|\gamma _{o,i}|^{2} \left( \frac{P_{t}}{n_{s}} \right) }{PL_{o} \left( I_{i}+\sigma ^{2}_{n} \right) }>T \right] \\&=E_{I_{i}} \left[ p \left( |\gamma _{o,i}|^{2}>\frac{Tn_{s}r_{o}^{\alpha }L_{0} (I_{i}+\sigma ^{2}_{n})}{P_{t}} \right) \right] \\&=E_{I_{i}} \left( e^{-\frac{n_{s}Tr_{o}^{\alpha }L_{0}}{\acute{\sigma }^{2}P_{t}}(I_{i}+\sigma ^{2}_{n}) } \right) =E_{I_{i}} \left( e^{-sI_{i}} \right) e^{-s\sigma ^{2}_{n} } , \end{aligned}$$

(25)

where $s\,\triangleq\,\frac{n_{s}Tr_{o}^{\alpha }L_{0}}{\acute{\sigma }^{2}P_{t}}.$ According to (2), we have

$$\begin{aligned} E_{I_{i}} \left( e^{-sI_{i}} \right) =\lim _{R\rightarrow \infty }e^ {\frac{-\lambda \pi R^{2}}{n_{h}} } \sum _{n=0}^{\infty }\frac{\left( \frac{\lambda \pi R^{2}}{n_{h}} \right) ^{n} E \left( e^ {-s\frac{P_{t}|\gamma _{j,i}|^{2}r_{j,i}^{-\alpha }}{L_{0}n_{s}} } \right) ^{n}}{n!}, \end{aligned}$$

where

$$\begin{aligned} E \left[ e^ {-s\frac{P_{t}|\gamma _{j,i}|^{2}r_{j,i}^{-\alpha }}{L_{0}n_{s}} } \right]&=E_{r_{j,i}} \left[ E_{|\gamma _{j,i}|^{2}} \left\{ e^ {-s\frac{P_{t}|\gamma _{j,i}|^{2}r_{j,i}^{-\alpha }}{L_{0}n_{s}} } \right\} |r_{j,i} \right] \\&=E_{r_{j,i}} \left[ \frac{1}{\acute{\sigma }^{2} \left( \frac{1}{\acute{\sigma }^{2}}+s\frac{P_{t}r_{j,i}^{-\alpha }}{L_{0}n_{s}} \right) } \right] \\&\overset{(a)}{=}\frac{1}{\pi R^{2}}\int _{0}^{R}\frac{2\pi u}{\left( 1+\acute{\sigma }^{2}s\frac{P_{t}u^{-\alpha }}{L_{0}n_{s}} \right) } du, \end{aligned}$$

which (a) is obtained based on (3). Therefore, we have

$$\begin{aligned} E_{I_{i}} \left( e^{-sI_{i} } \right)&=\lim _{R\rightarrow \infty }e^{\frac{-\lambda \pi R^{2}}{n_{h}}+\frac{2\lambda \pi }{n_{h}}\int _{0}^{R}\frac{u}{\left( 1+\acute{\sigma }^{2}s\frac{P_{t}u^{-\alpha }}{L_{0}n_{s}} \right) } du }=e^ {-\frac{2\pi ^{2}\lambda \left( \frac{\acute{\sigma }^{2}s P_{t}}{L_{0}n_{s}} \right) ^{\frac{2}{\alpha }}}{n_{h}\alpha \sin \left( \frac{2\pi }{\alpha } \right) } }. \end{aligned}$$

(26)

Finally, by substituting, (26) in (25) we obtain

$$\begin{aligned} p_{cov}=p ( SINR_{i}>T ) =e^ {-\frac{Tn_{s}L_{0}r_{o}^{\alpha }\sigma _{n}^{2}}{P_{t}\acute{\sigma }^{2}}-\frac{2\pi ^{2}\lambda r_{o}^{2}T^{\frac{2}{\alpha }}}{n_{h}\alpha \sin \left( \frac{2\pi }{\alpha } \right) } }. \end{aligned}$$

By considering $p_{out}=1-p_{cov}$, we have

$$\begin{aligned} p_{out}=1-e^ {-\frac{Tn_{s}L_{0}r_{o}^{\alpha }\sigma _{n}^{2}}{P_{t}\acute{\sigma }^{2}}-\frac{2\pi ^{2}\lambda r_{o}^{2}T^{\frac{2}{\alpha }}}{n_{h}\alpha \sin \left( \frac{2\pi }{\alpha } \right) } }. \end{aligned}$$

(27)