BER Analysis of Asynchronised Wireless Network in Presence of Non-Identically Distributed Interferers

Abstract

In this paper, bit error rate (BER) analysis of an asynchronised wireless network in the presence of non-identically distributed interferers employing the binary phase shift keying (BPSK) scheme is presented. While calculating the BER, the overlap of frames due to frequency reuse in the network is addressed carefully and a method is presented to calculate the index and the extent of overlap for the bit(s) of individual interfering signals. Taking into account the partial overlap of the bits, the probability density function (pdf) of effective signed fractional overlap variable is derived. The pdf of the resultant asynchronised interfering signal is calculated using an amplitude metric based approach. A closed form expression for BER is also derived. The effect of the number of concurrent transmissions (active users), interference range factor and size of deployment area on the network performance is investigated. Analytical results are verified through simulation studies.

Appendix

1.1 Details of Derivation of the PDF of RV $O_1^{\alpha }$ (2)

The overlap variable $O_1^{\alpha }$ as a function of RV $\alpha$ is defined as $O_1^{\alpha }$ = $\,\alpha - \left\lfloor \alpha \right\rfloor ; \,\,\, \forall \,\,\,\,\, b_1^I \, \ne \,\, 0$, as in (2). The pdf of RV $\alpha$ for type 1 interferers is $f_\alpha (x) = 1/N_b$, since RV $\alpha$ is uniformly distributed in the range $[0,\,N_b)$ for type 1 interferers. The pdf of $O_1^{\alpha }$ can be obtained using the method described as follows:

Figure 9 shows the variation of $O_1^{\alpha }$ with $\alpha$ (for convenience of plotting, range of $\alpha$ is set to [0, 5] only). It may be observed that for the range of interest of $\alpha$, $O_1^{\alpha } = \,\alpha - \left\lfloor \alpha \right\rfloor$ will have multiple roots. In such case, the pdf of $O_1^{\alpha }$ can be calculated as [20]

$$\begin{aligned} {f_{O_1^\alpha }}\left( y \right) = \sum \limits _k {{f_\alpha }\left( x \right) \,{{\left. {\left| {\frac{{dx}}{{dy}}} \right| \,} \right| }_{x = {\alpha _k}}}} \end{aligned}$$

(29)

where $\alpha _k$ are the multiple roots of (2). Now, using the $f_\alpha (x) = 1/N_b$, (29) can be solved to obtain ${f_{O_1^\alpha }}\left( y \right)$ as

$$\begin{aligned} {f_{O_1^\alpha }}\left( y \right) = \sum \limits _{k = 1}^{{N_b}} {\,\frac{1 /{N_b}}{1}} = 1 \end{aligned}$$

(30)

Fig. 9

Variation of overlap variable $O_1^\alpha$ with $\alpha$

1.2 Details of Derivation of the PDF of RV $W_j$ (23)

First, we define a new RV $\xi _j$ = $A_j^{\text {m}}$ ${\gamma _j^{s,T_1}}$. With the pdf of ${\gamma _j^{s,T_1}}$ calculated in (15), the pdf of RV $\xi _j$ can be easily calculated and we represent the same pdf by $f_{\xi _j}(\xi )$. The range of the new RV $\xi _j$ will be $[- A_j^{\text {m}}, A_j^{\text {m}}]$ Now, we provide some of the important steps to find the pdf of the effective interference term $W_j$ as follows:

After some simplification, the pdf of $W_j$ conditioned on $\xi _j$ can be written as

$$\begin{aligned} {f_{{W_j}}}\left( {w\left| \xi _j \right. } \right) = \frac{1}{\xi _j}\,{f_{{h_j}}}\left( {\frac{w}{\xi _j}} \right) \end{aligned}$$

(31)

where ${f_{{h_j}}}(.)$ is the pdf of channel fading parameter corresponding to the $j\text {th}$ effective interfering term. Removing the condition on $\xi _j$, we can write

$$\begin{aligned} {f_{{W_j}}}\left( w \right) = \int _{ - A_j^{m}}^{A_j^{m}} \frac{1}{\xi _j} {{f_{{h_j}}}\left( {\frac{w}{\xi _j}} \right) \,\,{f_{\xi _j}}\left( \xi \right) \,d \xi } \, \end{aligned}$$

(32)

For $\xi _j$ $\in$ $[- A_j^{\text {m}}, 0]$, $w$ requires to be less than zero to make the argument of the pdf of channel fading parameter positive. Therefore, for $w < 0$ ,

$$\begin{aligned} {f_{{W_j}}}\left( w \right) |_{\xi _j <0} = \int _{ - A_j^{m}}^{0} \frac{-1}{\xi _j} {{f_{{h_j}}}\left( {\frac{w}{\xi _j}} \right) \,\,{f_{\xi _j}}\left( \xi \right) \,d \xi } \, \end{aligned}$$

(33)

Equation  (33) can be further written as

$$\begin{aligned} {f_{{W_j}}}\left( w \right) |_{\xi _j <0}&= - \int \limits _{ - {A_j^\text {m}}}^0 {\frac{w}{{{\xi ^2}{\sigma _f^2}}}\,\exp \left( { - \frac{{{w^2}}}{{2{\xi ^2}{\sigma _f^2}}}} \right) \,\left[ {\frac{{\delta \left( {\xi + {A_j^\text {m}}} \right) }}{4}\,\, + \,\frac{1}{{4{A_j^\text {m}}}}} \right] \,\,d\xi } \nonumber \\&= \sqrt{\frac{\pi }{2}} \,\frac{1}{{4{A_j^\text {m}}\sigma _f}}\,\text {erfc}\left[ { - \frac{w}{{\sqrt{2} \,{A_j^\text {m}}\sigma _f}}} \right] \, - \,\frac{w}{{4{({A_j^\text {m}})^2}{\sigma _f^2}}}\,\exp \left[ { - \frac{{{w^2}}}{{2{({A_j^\text {m}})^2}{\sigma _f^2}}}} \right] \nonumber \\ \end{aligned}$$

(34)

For $\xi _j$ $\in$ $[0, A_j^{\text {m}}]$, $w$ has to be greater than zero to make the argument of the pdf of channel fading parameter positive. Therefore, for $w > 0$ also, we have

$$\begin{aligned} {f_{{W_j}}}\left( w \right) |_{\xi _j >0}&= \int \limits _0^{A_j^\text {m}} {\frac{w}{{{\xi ^2}{\sigma _f^2}}}\,\exp \left( { - \frac{{{w^2}}}{{2{\xi ^2}{\sigma _f^2}}}} \right) \,\left[ {\frac{{\delta \left( {\xi - {A_j^\text {m}}} \right) }}{4}\,\, + \,\frac{1}{{4{A_j^\text {m}}}}} \right] \,\,d\xi } \nonumber \\&= \sqrt{\frac{\pi }{2}} \,\frac{1}{{4{A_j^\text {m}}\sigma _f}}\,\text {erfc}\left[ {\frac{w}{{\sqrt{2} \,{A_j^\text {m}}\sigma _f}}} \right] \, + \,\frac{w}{{4{{(A_j^\text {m})}^2}{\sigma _f^2}}}\,\exp \left[ { - \frac{{{w^2}}}{{2{{(A_j^\text {m})}^2}{\sigma _f^2}}}} \right] \nonumber \\ \end{aligned}$$

(35)

Equation (34) and (35) jointly describe the complete pdf of the effective interfering term $W_j$, as given in (23).