Approximate Bit Error Rate of DPSK with Imperfect Phase Noise in TWDP Fading

Abstract

This study provides closed-form expressions for average bit error rate (ABER) of differential phase shift keying (DPSK) with phase error in two-wave with diffuse power (TWDP) fading. It is considered that the envelope of phase error is Gaussian distributed. The effect of phase synchronization on wireless system is studied for different values of TWDP fading parameters and phase error. The analytical results are evaluated to study the impact of phase error on the system performance. Also, the results are compared with the case of perfectly synchronized.

Appendix

The expression in (11) can be further solved by using identity given in ([14], 6.6.14.3) as:

$$\int\limits_{0}^{\infty } {\exp \left( { - \alpha x} \right)} I_{2v} \left( {2\sqrt {\beta x} } \right){\text{d}}x = \frac{1}{{\sqrt {\alpha \beta } }}\exp \left( {\frac{\beta }{2\alpha }} \right)\frac{{\Gamma \left( {v + 1} \right)}}{{\Gamma \left( {2v + 1} \right)}}M_{ - 1/2,v} \left( {\frac{\beta }{\alpha }} \right)$$

Now, (11) can be simplified as:

$$\begin{aligned} P_{e} & = \frac{\eta }{4}\sum\limits_{i = 1}^{L} {a_{i} } \sum\limits_{j = 0}^{1} {\exp \left( { - P_{2i - j} } \right)} \left( {1 - \frac{2}{C}} \right)^{ - 1/2} \exp \left( {\frac{{P_{2i - j} \eta }}{{2\left( {1 + \eta } \right)}}} \right) \\ & \quad \times \frac{1}{{\sqrt {\left( {1 + \eta } \right)P_{2i - j} \eta } }}M_{ - 1/2,0} \left( {\frac{{P_{2i - j} \eta }}{1 + \eta }} \right) \\ \end{aligned}$$

(13)

Use $M_{k,s} \left( z \right) = \exp \left( {\frac{ - z}{2}} \right)z^{s + 1/2} M\left( {s - k + \frac{1}{2};1 + 2s;z} \right)$ [14] in (13), and the final equation can be obtained as given in (12).