BER of Differentially Detected π/4-DQPSK with Selection Combining in Nakagami-m Fading

Abstract

Nakagami’s m distribution is a versatile statistical model to characterize small-scale multipath fading in wireless channels. On the other hand, selection combining (SC) is a widely practiced diversity technique to mitigate the detrimental effects of multipath fading. Thus, when SC is applied over Nakagami fading channel, the error performance improvement for any given modulation format is of considerable interest. Since the last decade, π/4-shifted differential quadrature phase shift keying (π/4-DQPSK) modulation has attracted much attention as it is used for high-capacity code division multiple access (CDMA) based digital cellular systems. One of the major reasons behind this is the provision for differential detection which allows production of low complexity mobile units. In this paper, we present analytical expressions for bit error rate (BER) of π/4-DQPSK modulation with L-branch SC diversity in Nakagami-m fading channels perturbed by additive white Gaussian noise (AWGN). The derived end expressions are in closed form and contain finite series of Gaussian hypergeometric function. This makes evaluation of error rates much more straightforward compared to earlier approaches that required single or even double numerical integration. Some special instances such as the nondiversity case and Rayleigh fading case are also investigated and plotted along with the main findings. For different fading parameter (m) values and for different diversity orders (L), simulated results are shown to be in excellent agreement with the derived analytical results. All the results are, however, limited to integer values of fading severity parameter m.

Appendix

1.1 Integrations Containing $Q\left( {a\sqrt x ,b\sqrt x } \right)$, x ⁿ, and exp (−mx)

The following integral after Simon et al. [1, Eq. (5.56)] or equivalently from [30, Eq. 3]

$$ I_{1} = \int\limits_{0}^{\infty } {x^{m - 1} \exp \left( { - {{p^{2} x} \mathord{\left/ {\vphantom {{p^{2} x} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)Q_{j} \left( {a\sqrt x ,b\sqrt x } \right)dx} $$

(A.1)

can be written as

$$ \begin{aligned} I_{1} = & \left({{\frac{2}{{p^{2}}}}} \right)^{m} \left({m - 1} \right)!\left[{1 + {\frac{{b^{2j}}}{{s^{j}}}}\sum\limits_{i = 0}^{m - 1} {\left({\begin{array}{*{20}c} {i + j} \\ i \\ \end{array}} \right)}} \right.\left({{\frac{{p^{2}}}{s}}} \right)^{i} \\& \times \left\{ {{\frac{{a^{2}}}{s}}{}_{2}F_{1} \left({{\frac{i + j + 1}{2}},{\frac{i + j + 2}{2}};\;j + 1;{\frac{{4a^{2} b^{2}}}{{s^{2}}}}} \right)} \right. \\& \left. {\left. {- {\frac{j}{i + j}}{}_{2}F_{1} \left({{\frac{i + j}{2}},{\frac{i + j + 1}{2}};\;j;\;{\frac{{4a^{2} b^{2}}}{{s^{2}}}}} \right)} \right\} } \right] \\ \end{aligned}$$

(A.2)

In (A.2) the variables {m, j} are arbitrary integers independent of each other and s = p ² + a ² + b ². A similar kind of integral may also be found in Okui’s paper [4].

Specializing for j = 1 and letting p ²/2 = t, m − 1 = n and $a, \, b = \sqrt {2 \pm \sqrt 2 }$ the integral becomes

$$ \begin{aligned} I_{2} =\,& {\frac{n!}{{t^{n + 1}}}}\left\{{1 + {\frac{1}{{2\left({t + 2} \right)^{2}}}}\sum\limits_{i = 0}^{n} {\left({i + 1} \right)} \left({{\frac{t}{t + 2}}} \right)^{i}} \right. \\& \times {}_{2}F_{1} \left({{\frac{i + 2}{2}},{\frac{i + 3}{2}};2;{\frac{2}{{\left({t + 2} \right)^{2}}}}} \right) - \left({{\frac{2 \mp \sqrt 2}{2t + 4}}} \right) \\ & \times \left. {\sum\limits_{i = 0}^{n} {\left({{\frac{t}{t + 2}}} \right)^{i} {}_{2}F_{1} \left({{\frac{i + 1}{2}},{\frac{i + 2}{2}};1;{\frac{2}{{\left({t + 2} \right)^{2}}}}} \right)}} \right\} \\ \end{aligned} $$

(A.3)

Now when an integral of the following form,

$$ \begin{aligned} I_{3} = & \int\limits_{0}^{\infty} {x^{n} \exp \left({- tx} \right)\left\{{Q_{1} \left({\sqrt {x\left({2 + \sqrt 2} \right)},\sqrt {x\left({2 - \sqrt 2} \right)}} \right)} \right.} \\& \left. {- Q_{1} \left({\sqrt {x\left({2 - \sqrt 2} \right)},\sqrt {x\left({2 + \sqrt 2} \right)}} \right)} \right\}dx \\ \end{aligned} $$

(A.4)

is evaluated, it results in a very compact form

$$ I_{3} = \sqrt 2 n!\sum\limits_{i = 0}^{n} {{\frac{{t^{i - n - 1} }}{{\left( {t + 2} \right)^{i + 1} }}}{}_{2}F_{1} \left[ {{\frac{i + 1}{2}},{\frac{i + 2}{2}};1;{\frac{2}{{\left( {t + 2} \right)^{2} }}}} \right]} $$

(A.5)

as some of the terms vanish due to the symmetrical nature of the integral I ₃. Further when n = 0, from (A.5) and by applying (9.121.1) [24] we have

$$ \int\limits_{0}^{\infty } {\exp \left( { - tx} \right)\left[ {Q_{1} \left( {\sqrt {ax} ,\sqrt {bx} } \right){ - }Q_{1} \left( {\sqrt {bx} ,\sqrt {ax} } \right)} \right]dx} = \frac{1}{t}\sqrt {{\frac{2}{{t^{2} + 4t + 2}}}} $$

(A.6)

where $\left\{ {a,b} \right\} = 2 \pm \sqrt 2$.