Novel Tight Closed-Form Bounds for the Symbol Error Rate of EGC and MRC Diversity Receivers Employing Linear Modulations Over $\alpha -\mu$ Fading

Abstract

In this paper, we propose novel lower and upper bounds on the average symbol error rate (SER) of the dual-branch maximal-ratio combining and equal-gain combining diversity receivers assuming independent branches. $M$-ary pulse amplitude modulation and $M$-ary phase shift keying schemes are employed and operation over the $\alpha -\mu$ fading channel is assumed. The proposed bounds are given in closed form and are very simple to calculate as they are composed of a double finite summation of basic functions that are readily available in the commercial software packages. Furthermore, the proposed bounds are valid for any combination of the parameters $\alpha$ and $\mu$ as well as $M$. Numerical results presented show that the proposed bounds are very tight when compared to the exact SER obtained via performing the exact integrations numerically making them an attractive much simpler alternative for SER evaluation studies.

Appendices

Appendix 1: Derivation of the Derivative of the Conditional SER for $M$-ary PSK

In $M$-ary PSK modulation, it is well-known that we have $M$ symbols whose complex baseband representations are spread evenly over a circle of radius $\sqrt{E_s}$ separated by an angle of $2\pi /M$ where $E_s$ is the symbol energy. That is, the transmitted signal $s$ can be one of $M$ possible symbols, i.e., $s\in \{s_0,\ldots ,s_{M-1}\}$ where $s_i=\sqrt{E_s}\exp (2Ji\pi /M)$ and $J=\sqrt{-1}$; see Fig. 7. At the receiver side, the received signal $y$ is given by $y = r s + n$ where $n$ is the complex baseband representations of noise and $r$ is the fading envelope. For additive white Gaussian noise (AWGN), $n$ follows the complex Gaussian distribution, i.e., $n\sim \mathcal {CN}(0,\sigma ^2)$ where $\sigma ^2$ is the noise power. Given a received signal $y$, a decision is made according to the following maximum a posteriori (MAP) rule:

$$\begin{aligned} \text {Decide }s_k\text { if }k&= \arg \max _{i=0,\ldots ,M-1} P(s=s_i|y)\end{aligned}$$

(28)

$$\begin{aligned}&= \arg \max _{i=0,\ldots ,M-1} f(y|s=s_i) P(s=s_i), \end{aligned}$$

(29)

where $f(y|s=s_i)$ is the conditional density of the received signal $y$ given that the transmitted symbol is $s_i$ and $P(s=s_i)$ is the a priori probability of transmitting the symbol $s_i$. It is usually assumed that all symbols have equal a priori probabilities and hence, the term $P(s=s_i)$ is dropped out from the last equation. Moreover, given $r$ and $s=s_i, y\sim \mathcal {CN}(rs_i,\sigma ^2), i=0,\ldots ,M-1$. Because of the symmetry of the $M$-ary PSK constellation, we can easily prove that the application of the above decision rule is equivalent to having the following decision region $D_i$ for the symbol $s_i$:

$$\begin{aligned} D_i = \left\{ y:\frac{(2i-1)\pi }{M}<\arg (y)<\frac{(2i+1)\pi }{M}\right\} \end{aligned}$$

where $\arg (y)$ denotes the phase angle of $y$. An illustration for the region $D_0$ is shown in Fig. 7. Hence, the conditional SER for $M$-ary PSK is given by

$$\begin{aligned} P(\text {error}|r)&= \sum _{i=0}^{M-1} P(y\notin D_i|r,s=s_i)P(s=s_i)\nonumber \\&= P(y\notin D_0|r,s=s_0) \end{aligned}$$

(30)

where the last line is derived using the fact that all symbols are equi-probable and the probabilities $P(y\notin D_i|s=s_i), i=0,\ldots ,M-1$ are equal because of the symmetry of the $M$-ary PSK constellation. Defining $y_1=\mathfrak {R}{(y)}$ and $y_2=\mathfrak {I}{(y)}$, the following expression for the conditional SER is obtained:

$$\begin{aligned} P(\text {error}|r)&= 1 - \frac{2}{\pi \sigma ^2}\int \limits _{y_2=0}^\infty \int \limits _{y_1=y_2\cot (\pi /M)}^\infty e^{-\frac{1}{\sigma ^2}\left( y_2^2+(y_1-r\sqrt{E_s})^2\right) }dy_1 dy_2\nonumber \\&= 1 - \frac{2}{\sqrt{\pi \sigma ^2}}\int \limits _{y_2=0}^\infty e^{-\frac{y_2^2}{\sigma ^2}} Q\left( \sqrt{2}\times \frac{y_2\cot (\pi /M)-r\sqrt{E_s}}{\sigma }\right) dy_2. \end{aligned}$$

(31)

Making the change of variables $u=y_2/\sigma$ and noting that the SNR $\zeta =r^2E_s/\sigma ^2$, we get

$$\begin{aligned} P(\text {error}|\zeta ) = 1 - \frac{2}{\sqrt{\pi }}\int \limits _{u=0}^\infty e^{-u^2} Q\left( \sqrt{2}u\cot (\pi /M)-\sqrt{2\zeta }\right) du \end{aligned}$$

(32)

Differentiating the last expression with respect to $\zeta$, we obtain

$$\begin{aligned} \frac{d}{d\zeta }P(\text {error}|\zeta ) = \frac{-1}{\pi \sqrt{\zeta }} \int \limits _{u=0}^\infty e^{-u^2} e^{-(u\cot (\pi /M)-\sqrt{\zeta })^2}du. \end{aligned}$$

(33)

After simple algebraic manipulations, we reach the following result:

$$\begin{aligned} \frac{d}{d\zeta }P(\text {error}|\zeta ) =-\frac{1}{\sqrt{\pi \zeta }}\sin (\pi /M)\exp \left( -\zeta \sin ^2(\pi /M)\right) \left[ 1-Q(\sqrt{2\zeta }\cos (\pi /M))\right] ,\nonumber \\ \end{aligned}$$

(34)

which concludes the proof of the lemma.

Fig. 7

Schematic diagram of the $M$-ary PSK constellation. The dotted area represents the decision region $D_0$

Appendix 2: CDF of the Sum of Two $\alpha -\mu$ Random Variables

In order to prove Theorem 1, we write the CDF in the form of a double integral and then apply a change of variables to simplify the integrand. Afterwards, we approximate the domain of integration to another simpler domain over which the integration can be done analytically. In particular, we have

$$\begin{aligned} F_{X}(x) = P(X\le x) = P(X_1+X_2\le x)= \int \limits _{x_1=0}^x\int \limits _{x_2=0}^{x-x_1} f_{X_1}(x_1) f_{X_2}(x_2) dx_1 dx_2, \end{aligned}$$

(35)

where $f_{X_1}(x)$ and $f_{X_2}(x)$ are given by (9). Performing the change of variables $u=\mu _1(x_1/\hat{x}_1)^{\alpha _1}$ and $v=\mu _2(x_2/\hat{x}_2)^{\alpha _2}$, we can easily derive that

$$\begin{aligned} F_{X}(x) = \int \int \limits _{D}\frac{u^{\mu _1-1}v^{\mu _2-1}e^{-u-v}}{\Gamma (\mu _1)\Gamma (\mu _2)}dudv, \end{aligned}$$

(36)

where the region $D$ is defined as

$$\begin{aligned} D&= \bigg \{(u,v):0\le u\le \mu _1\left( \frac{ x}{ \hat{x}_1}\right) ^{\alpha _1}\!\!,0\le v\le \mu _2\left( \frac{ x}{ \hat{x}_2}\right) ^{\alpha _2}\!\!, \nonumber \\&\quad \hat{x}_1\left( \frac{u}{\mu _1}\right) ^{1/\alpha _1}\!\!+\hat{x}_2\left( \frac{v}{\mu _2}\right) ^{1/\alpha _2}\!\!\le x\bigg \}. \end{aligned}$$

(37)

A sketch of the region $D$ corresponding to $F_{X}(x)$ is shown in Fig. 8. As expected, there is no closed-form expression for the integration in (36). However, we can approximate the region $D$ in (37) as a set of rectangles and perform the integration in (36) over those rectangles. Depending on how the rectangles are chosen, we can obtain either a lower bound or an upper bound on the CDF as will be shown in the sequel. The partitioning of the region $D$ is based on the fact that the curve

$$\begin{aligned} \hat{x}_1(u/\mu _1)^{1/\alpha _1}+\hat{x}_2(v/\mu _2)^{1/\alpha _2}= x,\quad 0\le u\le \mu _1\left( \frac{ x}{ \hat{x}_1}\right) ^{\alpha _1}, \end{aligned}$$

(38)

bounding the region $D$, can be parameterized as $u=\mu _1(kx/\hat{x}_1)^{\alpha _1}$ and $v=\mu _2((1-k)x/\hat{x}_2)^{\alpha _2}$ where $k\in [0,1]$ is the free parameter. Hence, in order to approximate the region $D$ as rectangles, we first determine the following set of points on its bounding curve:

$$\begin{aligned} \left( u_j(x),v_j(x)\right) = \left( \mu _1\left( \frac{j x}{K \hat{x}_1}\right) ^{\alpha _1}, {\mu _2\left( \frac{\left( K-j\right) x}{K \hat{x}_2}\right) ^{\alpha _2}}\right) ,\quad j=0,\ldots ,K, \end{aligned}$$

(39)

Fig. 8

$F_{X}(x)$ corresponds to the integration over the shaded area

where $K$ is the number of desired rectangles. Considering the set of rectangles illustrated in Fig. 9a, it is clear that the resulting CDF will represent a lower bound on the exact one. The $j{\text {th}}$ rectangle has the vertices: $(u_j(x),0), (u_{j+1}(x),0), (u_{j+1}(x),v_{j+1}(x))$, and $(u_{j}(x),v_{j+1}(x))$ ³. Performing the integration over these rectangles, $F_{X}(x)$ is found to be lower bounded by the finite series $L_s(x;K)$ given by

$$\begin{aligned} L_s(x;K) = \sum _{j=0}^{K-1} \left( \hat{\gamma }(\mu _1,u_{j+1}(x))-\hat{\gamma }(\mu _1,u_j(x))\right) \hat{\gamma }(\mu _2,v_{j+1}(x)), \end{aligned}$$

(40)

where $u_j(x)$ and $v_j(x)$ are given by (39). The $j{\text {th}}$-term in the series in (15) is the value of the integration of the integrand in (36) over the $j{\text {th}}$ rectangle. Similarly, if the rectangles are chosen as illustrated in Fig. 9b, an upper bound will then be obtained. The $j{\text {th}}$ rectangle of the upper-bound approximation has the vertices: $(u_j(x),0), (u_{j+1}(x),0), (u_{j+1}(x),v_{j}(x))$, and $(u_{j}(x),v_{j}(x))$. Consequently, $F_{X}(x)$ is found to be upper bounded by $U_s(x;K)$, given by

$$\begin{aligned} U_s(x;K) = \sum _{j=0}^{K-1} \left( \hat{\gamma }(\mu _1,u_{j+1}(x))-\hat{\gamma }(\mu _1,u_j(x))\right) \hat{\gamma }(\mu _2,v_{j}(x)), \end{aligned}$$

(41)

Hence, the exact CDF $F_{X}(x)$ is lower and upper bounded as

$$\begin{aligned} L_s(x;K)\le F_{X}(x) \le U_s(x;K). \end{aligned}$$

(42)

This completes the proof of the first part of Theorem 1. The proof of uniform convergence is provided in [5].

Fig. 9

Rectangular approximation for $F_{X}(x)$. a Lower bound approximation for $F_{X}(x)$. b Upper bound approximation for $F_{X}(x)$