Spectrum Efficiency Evaluation with Diversity Combining for Fading and Branch Correlation Impairments

Abstract

In this paper, closed-form expressions for capacities per unit bandwidth for fading channels with impairments due to Branch Correlation are derived for optimal power and rate adaptation, constant transmit power, channel inversion with fixed rate, and truncated channel inversion policies for maximal ratio combining diversity reception case. Closed-form expressions for system spectrum efficiency when employing different adaptation policies are derived. Analytical results show accurately that optimal power and rate adaptation policy provides the highest capacity over other adaptation policies. In the case of errors due to branch correlation, optimal power and rate adaptation policy provides the best results. All adaptation policies suffer no improvement in channel capacity as the branch correlation is increased. This fact is verified using various plots for different policies. With increase in branch correlation, capacity gains are significantly larger for optimal power and rate adaptation policy as compared to the other policies. The outage probability for branch correlation is also derived and analyzed using plots for the same.

Appendices

Appendix 1

The Spectrum Efficiency for OPRA policy is obtained by substituting (2) into (7) of [18] as

$$\begin{aligned} \frac{\mathop {\left\langle \hbox {C} \right\rangle }\nolimits _{\mathrm{OPRA}}^{\mathrm{(BC)}} }{\hbox {B}}&= \hbox { }\frac{1}{2\Gamma \varepsilon }\int \limits _{\gamma _0 }^\infty {\log _2 } \left( {\frac{\gamma }{\gamma _0 }} \right) \left[ \exp \left( {-\frac{\gamma }{\Gamma (1+\varepsilon )}} \right) -\exp \left( {-\frac{\gamma }{\Gamma (1-\varepsilon )}} \right) \right] d\gamma \end{aligned}$$

(28)

$$\begin{aligned}&= \frac{\gamma _0 \log _2 (e)}{2\Gamma \varepsilon }\int \limits _1^\infty {\log _e } \left( t \right) \exp \left( {-\frac{\gamma _0 }{\Gamma (1+\varepsilon )}t} \right) dt\nonumber \\&-\frac{\gamma _0 \log _2 (e)}{2\varepsilon \Gamma }\int \limits _1^\infty {\log _e (t)} \exp \left( {-\frac{\gamma _0 }{\Gamma (1-\varepsilon )}t} \right) dt. \end{aligned}$$

(29)

Substituting $\mu _1 =\frac{\gamma _0 }{\Gamma (1+\varepsilon )}$ and $\mu _2 =\frac{\gamma _0 }{\Gamma (1-\varepsilon )}$, $t=\frac{\gamma }{\gamma _0 }$ and $\hbox {dt} =\frac{d\gamma }{\gamma _0 }$ in (29), we have

$$\begin{aligned} \frac{\mathop {\left\langle \hbox {C} \right\rangle }\nolimits _{\mathrm{OPRA}}^{\mathrm{(BC)}} }{\hbox {B}}=\hbox { }\frac{\gamma _0 }{2\Gamma \varepsilon \log (2)}\int \limits _1^\infty {\log _2 } \left( t \right) \left[ {\exp \left( {-\mu _1 t} \right) -\exp \left( {-\mu _2 t} \right) } \right] dt. \end{aligned}$$

(30)

From Eq. (2) of section 4.331 on p. 567 in [20], we have $\int _1^\infty e^{-\mu t}\ln tdt=-\frac{1}{\mu }E_i (-\mu ),[\mathfrak {R}e\mu >0].$ Substituting this expression in (30), we obtain the capacity for OPRA policy in (6). The optimal policy suffers a probability of outage, $\hbox {P}_{\mathrm{out}}^{\mathrm{(BC)}}$ given by

$$\begin{aligned} \mathop {\hbox {P}}\nolimits _{\mathrm{out}}^{\mathrm{(BC)}} =\int \limits _0^{\gamma _0 } {f_2^{\mathrm{(BC)}} (\gamma )d\gamma } =1-\int \limits _{\gamma _0 }^\infty {f_2^{\mathrm{(BC)}} (\gamma )d\gamma }. \end{aligned}$$

(31)

Making change of variables in the integral of (31), where $\eta _1 =\frac{1}{\Gamma (1+\varepsilon )}$ and $\eta _2 =\frac{1}{\Gamma (1-\varepsilon )}$, we have

$$\begin{aligned} \mathop {\hbox {P}}\nolimits _{\mathrm{out}}^{\mathrm{(BC)}} =\int \limits _0^{\gamma _0 } {\frac{1}{2\Gamma \varepsilon }\left[ {\exp \left( {-\eta _1 \gamma } \right) -\exp \left( {-\eta _2 \gamma } \right) } \right] d\gamma }. \end{aligned}$$

(32)

Substituting $\int \nolimits _0^{\gamma _0 } {\exp (-\mu x)=\frac{1-\exp (-\gamma _0 \mu )}{\mu }}$ into (32) and simplifying, we have the outage probability expression as in (7). Substituting (2) into Eq. (29) of [18], we have

$$\begin{aligned} \frac{\mathop {\left\langle \hbox {C} \right\rangle }\nolimits _{\mathrm{ORA}}^{\mathrm{(BC)}} }{\hbox {B}}=\frac{\log _2 e}{2\varepsilon \Gamma }\int \limits _0^\infty {\log _e (1+\gamma )\left( {\exp \left( {-\frac{\gamma }{\Gamma (1+\varepsilon )}} \right) -\exp \left( {-\frac{\gamma }{\Gamma (1-\varepsilon )}} \right) } \right) } d\gamma . \end{aligned}$$

(33)

From Eq. (1) of section 4.337 on p. 568 in [20], we have $\int \nolimits _0^\infty \exp (-\mu x)\log _e (\beta +x)dx=\frac{1}{\mu }[\log _e (\beta )-e^{(\mu \beta )}E_i (-\beta \mu )],\;\forall \;[\left| {\arg \beta } \right| <\pi ,\mathfrak {R}e{}\mu >0].$

Substituting $\beta = 1, \eta _1 =\frac{1}{\Gamma (1+\varepsilon )},$ and $\eta _2 =\frac{1}{\Gamma (1-\varepsilon )}$ in (33), we have

$$\begin{aligned} \frac{\mathop {\left\langle \hbox {C} \right\rangle }\nolimits _{\mathrm{ORA}}^{\mathrm{(BC)}} }{\hbox {B}}=\frac{\log _2 e}{2\varepsilon \Gamma }\int \limits _0^\infty {\log _e (1+\gamma )\left( {\exp \left( {-\eta _1 \gamma } \right) -\exp \left( {-\eta _2 \gamma } \right) } \right) } d\gamma . \end{aligned}$$

(34)

Integrating the above equation, we obtain the expression for capacity for ORA policy as in (8).

Substituting (2) into Equation (46) of [18], we have

$$\begin{aligned} \frac{\mathop {\left\langle \hbox {C} \right\rangle }\nolimits _{\mathrm{CIFR}}^{\mathrm{(BC)}} }{\hbox {B}}=\log _2 \left[ {1+\frac{2\Gamma \varepsilon }{\int \nolimits _0^\infty {\frac{\left[ {\exp \left( {-\frac{\gamma }{\Gamma (1+\varepsilon )}} \right) -\exp \left( {-\frac{\gamma }{\Gamma (1-\varepsilon )}} \right) } \right] }{\gamma }d\gamma } }} \right] . \end{aligned}$$

(35)

Since the closed-form expression is not possible for $\int \nolimits _0^\infty {\frac{\exp \left( {-\gamma } \right) }{\gamma }d\gamma }$, we find the numerical limits for $\upgamma$ through simulation, and substitute the minimum and maximum values as limits to give

$$\begin{aligned} \frac{\mathop {\left\langle \hbox {C} \right\rangle }\nolimits _{\mathrm{CIFR}}^{\mathrm{(BC)}} }{\hbox {B}}=\log _2 \left[ {1+\frac{2\Gamma \varepsilon }{\int \nolimits _{\gamma _{\min } }^{\gamma _{\max } } {\frac{\left[ {\exp \left( {-\eta _1 \gamma } \right) -\exp \left( {-\eta _2 \gamma } \right) } \right] }{\gamma }d\gamma } }} \right] . \end{aligned}$$

(36)

Using Eq. (3) of Section 3.352 in [20], we obtain the expression for capacity of CIFR policy as in (9). Substituting (35) into Eq. (47) of [18], we have

$$\begin{aligned} \frac{\mathop {\left\langle \hbox {C} \right\rangle }\nolimits _{\mathrm{TIFR}}^{\mathrm{(BC)}} }{\hbox {B}}=\log _2 \left[ {1+\frac{2\Gamma \varepsilon }{\int \nolimits _{\gamma _0 }^\infty {\frac{\left[ {\exp \left( {-\frac{\gamma }{\Gamma (1+\varepsilon )}} \right) -\exp \left( {-\frac{\gamma }{\Gamma (1-\varepsilon )}} \right) } \right] }{\gamma }d\gamma } }} \right] . \end{aligned}$$

(37)

Making change of variables in the integral of (37), we have

$$\begin{aligned} \frac{\mathop {\left\langle \hbox {C} \right\rangle }\nolimits _{\mathrm{TIFR}}^{\mathrm{(BC)}} }{\hbox {B}}=\log _2 \left[ {1+\frac{2\Gamma \varepsilon }{\int \nolimits _{\gamma _0 }^\infty {\frac{\left[ {\exp \left( {-\eta _1 \gamma } \right) -\exp \left( {-\eta _2 \gamma } \right) } \right] }{\gamma }d\gamma } }} \right] . \end{aligned}$$

(38)

From Eq. (2) of section 3.352 on p. 343 in [20], we obtain the expression for capacity of TIFR policy as in (10).

Appendix 2

Figure 2a, b through Fig. 5a, b show the simulation of the system through the steps discussed below.

Step 1: The base for the simulation is to find out numerical instantaneous SNRs ($\upgamma )$. This is obtained from the CDF of the received instantaneous SNR ($\upgamma )$ for branch correlation at the output of two branch system given in (2), i.e.

$$\begin{aligned} \hbox {F}_2^{\mathrm{(BC)}} (\gamma )=1-\frac{1}{2\varepsilon }\left[ {(1+\varepsilon )\exp \left( {-\frac{\gamma }{\Gamma (1+\varepsilon )}} \right) -(1-\varepsilon )\exp \left( {-\frac{\gamma }{\Gamma (1-\varepsilon )}} \right) } \right] . \end{aligned}$$

(39)

Step 2: The CDF is equated to a uniform random number. Then, 10$^{6}$ Uniform random numbers are generated using rand command in MATLAB.

Step 3: The maximum and minimum values of U are found out.

Step 4: The maximum value of received SNR $(\upgamma _{\mathrm{max}})$ and the minimum value of received SNR $(\upgamma _{\mathrm{min}})$ are found for different values of Individual Branch SNRs $({\Gamma })$, different values of Correlation $(\upvarepsilon )$ and different number of diversity orders (M) as tabulated below.

M $=$ 2; $\bar{\gamma }=$ 5 dB				M $=$ 2; $\upvarepsilon = 0.5$
E	$\upgamma _{\mathrm{max}}$	$\upgamma _{\mathrm{min}}$	Range	$\bar{\gamma }$	$\upgamma _{\mathrm{max}}$	$\upgamma _{\mathrm{min}}$	Range
0.1	37	0.0375	36.9625	1	18	0.0135	17.9865
0.2	38	0.037	37.963	2	22	0.0165	21.9835
0.3	40	0.036	39.964	3	28	0.0207	27.9793
0.4	42	0.0354	41.9646	4	35	0.026	34.974

Step 5: Using the maximum and minimum values of instantaneous SNRs $(\upgamma )$, the capacity for all four policies is obtained for different values of ${\Gamma }, \upvarepsilon$ and M by approximating the integral as a numerical summation from $\upgamma _{\mathrm{min}}$ to $\upgamma _{\mathrm{max}}$ in steps of 0.001.

Step 6: The simulated capacity values for all four policies for different values of correlation, $\upvarepsilon$, at ${\Gamma } = 5 \hbox {dB}$ are compared with the corresponding analytical results for the above four cases. Figure 1a, b through Fig. 4a, b show the analytical versus simulated graphs for capacity for the case of impairments due to branch correlation errors.