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Spectral efficiency evaluation for selection combining diversity schemes under worst case of fading scenario.

1. Introduction

Wireless communication services, such as wireless personal area networks, satellite-terrestrial services, wireless mobile communication services, wireless local-area networks, and internet access have been growing at a rapid pace in recent years. These services require high data rate. Thus, channel capacity is of fundamental importance in the design of wireless communication systems as it determines the maximum achievable data rate of the system. Since wireless mobile channels are subjected to fading, which degrades the data rate performance. The channel capacity in fading environment can be improved by employing diversity combining and / or adaptive transmission schemes [1]-[5].

Diversity combining is known to be a powerful technique that can be used to combat fading in wireless mobile environment. Maximal ratio combining, equal gain combining and SC are most prevalent diversity combing techniques [3]-[4].

Adaptive transmission is another effective scheme that can be used to overcome fading. Adaptive transmission requires accurate channel estimation at the receiver and a reliable feedback path between the estimator and the transmitter [6]. There are four adaptation transmission schemes such as ORA, CIFR, optimum power and rate adaptation (OPRA) and truncated channel inversion with fixed Rate (TIFR) [6]-[8]. Numerous researchers have worked on the study of channel capacity over different fading channels. We discuss here some representative examples. Specifically, [3]-[4] discuss the channel capacity over correlated Nakagami-m (m [greater than or equal to] 1 & m < 1) fading channels under ORA and CIFR schemes with different diversity combining techniques. In [7], the channel capacity over uncorrelated Nakagamim (m [greater than or equal to] 1) fading channels with MRC and without diversity under different adaptive transmissions schemes was analyzed. Expressions for the capacity over uncorrelated Rayleigh fading channels with MRC and SC under different adaptive transmission schemes were obtained in [8]. An analytical performance study of the channel capacity for correlated generalized gamma fading channels with dual-branch SC under the different power and rate adaptation schemes was introduced in [9]. The channel capacity of Nakagami-m (m [greater than or equal to] 1) fading channel without diversity was derived in [10] for different adaptive transmission schemes. In [11], channel capacity of dual-branch SC and MRC systems over correlated Hoyt fading channels using different adaptive transmission schemes was presented. In [12], expression for the ergodic capacity of MRC over arbitrarily correlated Rician fading channels was derived. In [13], an expression for lower and upper bounds in the channel capacity expression for uncorrelated Rician and Hoyt fading channels with MRC using ORA scheme were obtained. The analytical study of the capacity under k - [micro] fading and Weibull fading channels with OPRA, ORA, CIFR and TIFR adaptation transmission schemes using different diversity systems was presented in [14]. In [15], an analytical performance study of the channel capacity for uncorrelated Nakagami-0.5 with dual-branch MRC using OPRA and TIFR was obtained. In [16], the channel capacity over correlated Nakagami-0.5 fading channels under OPRA and TIFR schemes with MRC was discussed. An analytical performance study of the channel capacity for uncorrelated Nakagami-0.5 fading channels with dual-branch SC under OPRA, and TIFR was introduced in [17]. The Nakagami-0.5 model has been widely used in general to study wireless mobile communication system performance. Results obtained for Nakagami-0.5 will have great practical usefulness, they will be of theoretical interest as a worst fading case. This paper fills this gap by presenting an analytical performance study of the channel capacity of dual-branch SC over uncorrelated Nakagami-0.5 fading channels using ORA, and CIFR schemes.

In this paper, SC has been considered which is one of the least complex diversity combining techniques [18].

The remainder of this paper is organized as follows: In Section 2, the channel model is defined. In Section 3, spectral efficiency of no diversity and dual-branch SC over Nakagami-0.5 fading channels are derived for ORA and CIFR schemes. In Section 4, several numerical results are presented and analyzed, whereas in Section 5, concluding remarks are given.

2. Channel Model

We assume slowly-varying Nakagami-m flat fading channel. The probability distribution function (pdf) of instantaneous received SNR([gamma]) of this fading channel is gamma distributed given by [7]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where m is the Nakagami fading parameter, which measures the amount of fading, [bar.[gamma]] is the average received SNR, and [GAMMA](.) is the gamma function. For different values of m, this expression simplifies to several important distributions describing fading models. Like m = 0.5 corresponds to the highest amount of fading, m = 1 corresponds to Rayleigh distribution, m [greater than or equal to] 1 corresponds to Rician distribution, and as m [right arrow] [infinity], the distribution converges to a nonfading AWGN from [19].

In case of no diversity the pdf under worst case of fading using (1) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Assuming independent branch signals and equal average received SNR, the pdf of the received SNR at the output of dual-branch SC under Nakagami-m fading channels is given by [19]-[20] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [bar.[gamma]] is the average received SNR, m (m [greater than or equal to] 0.5) is the fading parameter, and [Q.sub.m](.) is the Marcum Q-function, which can be represented, when m is not an integer, as given in [19]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [GAMMA][...] is the complementary incomplete gamma function.

As we consider worst case of fading, then by [21]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where erfc(.) is called complementary error function. So that

1 - erfc ([square root of (0.5[gamma]/[bar.[gamma]])) = erf)[square root of (0.5[gamma]/[bar.[gamma]]))

Hence, the pdf of dual-branch SC under worst case of fading using above mathematical transformation as given in [17]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

3. Spectral Efficiency

In this section, we present closed-form expressions for the spectral efficiency of uncorrelated Nakagami-0.5 fading channels with dual-branch SC and no diversity under ORA, and CIFR schemes. It is assumed that, for the above considered adaptation scheme, there exist perfect channel estimation and an error-free delayless feedback path, similar to the assumption made in [8].

3.1 ORA

The average channel capacity of fading channel with received SNR distribution [p.sub.[gamma]]([gamma]) under ORA scheme ([C.sub.ORA] [bit/sec]) is defined in [6] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where B [Hz] is the channel bandwidth.

In fact, (5) represents the capacity of the fading channel without transmitter feedback (i.e., with the channel fade level known at the receiver only).

3.1.1 Spectral efficiency in case of no diversity

Substituting (1) into (5), the average channel capacity becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The integral can be solved using partial integration as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let u = log(1 + [gamma])

then du = d[gamma]/1 + [gamma]

Now, let dv = exp (-0.5[gamma]/[bar.[gamma]])/[square root of [gamma] d[gamma]

After performing integral using [21], we obtain

v = [square root of [pi][bar.[gamma]] erf [[square root of 0.5[gamma]/[bar.[gamma]]]

Evaluating the above integral by using partial integration and after some mathematical transformation using [21]-[22], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [sup.2][F.sub.2](.,.;.,.;.) is the generalized hypergeometric function and erf i (.) is the imaginary error function.

Using that result, we obtain average channel capacity per unit bandwidth i.e. [C.sub.ORA]/B [bit/sec/Hz] said to be spectral efficiency as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where e is the Euler-Mascheroni constant having the value approximately equal to 0.577215665 given in [19].

3.1.2 Spectral efficiency in case of dual-branch SC

Substituting (4) into (5), the average channel capacity of dual-branch SC over uncorrelated Nakagami-0.5 fading channels is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

As we know that error function can be represented by [21] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Substituting (8) in (7), Cora after some mathematical transformation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using that result we obtain spectral efficiency i.e. [C.sub.ORA]/B [bit/sec/Hz] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The integral can be solved using partial integration as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let u = log(1 + [gamma])

then du = d[gamma]/1 + [gamma]

Now let dv = exp (-0.5 [gamma]/[bar.[gamma]]) [[gamma].sup.n] d[gamma]

After performing integral using [21], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Evaluating integral by using partial integral and some mathematical transformation using [21]-[22], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The computation of the spectral efficiency according to (10) requires the computation of an infinite series. To efficiently compute the series, we truncate the series, and present bounds for the spectral efficiency.

The spectral efficiency in (10) can be written as [[eta].sub.ORA] = [[eta].sub.ORA,N] + [[eta].sub.ORA,E], where [[eta].sub.ORA,N] is the expression in (10) with the infinite series truncated at the N th term as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [[eta].sub.ORA,E] is the truncation error resulting from truncating the infinite series in (10) at n = N.

The lower bound for [[eta].sub.ORA] is derived as

[[eta].sub.ORA] > [[eta].sub.ORA,N] + [[eta].sub.ORA,E]-low

where [[eta].sub.ORA,E-low], which is the lower bound of [[eta].sub.ORA,E]

The lower bound for the spectral efficiency can be derived by using the relationship between the area of the pdf and the expression of the spectral efficiency as discuss in [13].

As we know that area of pdf [p.sub.[gamma]] ([gamma]) is equal to unity.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Substituting (4) into (11), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

After integrating and using manipulation we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

And let

[DELTA][P.sub.N-1] = 4/[pi] x [(-1).sup.N]/(2N + 1) (13)

Then

[P.sub.N] = [P.sub.N-1] + [DELTA][P.sub.N-1]

Similarly, from (10) let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

And

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Dividing (15) by (13), yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Observing that ([DELTA[[eta].sub.ORA,N-1]/[DELTA][P.sub.N-1]) monotonically increases with increasing N, i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Hence, the spectral efficiency in (10) can be lower bounded [[eta].sub.ORA,E-low] by using (16) and (17) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

The upper bound for [[eta].sub.ORA] is derived as

[[eta].sub.ORA] < [[eta].sub.ORA,N] + [[eta].sub.ORA,E-up]

where [[eta].sub.ORA,E-up], which is the upper bound of [[eta].sub.ORA,E]

The expression in (9) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Let [a.sub.n] = 1/2n + 1,

Then [a.sub.n+1]/[a.sub.n] = 2n+1/2n+3 < 1

i.e. [a.sub.n] monotonically decreases with increase of n, therefore, [[eta].sub.ORA,E] can be upper bounded as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

After evaluating the integral (20) and some mathematical manipulations using [20]-[21], we obtain the upper bound

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Therefore, the spectral efficiency in (10) can be upper bounded as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

where [E.sub.1](.) is the exponential integral of first order.

Hence, the spectral efficiency is bounded using (18) and (22) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

The average channel capacity of fading channel with received SNR distribution [p.sub.[gamma]]([gamma]) under CIFR scheme ([C.sub.CIFR] [bit/sec]) is defined in [6] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Channel inversion with fixed rate is the least complex technique to implement, assuming good channel estimates are available at the transmitter and receiver.

3.2.1 Spectral Efficiency in case of no Diversity

The pdf for Nakagami-0.5 fading channel is given in (2) as

[p.sub.[gamma]]([gamma]) = exp(-0.5[gamma]/[bar.[gamma]])/[square root of 2[pi][bar.[gamma]][gamma]], [gamma] [greater than or equal to] 0

Hence,

[p.sub.[gamma]]([gamma])/[gamma] = exp(-0.5[gamma]/[bar.[gamma]])/[square root of 2[pi][bar.[gamma]][gamma]] x 1/[gamma], [gamma] [greater than or equal to] 0 (25)

Integrating (25) over an interval as shown below

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Evaluating integral by some manipulation using [21], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As we know that

erf(0) = 0, erf ([infinity]) = 1,

And

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Putting this value of integral in (24), we get

[C.sub.CIFR] = B [log.sub.2](1) = 0 (26)

So, the spectral efficiency with no diversity under Nakagami-0.5 fading channel is zero.

3.2.2 Spectral Efficiency in case of dual-branch SC

The pdf of dual-branch SC over uncorrelated Nakagami-0.5 fading channel is given in (4) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Integrating the (27) over an interval as shown below

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

After evaluating the integral using [21]-[22], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As we know that

[GAMMA](n,0) = [GAMMA](n)

And

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Putting this value of integral in (24), we get

[C.sub.CIFR] = B [log.sub.2](1) = 0 (28)

4. Numerical Results and Analysis

In this section, various performance evaluation results for the spectral efficiency have been obtained using dual-branch SC and no diversity under worst fading condition. These results also focus on spectral efficiency comparisons between the different adaptive transmission schemes.

In Fig. 1, the spectral efficiency under ORA scheme is plotted as a function of the average received SNR per branch [bar.[gamma]]. As expected, by increasing [bar.[gamma]] and/or employing diversity, spectral efficiency improves.

It is seen in Table. 1 that as the truncation error bounds becomes tighter as the truncation level, N, increases. It means that as the truncation level increases the difference between upper and lower bounds for each average received SNR per branch [bar.[gamma]] decreases and hence calculated spectral efficiency becomes more appropriate. That's why the infinite series in [[eta].sub.ORA] has been truncated at the 15th term to calculate the spectral efficiency for the Fig. 1.

In Fig. 2, the spectral efficiency under ORA scheme is plotted as a function of the average received SNR per branch [bar.[gamma]]. For comparison, the spectral efficiency of uncorrelated Rayleigh fading channels with dual-branch SC and without diversity, which was obtained in [8, Eq. (44)] and [8, Eq. (34)] respectively, is also presented in Fig. 2. As expected, as the channel fading conditions improves, i.e, m and / or [bar.[gamma]] increases, spectral efficiency improves. It is very interesting to observe that the spectral efficiency without diversity for [bar.[gamma]] [less than or equal to] -7.5 dB remains same as we move from worst fading condition to Rayleigh. Similarly, dual-branch SC for [bar.[gamma]] [less than or equal to] -2.5 dB, gives almost identical performance even we move from worst fading condition to Rayleigh.

Fig. 3 states that spectral efficiency versus average received SNR per branch g over Nakagami-0.5 fading channel remains zero as we go from no diversity to dual-branch SC under CIFR scheme.

In Fig. 4, the spectral efficiency under CIFR is plotted as a function of the average received SNR per branch [bar.[gamma]]. For comparison, the spectral efficiency of uncorrelated Rayleigh fading channels with dual-branch SC, which was obtained in [8, Eq. (52)], is also presented in Fig. 4. It is observed that, as the channel fading conditions improve, i.e, m and / or [bar.[gamma]] increases, spectral efficiency with no diversity remains zero. However, employing a dual-branch SC system improves the channel capacity as we go from worst case of fading conditions to Rayleigh fading conditions.

In Fig. 5, the spectral efficiency of uncorrelated Nakagami-0.5 fading channels with and without diversity is plotted as a function of [bar.[gamma]], considering ORA, OPRA, and TIFR adaptation schemes with the aid of (6), (23), [17, Eq. (27)], [17, Eq. (35)], [15, Eq. (8)], and [15, Eq. (22)]. It shows that, the spectral efficiency with no diversity under ORA scheme improves over TIFR for [bar.[gamma]] [greater than or equal to] 5 dB. It is also observed that the spectral efficiency with dual-branch SC under ORA scheme improves over TIFR for [bar.[gamma]] [greater than or equal to] 0 dB and OPRA scheme provides better efficiency under worst case of fading. It is also interesting to observe that for [bar.[gamma]] [less than or equal to] -7.5 dB ORA scheme with dual-branch SC gives inferior performance with respect to TIFR scheme without diversity.

5. Conclusions

In this paper, closed-form expressions for the spectral efficiency of dual-branch SC and no diversity under ORA and CIFR schemes have been obtained and analyzed. Error bounds have been derived for truncated infinite series under ORA. Numerical results illustrate that the bounds can be used effectively to determine the number of terms needed to achieve desirable level of accuracy. Results have been plotted, which show that by increasing [bar.[gamma]] and/or employing diversity, spectral efficiency improves under ORA scheme. It is also observed that the spectral efficiency under CIFR is zero under worst case of fading even when a dual-branch SC is utilized. It is important to note that the spectral efficiency using ORA scheme remains almost same even when fading conditions improve from Nakagami-0.5 to Rayleigh either under dual-branch SC for [bar.[gamma]] [less than or equal to] -2.5 dB or under no diversity for [bar.[gamma]] [less than or equal to] -7.5 dB. This paper finally concludes that under worst case of fading TIFR scheme is a better choice for low average received SNR and ORA scheme is for high average received SNR even employing diversity. Therefore it is recommended that under worst fading condition, ORA scheme is not always a better choice over TIFR.

References

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[3] S. Khatalin, J.P.Fonseka, "Channel capacity of dual-branch diversity systems over correlated Nakagami-m fading with channel inversion and fixed rate transmission scheme," IET Communications, Vol. 1, No.6, pp. 1161-1169, 2007.

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Mohammad Irfanul Hasan (1,2) and Sanjay Kumar (1)

(1) Department of Electronics and Communication Engineering, Birla Institute of Technology, Mesra, Ranchi, India

(2) Department of Electronics and Communication Engineering, Graphic Era University, Dehradun, India

irfanhasan25@rediffmail.com, skumar@bitmesra.ac.in

Table 1. Comparison of [[eta].sub.ORA,N],
[[eta].sub.ORA, E--up], and [[eta].sub.ORA, E--low] at
two different values of N for worst case of fading.

                                          N = 5

[bar.[gamma]]                        [[eta].sub.ORA,   [[eta].sub.ORA,
[dB]             [[eta].sub.ORA,N]       E--up]            E--low]

-10                  0.143708           0.061883        0.0581541295
-5                   0.411463           0.120879        0.1153123659
0                    0.977032           0.196274        0.1896116138
5                     1.89534            0.27954        0.2723395591
10                    3.10794            0.36565        0.3582797952
                      N = 15
                                          N = 5

[bar.[gamma]]                        [[eta].sub.ORA,   [[eta].sub.ORA,
[dB]             [[eta].sub.ORA,N]       E--up]            E--low]

-10                  0.164232           0.041359        0.04064235544
-5                   0.463140           0.069202        0.06833679633
0                     1.07295            0.10041        0.09942977640
5                     2.04205            0.13283        0.13180969046
10                    3.30793            0.16564       0.164623141056
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Date:Dec 1, 2015
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