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Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel
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Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel
1
2
Weiye Xu ∗ , Qingyun Wang
3
Q2
4
Q1 School of Communication Engineering, Nanjing Institute of Technology, Nanjing, China
5
a r t i c l e
6 19
i n f o
a b s t r a c t
7
Article history: Received 16 April 2015 Accepted 10 November 2015 Available online xxx
8 9 10 11 12
18
Keywords: Distributed antenna systems Channel capacity Outage capacity Nakagami channel Outage probability
20
1. Introduction
13 14 15 16 17
In this paper, the downlink capacity performance of distributed antenna systems (DAS) with antenna selection are investigated in composite Nakagami channel which take path loss, shadow fading and Nakagami fading into account. According to the performance analysis and using numerical integrations, the probability density function (PDF) and cumulative distribution function (CDF) of the effective output SNR of DAS are derived, respectively. Based on these results, tightly-approximate closed-form expressions of channel capacity and outage probability of DAS in composite Nakagami channel are further derived. These expressions include the ones under composite Rayleigh channel as special cases. Conditioned on the outage probability, a practical iterative algorithm based on Newton’s method for finding the outage capacity is proposed. To avoid iterative calculation, another approximate closed-form expression of outage capacity is also derived by utilizing the Gaussian distribution approximation. Simulation results show that the derived expressions have more accuracy than the existing ones, and can match the corresponding simulation well. Thus, the capacity performance of DAS in composite Nakagami channel can be evaluated effectively. © 2015 Published by Elsevier GmbH.
Distributed antenna system (DAS), as a promising technique for next generation wireless mobile communications, has received much attention due to its power and capacity advantages over the traditional co-located antenna system (CAS) [1,2]. Considering that channel capacity plays an important role in communication systems as a measure of the system performance, many academic works have been done to analyze the capacity performance of DAS, and proved that DAS can enhance the system capacity greatly [3–5]. The effect of maximal ratio combining based macrodiversity on the capacity of code division multiple access DAS uplink is studied in [3]. The downlink capacity of DAS over Rayleigh fading and Nakagami fading in multicell environment are analyzed in [4,5], respectively. The closed-form expressions of channel capacity are derived in the above two references, but the derived expression in [4] has minor error, while [5] needs iterative calculation. For this, an accurate closed-form expression of channel capacity of DAS downlink in Nakagami fading multicell environment is derived in [6]. However, the above literatures do not consider the effect of shadow fading on the performance. Based on this, the channel capacity of
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
Q3 Q2
∗ Corresponding author. Tel.: +86 2552169128. E-mail address: [email protected] (W. Xu).
DAS downlink is analyzed in [7] by using different cooperation strategies, but the influence of noise is neglected for analysis simplicity. In [8,9], by using a log-normal distribution to substitute the gamma-log-normal distribution, approximate channel capacities are derived for DAS downlink over shadowed fading channel, where Rayleigh fading and Nakagami fading are respectively considered, but the derived theoretical capacities are not accurate enough to reflect the actual values, and the analysis is limited in single receive antenna case. By assuming double-sided spatial correlation, an analytical lower bound of the ergodic capacity of DAS in composite Rayleigh fading channel is provided in [10], but the capacity is accurate in the low and high SNR regimes only. Considering that system outage probability is an important performance measure of communication systems, the outage performance of DAS in fading channel is studied in [11–14]. For the DAS uplink, the approximate outage probability is analyzed and derived for DAS in composite Rayleigh/lognormal fading channels [11,12], and in composite Nakagami fading channels [13], where the mobile terminal is equipped with one antenna only. In [14], a closed-form approximation of the outage probability for the DAS downlink and uplink is derived, and this expression will be tight and valid only when the rate or the number of nodes and antennas of the system become large. Although in all these studies, the channel capacity and outage probability are well analyzed, the obtained capacity expressions
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Please cite this article in press as: W. Xu, Q. Wang, Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.050
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(k)
where n represents the small-scale fading between RAn and the kth receive antenna of MT. For Nakagami-m fading channel, (k) (k) the amplitudes of {n }, {˛n } are modeled as independent Nakagami random variables of unit power with fading factor mn [15]. (k) The phase n is uniformly distributed over [0,2]. The Rayleigh distribution, which corresponds to m = mn = 1, is a special case of Nakagami-m distribution. For Nakagami fading channel, the PDF of (k) ˛n is given by
103
in composite fading channel are not accurate due to the usage of inaccurate approximation. Moreover, only limited work has been carried out for the outage performance analysis of distributed antenna system downlink. Especially, the outage capacity of distributed antenna system in composite Nakgagagi-m fading channels is not studied in the existing literatures. For this reason, we will study the capacity performance of distributed antenna system downlink with transmit antenna selection and multiple receive antennas in composite channel which considers path loss, lognormal shadowing and Nakagami fading, and focus on the derivation of the channel capacity and outage capacity of distributed antenna system over composite fading channel. In terms of the channel state information (CSI) and maximum SNR criterion, a selection transmission scheme is presented to maximize the effective SNR of the system, i.e., only the ‘best’ distributed antenna is selected for data transmission. Based on this scheme and the performance analysis, using some numerical integrations, the probability density function (PDF) and cumulative distribution function (CDF) of the effective SNR are respectively obtained. With the obtained CDF and PDF, tightly-approximate closed-form expression of channel capacity of distributed antenna system is derived. This expression has more accuracy than the existing one. To analyze the outage performance of the system well, we derive an accurate closed-form expression of outage probability for distributed antenna system under a given outage capacity as well. These theoretical expressions include the ones under composite Rayleigh channel as special cases. Besides, a practical iterative algorithm based on Newton method for calculating the outage capacity is proposed for a given outage probability. As a result, accurate outage capacity is obtained. Moreover, by utilizing the Gaussian distribution approximation, an approximate closed-form outage capacity is also derived. With these theoretical expressions, the capacity performance of distributed antenna system can be effectively evaluated. Simulation results verify the effectiveness of the theoretical analysis. The notations we use throughout this paper are as follows. Bold upper case and lower case letters denote matrices and column vectors, respectively. E{·} denotes the expectation. The superscripts (·)T and (·)H are used to stand for the transpose and Hermitian transpose, respectively.
104
2. Channel model and system model
ω exp f n ( ) = √ 2n
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102
105 106 107 108 109 110 111 112 113 114
115
In this paper, we consider a DAS downlink with multiple remote antennas (RAs) and multiple receive antennas in a single-cell environment, the remote antennas are distributed in the cell and connected with the central base station (BS, also named as RA0 ) via coaxial cables, fiber optics or radio links, where N RAs are considered, and the nth RA is denoted as RAn . The mobile terminal (MT) is equipped with K antennas. To improve the system performance, the antenna selection diversity scheme is employed for remote antennas. If RAn is selected to transmit the signals, the received signals at mobile terminal can be expressed as rn =
Pn hn b + z =
Pt hn
(1)
, . . ., hn T
(K) T
b+z
(1)
(k)
123
where r n = [rn (1) , rn (1) , . . ., rn (K) ] , r n is the received signal from (k) the kth antenna at MT, Pt is the transmit signal power. hn is the element of channel vector hn and denotes the composite channel fading coefficient between RAn and the kth antenna of MT. b is the transmitted signal from RAn with unity energy. z is the noise vector, whose elements are independent, identically distributed (i.i.d) complex Gaussian random variables with zero mean and variance (k) z2 . hn can be modeled as [8,9].
124
(k) hn
116 117 118 119 120 121 122
=
(k) n
Ln Sn
(2)
f
(k)
˛n
(˛) = 2(mm /˝)
mn
exp(−mn ˛2 /˝)˛2mn −1 / (mn )
(3)
(k) 2
where (.) is the Gamma function [16], ˝ = E{[˛n ] } = 1 is the average fading power. Ln and Sn denote the pass loss and shadowing effect between RAn and mobile terminal. The path loss term Ln is ˇ expressed as Ln = (d0 /dn ) n , where ˇn is the path loss exponent, d0 is the reference distance and dn represents the distance from RAn to the mobile terminal. The shadowing effect term Sn is log-normally distributed, and the mean of 10log10 Sn is assumed to be zero (in dB). As a result, the PDF of Sn can be expressed as [17] ω exp fSn (s) = √ 2n s
−
(10log10 s)
K
(k)
n = n
K
(4)
(2n2 )
k=1
(k)
˛n
2
= n
k=1
K
(k)
2
˛n
(k)
|n |2
(5)
(k)
−
(10log10 − n )2 (2n2 )
(6)
K
K (k) 2 (k) 2 (˛ ) = can be evaluated as n k=1 k=1 n f (u) = mn (mn u)mn K−1 exp(−mn u)/ (mn K)
n K mn K−1 mm mn n exp − x (mn K) xmn K
× exp
−
129 130 131 132
133
134 135 136 137 138 139 140 141
142
143 144 145 146
147
(10log10 x − n ) 2n2
2
148 149 150 151 152
153
154 155 156
(7)
With (6) and (7), the PDF of ␥n can be expressed as:
0
128
where n = 10log10 (Pt Ln /z2 ) is the mean of 10log10 n . From (3), using a variable transformation, the PDF of =
f n ( ) =
127
k=1
where n denotes the instantaneous SNR at the kth receive antenna, and n = Pt Ln Sn /z2 . According to the definition of Sn , n is also log-normally distributed. Thus, by a transformation of random variable, the PDF of
n can be given by
∞
126
2
where n (in dB) is the standard deviation of 10log10 Sn and ω = 10/log10. With (1) and (2), when single remote antenna n is selected, the effective SNR after maximum ratio combining can be obtained as
n =
125
157
158
ω √ 2n x
159
dx
(8)
160
161
Considering the difficulty of integration in (8), [8,9] resort to the log-normal distribution, and use it to approximate the Gammalog-normal distribution in (8) and derive the closed-form PDF of ␥n . Unfortunately, the derived PDF is not accurate in some cases. For this reason, we will recalculate the (8) using the numerical integration and the transformation of variable.
Please cite this article in press as: W. Xu, Q. Wang, Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.050
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Let t = (10log10 x − n )/ f n ( ) =
√ (Kmn )
Kmn −1 10−(
2n2 , (8) can be rewritten as
exp
−∞
168
+∞
n mKm n
W. Xu, Q. Wang / Optik xxx (2015) xxx–xxx
√ 2n t+ n )/10
Kmn
− 10
√ −( 2n t+ n )/10
This is a tightly approximate closed-form PDF of the effective SNR in composite Nakagami fading channel. When mn = 1 (n = 1,. . .,N), (12) and (14) are reduced to the PDF and CDF of the effective SNR in composite Rayleigh channel, i.e.,
mn
exp − t 2 dt
(9)
f ( ) ∼ =
N n=0
∼ =
169 170 171 172 173 174
1
√ (Kmn )
Hi exp ( − mn cni ) Kmn −1 (mn cni )Kmn
F n ( ) =
1 f n (v) dv ∼ =1− √ (Kmn )
Np
Hi (Kmn , mn cni )
i=1
where (·, ·) is the incomplete Gamma function [16].
178
3. Channel capacity analysis of distributed antenna systems
182 183
184
Ca = E C
= E log2 (1 + )
186 187 188 189 190 191 192
193
F ( ) =
N
n=0
n=0
F n ( ) ∼ =
1
1− √ (Kmn )
Np
Hi (Kmn , mn cni )
i=1
(12)
194
195 196
With (12), the PDF f( ) can be obtained as:
⎡
dF ( ) ⎣f n ( ) = d N
197
f ( ) =
n=0 198
199
F j ( )⎦
(13)
j=0,j = / n
Substituting (9) and (10) into (13) yields f ( ) ∼ =
N n=0
200
⎤
N
×
mn Kmn Hi e− mn cni ( cni )Kmn −1 cni √ (Kmn ) N
j=0,j = / n
1 1− √ Hi (Kmn , mn cni ) (Kmn ) N
i=1
(15)
207
n=0
1 1− √ Hi (K, cni ) (K) Np
208
(16)
209
i=1
210
mn cni √ ( mn cni )Kmn −1 (Kmn )
0
211
⎤
N
F n ( ) d ⎦
(17)
212
213
Np N
+∞
Ca ∼ =
Hi
⎡
=
Hi
n=0 i=1
Al log2
l=1
N
yKmn −1 √ (Kmn )
yl 1+ mn cni
(18)
mn cni
where F j ( ) is the CDF of ␥j as shown in (10), {Al } are the weights associated with the zeros {yl } of the one dimensional Nq th order Laguerre polynomial [18]. Eq. (18) is tightly approximate closedform expression of average channel capacity of distributed antenna systems in composite Nakagami fading channel, it has more accuracy than the existing expression in [9] because the latter employs the inaccurate approximate PDF for capacity derivation. It is shown that the derived (18) will be in agreement with the simulation. Besides, although [9] derives the capacity expression of distributed antenna systems in composite Nakagami channel, it is only suitable for single receive antenna case. For this, we use the approximate method in [9] to give the capacity expression under multiple receive antenna cases, i.e., Np N Hi
×
√ ˆ n )/10 2ˆ n ti +
√ log2 (1 + 10(
N
1 − 0.5erfc
j=0,j = / n
216
n −1 yKm l √ (Kmn )
y l
F j
j=0,j = / n
n=0 i=1
(14)
215
cni mn
Nq N Np
×
y mn cni
⎤ y ⎦ e−y dy F j
j=0,j = / n
Ca ∼ =
log2 1 +
214
0
N
×⎣
Np
i=1
205
Utilizing the transformation of variable y = mn cni and Gaussian–Laguerre integration in [18], (17) can be changed to
(11)
where C = log2 (1 + ) is the channel capacity, and f( ) is PDF of effective SNR at the receiver. Based on the analysis in Section 2, the selective diversity scheme is applied to the transmitter, i.e., only one ‘best’ RA is selected for transmission to maximize the effective SNR. Thus, we have that = max{ 0 , 1 , . . ., N }. Considering large space among remote antennas, it is reasonable to assume that { n } are independent. Thus, with (10), we can evaluate the CDF of , F( ), as follows: N
F n ( ) ∼ =
n=0
0 185
n=0 i=1
f ( ) log2 (1 + ) d
204
i=1
N
n=0 i=1
203
j=0,j = / n
∞ =
N
× e− mn cni
Considering that the capacity is an important performance measure of the communication systems in fading channel, we will give the capacity analysis of the distributed antenna systems over a composite Nakagami fading channel in this section. The average channel capacity can be expressed as
1 1− √ Hi (K, cni ) (K)
N
(10)
202
206
i=1
and F ( ) =
Hi e− cni ( cni )K−1 cni
⎡+∞ N Np Ca ∼ Hi ⎣ log2 (1 + ) =
177
181
Np
Substituting (15) into (11) gives:
176
180
N j=0,j = / n
0
179
1
√ (K)
×
i=1
In the above derivation, the Gauss–Hermite integration [18] is √ utilized, and cni = 10−( 2n ti + n )/10 . ti and Hi are the base point and weight factor of the Np -order Hermite polynomial, respectively. Eq. (9) is tightly approximate PDF, and has more accuracy than the existing approximate PDFs [8,9]. From (9), the CDF of ␥n can be calculated as
175
Np
3
√
)
2ˆ n ti + ˆn − ˆj √ 2ˆ j
217 218 219 220 221 222 223 224 225 226 227 228 229
230
(19)
201
231
232
Please cite this article in press as: W. Xu, Q. Wang, Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.050
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233 234 235 236 237 238 239
240
where ˆ n = n + ω[ (Kmn ) − log(mn )], ˆ n2 = n2 + ω2 ς(2, Kmn ), (·) is the Euler psi function, ς(·, ·) is the Reimann’s zeta function, and erfc{.} denotes the complementary error function [16]. For mn = 1, (19) is reduced to (16) in [9]. Substituting mn = 1 into (18), we can obtain tightly approximate closed-form expression of average channel capacity of DAS in composite Rayleigh channel, i.e., Ca ∼ =
Np Nq N
Hi
n=0 i=1 241 242 243 244
245
Al log2
l=1
y 1+ l cni
yK−1 l
√ (K)
N
F j
y l
cni
j=0,j = / n
G (C o ) = F (2Co − 1) − ε ∼ =
N n=0
Hence, the derived capacity (18) includes the capacity under Rayleigh channel as a special case. Eq. (20) has more accuracy than the existing expression in [8] since the later uses inaccurate PDF to derive the capacity.
Np
∂G (Co ) = f (2Co − 1) 2Co log 2 > 0 ∂Co
247 248 249 250 251 252 253 254 255 256 257 258 259
4.1. Outage probability
Np N 2Co log 2
√ (Kmn )
×
2Co −1 260
Co
Po = Pr (C ≤ Co ) = Pr ( ≤ 2
Co
− 1) =
f ( ) d = F (2
− 1)
j=0,j = / n
(21)
Kmn −1
Hi e− mn cni 2Co − 1
280
281 282 283 284
(mn cni )Kmn
i=1
1 1− √ Hi Kmn , 2Co − 1 mn cni (Kmn ) N
285
i=1
(26)
286
287
From (24), we have:
288
G (0) = F (0) − ε = 0 − ε < 0, and G (+∞) =
289
lim G (C o ) = 1 − ε > 0
Co →+∞
(27)
290
291
This is because outage probability ε ∈ (0, 1). According to the above results, and considering the outage capacity Co > 0, the equation G(Co ) = 0 in (24) will have a unique solution. There are many methods such as bisection method for finding the root of a strictly monotonic function. We propose to use Newton’s method to find the root iteratively because it has the quadratic convergence rate. Newton’s method is described as follows:
0 261
(25)
G (C o )
n=0
The outage capacity and probability are also important performance measure indicators of communication systems in fading channels [11–13]. However, the related research is relatively less. Especially the outage capacity analysis is much less. Based on the reason above, we will give the outage performance analysis of the distributed antenna systems in a composite Nakagami fading channel, and derive the closed-form outage probability and capacity. Because the channel capacity is a random variable, it is meaningful to consider its statistical distribution. A useful measure of statistical characteristic is the outage capacity [19]. For the given outage capacity Co , Po is defined as the outage probability that channel capacity C = log2 (1 + ) falls below Co . Thus, the outage probability can be expressed as
279
where f( ) is the PDF of as shown in (14). Considering that 2Co , log 2 and f (2Co − 1) are all positive, the derivative G (Co ) in (25) is also positive. So G(Co ) is a strictly monotonically increasing function of Co . Substituting (14) into (25) yields
N
246
277
278
Differentiating G(Co ) with respect to Co yields
∼ =
4. Outage capacity analysis of DAS
1 1− √ Hi (Kmn , (2Co − 1) mn cni ) − ε (24) (Kmn ) i=1
G (C o ) = (20)
276
(u+1)
Co
=
(u) Co
−G
(u) Co
(u)
G Co
(28)
292 293 294 295 296 297 298
299
262 263
264
265
Substituting (12) into (21) yields Po = F (2 ∼ =
N
n=0 266 267 268 269
270
272 273 274 275
(u)
− 1)
1 1− √ (Kmn )
Np
Hi (Kmn , (2Co − 1) c ni mn )
(22)
i=1
Eq. (22) is a tightly approximate closed-form expression of outage probability of DAS in composite Nakagami fading channel, which is shown to agree the simulation well. When mn = 1, (22) is reduced to the outage probability of DAS in composite Rayleigh channel, i.e., P0 ∼ =
N n=0
271
Co
1 1− √ Hi (K, (2Co − 1) c ni ) (K) Np
i=1
4.2. Outage capacity In this subsection, the outage capacity performance is analyzed. For a given outage probability ε = Po , a practical iterative algorithm based on Newton’s method is proposed to find the outage capacity Co . The outage capacity from (22) can be expressed as finding the root of G(Co ) = 0 with
300 301 302 303 304 305 306
+∞
(log2 (1 + ))2 f ( )d
E(C 2 ) = 0
(23)
(u)
where á Co is the outage capacity value at the uth iteration. G(Co ) (u) and G (Co ) are computed by (24) and (26), respectively. With (28), the accurate outage capacity may be obtained, but iterative calculation is required. For this reason, we will derive an approximate closed-form expression of outage capacity in the following. Using (14) and (18), the second moment of the channel capacity C can be calculated as
∼ =
Np Nq N n=0 i=1
×
Al log 2 1 +
Hi
l=1
ylKmn −1 √ (Kmn )
N
F j
j=0,j = / n
yl mn cni
2
(29)
y l mn cni
From (18) and (29), the variance of C can be evaluated as
2
Vc = E C
2
− [E (C)]2 = E C
307
− Ca2
308
(30)
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With the mean value (18) and the variance (30), using a Gaussian distribution approximation, the CDF of the channel capacity C may be approximated as
c 313
F (c) ≈ −∞
1 2Vc
5
d
c − Ca
(31)
318
Co ≈
2Vc erfc −1 2 (1 − ε) + Ca
(32)
328
329
5. Simulation results
324 325 326 327
330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345
In this section, we use the derived theoretical formulae and computer simulation to evaluate the capacity of the distributed antenna systems over composite channels, where path loss, Nakagami-m fading and lognormal shadowing are all considered. In simulation, the cell shape is assumed to be circle with radius R. The BS (RA0 ) is in the center of the cell, and the N RAs are uniformly distributed over a circle with radius r. The Monte-Carlo method is employed for simulation. Different Nakagami factor m, path loss exponent ˇ and receive antenna K are considered for comparison. The polar coordinate of the ith RA is (r, 2(n − 1)/N, n = 1,2,. . .,N. The main parameters are listed as: N = 6; r = 2R/3; the reference distance d0 = 80 m; radius of the cell R = 103 m; and standard deviation = n = 6 dB. The simulation results are illustrated in Figs. 1–6. In Figs. 1 and 2, we plot the theoretical average channel capacity and corresponding simulation of DAS with different Nakagami factors over composite fading channel, where m = mn = 1, 3. Eq. (18)
0
0
7
6
5
wiht m=1(Simu.) with m=1(theory1) with m=1(theory2) wiht m=3(Simu.) with m=3(theory1) with m=3(theory2)
10
15
20
25
30
Fig. 2. Average channel capacity of DAS with different receive antennas and Nakagami factors (ˇ = 3).
7
6
5
DAS DAS DAS DAS DAS DAS
with m=1(simu.) with m=1(theory4) with m=1(theory3) with m=3(simu.) with m=3(theory4) with m=3(theory2)
4 Po=0.3 3
2 Po=0.15 1
0 0
5
10
15
20
25
30
SNR (db)
Fig. 3. Outage capacity of DAS with different outage probability and Nakagami factors (K = 2, ˇ = 3).
7
6
5 DAS DAS DAS DAS DAS DAS
5
SNR (db)
Co(bit/s/Hz)
323
DAS DAS DAS DAS DAS DAS
with Po=0.3(simu.) with Po=0.3(theory4) with Po=0.3(theory3) with Po=0.15(simu.) with Po=0.15(theory4) with Po=0.15(theory3)
4
β=2
3
β=3 2
Ca (bit/s/Hz)
322
1
Co(bit/s/Hz)
where ε is the given outage probability, erfc−1 {.} denotes the inverse complementary error function. Eq. (32) is an approximate closed-form expression of outage capacity, and has the value close to the simulation. From (32) and (23), it is found that Co will increase (decrease) as ε increases (decreases), which accords with the existing knowledge. Hence, the derived theoretical formulae will provide effective methods to assess the outage performance of distributed antenna systems over composite Nakagami channel. Moreover, (32) is relatively simpler because it has closed-form calculation and does not need any iterative calculation.
321
K=2 3 K=1
By setting the above F(c) = ε in (31), the outage capacity can be obtained as
320
4
2
317
319
wiht m=1(Simu.) with m=1(theory1) with m=1(theory2) wiht m=3(Simu.) with m=3(theory1) with m=3(theory2)
2Vc
315 316
DAS DAS DAS DAS DAS DAS
6
= 1 − 0.5erfc
314
( − Ca ) 2Vc
−
exp
2
7
Ca (bit/s/Hz)
310
5
4
1 3
0 0
β=2 2
β=4
0
5
10
15
20
10
15 SNR (db)
20
25
30
Fig. 4. Outage capacity of DAS with different path loss exponent and outage probabilities (m = 3, K = 1).
1
0
5
25
30
SNR (db)
Fig. 1. Average channel capacity of DAS with different path loss exponent and Nakagami factors (K = 1).
is adopted for theoretical capacity calculation (referred as ‘theory 1’), and the existing theoretical formula [9] (i.e., (19), referred as ‘theory 2’) is employed for comparison. In Fig. 1, single receive antenna (K = 1) and different path loss exponents (ˇ = ˇn = 2,4) are
Please cite this article in press as: W. Xu, Q. Wang, Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.050
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10
β=4
-1
Outage Probability
10
β=3
-2
10
-3
10
DAS DAS DAS DAS
with m=1(Simu.) with m=1(theory) with m=2(Simu.) with m=2(theory)
-4
10
10
15
20 SNR (db)
25
30
Fig. 5. Outage probability of DAS with different Nakagami factor and path loss exponents (K = 2).
Fig. 6. Outage probability of DAS with different receive antennas (m = 2, ˇ = 3).
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considered. From Fig. 1, it is found that the theoretical values from ‘theory 1’ agree well with the simulated ones for different m and ˇ, while the theoretical values from ‘theory 2’ have some difference with the corresponding simulations, and the difference will be more obvious when the Nakagami factor m is smaller. This is because the latter adopts the log-normal distribution to replace the Gamma-log-normal distribution for deriving the PDF, which brings about the inaccuracy of the channel capacity in some cases. Besides, the capacity of DAS with ˇ = 2 is obviously higher than that with ˇ = 4 due to smaller path loss. The capacity under Nakagami factor m = 3 is higher than that with m = 1, this is because the fading severity decreases as the Nakagami factor m increases. The above results indicate that the derived expression of average channel capacity for DAS is effective. In Fig. 2, we consider different receive antennas (K = 1,2) and different Nakagami factor (m = 1,3), path loss exponent ˇ = 3. As shown in Fig. 2, the derived ‘theory 1’ is still good agreement with the corresponding simulation, whereas the ‘theory 2’ still has the difference with the simulation, especially in the case of single receive antenna. Besides, the systems with three receive antennas outperform those with single receive antenna because the former has greater diversity, and the capacity of DAS with m = 3 is higher than that with m = 2 due to better fading case. The above results further show that the derived channel capacity is valid and reasonable.
In Figs. 3 and 4, we plot the theoretical outage capacity and corresponding simulation of DAS with different outage probabilities, where Po is set as 0.15 and 0.3, respectively. The accurate and approximate outage capacities are, respectively, computed by (28) and (32), and they are referred as ‘theory 3’ and ‘theory 4’, respectively. In Fig. 3, different Nakagami factor (m = 1,3), two receive antennas (K = 2) and path loss exponent (ˇ = 3) are considered. It is shown in Fig. 3 that theoretical values are all close to the corresponding simulated ones. Especially, the ‘theory 3’ from the Newton’s method has more accuracy than the ‘theory 4’ from Gaussian distribution approximation, but the latter does not need iterative calculation. It is observed that outage capacity of DAS with Po = 0.3 is higher than that with Po = 0.15 since larger outage probability is permitted to happen, which accords with the theoretical analysis in Section 4. Besides, the outage capacity under m = 3 is higher than that under m = 1 because of better fading case, as expected. Compared with Fig. 1, it is found the outage capacity in Fig. 3 is lower than the average channel capacity in Fig. 1 under the same case due to the constraint of outage probability. In Fig. 4, we consider different path loss exponents (ˇ = 2,3), the receive antenna number K = 1, Nakagami factor m = 3. From Fig. 4, we can find the results similar to Fig. 3, that is, the derived theoretical outage capacity can match the corresponding simulation well, only some little differences are observed for ‘theory 4’, and the outage capacity of DAS with large Po is higher than that with small Po . Besides, the outage capacity with ˇ = 3 is lower than that with ˇ = 2 since the former experiences larger path loss, which agrees with the existing knowledge. By comparing Figs. 3 and 4, it is found that the outage capacity with two receive antennas is higher than that with single receive antenna under the same Po and ˇ since multiple antennas are employed, as expected. The above results indicate that the derived outage capacity is effective. In Figs. 5 and 6, we plot the theoretical outage probability and corresponding simulation of DAS, and the theoretical outage probability is calculated by (22). In Fig. 5, different path loss exponents (ˇ = 3,4) and Nakagami factors (m = 1,2) are considered, outage capacity Co = 2 bit/s/Hz and K = 2. From Fig. 5, we can see that the theoretical outage probability and the corresponding simulation result are very close to each other. Due to larger path loss, the outage probability of DAS with ˇ = 4 is larger than that with ˇ = 3 as expected. It is observed that the outage probability of DAS with Nakagami factor m = 2 is higher than that with m = 1. The above results show that the derived outage probability is valid. In Fig. 6, we consider different receive antennas (K = 1, 2) and outage capacities (Co = 2, 4 bit/s/Hz), path loss exponent ˇ = 3 and Nakagami factor m = 2. As shown in Fig. 6, the theoretical outage probability is good agreement with the corresponding simulation, and the outage probability of DAS with two receive antennas is lower than that with single receive antenna since the former has greater diversity than the latter. Besides, the outage probability under Co = 4 bit/s/Hz is larger than that under Co = 2 bit/s/Hz. This is because larger outage capacity is required. As a result, higher outage probability happens. The above results further testify that the derived outage capacity and probability are valid for the outage performance evaluation.
6. Conclusion We have studied the capacity performance of distributed antenna systems over composite Nakagami-m fading channel, and the channel capacity and outage capacity as well as outage probability are all analyzed. According to the performance analysis in composite fading channel, using some numerical integration operations, the PDF and CDF of the effective SNR are respectively derived. Using these results, the tightly approximate closed-form expressions of average channel capacity are obtained. This expression
Please cite this article in press as: W. Xu, Q. Wang, Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.050
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has more accuracy than the existing one, and has the values very close to the simulated ones. For a given outage capacity, a tightly approximate closed-form expression of outage probability is also derived, and Newton’s method is proposed to find the outage capacity for a given outage probability. Besides, to avoid the iterative calculation of the outage capacity, an approximate closed-form expression of outage capacity is also derived by using the Gaussian distribution approximation. Simulation results indicate that the theoretical analysis are in agreement with the corresponding simulations, and the system performance can be effectively improved as the receive antenna number increases and/or Nakagami factor m increases and/or path loss exponent decreases. As a result, the derived theoretical expression can provide good performance evaluation for distributed antenna systems over composite Nakagami fading channel, and avoid the conventional requirement for Monte Carlo simulation.
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Acknowledgment
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This work is supported by Special Research Found of Nanjing Institute of Technology (ZKJ201202). References [1] H. Robert, P. Steven, W. Yi, A current perspective on distributed antenna systems for the downlink of cellular systems, IEEE Commun. Mag. 51 (2013) 161–167. [2] J.-B. Wang, J.-Y. Wang, M. Chen, Downlink system capacity analysis in distributed antenna systems, Wireless Pers. Commun. 67 (2012) 631–645. [3] L. Dai, S. Zhou, Y. Yao, Capacity analysis in CDMA distributed antenna systems, IEEE Trans. Wireless Commun. 4 (2005) 2613–2620. [4] W. Choi, J.D. Andrews, Downlink performance and capacity of distributed antenna systems in a multicell environment, IEEE Trans. Wireless Commun. 6 (2007) 69–77.
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[5] Y. Liu, J. Liu, H. Chen, et al., Downlink performance analysis of distributed Q4 antenna systems, Proc. IEEE Wireless Commun. Signal Proc. (2011) 1–5. [6] X. Yu, Y. Yang, M. Li, M. Chen, Capacity analysis of distributed antenna systems in MIMO Nakagami fading multicell environment, Ann. Telecommun. 67 (2012) 589–595. [7] J. Park, E. Song, W. Sung, Capacity analysis for distributed antenna systems using cooperative transmission schemes in fading channels, IEEE Trans. Wireless Commun. 8 (2009) 586–592. [8] J.Y. Wang, J.B. Wang, X.Y. Dang, et al., System capacity analysis of downlink distributed antenna systems over composite channels, in: Proc. IEEE International Conference on Communication Technology, 2010, pp. 1076–1079. [9] H.-M. Chen, M. Chen, Capacity of the distributed antenna systems over shadowed fading channels, in: Proc. the IEEE Vehicular Technology Conference, 2009, pp. 1–4. [10] M. Matthaiou, N.D. Chatzidiamantis, G.K. Karagiannidis, A new lower bound on the ergodic capacity of distributed MIMO systems, IEEE Signal Process Lett. 8 (2011) 227–230. [11] W. Roh, A. Paulraj, Outage performance of the distributed antenna systems in a composite fading channel, in: Proc. IEEE Vehicular Technology Conference, 2002, pp. 1520–1524. [12] J.-Y. Wang, J.-B. Wang, M. Chen, et al., System outage probability analysis of uplink distributed antenna systems over a composite channel, in: Proc. IEEE 73rd Vehicular Technology Conference, 2011, pp. 1–5. [13] H.-M. Chen, J.-B. Wang, M. Chen, Outage performance of distributed antenna systems over shadowed Nakagami-m fading channels, Eur. Trans. Telecommun. 20 (2009) 531–535. [14] F. Héliot, R. Hoshyar, R. Tafazolli, A closed-form approximation of the outage probability for distributed MIMO systems, in: Proc. IEEE 10th Workshop on Signal Processing Advances in Wireless Communications, 2009, pp. 529–533. [15] M.S. Alouini, A.J. Goldsmith, Adaptive modulation over Nakagami fading channels, Wirel. Pers. Commun. 13 (2000) 119–143. [16] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, seventh Q5 ed., Academic Press, San Diego, CA, 2007. [17] M.K. Simon, M.S. Alouini, Digital Communication over Fading Channels, Wiley, New York, NY, 2005. [18] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Courier Dover Publications, Q6 2012. [19] E.G. Larsson, P. Stoica, Space-Time Block Coding for Wireless Communications, Cambridge Univ. Press, Cambridge, U.K., 2003.
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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel
1
2
Weiye Xu ∗ , Qingyun Wang
3
Q2
4
Q1 School of Communication Engineering, Nanjing Institute of Technology, Nanjing, China
5
a r t i c l e
6 19
i n f o
a b s t r a c t
7
Article history: Received 16 April 2015 Accepted 10 November 2015 Available online xxx
8 9 10 11 12
18
Keywords: Distributed antenna systems Channel capacity Outage capacity Nakagami channel Outage probability
20
1. Introduction
13 14 15 16 17
In this paper, the downlink capacity performance of distributed antenna systems (DAS) with antenna selection are investigated in composite Nakagami channel which take path loss, shadow fading and Nakagami fading into account. According to the performance analysis and using numerical integrations, the probability density function (PDF) and cumulative distribution function (CDF) of the effective output SNR of DAS are derived, respectively. Based on these results, tightly-approximate closed-form expressions of channel capacity and outage probability of DAS in composite Nakagami channel are further derived. These expressions include the ones under composite Rayleigh channel as special cases. Conditioned on the outage probability, a practical iterative algorithm based on Newton’s method for finding the outage capacity is proposed. To avoid iterative calculation, another approximate closed-form expression of outage capacity is also derived by utilizing the Gaussian distribution approximation. Simulation results show that the derived expressions have more accuracy than the existing ones, and can match the corresponding simulation well. Thus, the capacity performance of DAS in composite Nakagami channel can be evaluated effectively. © 2015 Published by Elsevier GmbH.
Distributed antenna system (DAS), as a promising technique for next generation wireless mobile communications, has received much attention due to its power and capacity advantages over the traditional co-located antenna system (CAS) [1,2]. Considering that channel capacity plays an important role in communication systems as a measure of the system performance, many academic works have been done to analyze the capacity performance of DAS, and proved that DAS can enhance the system capacity greatly [3–5]. The effect of maximal ratio combining based macrodiversity on the capacity of code division multiple access DAS uplink is studied in [3]. The downlink capacity of DAS over Rayleigh fading and Nakagami fading in multicell environment are analyzed in [4,5], respectively. The closed-form expressions of channel capacity are derived in the above two references, but the derived expression in [4] has minor error, while [5] needs iterative calculation. For this, an accurate closed-form expression of channel capacity of DAS downlink in Nakagami fading multicell environment is derived in [6]. However, the above literatures do not consider the effect of shadow fading on the performance. Based on this, the channel capacity of
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Q3 Q2
∗ Corresponding author. Tel.: +86 2552169128. E-mail address: [email protected] (W. Xu).
DAS downlink is analyzed in [7] by using different cooperation strategies, but the influence of noise is neglected for analysis simplicity. In [8,9], by using a log-normal distribution to substitute the gamma-log-normal distribution, approximate channel capacities are derived for DAS downlink over shadowed fading channel, where Rayleigh fading and Nakagami fading are respectively considered, but the derived theoretical capacities are not accurate enough to reflect the actual values, and the analysis is limited in single receive antenna case. By assuming double-sided spatial correlation, an analytical lower bound of the ergodic capacity of DAS in composite Rayleigh fading channel is provided in [10], but the capacity is accurate in the low and high SNR regimes only. Considering that system outage probability is an important performance measure of communication systems, the outage performance of DAS in fading channel is studied in [11–14]. For the DAS uplink, the approximate outage probability is analyzed and derived for DAS in composite Rayleigh/lognormal fading channels [11,12], and in composite Nakagami fading channels [13], where the mobile terminal is equipped with one antenna only. In [14], a closed-form approximation of the outage probability for the DAS downlink and uplink is derived, and this expression will be tight and valid only when the rate or the number of nodes and antennas of the system become large. Although in all these studies, the channel capacity and outage probability are well analyzed, the obtained capacity expressions
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(k)
where n represents the small-scale fading between RAn and the kth receive antenna of MT. For Nakagami-m fading channel, (k) (k) the amplitudes of {n }, {˛n } are modeled as independent Nakagami random variables of unit power with fading factor mn [15]. (k) The phase n is uniformly distributed over [0,2]. The Rayleigh distribution, which corresponds to m = mn = 1, is a special case of Nakagami-m distribution. For Nakagami fading channel, the PDF of (k) ˛n is given by
103
in composite fading channel are not accurate due to the usage of inaccurate approximation. Moreover, only limited work has been carried out for the outage performance analysis of distributed antenna system downlink. Especially, the outage capacity of distributed antenna system in composite Nakgagagi-m fading channels is not studied in the existing literatures. For this reason, we will study the capacity performance of distributed antenna system downlink with transmit antenna selection and multiple receive antennas in composite channel which considers path loss, lognormal shadowing and Nakagami fading, and focus on the derivation of the channel capacity and outage capacity of distributed antenna system over composite fading channel. In terms of the channel state information (CSI) and maximum SNR criterion, a selection transmission scheme is presented to maximize the effective SNR of the system, i.e., only the ‘best’ distributed antenna is selected for data transmission. Based on this scheme and the performance analysis, using some numerical integrations, the probability density function (PDF) and cumulative distribution function (CDF) of the effective SNR are respectively obtained. With the obtained CDF and PDF, tightly-approximate closed-form expression of channel capacity of distributed antenna system is derived. This expression has more accuracy than the existing one. To analyze the outage performance of the system well, we derive an accurate closed-form expression of outage probability for distributed antenna system under a given outage capacity as well. These theoretical expressions include the ones under composite Rayleigh channel as special cases. Besides, a practical iterative algorithm based on Newton method for calculating the outage capacity is proposed for a given outage probability. As a result, accurate outage capacity is obtained. Moreover, by utilizing the Gaussian distribution approximation, an approximate closed-form outage capacity is also derived. With these theoretical expressions, the capacity performance of distributed antenna system can be effectively evaluated. Simulation results verify the effectiveness of the theoretical analysis. The notations we use throughout this paper are as follows. Bold upper case and lower case letters denote matrices and column vectors, respectively. E{·} denotes the expectation. The superscripts (·)T and (·)H are used to stand for the transpose and Hermitian transpose, respectively.
104
2. Channel model and system model
ω exp f n ( ) = √ 2n
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105 106 107 108 109 110 111 112 113 114
115
In this paper, we consider a DAS downlink with multiple remote antennas (RAs) and multiple receive antennas in a single-cell environment, the remote antennas are distributed in the cell and connected with the central base station (BS, also named as RA0 ) via coaxial cables, fiber optics or radio links, where N RAs are considered, and the nth RA is denoted as RAn . The mobile terminal (MT) is equipped with K antennas. To improve the system performance, the antenna selection diversity scheme is employed for remote antennas. If RAn is selected to transmit the signals, the received signals at mobile terminal can be expressed as rn =
Pn hn b + z =
Pt hn
(1)
, . . ., hn T
(K) T
b+z
(1)
(k)
123
where r n = [rn (1) , rn (1) , . . ., rn (K) ] , r n is the received signal from (k) the kth antenna at MT, Pt is the transmit signal power. hn is the element of channel vector hn and denotes the composite channel fading coefficient between RAn and the kth antenna of MT. b is the transmitted signal from RAn with unity energy. z is the noise vector, whose elements are independent, identically distributed (i.i.d) complex Gaussian random variables with zero mean and variance (k) z2 . hn can be modeled as [8,9].
124
(k) hn
116 117 118 119 120 121 122
=
(k) n
Ln Sn
(2)
f
(k)
˛n
(˛) = 2(mm /˝)
mn
exp(−mn ˛2 /˝)˛2mn −1 / (mn )
(3)
(k) 2
where (.) is the Gamma function [16], ˝ = E{[˛n ] } = 1 is the average fading power. Ln and Sn denote the pass loss and shadowing effect between RAn and mobile terminal. The path loss term Ln is ˇ expressed as Ln = (d0 /dn ) n , where ˇn is the path loss exponent, d0 is the reference distance and dn represents the distance from RAn to the mobile terminal. The shadowing effect term Sn is log-normally distributed, and the mean of 10log10 Sn is assumed to be zero (in dB). As a result, the PDF of Sn can be expressed as [17] ω exp fSn (s) = √ 2n s
−
(10log10 s)
K
(k)
n = n
K
(4)
(2n2 )
k=1
(k)
˛n
2
= n
k=1
K
(k)
2
˛n
(k)
|n |2
(5)
(k)
−
(10log10 − n )2 (2n2 )
(6)
K
K (k) 2 (k) 2 (˛ ) = can be evaluated as n k=1 k=1 n f (u) = mn (mn u)mn K−1 exp(−mn u)/ (mn K)
n K mn K−1 mm mn n exp − x (mn K) xmn K
× exp
−
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133
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142
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147
(10log10 x − n ) 2n2
2
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153
154 155 156
(7)
With (6) and (7), the PDF of ␥n can be expressed as:
0
128
where n = 10log10 (Pt Ln /z2 ) is the mean of 10log10 n . From (3), using a variable transformation, the PDF of =
f n ( ) =
127
k=1
where n denotes the instantaneous SNR at the kth receive antenna, and n = Pt Ln Sn /z2 . According to the definition of Sn , n is also log-normally distributed. Thus, by a transformation of random variable, the PDF of
n can be given by
∞
126
2
where n (in dB) is the standard deviation of 10log10 Sn and ω = 10/log10. With (1) and (2), when single remote antenna n is selected, the effective SNR after maximum ratio combining can be obtained as
n =
125
157
158
ω √ 2n x
159
dx
(8)
160
161
Considering the difficulty of integration in (8), [8,9] resort to the log-normal distribution, and use it to approximate the Gammalog-normal distribution in (8) and derive the closed-form PDF of ␥n . Unfortunately, the derived PDF is not accurate in some cases. For this reason, we will recalculate the (8) using the numerical integration and the transformation of variable.
Please cite this article in press as: W. Xu, Q. Wang, Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.050
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Let t = (10log10 x − n )/ f n ( ) =
√ (Kmn )
Kmn −1 10−(
2n2 , (8) can be rewritten as
exp
−∞
168
+∞
n mKm n
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√ 2n t+ n )/10
Kmn
− 10
√ −( 2n t+ n )/10
This is a tightly approximate closed-form PDF of the effective SNR in composite Nakagami fading channel. When mn = 1 (n = 1,. . .,N), (12) and (14) are reduced to the PDF and CDF of the effective SNR in composite Rayleigh channel, i.e.,
mn
exp − t 2 dt
(9)
f ( ) ∼ =
N n=0
∼ =
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1
√ (Kmn )
Hi exp ( − mn cni ) Kmn −1 (mn cni )Kmn
F n ( ) =
1 f n (v) dv ∼ =1− √ (Kmn )
Np
Hi (Kmn , mn cni )
i=1
where (·, ·) is the incomplete Gamma function [16].
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3. Channel capacity analysis of distributed antenna systems
182 183
184
Ca = E C
= E log2 (1 + )
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193
F ( ) =
N
n=0
n=0
F n ( ) ∼ =
1
1− √ (Kmn )
Np
Hi (Kmn , mn cni )
i=1
(12)
194
195 196
With (12), the PDF f( ) can be obtained as:
⎡
dF ( ) ⎣f n ( ) = d N
197
f ( ) =
n=0 198
199
F j ( )⎦
(13)
j=0,j = / n
Substituting (9) and (10) into (13) yields f ( ) ∼ =
N n=0
200
⎤
N
×
mn Kmn Hi e− mn cni ( cni )Kmn −1 cni √ (Kmn ) N
j=0,j = / n
1 1− √ Hi (Kmn , mn cni ) (Kmn ) N
i=1
(15)
207
n=0
1 1− √ Hi (K, cni ) (K) Np
208
(16)
209
i=1
210
mn cni √ ( mn cni )Kmn −1 (Kmn )
0
211
⎤
N
F n ( ) d ⎦
(17)
212
213
Np N
+∞
Ca ∼ =
Hi
⎡
=
Hi
n=0 i=1
Al log2
l=1
N
yKmn −1 √ (Kmn )
yl 1+ mn cni
(18)
mn cni
where F j ( ) is the CDF of ␥j as shown in (10), {Al } are the weights associated with the zeros {yl } of the one dimensional Nq th order Laguerre polynomial [18]. Eq. (18) is tightly approximate closedform expression of average channel capacity of distributed antenna systems in composite Nakagami fading channel, it has more accuracy than the existing expression in [9] because the latter employs the inaccurate approximate PDF for capacity derivation. It is shown that the derived (18) will be in agreement with the simulation. Besides, although [9] derives the capacity expression of distributed antenna systems in composite Nakagami channel, it is only suitable for single receive antenna case. For this, we use the approximate method in [9] to give the capacity expression under multiple receive antenna cases, i.e., Np N Hi
×
√ ˆ n )/10 2ˆ n ti +
√ log2 (1 + 10(
N
1 − 0.5erfc
j=0,j = / n
216
n −1 yKm l √ (Kmn )
y l
F j
j=0,j = / n
n=0 i=1
(14)
215
cni mn
Nq N Np
×
y mn cni
⎤ y ⎦ e−y dy F j
j=0,j = / n
Ca ∼ =
log2 1 +
214
0
N
×⎣
Np
i=1
205
Utilizing the transformation of variable y = mn cni and Gaussian–Laguerre integration in [18], (17) can be changed to
(11)
where C = log2 (1 + ) is the channel capacity, and f( ) is PDF of effective SNR at the receiver. Based on the analysis in Section 2, the selective diversity scheme is applied to the transmitter, i.e., only one ‘best’ RA is selected for transmission to maximize the effective SNR. Thus, we have that = max{ 0 , 1 , . . ., N }. Considering large space among remote antennas, it is reasonable to assume that { n } are independent. Thus, with (10), we can evaluate the CDF of , F( ), as follows: N
F n ( ) ∼ =
n=0
0 185
n=0 i=1
f ( ) log2 (1 + ) d
204
i=1
N
n=0 i=1
203
j=0,j = / n
∞ =
N
× e− mn cni
Considering that the capacity is an important performance measure of the communication systems in fading channel, we will give the capacity analysis of the distributed antenna systems over a composite Nakagami fading channel in this section. The average channel capacity can be expressed as
1 1− √ Hi (K, cni ) (K)
N
(10)
202
206
i=1
and F ( ) =
Hi e− cni ( cni )K−1 cni
⎡+∞ N Np Ca ∼ Hi ⎣ log2 (1 + ) =
177
181
Np
Substituting (15) into (11) gives:
176
180
N j=0,j = / n
0
179
1
√ (K)
×
i=1
In the above derivation, the Gauss–Hermite integration [18] is √ utilized, and cni = 10−( 2n ti + n )/10 . ti and Hi are the base point and weight factor of the Np -order Hermite polynomial, respectively. Eq. (9) is tightly approximate PDF, and has more accuracy than the existing approximate PDFs [8,9]. From (9), the CDF of ␥n can be calculated as
175
Np
3
√
)
2ˆ n ti + ˆn − ˆj √ 2ˆ j
217 218 219 220 221 222 223 224 225 226 227 228 229
230
(19)
201
231
232
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where ˆ n = n + ω[ (Kmn ) − log(mn )], ˆ n2 = n2 + ω2 ς(2, Kmn ), (·) is the Euler psi function, ς(·, ·) is the Reimann’s zeta function, and erfc{.} denotes the complementary error function [16]. For mn = 1, (19) is reduced to (16) in [9]. Substituting mn = 1 into (18), we can obtain tightly approximate closed-form expression of average channel capacity of DAS in composite Rayleigh channel, i.e., Ca ∼ =
Np Nq N
Hi
n=0 i=1 241 242 243 244
245
Al log2
l=1
y 1+ l cni
yK−1 l
√ (K)
N
F j
y l
cni
j=0,j = / n
G (C o ) = F (2Co − 1) − ε ∼ =
N n=0
Hence, the derived capacity (18) includes the capacity under Rayleigh channel as a special case. Eq. (20) has more accuracy than the existing expression in [8] since the later uses inaccurate PDF to derive the capacity.
Np
∂G (Co ) = f (2Co − 1) 2Co log 2 > 0 ∂Co
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4.1. Outage probability
Np N 2Co log 2
√ (Kmn )
×
2Co −1 260
Co
Po = Pr (C ≤ Co ) = Pr ( ≤ 2
Co
− 1) =
f ( ) d = F (2
− 1)
j=0,j = / n
(21)
Kmn −1
Hi e− mn cni 2Co − 1
280
281 282 283 284
(mn cni )Kmn
i=1
1 1− √ Hi Kmn , 2Co − 1 mn cni (Kmn ) N
285
i=1
(26)
286
287
From (24), we have:
288
G (0) = F (0) − ε = 0 − ε < 0, and G (+∞) =
289
lim G (C o ) = 1 − ε > 0
Co →+∞
(27)
290
291
This is because outage probability ε ∈ (0, 1). According to the above results, and considering the outage capacity Co > 0, the equation G(Co ) = 0 in (24) will have a unique solution. There are many methods such as bisection method for finding the root of a strictly monotonic function. We propose to use Newton’s method to find the root iteratively because it has the quadratic convergence rate. Newton’s method is described as follows:
0 261
(25)
G (C o )
n=0
The outage capacity and probability are also important performance measure indicators of communication systems in fading channels [11–13]. However, the related research is relatively less. Especially the outage capacity analysis is much less. Based on the reason above, we will give the outage performance analysis of the distributed antenna systems in a composite Nakagami fading channel, and derive the closed-form outage probability and capacity. Because the channel capacity is a random variable, it is meaningful to consider its statistical distribution. A useful measure of statistical characteristic is the outage capacity [19]. For the given outage capacity Co , Po is defined as the outage probability that channel capacity C = log2 (1 + ) falls below Co . Thus, the outage probability can be expressed as
279
where f( ) is the PDF of as shown in (14). Considering that 2Co , log 2 and f (2Co − 1) are all positive, the derivative G (Co ) in (25) is also positive. So G(Co ) is a strictly monotonically increasing function of Co . Substituting (14) into (25) yields
N
246
277
278
Differentiating G(Co ) with respect to Co yields
∼ =
4. Outage capacity analysis of DAS
1 1− √ Hi (Kmn , (2Co − 1) mn cni ) − ε (24) (Kmn ) i=1
G (C o ) = (20)
276
(u+1)
Co
=
(u) Co
−G
(u) Co
(u)
G Co
(28)
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299
262 263
264
265
Substituting (12) into (21) yields Po = F (2 ∼ =
N
n=0 266 267 268 269
270
272 273 274 275
(u)
− 1)
1 1− √ (Kmn )
Np
Hi (Kmn , (2Co − 1) c ni mn )
(22)
i=1
Eq. (22) is a tightly approximate closed-form expression of outage probability of DAS in composite Nakagami fading channel, which is shown to agree the simulation well. When mn = 1, (22) is reduced to the outage probability of DAS in composite Rayleigh channel, i.e., P0 ∼ =
N n=0
271
Co
1 1− √ Hi (K, (2Co − 1) c ni ) (K) Np
i=1
4.2. Outage capacity In this subsection, the outage capacity performance is analyzed. For a given outage probability ε = Po , a practical iterative algorithm based on Newton’s method is proposed to find the outage capacity Co . The outage capacity from (22) can be expressed as finding the root of G(Co ) = 0 with
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+∞
(log2 (1 + ))2 f ( )d
E(C 2 ) = 0
(23)
(u)
where á Co is the outage capacity value at the uth iteration. G(Co ) (u) and G (Co ) are computed by (24) and (26), respectively. With (28), the accurate outage capacity may be obtained, but iterative calculation is required. For this reason, we will derive an approximate closed-form expression of outage capacity in the following. Using (14) and (18), the second moment of the channel capacity C can be calculated as
∼ =
Np Nq N n=0 i=1
×
Al log 2 1 +
Hi
l=1
ylKmn −1 √ (Kmn )
N
F j
j=0,j = / n
yl mn cni
2
(29)
y l mn cni
From (18) and (29), the variance of C can be evaluated as
2
Vc = E C
2
− [E (C)]2 = E C
307
− Ca2
308
(30)
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With the mean value (18) and the variance (30), using a Gaussian distribution approximation, the CDF of the channel capacity C may be approximated as
c 313
F (c) ≈ −∞
1 2Vc
5
d
c − Ca
(31)
318
Co ≈
2Vc erfc −1 2 (1 − ε) + Ca
(32)
328
329
5. Simulation results
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In this section, we use the derived theoretical formulae and computer simulation to evaluate the capacity of the distributed antenna systems over composite channels, where path loss, Nakagami-m fading and lognormal shadowing are all considered. In simulation, the cell shape is assumed to be circle with radius R. The BS (RA0 ) is in the center of the cell, and the N RAs are uniformly distributed over a circle with radius r. The Monte-Carlo method is employed for simulation. Different Nakagami factor m, path loss exponent ˇ and receive antenna K are considered for comparison. The polar coordinate of the ith RA is (r, 2(n − 1)/N, n = 1,2,. . .,N. The main parameters are listed as: N = 6; r = 2R/3; the reference distance d0 = 80 m; radius of the cell R = 103 m; and standard deviation = n = 6 dB. The simulation results are illustrated in Figs. 1–6. In Figs. 1 and 2, we plot the theoretical average channel capacity and corresponding simulation of DAS with different Nakagami factors over composite fading channel, where m = mn = 1, 3. Eq. (18)
0
0
7
6
5
wiht m=1(Simu.) with m=1(theory1) with m=1(theory2) wiht m=3(Simu.) with m=3(theory1) with m=3(theory2)
10
15
20
25
30
Fig. 2. Average channel capacity of DAS with different receive antennas and Nakagami factors (ˇ = 3).
7
6
5
DAS DAS DAS DAS DAS DAS
with m=1(simu.) with m=1(theory4) with m=1(theory3) with m=3(simu.) with m=3(theory4) with m=3(theory2)
4 Po=0.3 3
2 Po=0.15 1
0 0
5
10
15
20
25
30
SNR (db)
Fig. 3. Outage capacity of DAS with different outage probability and Nakagami factors (K = 2, ˇ = 3).
7
6
5 DAS DAS DAS DAS DAS DAS
5
SNR (db)
Co(bit/s/Hz)
323
DAS DAS DAS DAS DAS DAS
with Po=0.3(simu.) with Po=0.3(theory4) with Po=0.3(theory3) with Po=0.15(simu.) with Po=0.15(theory4) with Po=0.15(theory3)
4
β=2
3
β=3 2
Ca (bit/s/Hz)
322
1
Co(bit/s/Hz)
where ε is the given outage probability, erfc−1 {.} denotes the inverse complementary error function. Eq. (32) is an approximate closed-form expression of outage capacity, and has the value close to the simulation. From (32) and (23), it is found that Co will increase (decrease) as ε increases (decreases), which accords with the existing knowledge. Hence, the derived theoretical formulae will provide effective methods to assess the outage performance of distributed antenna systems over composite Nakagami channel. Moreover, (32) is relatively simpler because it has closed-form calculation and does not need any iterative calculation.
321
K=2 3 K=1
By setting the above F(c) = ε in (31), the outage capacity can be obtained as
320
4
2
317
319
wiht m=1(Simu.) with m=1(theory1) with m=1(theory2) wiht m=3(Simu.) with m=3(theory1) with m=3(theory2)
2Vc
315 316
DAS DAS DAS DAS DAS DAS
6
= 1 − 0.5erfc
314
( − Ca ) 2Vc
−
exp
2
7
Ca (bit/s/Hz)
310
5
4
1 3
0 0
β=2 2
β=4
0
5
10
15
20
10
15 SNR (db)
20
25
30
Fig. 4. Outage capacity of DAS with different path loss exponent and outage probabilities (m = 3, K = 1).
1
0
5
25
30
SNR (db)
Fig. 1. Average channel capacity of DAS with different path loss exponent and Nakagami factors (K = 1).
is adopted for theoretical capacity calculation (referred as ‘theory 1’), and the existing theoretical formula [9] (i.e., (19), referred as ‘theory 2’) is employed for comparison. In Fig. 1, single receive antenna (K = 1) and different path loss exponents (ˇ = ˇn = 2,4) are
Please cite this article in press as: W. Xu, Q. Wang, Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.050
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6 0
10
β=4
-1
Outage Probability
10
β=3
-2
10
-3
10
DAS DAS DAS DAS
with m=1(Simu.) with m=1(theory) with m=2(Simu.) with m=2(theory)
-4
10
10
15
20 SNR (db)
25
30
Fig. 5. Outage probability of DAS with different Nakagami factor and path loss exponents (K = 2).
Fig. 6. Outage probability of DAS with different receive antennas (m = 2, ˇ = 3).
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considered. From Fig. 1, it is found that the theoretical values from ‘theory 1’ agree well with the simulated ones for different m and ˇ, while the theoretical values from ‘theory 2’ have some difference with the corresponding simulations, and the difference will be more obvious when the Nakagami factor m is smaller. This is because the latter adopts the log-normal distribution to replace the Gamma-log-normal distribution for deriving the PDF, which brings about the inaccuracy of the channel capacity in some cases. Besides, the capacity of DAS with ˇ = 2 is obviously higher than that with ˇ = 4 due to smaller path loss. The capacity under Nakagami factor m = 3 is higher than that with m = 1, this is because the fading severity decreases as the Nakagami factor m increases. The above results indicate that the derived expression of average channel capacity for DAS is effective. In Fig. 2, we consider different receive antennas (K = 1,2) and different Nakagami factor (m = 1,3), path loss exponent ˇ = 3. As shown in Fig. 2, the derived ‘theory 1’ is still good agreement with the corresponding simulation, whereas the ‘theory 2’ still has the difference with the simulation, especially in the case of single receive antenna. Besides, the systems with three receive antennas outperform those with single receive antenna because the former has greater diversity, and the capacity of DAS with m = 3 is higher than that with m = 2 due to better fading case. The above results further show that the derived channel capacity is valid and reasonable.
In Figs. 3 and 4, we plot the theoretical outage capacity and corresponding simulation of DAS with different outage probabilities, where Po is set as 0.15 and 0.3, respectively. The accurate and approximate outage capacities are, respectively, computed by (28) and (32), and they are referred as ‘theory 3’ and ‘theory 4’, respectively. In Fig. 3, different Nakagami factor (m = 1,3), two receive antennas (K = 2) and path loss exponent (ˇ = 3) are considered. It is shown in Fig. 3 that theoretical values are all close to the corresponding simulated ones. Especially, the ‘theory 3’ from the Newton’s method has more accuracy than the ‘theory 4’ from Gaussian distribution approximation, but the latter does not need iterative calculation. It is observed that outage capacity of DAS with Po = 0.3 is higher than that with Po = 0.15 since larger outage probability is permitted to happen, which accords with the theoretical analysis in Section 4. Besides, the outage capacity under m = 3 is higher than that under m = 1 because of better fading case, as expected. Compared with Fig. 1, it is found the outage capacity in Fig. 3 is lower than the average channel capacity in Fig. 1 under the same case due to the constraint of outage probability. In Fig. 4, we consider different path loss exponents (ˇ = 2,3), the receive antenna number K = 1, Nakagami factor m = 3. From Fig. 4, we can find the results similar to Fig. 3, that is, the derived theoretical outage capacity can match the corresponding simulation well, only some little differences are observed for ‘theory 4’, and the outage capacity of DAS with large Po is higher than that with small Po . Besides, the outage capacity with ˇ = 3 is lower than that with ˇ = 2 since the former experiences larger path loss, which agrees with the existing knowledge. By comparing Figs. 3 and 4, it is found that the outage capacity with two receive antennas is higher than that with single receive antenna under the same Po and ˇ since multiple antennas are employed, as expected. The above results indicate that the derived outage capacity is effective. In Figs. 5 and 6, we plot the theoretical outage probability and corresponding simulation of DAS, and the theoretical outage probability is calculated by (22). In Fig. 5, different path loss exponents (ˇ = 3,4) and Nakagami factors (m = 1,2) are considered, outage capacity Co = 2 bit/s/Hz and K = 2. From Fig. 5, we can see that the theoretical outage probability and the corresponding simulation result are very close to each other. Due to larger path loss, the outage probability of DAS with ˇ = 4 is larger than that with ˇ = 3 as expected. It is observed that the outage probability of DAS with Nakagami factor m = 2 is higher than that with m = 1. The above results show that the derived outage probability is valid. In Fig. 6, we consider different receive antennas (K = 1, 2) and outage capacities (Co = 2, 4 bit/s/Hz), path loss exponent ˇ = 3 and Nakagami factor m = 2. As shown in Fig. 6, the theoretical outage probability is good agreement with the corresponding simulation, and the outage probability of DAS with two receive antennas is lower than that with single receive antenna since the former has greater diversity than the latter. Besides, the outage probability under Co = 4 bit/s/Hz is larger than that under Co = 2 bit/s/Hz. This is because larger outage capacity is required. As a result, higher outage probability happens. The above results further testify that the derived outage capacity and probability are valid for the outage performance evaluation.
6. Conclusion We have studied the capacity performance of distributed antenna systems over composite Nakagami-m fading channel, and the channel capacity and outage capacity as well as outage probability are all analyzed. According to the performance analysis in composite fading channel, using some numerical integration operations, the PDF and CDF of the effective SNR are respectively derived. Using these results, the tightly approximate closed-form expressions of average channel capacity are obtained. This expression
Please cite this article in press as: W. Xu, Q. Wang, Channel capacity of distributed antenna systems downlink over composite Nakagami fading channel, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.050
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has more accuracy than the existing one, and has the values very close to the simulated ones. For a given outage capacity, a tightly approximate closed-form expression of outage probability is also derived, and Newton’s method is proposed to find the outage capacity for a given outage probability. Besides, to avoid the iterative calculation of the outage capacity, an approximate closed-form expression of outage capacity is also derived by using the Gaussian distribution approximation. Simulation results indicate that the theoretical analysis are in agreement with the corresponding simulations, and the system performance can be effectively improved as the receive antenna number increases and/or Nakagami factor m increases and/or path loss exponent decreases. As a result, the derived theoretical expression can provide good performance evaluation for distributed antenna systems over composite Nakagami fading channel, and avoid the conventional requirement for Monte Carlo simulation.
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Acknowledgment
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This work is supported by Special Research Found of Nanjing Institute of Technology (ZKJ201202). References [1] H. Robert, P. Steven, W. Yi, A current perspective on distributed antenna systems for the downlink of cellular systems, IEEE Commun. Mag. 51 (2013) 161–167. [2] J.-B. Wang, J.-Y. Wang, M. Chen, Downlink system capacity analysis in distributed antenna systems, Wireless Pers. Commun. 67 (2012) 631–645. [3] L. Dai, S. Zhou, Y. Yao, Capacity analysis in CDMA distributed antenna systems, IEEE Trans. Wireless Commun. 4 (2005) 2613–2620. [4] W. Choi, J.D. Andrews, Downlink performance and capacity of distributed antenna systems in a multicell environment, IEEE Trans. Wireless Commun. 6 (2007) 69–77.
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[5] Y. Liu, J. Liu, H. Chen, et al., Downlink performance analysis of distributed Q4 antenna systems, Proc. IEEE Wireless Commun. Signal Proc. (2011) 1–5. [6] X. Yu, Y. Yang, M. Li, M. Chen, Capacity analysis of distributed antenna systems in MIMO Nakagami fading multicell environment, Ann. Telecommun. 67 (2012) 589–595. [7] J. Park, E. Song, W. Sung, Capacity analysis for distributed antenna systems using cooperative transmission schemes in fading channels, IEEE Trans. Wireless Commun. 8 (2009) 586–592. [8] J.Y. Wang, J.B. Wang, X.Y. Dang, et al., System capacity analysis of downlink distributed antenna systems over composite channels, in: Proc. IEEE International Conference on Communication Technology, 2010, pp. 1076–1079. [9] H.-M. Chen, M. Chen, Capacity of the distributed antenna systems over shadowed fading channels, in: Proc. the IEEE Vehicular Technology Conference, 2009, pp. 1–4. [10] M. Matthaiou, N.D. Chatzidiamantis, G.K. Karagiannidis, A new lower bound on the ergodic capacity of distributed MIMO systems, IEEE Signal Process Lett. 8 (2011) 227–230. [11] W. Roh, A. Paulraj, Outage performance of the distributed antenna systems in a composite fading channel, in: Proc. IEEE Vehicular Technology Conference, 2002, pp. 1520–1524. [12] J.-Y. Wang, J.-B. Wang, M. Chen, et al., System outage probability analysis of uplink distributed antenna systems over a composite channel, in: Proc. IEEE 73rd Vehicular Technology Conference, 2011, pp. 1–5. [13] H.-M. Chen, J.-B. Wang, M. Chen, Outage performance of distributed antenna systems over shadowed Nakagami-m fading channels, Eur. Trans. Telecommun. 20 (2009) 531–535. [14] F. Héliot, R. Hoshyar, R. Tafazolli, A closed-form approximation of the outage probability for distributed MIMO systems, in: Proc. IEEE 10th Workshop on Signal Processing Advances in Wireless Communications, 2009, pp. 529–533. [15] M.S. Alouini, A.J. Goldsmith, Adaptive modulation over Nakagami fading channels, Wirel. Pers. Commun. 13 (2000) 119–143. [16] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, seventh Q5 ed., Academic Press, San Diego, CA, 2007. [17] M.K. Simon, M.S. Alouini, Digital Communication over Fading Channels, Wiley, New York, NY, 2005. [18] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Courier Dover Publications, Q6 2012. [19] E.G. Larsson, P. Stoica, Space-Time Block Coding for Wireless Communications, Cambridge Univ. Press, Cambridge, U.K., 2003.
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