Analysis of envelope correlation on performance of MRC in correlated Rician-fading channels

Int. J. Electron. Commun. (AEÜ) 69 (2015) 937–942

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International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue

Analysis of envelope correlation on performance of MRC in correlated Rician-fading channels Zhuwei Wang a,b,∗ a b

College of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100124, PR China Department of Electrical and Engineering, Columbia University, New York 10027, NY, USA

a r t i c l e

i n f o

Article history: Received 3 March 2014 Accepted 21 February 2015 Keywords: Envelope correlation Correlated Rician-fading Maximal-ratio combined System performance

a b s t r a c t A simple envelope correlation expression of two maximal-ratio combined (MRC) signals in correlated diversity Rician-fading channels is derived in high signal-to-noise ratios (SNRs). In addition, the relationship between the system performance and envelope correlation is also obtained, which indicates that it is much easier to evaluate the system performance using the envelope correlation, since the effects of parameters such as the Rician factor and all channel attenuations are able to be replaced by the envelope correlation. The validity of the analytical relationship is verified by numerical simulations using a correlated Rician-fading emulator. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction Maximal ratio combining (MRC) diversity has been shown to be an effective way to combat multipath fading and interference in various radio communication systems such as MIMO systems, spectrum diversity systems, and satellite communications [1,6,7,9,10]. Unfortunately, channel correlation among the diversity branches would yield a non-negligible degradation [2]. Therefore, researches on the correlation property of MRC outputs and performance of the diversity scheme over correlated fading channels are growing rapidly in radio communication systems. The channel envelope correlation in the narrow-band system is studied as early as in [3], and then developed over correlated Rician-fading channels in [4]. The power correlation is addressed in [5], and, neglecting the noise, the envelope auto-correlation is analyzed in [6] for the MRC outputs in correlated Rician-fading channels. Recently, [7] investigate the statistical properties of correlated Rician-fading channels, and also analyze the performance of MRC outputs. On the other hand, relevant researches on the evaluation of system performance in correlated fading channels include [8–18]. Closed-form expressions for the ergodic capacity and outage probability of MRC diversity are given in the case of correlated Rayleigh-fading channels in [8,9] and correlated Rician-fading

channels in [10–14]. While [15–18] propose the approximate expressions of system performances in some other correlated fading channels. However, the analytical expressions in above works are always too complex to apply in the practical system, and there is still no relationship found between the envelope correlation and system performance. In this paper, we address the cross-correlation of MRC outputs considering the noise in correlated Rician-fading channels, and a simple envelope correlation formula is obtained. In addition, the relationship between the envelope correlation and the system performance is derived, which shows that the evaluations of the outage probability and ergodic capacity can be simplified since the envelope correlation is able to replace the effect of parameters such as channel attenuations of all branches and Rician factor, since, in general, the envelop correlation can be directly obtained by the received signals, while the Rician factor and channel attenuations all need the redundant signals to be estimated. The remainder of this paper is organized as follows. The expression for the envelope correlation is derived and its properties are presented in Section 2. We then investigate the relationship between the envelope correlation and system performance in Section 3. Numerical results and conclusions are given in Sections 4 and 5, respectively. 2. The envelope correlation

∗ Correspondence to: College of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100124, PR China. Tel.: +86 1067396130. E-mail address: [email protected] http://dx.doi.org/10.1016/j.aeue.2015.02.008 1434-8411/© 2015 Elsevier GmbH. All rights reserved.

In this section, we first derive the envelope correlation between two MRC outputs, and then analyze the properties of the envelope correlation in several special cases.

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2.1. Envelope correlation derivation The MRC diversity receiver is employed over correlated Ricianfading channels, and J resolvable branches are assumed to be at each carrier. Then, the received signal on the kth branch at carrier i is given by ri,k =



Ps S˛i,k +



Pn ni,k ,

(1)

where ni,k is the Gaussian noise with unit variance, S is the normalized transmit signal that |S| = 1, Ps and Pn are the signal and noise ¯ i,k ) denotes the powers, respectively, ˛i,k = (xi,k + mi,k ) + j(¯xi,k + m Rician-fading channel with scatter part xi,k + jx¯ i,k and direct path ¯ i,k . In general, for a given carrier, the total power of fading mi,k + jm

J

E[|˛i,k |2 ] = 1, branches are assumed to be normalized so that k=1 where E[ · ] is the expectation function. In general, the corresponding branches of different carriers are set to be correlated. That is [3],

where Var ( · ) denotes the variance function, ϕk = m1,k m2,k + ¯ 1,k m2,k − m1,k m ¯ 2,k , and ϕ ¯ 2,k . ¯ 1,k m ¯k = m m Considering the identical distributed branches as in [4,5] that ¯ 1,k = m ¯ 2,k = m ¯ k ,  = 0 and  1,k =  2,k =  k , the m1,k = m2,k = mk , m envelope correlation in (6) is simplified as

J

=

h4 k=1 k , J + 2 (1 + 2K) k=1 h4k

2 (1 + 2K) (1 + K)

2



2 where = 1 + 42 f 2 rms



−1

(7)

, = Ps /Pn is the average SNR,

¯ 2k + 2k2 is the channel attenuation, and K = (m2k + hk = m2k + m ¯ 2k )/2k2 is the Rician factor. m 2.2. Special cases In this subsection, the properties of envelope correlation are investigated in several special cases.

Cov[x1,k , x2,k ] = Cov[¯x1,k , x¯ 2,k ] = ˇk , Cov[x1,k , x¯ 2,k ] = −Cov[¯x1,k , x2,k ] = −2frms ˇk , ˇk =

1,k 2,k J0 (2fm ) 2 1 + 42 f 2 rms

,

(2)

k = 1, 2, . . ., J,

2 is the variwhere Cov[·] denotes the covariance function, i,k ance value of xi,k (or x¯ i,k ), f and fm are the frequency separation and maximal Doppler shift, respectively, J (·) is the  order Bessel function of the first kind, and  rms and  are the root mean square delay spread and time delay between two corresponding branches, respectively. The perfect channel estimation is assumed to be available at the receiver. Then, the MRC output at ith carrier can be written as

ri =

J 

ri,k ˛∗i,k =

k=1

J  

Ps S|˛i,k |2 +



Pn ni,k ˛∗i,k ,

(3)

k=1

where ( · )∗ denotes the conjugate. Considering the high SNR, the envelope of the desired signal is always much larger than that of the noise, that is

  J   J           Pr  Ps S|˛i,k |2    Pn ni,k ˛∗i,k  → 1,     k=1

(4)

k=1

where Pr { · } represents the probability function.

2.2.1. Noise neglect If the effect of noise is eliminated, that is → ∞. Then, the envelope correlation in (7) is simplified as  = .

(8)

From (8), we observe that the envelope correlation is directly determined by the frequency separation and the root mean square delay spread, which means that the effect of the multiple path can be ignored when the average SNR is extremely high. On the other hand, → ∞ indicates that  K, which allows the effect of the direct path to be eliminated. Thus, the envelop correlation in the case of noise neglect can be simplified as the same as the singlebranch case over the correlated Rayleigh fading channel [3]. 2.2.2. K 

Consider the case K  , we have

 

=O

K

 ,

(9)

which means that the envelope correlation will be reduced largely when the direct path introduces the major contribution to the received signal. This is reasonable because there is no envelope correlation between the direct paths.

Lemma 1. x ∈ R and x > 0, and y is a complex number. If x  |y|, it has |x + y| ≈ x + Re{y}.

2.2.3. → 0 or → 1 When → 0, we get  → 0. While → 1, we have

Based on (3), (4), and Lemma 1, the envelope of MRC output is derived as

=

|ri | = |ri S ∗ | ≈

J  

Ps |˛i,k |2 + Re

k=1

 J 



Pn ni,k ˛∗i,k S ∗

.

(5)

k=1

Using the approximation in (5), from (2), the envelope correlation between two MRC outputs is able to deduce as



=

=

Cov[|r1 |, |r2 |]

J

h4 k=1 k . J (1 + K)2 + 2 (1 + 2K) k=1 h4k 2 (1 + 2K)



(10)

which indicates that the envelope correlation must be extremely small when → 0, while it might be not large when → 1. Actually, in a real communication system,  rms is a given system parameter, which means that → 0 equals to the case f→ ∞, while → 1 equals to the case f → 0. Thus, we can see that the envelop correlation can be reduced when the larger frequency separate is chosen.

1 [Var(|r1 |)Var(|r J 2 |)] 2  Ps ˇk 1,k 2,k J0 (2fm ) + 2frms ϕ ¯ k + ϕk

3. The effect of envelope correlation

k=1

2 J Pn i=1

k=1

8

2 ¯ 2i,k 2i,k + m2i,k + m



2 2 ¯ 2i,k + Ps i,k i,k + m2i,k + m

 2 1

,

(6)

In this section, we investigate the effect of envelope correlation on system performance such as ergodic capacity and outage probability, and find out that some system parameters can be replaced by the envelope correlation to evaluate the system performance.

Z. Wang / Int. J. Electron. Commun. (AEÜ) 69 (2015) 937–942

3.1. Ergodic capacity analysis

where

We first derive the expression of ergodic capacity, and then find out the relationship between the ergodic capacity and envelope correlation. The channel capacity is given by

=

C = log2 (1 + ␣H ␣),

(11)

where ( · )H is the Hermitian transposition, and ˛ = [˛1,1 , . . ., ˛1,J , ˛2,1 , . . ., ˛2,J ] is a joint 2J-dimension complex Gaussian vector that ˛ ∼ CN2J (m␣ , R␣ ), where m␣ is the complex value of direct paths, and R␣ is derived as a Hermitian matrix with full rank that

⎡

2 21,1

⎢ ⎢0 ⎢ ⎢. ⎢. ⎢. ⎢ ⎢0 ⎢ ⎢ R␣ = ⎢ 2ˇ1 ∗ ⎢ ⎢0 ⎢ ⎢ ⎢. ⎢ .. ⎢ ⎢0 ⎣

0

···

0

2ˇ1 

0

···

2 21,2

···

0

0

2ˇ2 

···

.. .

..

.. .

.. .

.. .

..

2 21,J

0

0

···

0

2 22,1

0

··· ···

.

···

0

···

0 ∗

.

···

0

0

2 22,2

.. .

..

.. .

.. .

.. .

..

0

···

2ˇJ ∗

0

0

···

2ˇ2

.

.

0

⎤

⎥ ⎥ ⎥ ⎥ .. ⎥ ⎥ . ⎥ 2ˇJ  ⎥ ⎥ ⎥ 0 ⎥, ⎥ ⎥ 0 ⎥ ⎥ ⎥ .. ⎥ . ⎥ 2 ⎥ 22,J ⎦ 0

(12)

where  = 1 + j2f rms . According to the properties of m␣ and R␣ , 1 + ␣H ␣ can be approximated by a single chi-square variable with different degrees of freedom and an adequate scaling factor [10], which is given by 1 + ␣H ␣ ≈ t 2 ( , 0),

(13)

where 2 ( · ) is a noncentral chi-square variable, and t and subject to

2 1 + 2

2

(14)

From (17), (18) and (19), we observe that the effects of Rician factor and channel attenuations of all branches are replaced by the envelope correlation to evaluate the ergodic capacity. In general, the envelope correlation is able to be directly derived based on the received signal at the receiver, while the Rician factor and channel attenuations all need the redundant signals to be estimated. Therefore, it is much easier to obtain the envelope correlation than all channel attenuations and Rician factor, which largely simplifies the ergodic capacity analysis if using the envelope correlation. 3.2. Outage probability analysis We first derive the expression of outage probability, and then find out the relationship with the envelope correlation. From (3), the SNR of the sum of MRC outputs is given by

, 

SNR =

(20)

where

(␣H n) ␣H n

=



␣H ␣

1 1 2 − + .

3 2 15 4

(15)

Based on (11), (13) and Lemma 2, the ergodic capacity can be derived as E(C) ≈ log2 t −

1 ln 2

H

2

, (21)

n = [n1,1 , . . ., n1,J , n2,1 , . . ., n2,J ].

We observe that n is a joint 2J-dimension complex Gaussian vector, whose pdf is n ∼ CN2J (0, I). Below, we analyze the pdf of . Define  = ␣H n, whose properties are derived as E[] = E[T ] = 0,

 1  2 2 ¯ 2i,k + 2i,k = mi,k + m . 2 2

E[H ]

J

(22)

1 1 2 + −

3 2 15 4

According to the properties of  in (22),  can be seen as a complex Wishart distribution [20], whose pdf is given by f|␣ (ϕ|␣) = −

[19]. If x ∼ 2 ( , 0), we have



(19)

.

i=1 k=1

2t 2 = 2 (traceR␣ 2 + 2m␣ H R␣ m␣ ).

E(ln x) ≈ ln −

( − )

(1 + )

t = 1 + (traceR␣ + m␣ H m␣ ),

Lemma 2.

939



d −␣H ␣ϕ e . dϕ

From (20) and (23), the outage probability can be derived as [9] Pout

= Pr{SNR < 0 }

=



exp −m␣ H R␣ −1 m␣ + m␣ H R␣ −1/2 I + ıR␣ R␣ −1/2 m␣

det I + ıR␣

(16)

.

(23)





=

1 + 2

2

(1 + K)

2

J

2 (1 + 2K)(1 + )

h4 k=1 k

(17)

.

Then, based on (7), (16) and (17), the relationship between the envelope correlation and ergodic capacity can be written as 1 E(C) ≈ log2 (1 + 2 ) − ln 2



,

(24)

From (12) and (14), we get t = 1 + 2 ,



1 2 1 − +

3 2 15 4



,

(18)

where ı = PS /0 PN , and 0 is the threshold. Considering the identical distributed branches, the outage probability in (24) is simplified as Pout =

 J 

(1 + K)2 (1 + K)2 + 4(1 + K)ıh2i + 4(1 − )ı2 h4i

⎡ J  × exp ⎣ i=1



j=1

−4Kıh2j [2(1 − )ı˛2j + (1 + K)] (1 + K)2 + 4(1 + K)ıh2j + 4(1 − )ı2 h4j

⎤ ⎦.

(25)

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Z. Wang / Int. J. Electron. Commun. (AEÜ) 69 (2015) 937–942

1 0.9 0.8 0.7

ECC

0.6 0.5 0.4 Proposed (ξ = 20dB, Δf = 0MHz) Simulation (ξ = 20dB, Δf = 0MHz)

0.3

Proposed (ξ = 20dB, Δf = 0.5MHz) Simulation (ξ = 20dB, Δf = 0.5MHz)

0.2

Proposed (ξ = 10dB, Δf = 0MHz) Simulation (ξ = 10dB, Δf = 0MHz)

0.1

Proposed (ξ = 10dB, Δf = 0.5MHz) Simulation (ξ = 10dB, Δf = 0.5MHz)

0 0

2

4

6

8

10

12

14

16

18

20

Rician Factor (K) Fig. 1. Comparison between the approximation and simulation envelope correlations.

In the practical system, it is reasonable to assume K  1 and ı = O(1). Since h2j < 1, we have ıh2j /(K + 1)  1. Then, from (7) and (25), the outage probability can be approximated as

 Pout ≈

1−

2ı 1+K



 × exp

−2ı +

(1 + )ı2 2 ( − )

the outage probability, which also indicates that it can largely simplify the outage probability analysis if directly using the envelope correlation.

 ,

(26)

where the Taylor formula that (1 − x)−1 ≈ 1 + x − x2 when 0 < x  1 is used. From (25) and (26), it can be seen that the envelope correlation can replace the effects of all channel attenuations to evaluate

4. Simulation results In this section, numerical simulations are presented to illustrate the properties of the envelope correlation and its effects on the system performance. In the correlated Rician-fading channels, two carriers are assumed to have the same carrier frequency 900 MHz, and each has two resolvable branches. The time delay

6

5

Ergodic Capacity

4

3

Proposed (ξ = 15dB, Δf = 0.5MHz)

2

Simulation (ξ = 15dB, Δf = 0.5MHz) Proposed (ξ = 15dB, Δf = 0MHz) Simulation (ξ = 15dB, Δf = 0MHz)

1

Proposed (ξ = 10dB, Δf = 0.5MHz) Simulation (ξ = 10dB, Δf = 0.5MHz)

0

Proposed (ξ = 10dB, Δf = 0MHz) Simulation (ξ = 10dB, Δf = 0MHz) Proposed (ξ = 5dB, Δf = 0.5MHz)

−1

Simulation (ξ = 5dB, Δf = 0.5MHz) Proposed (ξ = 5dB, Δf = 0MHz)

−2

Simulation (ξ = 5dB, Δf = 0MHz)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Envelope Correlation Fig. 2. Relationship between the envelope correlation and the ergodic capacity.

0.9

1

Z. Wang / Int. J. Electron. Commun. (AEÜ) 69 (2015) 937–942

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0

10

Proposed (ξ = 15dB, Δf = 0.5MHz) Simulation (ξ = 15dB, Δf = 0.5MHz) Proposed (ξ = 15dB, Δf = 0MHz) Simulation (ξ = 15dB, Δf = 0MHz) Proposed (ξ = 10dB, Δf = 0.5MHz) Simulation (ξ = 10dB, Δf = 0.5MHz)

Outage Probability

Proposed (ξ = 10dB, Δf = 0MHz) Simulation (ξ = 10dB, Δf = 0MHz) Proposed (ξ = 5dB, Δf = 0.5MHz) Simulation (ξ = 5dB, Δf = 0.5MHz) Proposed (ξ = 5dB, Δf = 0MHz)

−1

10

Simulation (ξ = 5dB, Δf = 0MHz)

−2

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Envelope Correlation Fig. 3. Relationship between the envelope correlation and the outage probability.

spread and mobile speed are set to be 0.3 ␮s and 30 m/s, respectively. Fig. 1 shows the properties of the envelope correlation under various Rician factors, SNRs and frequency separations. From Fig. 1, the proposed approximation envelope correlation and the real simulation values are in the excellent agreement. We observe that the value of the envelope correlation is increasing with the SNR increasing, while decreasing with larger Rician factor and frequency separation. All these results match well with (7). The effect of the envelope correlation on the ergodic capacity is presented in Fig. 2, where the Rician factor is chosen from 0.01 to 100, and the ratio of two resolvable branches is set from 1 to 100. We observe that the relationship between the envelope correlation and the ergodic capacity matches well between the proposed formula and the real system simulation, and the envelope correlation, especially the high correlation, introduces system degradation. It also can be seen that the relationship between the envelope correlation and ergodic capacity is constant with various channel attenuations and Rician factors, which accords well with (18) and indicates that it is easier to use the envelope correlation to evaluate the ergodic capacity. In addition, the envelope correlation decreases with the frequency separate increasing, which means that the effect of the channel correlation can be reduce when the larger frequency separate is chosen. Fig. 3 shows the effect of the envelope correlation on the outage probability, where the ratio of two resolvable branches is also chosen from 1 to 100. We also obtain the same results as Fig. 2 that the envelope correlation degrades the outage probability, and all channel attenuations are not able to affect the relationship between the envelope correlation and outage probability. This matches well with (26), and it is easier to predict the outage probability using the envelope correlation. 5. Conclusions In the paper, considering the noise, a simple approximation expression of the envelope correlation between two MRC outputs is

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