The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'07)

AN IMPORTANT CHARACTER FOR THE ENVELOPE CORRELATION COEFFICIENT OF MRC SIGNALS OVER CORRELATED RICIAN FADING CHANNELS Lihan Liu Beijing University of Posts and Telecommunications Beijing, P. R. China

Yangbin Pu West Branch of Zhejiang University of Technology Zhejiang, P. R. China

ABSTRACT Envelope correlation coefficient (ECC) of the maximal ratio combining (MRC) output in correlated Rican fading channels is discussed in this paper in a close form, which is similar to that of single path conditions in stationary environment. Besides, with minor approximation we get the analytical formula outage probability. On this base, a simple relationship between ECC and the outage probability is derived, which indicates that ECC can be used to make predictions of outage probability. I.

INTRODUCTION

Recently, Multicarrier direct Sequence Code Division Multiple Access (MC DS-CDMA) concept has been a hotpot in the research of next generation communication systems due to the diversity gain and bandwidth efficiency [1][2]. Unfortunately, high channel correlation introduce sever performance degradation. Besides, investigation of the correlation property of MRC signals is of great significance, because maximal ratio combining (MRC) represents a theoretical optimal receive algorithm over fading channels as a diversity scheme in communication systems. So far, many performance analysis results for MRC in correlated fading channels with multiple correlated paths have been explored [3]-[6], such as capacity, bit error rate (BER) and outage probability. Jakes [7] and Lee [8] studied channel envelope correlation coefficient (ECC) for two narrow-band signals with different carrier frequencies. Then Karasawa developed a general formula for channel envelope correlation in Rician fading channel [9]. More recently, some closedform expressions are provided by Mathiopoulos [10] for the envelope auto-correlation of MRC signals in correlated Nakagami fading environments. In this paper, we developed the conclusion of [10] to crosscorrelation of MRC signals, and deduced a close form formula of cross-correlation of ECC. It analysed the signal correlation under correlated Rician fading environments. Moreover, the outage probability is also derived by a simple approximated expression, which is absent in previous papers. In addition, we set up the relationship between ECC and the outage probability, which indicates that ECC can be effectively used in predicting outage probability. The reminder of this paper is arranged as follows. In Section II, system model is presented, and two types ECC expressions of MRC signals have been derived in Section III. In Section IV, an exact formula of outage probability is proposed, the effectiveness of which is validated by simulations in section IV. At last, some conclusions come in

1-4244-1144-0/07/$25.00 ©2007 IEEE

Zongrui Ding National Laboratory for Information Science and Technology, Tsinghua University Beijing, P. R. China

Section V. II. SYSTEM MODEL Consider a general frequency diversity synchronous system with single resolvable path in correlated Rayleigh fading channel. The complex channel gain hk at carrier frequency f k can be expressed as hk = xk + jyk k = 1, 2

(1)

where xk and yk denote the real and imaginary parts of the scatter components with zero-mean Gaussian random variables and variance σ k2 . We can deduce the relation between them as follows [7] E [ x1 x2 ] = E [ y1 y2 ] =

σ k2

1 + (ωτ rms )

E [ x1 y2 ] = − E [ x2 y1 ] =

2

(2)

−ωτ rmsσ k2

1 + (ωτ rms )

2

where ω = 2πΔf , and Δf is the frequency separation. τ rms is

the root mean square (rms) delay spread, and E [⋅] denotes expectation operation. The corresponding envelopes of two carrier frequencies are expressed by r1 = x12 + y12 r2 = x22 + y22

(3)

We can represent the correlation coefficient of the two envelopes as [7]

ρ=

=

Cov [ r1 , r2 ] var [ r1 ] var [ r2 ] 1 1 + (ωτ rms )

2

(4)

We can see that envelopes correlation coefficient (ECC) of the single received path is influenced by the frequency separation and rms delay spread. III. ENVELOPE CORRELATION COEFFICIENT Consider a correlated Rician fading channel with L resolvable paths in each carrier. The complex channel gain on

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'07)

Δτ = 0

the l th path at carrier f k can be expressed as

hk ,l = xk ,l + jyk ,l

k = 1, 2, l =1, 2,..., L

(5)

where xk ,l and yk ,l denote the Gaussian distributed real and imaginary components with mean value mxk ,l and myk ,l , and

mx1,l = mx2,l , m y1,l = m y2,l

σ 1,l = σ 2,l = σ l , l =1, 2,..., L Then (10) can be simplified to

with same variance σ k ,l , respectively. And mxk ,l and m yk ,l denote the real and imaginary paths of the complex LOS component. Without loss of generality, all fading paths of each carrier are assumed to be normalized. Assuming only the two corresponding paths of different carries are correlated and the time delay between them is Δτ . Based on (2), we deduce the correlation properties of resolvable paths as follows cov ⎡⎣ x1,l , x2,l ⎤⎦ = cov ⎡⎣ y1,l , y2,l ⎤⎦ =

σ 1,lσ 2,l J 0 ( 2π f m Δτ ) 1 + (ωτ rms )

2

= κl

where f m denotes the maximal Doppler shift and J 0 ( ⋅) is the zeroth order Bessel function of the first kind. After despreading and demodulation, the received signal of the l th path at carrier f k is L

l =1

(7)

l =1

where PS and Pn denote the average signal and noise power, respectively. S represents the transmitted signal. nk ,l represents the Gaussian noise component with zero mean and unit variance. Then the MRC signals at carrier f k can be expressed as L

L

Rk = ∑ PS S hk ,l + ∑ Pn nk ,l hk∗,l 2

l =1

1 1 + (ωτ rms )

(12)

2

From (4) and (12), we can see that the ECC of two received signals with no diversity and the ECC of MRC signals with diversity are the same, namely

ρ = ρ′ =

1 1 + (ωτ rms )

(13)

2

IV. OUTAGE PROBABILITY (6)

L

ρ′ =

The character of (13) is an important usefulness for ECC of MRC signals in diversity system.

cov ⎡⎣ x1,l , y2,l ⎤⎦ = − cov ⎡⎣ y1,l , x2,l ⎤⎦ = −ωτ rmsκ l

rk = ∑ PS Shk ,l + ∑ Pn nk ,l

(11)

(8)

l =1

where the superscript ( ⋅) denotes conjugate.

The signal to noise ratio (SNR) of the of MRC signal at the receiver is expressed to be

γ=

PS ( h H h )

2

PZ ( h H n )( h H n )

(14)

H

where the superscript H denotes the conjugate and transpose. n = ⎡⎣ n1,1 , n1,2 ,L , n1, L , n2,1 , n2,2 ,L , n2, L ⎤⎦ follows the joint 2 L dimensional zero-mean complex Gaussian distribution with covariance matrix R Z = I , so we can write n CN 2 L (0, I ) . h = ⎣⎡ h1,1 , h1,2 ,L , h1, L , h2,1 , h2,2 ,L , h2, L ⎦⎤ , whose elements are

complex Gaussian distributed random variables, can be described as h CN 2 L (ν h , R h ) , where ν h denotes the LOS component. We define U = ( h H n )( h H n )

H

(h h ) H

2

(15)

*

If constant envelope modulation is used, namely S = 1 , the envelope of MRC signals ignoring the noise can be expressed as Rk = PS ∑ ( x + y L

l =1

2 k ,l

2 k ,l

)

(9)

Then we can obtain the ECC of the MRC received signals as follows

ρ′ =

Cov ⎣⎡ R1 , R2 ⎦⎤ var ⎡⎣ R1 ⎤⎦ var ⎡⎣ R2 ⎤⎦

(10)

In particular, for a stationary environment, corresponding paths have identical statistical characteristic. Therefore

Because all entries of n are i.i.d, h H n is a zero-mean complex Gaussian random with a fixed h . Then the probability density function (pdf) of U is the complex Wishart distribution given by [11, eq. (9)] fU|h ( u | h ) = ς e −ς u

(16)

where ς = h H h , then fU ( u ) = Eh ⎡⎣ fU|h ( u |h ) ⎤⎦

d = Eς ⎡⎣ fU|ς ( u ) ⎤⎦ = − E ( e −ς u ) du

(17)

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'07)

Assuming that the signal to noise protection ratio is Λ , the outage probability is simply Pout = Pr ( γ < Λ ) = Pr (U > Λ 0 ) =∫

∞

Λ0

where Λ 0 = PS

( Pn Λ ) = ξ

(

fU ( u ) du = E e −Λ0ς

)

(18)

Λ , and ξ = PS Pn is the average

SNR per symbol. Besides, The expected value of exp(−ς u ) simply equals the characteristic function (CHF) of ς with transform variable −u , namely ϕς ( −u ) . It follows that [6]

E ( e−ς u ) = ϕς ( −u ) =

Fig. 1. Two ECC types versus Rician factor in correlated Rician fading channels.

1 exp ( −ν hH R −h1ν h det ( I + uR h )

+ν hH R −h1/ 2 ( I + uR h ) R h−1/ 2ν h )

(19)

Using (18) and (19), we can deduce the outage probability as follows Pout = E e−Λ0ς

(

=

)

1 exp ( −ν hH R −h1ν h det ( I + Λ 0 R h ) +ν hH R −h1/ 2 ( I + Λ 0 R h ) R h−1/ 2ν h )

(20)

Based on (6), R h can be deduced as follows ⎡ A1 ⎢0 ⎢ ⎢M ⎢ ⎢0 ⎢0 Rh = ⎢ * ⎢ B1 ⎢0 ⎢ ⎢M ⎢ ⎢0 ⎢⎣ 0

0 L A2

0 0

B1 0

AL −1

M 0

M 0

0 0

L

0

L

0

O 0

0 B2

L

0

AL

0

0

L

0

0

AL +1

0

0

0

0

AL + 2

* L −1

M 0

M 0

0

* L

0

0

O 0 0

B L

0

B

0 0

O

0 B2*

L

0 0

BL −1

0 O A2 L −1 L

0

0 ⎤ 0 ⎥⎥ M ⎥ ⎥ 0 ⎥ BL ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ M ⎥ ⎥ 0 ⎥ A2 L ⎥⎦

(21) where Al = 2σ 1,2l ,

AL + l = 2σ 2,2 l

and Bl = 2κl (1+ jωτrms ) ,

l = 1,L L . For a stationary environment, based on (20) and (21), we can deduce the outage probability as follows L

1 2 + Λ + 1 4 4 σ (1 − ρ ′ ) Λ 02σ i4 i =1 0 i

Pout = ∏

⎡ L 4 K Λ 0σ 2j ( 2 ( ρ ′ − 1) Λ 0σ 2j − 1) ⎤ ⎥ × exp ⎢ ∑ 2 2 4 ⎢⎣ j =1 1 + 4Λ 0σ j + 4 (1 − ρ ′ ) Λ 0σ j ⎥⎦

(

where ρ ′ = 1 1 + (ωτ rms )

2

)

(22)

denotes the ECC of the MRC

signals. From (12) and (22), we can obtain that ECC of the MRC signals can be used to predict the outage probability. It is obvious from (22) that the relationship between ECC and the outage probability is simple for correlated Rician channels. When using ECC to valuate outage probability, the outage probability is a function of K , α l and ρ ′ , which indicates that the influence of Δf , τ rms are replaced by ρ ′ . This means the relationship between ECC and the outage probability is irrespective of the frequency separation and rms delay spread, which is an important characteristic for ECC. V. SIMULATION RESULTS In order to demonstrate the validity of (12), (20) and (22), we simulate the case that each carrier has two resolvable paths. Monte Carlo simulation is performed in a correlated Rician fading simulator and the parameters are listed in Table 1.

Table 1 Simulation parameters Channel Rician channel Carrier frequency 900MHz Mobile speed 75km/h Power ratio between two paths 2dB Time delay spread 0.25μs We compare the ECC of two received signals with no diversity with the ECC of MRC signals with diversity in Fig. 1. The rms delay spread is set to be 0.25 μ s , and the frequency separation ranges form 0 to 2MHz. It can be easily observed that the two types ECC almost the same, which is according to (13). We also can see that ECC decreases as the frequency separation becomes larger, and decreasing rms

The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'07)

Fig. 2. Outage probability versus average SNR in correlated Rician fading channels.

delay spread leads to larger ECC values. Fig. 2 simulates outage performance versus average SNR with different power protection ratio Λ =12dB and 14dB and different Rician factor K =10 and 20. The outage probability becomes lower with average SNR increasing. In addition, larger Rician factor also leads lower outage probability. Fig. 3 demonstrates the relationship between ECC and the outage probability with frequency separation and rms delay spread changing. The power protection ratio Λ =12dB, and the Rician factor K =10. It can be easily observed that the outage probability is a monotone increasing function of ECC, and the outage performance tends to be affected by ECC more severely in lower outage probability environments. This means that channel correlation degraded the diversity gains. Besides, the function between ECC and outage probability is the same with the different frequency separation and rms delay spread, in other words, the influence of frequency separation and rms delay spread are replaced by ECC of MRC signals. VI. CONCLUSION With growing demand for high data rate communication, MC DS-CDMA received increasing attention in the research of next generation communication systems for its advantages in terms of diversity gain and bandwidth efficiency. In this paper, we derived ECC expression of MRC signals in a close form in correlated Rician channels. Moreover, an expression is also deduced with minor approximation. Based on this, we find the relationship between ECC and outage probability, which indicates ECC can be utilized to quantitatively predict system performance REFERENCES [1]

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Fig. 3. Outage probability versus ECC in correlated Rician fading channels. [4]

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