Exact Crossing Rates of Dual Diversity over ... - Semantic Scholar
IEEE COMMUNICATIONS LETTERS, VOL. 10, NO. 1, JANUARY 2006
37
Exact Crossing Rates of Dual Diversity over Unbalanced Correlated Rayleigh Channels Jos´e Cˆandido S. Santos Filho, Gustavo Fraidenraich, and Michel D. Yacoub
Abstract— General exact expressions for the level crossing rate of dual-branch selection, equal-gain, and maximal-ratio combiners operating over unbalanced correlated Rayleigh channels are derived. Sample numerical results are presented by specializing the general expressions to a space-diversity system with horizontally spaced omnidirectional antennas at the mobile station. It is observed that, whereas power imbalance is invariably prejudicial, fading correlation may be advantageous with closely spaced antennas arranged parallel to the direction of motion. Index Terms— Correlation, diversity methods, level-crossing problems, Rayleigh channels.
I. I NTRODUCTION
T
HE LEVEL crossing rate (LCR) and the average fade duration (AFD) are widely-used as performance measures of wireless diversity systems. However, although the branch signals may be correlated and non-identically distributed in practical systems [1]–[3], the literature on LCR and AFD of diversity techniques over non-identical correlated fading is rather scarce. Pioneering, well-established work on this issue was carried out by Adachi et al. [1] for dual-branch selection (SC), equal-gain (EGC), and maximal-ratio combining (MRC) over identical correlated Rayleigh channels. The non-identical correlated Rayleigh case was addressed in [2] for two-, three-, and four-branch MRC. More recently, [3] presented a unified treatment for the LCR and the AFD of M branch SC over unbalanced correlated Rayleigh, Ricean, and Nakagami-m channels. In this letter, we extend the results of [1] by providing exact expressions for the LCR of dual-branch SC, EGC, and MRC operating over unbalanced correlated Rayleigh channels. As for the AFD, it is well known to be the ratio between the cumulative distribution function (CDF) and the LCR. Since the CDFs of MRC and SC over nonidentical correlated Rayleigh channels are given, for instance, in [2] and [3], and since the corresponding CDF of EGC can be obtained as a two-fold integration [4, Eq. (10)] of the general bivariate Rayleigh distribution [5], general AFD expressions are directly obtainable from the known CDFs and from our new LCR results. For brevity, the AFD discussion is omitted in the present work. Manuscript received July 11, 2005. The associate editor coordinating the review of this letter and approving it for publication was Dr. Rick Blum. The authors are with the Dept. of Communications, School of Electrical and Computer Engineering, University of Campinas, Brazil (e-mail: {candido, gf, michel}@decom.fee.unicamp.br). Digital Object Identifier 10.1109/LCOMM.2006.01020.
II. T HE L EVEL C ROSSING R ATE The normalized received signal at the ith antenna (i = 1, 2) can be represented in complex form as (1) Zi = (Ri / Ωi ) exp(jΘi ) where Zi is a zero-mean, unit-variance complex Gaussian random variable (RV), the envelope Ri is a Rayleigh RV with mean power Ωi = E[Ri2 ], and the phase 0 ≤ Θi ≤ 2π is uniform. The combiner output envelope R for each scheme is ⎧ ) SC ⎨ max (R1 , R2√ (R + R ) / 2 EGC (2) R= 1 2 ⎩ 2 R1 + R22 MRC ∞ The LCR of R is given as NR (r) = 0 rf ˙ R,R ˙ r)dr, ˙ ˙ (r, (·, ·) is the joint probability density function (PDF) where fR,R ˙ ˙ Next, NR (r) shall be attained of R and its time derivative R. by means of the joint and conditional PDFs fR1 ,R2 ,Θ12 (·, ·, ·) and fR|R ˙ 1 ,R2 ,Θ12 (·|·, ·, ·), where Θ12 Θ2 − Θ1 . The former is known to be given by [5] 2r1 r2 fR1 ,R2 ,Θ12 (r1 , r2 , θ12 ) = Ω1 Ω2 π(1 − |ρ12 |2 ) ⎤ ⎡ 2 r1 r22 √2r1 r2 Re[ρ∗ ejθ12 ] + − 12 Ω Ω2 Ω1 Ω2 ⎦ (3) × exp ⎣− 1 1 − |ρ12 |2 where ρ12 (τ ) = E[Z1∗ (t)Z2 (t + τ )] is the crosscorrelation coefficient between Z1 (t) e Z2 (t) and ρ12 = ρ12 (0). In order ˙ to obtain fR|R ˙ 1 ,R2 ,Θ12 (·|·, ·, ·), we write the time derivative Ri of Ri in terms of the time derivative Z˙ i of Zi as R˙ i = Ωi Re[Z˙ i e−jΘi ] (4) so that, conditioned on Z = {Z1 , Z2 }, R˙ 1 and R˙ 2 are Gaussian RVs. (Re[·] denotes the real part.) Knowing that
˙ ˙ Cov Zi , Zj∗ |Z = 0, ∀ i, j ∈ {1, 2}, the conditional mean, variance, and covariance of R˙ 1 and R˙ 2 can be found from (4) as
E R˙ i |Z = Ωi Re E Z˙ i |Z e−jθi
Ω
i Var Z˙ i |Z (5) Var R˙ i |Z = 2 √
∗ Ω1 Ω2 Re Cov Z˙ 1 , Z˙ 2 |Z ejθ12 Cov R˙ 1 , R˙ 2 |Z = 2 In the above, Var[·] and Cov[·, ·] denote variance
and covari ance, respectively. The statistics E Z˙ i |Z , Var Z˙ i |Z , and
Cov Z˙ 1 , Z˙ 2 |Z required to evaluate (5) have been obtained
c 2006 IEEE 1089-7798/06$20.00
38
IEEE COMMUNICATIONS LETTERS, VOL. 10, NO. 1, JANUARY 2006
NR(r)/fD
1
10
α=0 α = π/2 IID
α=0 α = π/2 IID
0.1
Ω1 = Ω2 = 0.5
0.01
1E-3 -25
-20
-15
-5
0
1E-3 -25
5
Fig. 3.
α=0 α = π/2 IID
-15
-10
-5
0
5
/ √ Ω1 + Ω2 , dB
LCR of MRC (d/λ = 0.2).
Ω1 = Ω2 . Now, from (3), (7), (8), and (9), and using [1, Eq. (8)] and [4, Eq. (17)], exact expressions for the LCR of dual-branch SC, EGC, and MRC operating over unbalanced correlated Rayleigh channels can be attained as in (10), where erfc(·) is the complementary error function.
Ω1 = 0.1, Ω2 = 0.9
NR(r)/fD
-20
r
1
0.1
III. N UMERICAL R ESULTS
1E-3 -25
The derivation above is general and can be applied to any type of diversity. In this section, sample numerical results are obtained by specializing (10) to a space-diversity system with horizontally spaced omnidirectional antennas at the mobile station. In this case, for isotropic scattering, it is known that [1]
Ω1 = Ω2 = 0.5
0.01
-20
-15
-10
r
Fig. 2.
Ω1 = Ω2 = 0.5
/ √ Ω1 + Ω2 , dB
LCR of SC (d/λ = 0.2). 10
Ω1 = 0.1, Ω2 = 0.9 0.1
0.01
-10
r
Fig. 1.
1
Ω1 = 0.1, Ω2 = 0.9
NR(r)/fD
10
-5
0
5
/ √ Ω + Ω , dB 1
ρ11 (τ ) = J0 (2πfD τ ) 2 2 ρ12 (τ ) = J0 2π (fD τ ) + (d/λ) − 2fD τ (d/λ) cos α
2
LCR of EGC (d/λ = 0.2).
ρ¨11 = −2(πfD )2
2
in [1, Eq. (25), 2σ = 1]. Note from ⎧ R˙ , R1 ≥R2 and R˙ 2 , R1 < R2 ⎪ ⎪ ⎨ 1 √ R˙ 1 + R˙ 2 / 2 R˙ =
⎪ ⎪ ⎩ R1 R˙ 1 + R2 R˙ 2 / R2 + R2 1
2
SC EGC
(6)
MRC
that R˙ is also a Gaussian RV conditioned on Z, whose conditional mean m(r ˙ 1 , r2 , θ12 ) and conditional variance σ˙ 2 (r1 , r2 , θ12 ) can be calculated by means of (5) and [1, Eq. (25), 2σ 2 = 1] as in (7) and (8), after lengthy algebraic manipulations, where ρij (τ ) = E[Zi∗ (t)Zj (t + τ )] is the correlation coefficient between Zi (t) and Zj (t), d d2 ρij (τ ), ρ¨ij (τ ) = dτ ρij = ρij (0), ρ˙ ij (τ ) = dτ 2 ρij (τ ), ρ˙ ij = ρ˙ ij (0), and ρ¨ij = ρ¨ij (0). Then fR|R ˙ 1 , r2 , θ12 ) = ˙ 1 ,R2 ,Θ12 (r|r
1 2π σ˙ 2 (r1 , r2 , θ12 )
[r˙ − m(r ˙ 1 , r2 , θ12 )]2 × exp − (9) 2σ˙ 2 (r1 , r2 , θ12 )
We emphasize that (7) and (8) are important new general expressions which specialize to [1, Eqs. (26) and (27)] for
ρ12 = J0 (2πd/λ) ρ˙ 12 = 2πfD J1 (2πd/λ) cos α (11) d J1 2π λ d ρ¨12 = (2πfD )2 cos 2α − J0 2π cos2 α d λ 2π λ (12) where Jk (·) is the Bessel Function of the first kind and kth order, fD = v/λ is the maximum Doppler shift in Hz for a vehicle speed v and a carrier wavelength λ, d is the antenna spacing, and 0 ≤ α ≤ π/2 is the angle between the antenna axis and the vehicle speed. Figs. 1, 2, and 3 depict the influence of fading correlation and power imbalance on the LCR of SC, EGC, and MRC, respectively. A small antenna spacing d/λ = 0.2 is used, so that ρ12 = 0.6425. For comparison, the independent identically distributed (IID) scenario is also plotted. It can be seen in the figures that power imbalance always deteriorates the performance by increasing the LCR for deep fades. On the other hand, in comparison to the independent case, the introduction of fading correlation is observed to be prejudicial for α = π/2 but advantageous for α = 0. In general, it has been observed that, whereas power
SANTOS FILHO et al.: EXACT CROSSING RATES OF DUAL DIVERSITY OVER UNBALANCED CORRELATED RAYLEIGH CHANNELS
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ m(r ˙ 1 , r2 , θ12 ) =
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
√
39
√
r1 Re[ρ˙ ∗ 12 ρ12 ]−
jθ12 jθ12 Ω1 /Ω2 r2 Re[ρ˙ ∗ −r2 Re[ρ˙ ∗ Ω2 /Ω1 r1 Re[ρ˙ ∗ ] ] 12 e 12 ρ12 ]+ 12 e , r1 ≥ r2 and , 2 1−|ρ12 |2 1−|ρ | 12
√ √ ∗ jθ12 Re (r1 −r2 )Re[ρ˙ ∗ ρ ]+ Ω /Ω r − Ω /Ω r ρ ˙ e [ 12 ] 2 1 1 1 2 2 12 12 √ 2 1−|ρ12 |2 )
√ ( √ (r12 −r22 )Re[ρ˙∗12 ρ12 ]+ Ω2 /Ω1 − Ω1 /Ω2 r1 r2 Re[ρ˙∗12 ejθ12 ]
√
r1 < r2
EGC MRC
r12 +r22 (1−|ρ12 |2 )
⎧ |ρ˙ 12 |2 |ρ˙ 12 |2 ⎪ − Ω21 ρ¨11 + 1−|ρ , r1 ≥ r2 and − Ω22 ρ¨11 + 1−|ρ , r1 < r2 2 2 ⎪ | | ⎪ 12 12
√ ∗ ⎨ ∗ 2 2 ρ12 (ρ˙ 12 ) |ρ˙ 12 | Ω1 Ω2 2) jθ12 − (Ω1 +Ω + Re ρ ¨ + e ρ ¨ − σ˙ 2 (r1 , r2 , θ12 ) = 2 2 11 12 4 2 1−|ρ12 | ⎪ √ ∗
1−|ρ12 | ∗
⎪ 2 2 2 2 ⎪ jθ12 ⎩ − Ω1 r12+Ω22r2 ρ¨11 + |ρ˙12 | 2 − Ω12Ω2 r21 r2 Re ρ¨12 + ρ12 (ρ˙12 )2 e r1 +r2 1−|ρ12 | 1−|ρ12 | 2(r1 +r2 )
EGC
⎧ 2π r ⎪ N (r, r2 , θ12 )fR1 ,R2 ,Θ12 (r, r2 , θ12 )dr2 dθ12 ⎪ 0 0 R1 ,R2 ,Θ12 ⎪ ⎪ ⎨ + 2π r NR1 ,R2 ,Θ12 (r1 , r, θ12 )fR1 ,R2 ,Θ12 (r1 , r, θ12 )dr1 dθ12 0 √ √ √ 2π0 √2r NR (r) = 2NR1 ,R2 ,Θ12 ( 2r − r , r , θ )f ( 2r ⎪ 2 2 12 R ,R ,Θ 1 2 12 ⎪ 0 0 − r2 , r2 , θ12 )dr2 dθ12 ⎪ ⎪ 2π r √ r N 2 2 2 2 ⎩ R1 ,R2 ,Θ12 ( r − r2 , r2 , θ12 )fR1 ,R2 ,Θ12 ( r − r2 , r2 , θ12 )dr2 dθ12 2 0 0 2
SC
r −r2
NR1 ,R2 ,Θ12 (r1 , r2 , θ12 )
SC
(7) SC (8)
MRC
EGC MRC
(10)
∞
rf ˙ R|R ˙ 1 , r2 , θ12 )dr˙ ˙ 1 ,R2 ,Θ12 (r|r 2 π m(r 1 m(r ˙ 1 , r2 , θ12 ) ˙ 1 , r2 , θ12 ) 1 m(r ˙ 1 , r2 , θ12 ) σ(r ˙ 1 , r2 , θ12 ) √ erfc − √ exp − + = 2 σ(r ˙ 1 , r2 , θ12 ) 2 σ(r ˙ 1 , r2 , θ12 ) ˙ 1 , r2 , θ12 ) 2π 2 σ(r 0
imbalance is invariably deleterious, fading correlation may have both positive and negative effects on the performance, depending on d and α. Such a dual effect of fading correlation has also been reported in [1] but for identical channels.
using closely spaced antennas arranged parallel to the direction of the vehicle motion.
IV. C ONCLUSION
[1] F. Adachi, M. T. Feeney, and J. D. Parsons, “Effects of correlated fading on level crossing rates and average fade durations with predetection diversity reception,” Proc. IEE, vol. 135, pp. 11-17, Feb. 1988. [2] X. Dong and N. C. Beaulieu, “Average level crossing rate and fade duration of maximal ratio diversity in unbalanced and correlated channels,” in Proc. IEEE Wireless Communications and Networking Conference, vol. 2, pp. 762-767, Mar. 2002. [3] L. Yang and M.-S. Alouini, “An exact analysis of the impact of fading correlation on the average level crossing rate and average outage duration of selection combining,” in Proc. IEEE Vehicular Technology Conference, vol. 1, pp. 241-245, Apr. 2003.
General exact expressions for the level crossing rate of dualdiversity selection, equal-gain, and maximal-ratio combiners operating on unbalanced correlated Rayleigh channels were derived. Sample numerical results were presented by specializing the general expressions to a space-diversity system with horizontally spaced omnidirectional antennas at the mobile station. It has been observed that, whereas power imbalance is invariably deleterious by increasing the LCR for deep fades, fading correlation may be advantageous depending on the antenna spacing and on the angle between the antenna axis and the direction of the vehicle motion. In particular, considerable improvement over the independent case can be achieved by
R EFERENCES
[4] G. Fraidenraich, J. C. S. Santos Filho, and M. D. Yacoub, “Second-order statistics of maximal-ratio and equal-gain combining in Hoyt fading,” IEEE Commun. Lett., vol. 9, pp. 19-21, Jan. 2005. [5] W. B. Davenport, Jr. and W. L. Root, An Introduction to the Theory of Random Signals and Noise. New York: McGraw-Hill, 1958.
37
Exact Crossing Rates of Dual Diversity over Unbalanced Correlated Rayleigh Channels Jos´e Cˆandido S. Santos Filho, Gustavo Fraidenraich, and Michel D. Yacoub
Abstract— General exact expressions for the level crossing rate of dual-branch selection, equal-gain, and maximal-ratio combiners operating over unbalanced correlated Rayleigh channels are derived. Sample numerical results are presented by specializing the general expressions to a space-diversity system with horizontally spaced omnidirectional antennas at the mobile station. It is observed that, whereas power imbalance is invariably prejudicial, fading correlation may be advantageous with closely spaced antennas arranged parallel to the direction of motion. Index Terms— Correlation, diversity methods, level-crossing problems, Rayleigh channels.
I. I NTRODUCTION
T
HE LEVEL crossing rate (LCR) and the average fade duration (AFD) are widely-used as performance measures of wireless diversity systems. However, although the branch signals may be correlated and non-identically distributed in practical systems [1]–[3], the literature on LCR and AFD of diversity techniques over non-identical correlated fading is rather scarce. Pioneering, well-established work on this issue was carried out by Adachi et al. [1] for dual-branch selection (SC), equal-gain (EGC), and maximal-ratio combining (MRC) over identical correlated Rayleigh channels. The non-identical correlated Rayleigh case was addressed in [2] for two-, three-, and four-branch MRC. More recently, [3] presented a unified treatment for the LCR and the AFD of M branch SC over unbalanced correlated Rayleigh, Ricean, and Nakagami-m channels. In this letter, we extend the results of [1] by providing exact expressions for the LCR of dual-branch SC, EGC, and MRC operating over unbalanced correlated Rayleigh channels. As for the AFD, it is well known to be the ratio between the cumulative distribution function (CDF) and the LCR. Since the CDFs of MRC and SC over nonidentical correlated Rayleigh channels are given, for instance, in [2] and [3], and since the corresponding CDF of EGC can be obtained as a two-fold integration [4, Eq. (10)] of the general bivariate Rayleigh distribution [5], general AFD expressions are directly obtainable from the known CDFs and from our new LCR results. For brevity, the AFD discussion is omitted in the present work. Manuscript received July 11, 2005. The associate editor coordinating the review of this letter and approving it for publication was Dr. Rick Blum. The authors are with the Dept. of Communications, School of Electrical and Computer Engineering, University of Campinas, Brazil (e-mail: {candido, gf, michel}@decom.fee.unicamp.br). Digital Object Identifier 10.1109/LCOMM.2006.01020.
II. T HE L EVEL C ROSSING R ATE The normalized received signal at the ith antenna (i = 1, 2) can be represented in complex form as (1) Zi = (Ri / Ωi ) exp(jΘi ) where Zi is a zero-mean, unit-variance complex Gaussian random variable (RV), the envelope Ri is a Rayleigh RV with mean power Ωi = E[Ri2 ], and the phase 0 ≤ Θi ≤ 2π is uniform. The combiner output envelope R for each scheme is ⎧ ) SC ⎨ max (R1 , R2√ (R + R ) / 2 EGC (2) R= 1 2 ⎩ 2 R1 + R22 MRC ∞ The LCR of R is given as NR (r) = 0 rf ˙ R,R ˙ r)dr, ˙ ˙ (r, (·, ·) is the joint probability density function (PDF) where fR,R ˙ ˙ Next, NR (r) shall be attained of R and its time derivative R. by means of the joint and conditional PDFs fR1 ,R2 ,Θ12 (·, ·, ·) and fR|R ˙ 1 ,R2 ,Θ12 (·|·, ·, ·), where Θ12 Θ2 − Θ1 . The former is known to be given by [5] 2r1 r2 fR1 ,R2 ,Θ12 (r1 , r2 , θ12 ) = Ω1 Ω2 π(1 − |ρ12 |2 ) ⎤ ⎡ 2 r1 r22 √2r1 r2 Re[ρ∗ ejθ12 ] + − 12 Ω Ω2 Ω1 Ω2 ⎦ (3) × exp ⎣− 1 1 − |ρ12 |2 where ρ12 (τ ) = E[Z1∗ (t)Z2 (t + τ )] is the crosscorrelation coefficient between Z1 (t) e Z2 (t) and ρ12 = ρ12 (0). In order ˙ to obtain fR|R ˙ 1 ,R2 ,Θ12 (·|·, ·, ·), we write the time derivative Ri of Ri in terms of the time derivative Z˙ i of Zi as R˙ i = Ωi Re[Z˙ i e−jΘi ] (4) so that, conditioned on Z = {Z1 , Z2 }, R˙ 1 and R˙ 2 are Gaussian RVs. (Re[·] denotes the real part.) Knowing that
˙ ˙ Cov Zi , Zj∗ |Z = 0, ∀ i, j ∈ {1, 2}, the conditional mean, variance, and covariance of R˙ 1 and R˙ 2 can be found from (4) as
E R˙ i |Z = Ωi Re E Z˙ i |Z e−jθi
Ω
i Var Z˙ i |Z (5) Var R˙ i |Z = 2 √
∗ Ω1 Ω2 Re Cov Z˙ 1 , Z˙ 2 |Z ejθ12 Cov R˙ 1 , R˙ 2 |Z = 2 In the above, Var[·] and Cov[·, ·] denote variance
and covari ance, respectively. The statistics E Z˙ i |Z , Var Z˙ i |Z , and
Cov Z˙ 1 , Z˙ 2 |Z required to evaluate (5) have been obtained
c 2006 IEEE 1089-7798/06$20.00
38
IEEE COMMUNICATIONS LETTERS, VOL. 10, NO. 1, JANUARY 2006
NR(r)/fD
1
10
α=0 α = π/2 IID
α=0 α = π/2 IID
0.1
Ω1 = Ω2 = 0.5
0.01
1E-3 -25
-20
-15
-5
0
1E-3 -25
5
Fig. 3.
α=0 α = π/2 IID
-15
-10
-5
0
5
/ √ Ω1 + Ω2 , dB
LCR of MRC (d/λ = 0.2).
Ω1 = Ω2 . Now, from (3), (7), (8), and (9), and using [1, Eq. (8)] and [4, Eq. (17)], exact expressions for the LCR of dual-branch SC, EGC, and MRC operating over unbalanced correlated Rayleigh channels can be attained as in (10), where erfc(·) is the complementary error function.
Ω1 = 0.1, Ω2 = 0.9
NR(r)/fD
-20
r
1
0.1
III. N UMERICAL R ESULTS
1E-3 -25
The derivation above is general and can be applied to any type of diversity. In this section, sample numerical results are obtained by specializing (10) to a space-diversity system with horizontally spaced omnidirectional antennas at the mobile station. In this case, for isotropic scattering, it is known that [1]
Ω1 = Ω2 = 0.5
0.01
-20
-15
-10
r
Fig. 2.
Ω1 = Ω2 = 0.5
/ √ Ω1 + Ω2 , dB
LCR of SC (d/λ = 0.2). 10
Ω1 = 0.1, Ω2 = 0.9 0.1
0.01
-10
r
Fig. 1.
1
Ω1 = 0.1, Ω2 = 0.9
NR(r)/fD
10
-5
0
5
/ √ Ω + Ω , dB 1
ρ11 (τ ) = J0 (2πfD τ ) 2 2 ρ12 (τ ) = J0 2π (fD τ ) + (d/λ) − 2fD τ (d/λ) cos α
2
LCR of EGC (d/λ = 0.2).
ρ¨11 = −2(πfD )2
2
in [1, Eq. (25), 2σ = 1]. Note from ⎧ R˙ , R1 ≥R2 and R˙ 2 , R1 < R2 ⎪ ⎪ ⎨ 1 √ R˙ 1 + R˙ 2 / 2 R˙ =
⎪ ⎪ ⎩ R1 R˙ 1 + R2 R˙ 2 / R2 + R2 1
2
SC EGC
(6)
MRC
that R˙ is also a Gaussian RV conditioned on Z, whose conditional mean m(r ˙ 1 , r2 , θ12 ) and conditional variance σ˙ 2 (r1 , r2 , θ12 ) can be calculated by means of (5) and [1, Eq. (25), 2σ 2 = 1] as in (7) and (8), after lengthy algebraic manipulations, where ρij (τ ) = E[Zi∗ (t)Zj (t + τ )] is the correlation coefficient between Zi (t) and Zj (t), d d2 ρij (τ ), ρ¨ij (τ ) = dτ ρij = ρij (0), ρ˙ ij (τ ) = dτ 2 ρij (τ ), ρ˙ ij = ρ˙ ij (0), and ρ¨ij = ρ¨ij (0). Then fR|R ˙ 1 , r2 , θ12 ) = ˙ 1 ,R2 ,Θ12 (r|r
1 2π σ˙ 2 (r1 , r2 , θ12 )
[r˙ − m(r ˙ 1 , r2 , θ12 )]2 × exp − (9) 2σ˙ 2 (r1 , r2 , θ12 )
We emphasize that (7) and (8) are important new general expressions which specialize to [1, Eqs. (26) and (27)] for
ρ12 = J0 (2πd/λ) ρ˙ 12 = 2πfD J1 (2πd/λ) cos α (11) d J1 2π λ d ρ¨12 = (2πfD )2 cos 2α − J0 2π cos2 α d λ 2π λ (12) where Jk (·) is the Bessel Function of the first kind and kth order, fD = v/λ is the maximum Doppler shift in Hz for a vehicle speed v and a carrier wavelength λ, d is the antenna spacing, and 0 ≤ α ≤ π/2 is the angle between the antenna axis and the vehicle speed. Figs. 1, 2, and 3 depict the influence of fading correlation and power imbalance on the LCR of SC, EGC, and MRC, respectively. A small antenna spacing d/λ = 0.2 is used, so that ρ12 = 0.6425. For comparison, the independent identically distributed (IID) scenario is also plotted. It can be seen in the figures that power imbalance always deteriorates the performance by increasing the LCR for deep fades. On the other hand, in comparison to the independent case, the introduction of fading correlation is observed to be prejudicial for α = π/2 but advantageous for α = 0. In general, it has been observed that, whereas power
SANTOS FILHO et al.: EXACT CROSSING RATES OF DUAL DIVERSITY OVER UNBALANCED CORRELATED RAYLEIGH CHANNELS
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ m(r ˙ 1 , r2 , θ12 ) =
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
√
39
√
r1 Re[ρ˙ ∗ 12 ρ12 ]−
jθ12 jθ12 Ω1 /Ω2 r2 Re[ρ˙ ∗ −r2 Re[ρ˙ ∗ Ω2 /Ω1 r1 Re[ρ˙ ∗ ] ] 12 e 12 ρ12 ]+ 12 e , r1 ≥ r2 and , 2 1−|ρ12 |2 1−|ρ | 12
√ √ ∗ jθ12 Re (r1 −r2 )Re[ρ˙ ∗ ρ ]+ Ω /Ω r − Ω /Ω r ρ ˙ e [ 12 ] 2 1 1 1 2 2 12 12 √ 2 1−|ρ12 |2 )
√ ( √ (r12 −r22 )Re[ρ˙∗12 ρ12 ]+ Ω2 /Ω1 − Ω1 /Ω2 r1 r2 Re[ρ˙∗12 ejθ12 ]
√
r1 < r2
EGC MRC
r12 +r22 (1−|ρ12 |2 )
⎧ |ρ˙ 12 |2 |ρ˙ 12 |2 ⎪ − Ω21 ρ¨11 + 1−|ρ , r1 ≥ r2 and − Ω22 ρ¨11 + 1−|ρ , r1 < r2 2 2 ⎪ | | ⎪ 12 12
√ ∗ ⎨ ∗ 2 2 ρ12 (ρ˙ 12 ) |ρ˙ 12 | Ω1 Ω2 2) jθ12 − (Ω1 +Ω + Re ρ ¨ + e ρ ¨ − σ˙ 2 (r1 , r2 , θ12 ) = 2 2 11 12 4 2 1−|ρ12 | ⎪ √ ∗
1−|ρ12 | ∗
⎪ 2 2 2 2 ⎪ jθ12 ⎩ − Ω1 r12+Ω22r2 ρ¨11 + |ρ˙12 | 2 − Ω12Ω2 r21 r2 Re ρ¨12 + ρ12 (ρ˙12 )2 e r1 +r2 1−|ρ12 | 1−|ρ12 | 2(r1 +r2 )
EGC
⎧ 2π r ⎪ N (r, r2 , θ12 )fR1 ,R2 ,Θ12 (r, r2 , θ12 )dr2 dθ12 ⎪ 0 0 R1 ,R2 ,Θ12 ⎪ ⎪ ⎨ + 2π r NR1 ,R2 ,Θ12 (r1 , r, θ12 )fR1 ,R2 ,Θ12 (r1 , r, θ12 )dr1 dθ12 0 √ √ √ 2π0 √2r NR (r) = 2NR1 ,R2 ,Θ12 ( 2r − r , r , θ )f ( 2r ⎪ 2 2 12 R ,R ,Θ 1 2 12 ⎪ 0 0 − r2 , r2 , θ12 )dr2 dθ12 ⎪ ⎪ 2π r √ r N 2 2 2 2 ⎩ R1 ,R2 ,Θ12 ( r − r2 , r2 , θ12 )fR1 ,R2 ,Θ12 ( r − r2 , r2 , θ12 )dr2 dθ12 2 0 0 2
SC
r −r2
NR1 ,R2 ,Θ12 (r1 , r2 , θ12 )
SC
(7) SC (8)
MRC
EGC MRC
(10)
∞
rf ˙ R|R ˙ 1 , r2 , θ12 )dr˙ ˙ 1 ,R2 ,Θ12 (r|r 2 π m(r 1 m(r ˙ 1 , r2 , θ12 ) ˙ 1 , r2 , θ12 ) 1 m(r ˙ 1 , r2 , θ12 ) σ(r ˙ 1 , r2 , θ12 ) √ erfc − √ exp − + = 2 σ(r ˙ 1 , r2 , θ12 ) 2 σ(r ˙ 1 , r2 , θ12 ) ˙ 1 , r2 , θ12 ) 2π 2 σ(r 0
imbalance is invariably deleterious, fading correlation may have both positive and negative effects on the performance, depending on d and α. Such a dual effect of fading correlation has also been reported in [1] but for identical channels.
using closely spaced antennas arranged parallel to the direction of the vehicle motion.
IV. C ONCLUSION
[1] F. Adachi, M. T. Feeney, and J. D. Parsons, “Effects of correlated fading on level crossing rates and average fade durations with predetection diversity reception,” Proc. IEE, vol. 135, pp. 11-17, Feb. 1988. [2] X. Dong and N. C. Beaulieu, “Average level crossing rate and fade duration of maximal ratio diversity in unbalanced and correlated channels,” in Proc. IEEE Wireless Communications and Networking Conference, vol. 2, pp. 762-767, Mar. 2002. [3] L. Yang and M.-S. Alouini, “An exact analysis of the impact of fading correlation on the average level crossing rate and average outage duration of selection combining,” in Proc. IEEE Vehicular Technology Conference, vol. 1, pp. 241-245, Apr. 2003.
General exact expressions for the level crossing rate of dualdiversity selection, equal-gain, and maximal-ratio combiners operating on unbalanced correlated Rayleigh channels were derived. Sample numerical results were presented by specializing the general expressions to a space-diversity system with horizontally spaced omnidirectional antennas at the mobile station. It has been observed that, whereas power imbalance is invariably deleterious by increasing the LCR for deep fades, fading correlation may be advantageous depending on the antenna spacing and on the angle between the antenna axis and the direction of the vehicle motion. In particular, considerable improvement over the independent case can be achieved by
R EFERENCES
[4] G. Fraidenraich, J. C. S. Santos Filho, and M. D. Yacoub, “Second-order statistics of maximal-ratio and equal-gain combining in Hoyt fading,” IEEE Commun. Lett., vol. 9, pp. 19-21, Jan. 2005. [5] W. B. Davenport, Jr. and W. L. Root, An Introduction to the Theory of Random Signals and Noise. New York: McGraw-Hill, 1958.
