On the MIMO Channel Capacity for the Nakagami ... - Semantic Scholar
1
On the MIMO Channel Capacity for the Nakagami-m Channel Gustavo Fraidenraich, Student Member, IEEE, Olivier L´evˆeque Member, IEEE, and John M. Cioffi, Fellow, IEEE
Abstract— This paper presents the MIMO channel capacity over the Nakagami-m fading channel. The joint eigenvalue density function of W = HH† , where H is the channel matrix, is derived in closed form for H (2×2) and any integer values of m, as well as for H (2×3) with m = 2 and m = 3. The marginal eigenvalue distribution of W is also derived in a closed-form expression. All the results are validated by numerical Monte Carlo simulations and are in excellent agreement. Index Terms— Fading distributions, Rayleigh distribution, Nakagami-m distribution, Eigenvalue distribution, MIMO channels.
I. I NTRODUCTION It has been acknowledged in recent years that the use of multiple inputs and multiple outputs (MIMO) can potentially provide large spectral efficiency for wireless communications in the presence of multipath fading environments. In the papers presented by Winters [1], Foschini [2] and Telatar [3], the capacity was in particular shown to scale linearly with the number of antennas. In most previous research on MIMO capacity, the channel fading is assumed to be Rayleigh distributed. Of course, the Rayleigh fading model is known to be a reasonable assumption for the fading encountered in many wireless communications systems. Nevertheless, many measurements campaigns [4], [5] show that the Nakagami-m distribution provides a much better fitting for the fading channel distribution. In fact, since the Nakagami-m distribution has one more free parameter, it allows for more flexibility. It moreover contains the Rayleigh distribution (m = 1), the one-sided Gaussian distribution (m → 0.5), and the uniform distribution on the unit circle1 (m → ∞) as special (extreme) cases. Gustavo Fraidenraich and John M. Cioffi are with the Electrical Engineering Department, Stanford University, Stanford, CA 94305 USA. The work of Gustavo Fraidenraich was supported by Cnpq (Brazil) Grant Nr 200869/2005-1. Olivier L´evˆeque is now with the Ecole Polytechnique F´ed´erale de Lausanne, Switzerland, but part of his work was supported by Swiss NSF grant Nr PA002-108987, when he was with the Electrical Engineering Department at Stanford University. 1 When the uniform phase distribution is considered
The Nakagami-m distribution is a general, but approximate solution to the random phase problem [6]. The exact solution to this problem involves the knowledge of the distribution and the correlations of all of the partial waves composing the total signal and becomes infeasible due to its complexity [7]. This has been circumvented by Nakagami [6] who, through empirical methods based on field measurements followed by a curve-fitting process, obtained the approximate distribution. Since the publication of [3], other distributions have been considered, as in [8] for the Ricean case, and recently [9] addressed the Hoyt distribution case. A more realistic MIMO Rayleigh channel including correlation has been addressed in [10]. The Nakagami-m distribution has also been addressed for some special cases as in [11] for the SIMO (single input multiple output) and MISO (multiple input single output) cases, and in [12] for the keyhole channel. The MIMO channel capacity can be computed by means of the joint eigenvalue density function (JEDF) of the matrix W = HH† , where † denotes the complex conjugate transpose. In the classical Rayleigh fading model, the entries of H are assumed to be i.i.d. zero mean complex Gaussian, resulting in a matrix W which is Wishart distributed. This model takes advantages of the numerous results provided by the literature [13], [14]. Unfortunately, as one departs from this model, there is not too much that can be said. The result presented here follows the sequence of decompositions H = LQ, W = LL† and W = S∆S† . The decomposition through the unitary matrices Q provides a way to integrate over the volume dQ, which W = HH† does not provide. Denoting dH as dH = J dL dQ where J is the Jacobian of the transformation, the distribution of p(H) does not depend on Q for the Rayleigh case. Hence, the integration on the Stiefel manifold can be computed in closed form [15]. Unfortunately, for the Nakagami-m case, p(H) depends on Q, so a larger number of variables has to be integrated. To be more accurate, 2rt − r2 integrations over real variables are necessary (the matrix H has 2rt real variables and L has r2 real variables, so Q has 2rt − r2 variables).
2
For either m = 1 or m = 0.5, the resulting integral is simply the volume of the Stiefel manifold. This paper presents the MIMO channel capacity over the Nakagami-m fading channel for H with dimensions 2 × 2 and 2 × 3. Channel state information is assumed at the receiver side only. In this case, it is shown that the uniform power distribution across the transmitting antennas achieves the channel capacity. Assuming that the entries of H are i.i.d. with Nakagami-m distributed envelopes and uniform phases, an elegant and simple expression for the JEDF of W is derived. The paper is organized as follows: Sec. II introduces the Nakagami distribution in more detail. Sec. III defines channel capacity, and Sec. IV presents the main result for H 2×2 and any integer m. Sec. V provides the result for H 2 × 3, and m = 2 or m = 3. Sec. VI presents the asymptotic result. Sec. VII presents some numerical simulations and Sec. VIII draws the conclusion.
II. T HE NAKAGAMI -m DISTRIBUTION The entries of the r ×t channel matrix H are assumed to be i.i.d. and distributed as Z = R exp(jΘ)
(1)
where the phase Θ is uniformly distributed and independent of the envelope R. R is in turn given by R2 =
m X
Xi2 + Yi2
(2)
i=1
where Xi and Yi are i.i.d. zero mean Gaussian distributed with variance Ω/2m. The distribution of R is therefore the Nakagami-m distribution [6] given by 2mm r2m−1 mr2 p(r) = exp − (3) Ωm Γ(m) Ω where Ω = E R2 , m = E[R2 ]2 /Var[R2 ], and Γ(·) denotes the Euler Gamma function. In addition, p(r, θ) = 1 , yielding p(r) 2π mr2 mm r2m−1 (4) exp − p(r, θ) = π Ωm Γ(m) Ω Note that p(r, θ) reduces to the Rayleigh distribution for m = 1 and√ to the uniform distribution on the circle of radius Ω for m → ∞. The above family of distributions therefore allows to interpolate between the classical Rayleigh distribution and the “pure random phase” distribution. Using the standard polar-rectangular transformation, the joint distribution of the real and imaginary parts of
Z is given by p(x, y) = p(x, y) =
p(r,θ) r ,
thus 2 m−1
mm x2 + y π Ωm Γ(m)
e−
m(x2 +y 2 ) Ω
(5)
III. MIMO C HANNEL C APACITY The following MIMO single-user Gaussian channel is considered, with t antennas at the transmitter and r antennas at the receiver: y = Hx + n
(6)
H is the r × t channel matrix with i.i.d. entries hij , each distributed as the random variable Z defined in (1). The vector y ∈ C r , x ∈ Ct , and n is zero-mean complex Gaussian noise with E n n† = I. In addition, a total transmit power constraint E[x† x] ≤ P is assumed. For a given input covariance matrix Σ, the MIMO instantaneous capacity is given by C(Σ) = log2 det I + HΣH† (7)
and the MIMO channel capacity in the absence of channel knowledge at the transmitter is given by [3] C=
sup Σ≥0:tr[Σ]≤P
E [C(Σ)]
(8)
Since the entries hij are i.i.d. and the distribution of hij is the same as that of −hij for all i, j , one obtains from [16, Corollary 1b] that the uniform power allocation over the t transmit antennas achieves the channel capacity. The capacity is therefore given by P (9) C = E log2 det I + W t where W=
IV. M AIN
HH† r ≤ t H† H r > t
RESULT FOR
(10)
H2×2
Theorem 1: The JEDF of the 2×2 matrix W = HH† , where the entries of the 2 × 2 matrix H are i.i.d. with Nakagami-m envelope and uniform phase, is given by p(λ1 , λ2 ) = K22 e−
m(λ1 +λ2 ) Ω
where Kij =
(λ1 − λ2 )2 F (λ1 , λ2 ) (11)
mm πΩm Γ(m)
i j
,
(12)
and F (λ1 , λ2 ) is given by (13), and f (i1 , i2 , k1 , k2 ) is given by (14). In order to gain some intuition on the above result, note that the function F (λ1 , λ2 ) is a homogeneous polynomial of order 4(m − 1). In the Rayleigh case
3
i1 X i2 m−1 X X X m−1 1 π4 F (λ1 , λ2 ) = f (i1 , i2 , k1 , k2 ) (λ1 λ2 ) 2 (i1 +i2 −2(k1 +k2 ))+m−1 (λ1 −λ2 )−i1 −i2 +2(k1 +k2 +m−1) Γ(4m − 2) i1 =0 i2 =0 k1 =0 k2 =0 (13)
i1 k1
i2 k2
m−1 i1
m−1 i2
(−1)−i1 −2i2 +3m+1 2−2(k1 +k2 +1) Γ − i21 − i22 + m − 21 f (i1 , i2 , k1 , k2 ) = Γ − i21 − i22 + m Γ − i21 − i22 + k1 + k2 + m + 12 i1 i2 i1 i2 i1 i2 × Γ − − + k1 + k2 + m Γ − + + k1 − k2 + 2m − 1 Γ − − k1 + k2 + 2m − 1 (14) 2 2 2 2 2 2 0.6 Simulated Theoretical
1
)
0.4
p(
m m=10 0.2
m=1
m=2
0.0 0
2
4
6
1
Fig. 1. The Nakagami eigenvalue distribution function for H (2×2) and m = 1, 2, 10, 100 (Ω = 1).
(m = 1), F is a constant and one recovers the classical JEDF of a Wishart matrix [13]: 1 − (λ1 +λ2 ) Ω p(λ1 , λ2 ) = e (λ1 − λ2 )2 (15) 2Ω4 The effect of the Nakagami-m envelope distribution on the JEDF is therefore expressed by the polynomial F (λ1 , λ2 ). To illustrate the effect of the parameter m, Fig. 1 shows the resulting marginal eigenvalue distribution p(λ) for m = 1, 2, 10, ∞ (see also Corollary 2). It can be observed that as the parameter m increases, the eigenvalues concentrate on the interval [0, 4Ω], with higher probability on the boundary of the interval. It can be also seen in Fig. 1 that the result is in perfect agreement with Monte-Carlo simulations. Proof: In order to get the JEDF of W, the following steps need to be performed: 1) the joint distribution of H can be easily found, since its entries are i.i.d.; 2) the matrix H is decomposed as H = LQ, where L is a
complex lower triangular matrix with real positive diag onals and Q is a complex unitary matrix QQ† = I ; 3) therefore, W = LL† ; 4) finally, performing the eigenvalue decomposition W = SΛS† , one obtains the JEDF of W. Since the entries of the channel matrix H are independent, their joint distribution is given by ! 2 Y m tr HH† |hij |2(m−1) p (H) = K22 exp − Ω i,j=1 (16) Using the LQ decomposition, the matrix H can be written as H = LQ, where the matrix Q is given as [17] ejφ2 sin(θ) ejφ1 cos(θ) (17) Q= −ej(φ3 −φ2 ) sin(θ) ej(φ3 −φ1 ) cos(θ) and the variables are defined in the following range 0 ≤ φ1 , φ2 , φ3 ≤ 2π , 0 ≤ θ ≤ π/2. The matrix L is given by l11 0 (18) L= l21R + j l21I l22 The Jacobian of this transformation is given by J = 3 l l11 22 sin(θ) cos(θ), so the joint probability density function (PDF) of L and Q is given by m tr(L L† ) − K22 Ω 4m−1 p (L, Q) = 2m−1 e l11 l22 sin2m−1 (2θ) 2 m−1 2 × l22 cos2 (θ) + T1L sin2 (θ) − T2L sin (2θ) m−1 2 × T1L cos2 (θ) + l22 sin2 (θ) + T2L sin(2θ) (19)
„
«
where
2 2 T1L = l21I + l21R
and
(20)
T2L = l22 (l21I sin (φ1 + φ2 − φ3 ) − l21R cos (φ1 + φ2 − φ3 )) (21) Since m is integer, it is possible to use the classical
4
m = 2, the JEDF (11) specializes to
binomial expansion in (19). Therefore „
m tr L L†
«
2(λ1 +λ2 )
( ) − 4e− Ω K22 Ω 4m−1 (λ1 − λ2 )2 p(λ1 , λ2 ) = p (L, Q) = 2m−1 e l11 l22 225π 3 Ω8 2 i1 m−1 i2 m−1 × λ41 + λ2 λ31 + 26λ22 λ21 + λ32 λ1 + λ42 (29) X X X X m−1 i1 m−1 × i2 k1 i1 In the same way for m = 3, the following is obtained i =0 k =0 i2 =0 k2 =0 1 1 3(λ1 +λ2 ) i2 k1 +k2 2m−2−i1 −i2 2(i2 −k2 +i1 −k1 ) l22 × (−1)m−1−i2 T1L T2L 6561e− Ω (λ1 − λ2 )2 k2 p(λ1 , λ2 ) = 1254400π 3 Ω12 2(k1 +i2 −k2 ) 2(i1 −k1 +k2 ) 4m−3−i1 −i2 8 7 ×cos (θ) sin (θ) sin(2θ) λ1 + λ2 λ1 + 37λ22 λ61 + 37λ32 λ51 + 478λ42 λ41 (22) +37λ52 λ31 + 37λ62 λ21 + λ72 λ1 + λ82 (30) Integrating now over θ, φ1 , φ2 and φ3 , the distribution The case m → ∞ requires a slightly different analysis, of p(L) is obtained as (23). The next transformation is which leads to the result below. given by Theorem 3: The marginal eigenvalue distribution of w1 w3 − jw4 † W = LL = (24) the 2 × 2 matrix W = HH† , where H (2 × 2) has w3 + jw4 w2 i.i.d. entries with Nakagami-∞ envelope and uniform where the Jacobian of this change is given by J = phase, is given by 3 l . Using this transformation, the distribution of 4l11 22 1 p(λ) = √ 10
On the MIMO Channel Capacity for the Nakagami-m Channel Gustavo Fraidenraich, Student Member, IEEE, Olivier L´evˆeque Member, IEEE, and John M. Cioffi, Fellow, IEEE
Abstract— This paper presents the MIMO channel capacity over the Nakagami-m fading channel. The joint eigenvalue density function of W = HH† , where H is the channel matrix, is derived in closed form for H (2×2) and any integer values of m, as well as for H (2×3) with m = 2 and m = 3. The marginal eigenvalue distribution of W is also derived in a closed-form expression. All the results are validated by numerical Monte Carlo simulations and are in excellent agreement. Index Terms— Fading distributions, Rayleigh distribution, Nakagami-m distribution, Eigenvalue distribution, MIMO channels.
I. I NTRODUCTION It has been acknowledged in recent years that the use of multiple inputs and multiple outputs (MIMO) can potentially provide large spectral efficiency for wireless communications in the presence of multipath fading environments. In the papers presented by Winters [1], Foschini [2] and Telatar [3], the capacity was in particular shown to scale linearly with the number of antennas. In most previous research on MIMO capacity, the channel fading is assumed to be Rayleigh distributed. Of course, the Rayleigh fading model is known to be a reasonable assumption for the fading encountered in many wireless communications systems. Nevertheless, many measurements campaigns [4], [5] show that the Nakagami-m distribution provides a much better fitting for the fading channel distribution. In fact, since the Nakagami-m distribution has one more free parameter, it allows for more flexibility. It moreover contains the Rayleigh distribution (m = 1), the one-sided Gaussian distribution (m → 0.5), and the uniform distribution on the unit circle1 (m → ∞) as special (extreme) cases. Gustavo Fraidenraich and John M. Cioffi are with the Electrical Engineering Department, Stanford University, Stanford, CA 94305 USA. The work of Gustavo Fraidenraich was supported by Cnpq (Brazil) Grant Nr 200869/2005-1. Olivier L´evˆeque is now with the Ecole Polytechnique F´ed´erale de Lausanne, Switzerland, but part of his work was supported by Swiss NSF grant Nr PA002-108987, when he was with the Electrical Engineering Department at Stanford University. 1 When the uniform phase distribution is considered
The Nakagami-m distribution is a general, but approximate solution to the random phase problem [6]. The exact solution to this problem involves the knowledge of the distribution and the correlations of all of the partial waves composing the total signal and becomes infeasible due to its complexity [7]. This has been circumvented by Nakagami [6] who, through empirical methods based on field measurements followed by a curve-fitting process, obtained the approximate distribution. Since the publication of [3], other distributions have been considered, as in [8] for the Ricean case, and recently [9] addressed the Hoyt distribution case. A more realistic MIMO Rayleigh channel including correlation has been addressed in [10]. The Nakagami-m distribution has also been addressed for some special cases as in [11] for the SIMO (single input multiple output) and MISO (multiple input single output) cases, and in [12] for the keyhole channel. The MIMO channel capacity can be computed by means of the joint eigenvalue density function (JEDF) of the matrix W = HH† , where † denotes the complex conjugate transpose. In the classical Rayleigh fading model, the entries of H are assumed to be i.i.d. zero mean complex Gaussian, resulting in a matrix W which is Wishart distributed. This model takes advantages of the numerous results provided by the literature [13], [14]. Unfortunately, as one departs from this model, there is not too much that can be said. The result presented here follows the sequence of decompositions H = LQ, W = LL† and W = S∆S† . The decomposition through the unitary matrices Q provides a way to integrate over the volume dQ, which W = HH† does not provide. Denoting dH as dH = J dL dQ where J is the Jacobian of the transformation, the distribution of p(H) does not depend on Q for the Rayleigh case. Hence, the integration on the Stiefel manifold can be computed in closed form [15]. Unfortunately, for the Nakagami-m case, p(H) depends on Q, so a larger number of variables has to be integrated. To be more accurate, 2rt − r2 integrations over real variables are necessary (the matrix H has 2rt real variables and L has r2 real variables, so Q has 2rt − r2 variables).
2
For either m = 1 or m = 0.5, the resulting integral is simply the volume of the Stiefel manifold. This paper presents the MIMO channel capacity over the Nakagami-m fading channel for H with dimensions 2 × 2 and 2 × 3. Channel state information is assumed at the receiver side only. In this case, it is shown that the uniform power distribution across the transmitting antennas achieves the channel capacity. Assuming that the entries of H are i.i.d. with Nakagami-m distributed envelopes and uniform phases, an elegant and simple expression for the JEDF of W is derived. The paper is organized as follows: Sec. II introduces the Nakagami distribution in more detail. Sec. III defines channel capacity, and Sec. IV presents the main result for H 2×2 and any integer m. Sec. V provides the result for H 2 × 3, and m = 2 or m = 3. Sec. VI presents the asymptotic result. Sec. VII presents some numerical simulations and Sec. VIII draws the conclusion.
II. T HE NAKAGAMI -m DISTRIBUTION The entries of the r ×t channel matrix H are assumed to be i.i.d. and distributed as Z = R exp(jΘ)
(1)
where the phase Θ is uniformly distributed and independent of the envelope R. R is in turn given by R2 =
m X
Xi2 + Yi2
(2)
i=1
where Xi and Yi are i.i.d. zero mean Gaussian distributed with variance Ω/2m. The distribution of R is therefore the Nakagami-m distribution [6] given by 2mm r2m−1 mr2 p(r) = exp − (3) Ωm Γ(m) Ω where Ω = E R2 , m = E[R2 ]2 /Var[R2 ], and Γ(·) denotes the Euler Gamma function. In addition, p(r, θ) = 1 , yielding p(r) 2π mr2 mm r2m−1 (4) exp − p(r, θ) = π Ωm Γ(m) Ω Note that p(r, θ) reduces to the Rayleigh distribution for m = 1 and√ to the uniform distribution on the circle of radius Ω for m → ∞. The above family of distributions therefore allows to interpolate between the classical Rayleigh distribution and the “pure random phase” distribution. Using the standard polar-rectangular transformation, the joint distribution of the real and imaginary parts of
Z is given by p(x, y) = p(x, y) =
p(r,θ) r ,
thus 2 m−1
mm x2 + y π Ωm Γ(m)
e−
m(x2 +y 2 ) Ω
(5)
III. MIMO C HANNEL C APACITY The following MIMO single-user Gaussian channel is considered, with t antennas at the transmitter and r antennas at the receiver: y = Hx + n
(6)
H is the r × t channel matrix with i.i.d. entries hij , each distributed as the random variable Z defined in (1). The vector y ∈ C r , x ∈ Ct , and n is zero-mean complex Gaussian noise with E n n† = I. In addition, a total transmit power constraint E[x† x] ≤ P is assumed. For a given input covariance matrix Σ, the MIMO instantaneous capacity is given by C(Σ) = log2 det I + HΣH† (7)
and the MIMO channel capacity in the absence of channel knowledge at the transmitter is given by [3] C=
sup Σ≥0:tr[Σ]≤P
E [C(Σ)]
(8)
Since the entries hij are i.i.d. and the distribution of hij is the same as that of −hij for all i, j , one obtains from [16, Corollary 1b] that the uniform power allocation over the t transmit antennas achieves the channel capacity. The capacity is therefore given by P (9) C = E log2 det I + W t where W=
IV. M AIN
HH† r ≤ t H† H r > t
RESULT FOR
(10)
H2×2
Theorem 1: The JEDF of the 2×2 matrix W = HH† , where the entries of the 2 × 2 matrix H are i.i.d. with Nakagami-m envelope and uniform phase, is given by p(λ1 , λ2 ) = K22 e−
m(λ1 +λ2 ) Ω
where Kij =
(λ1 − λ2 )2 F (λ1 , λ2 ) (11)
mm πΩm Γ(m)
i j
,
(12)
and F (λ1 , λ2 ) is given by (13), and f (i1 , i2 , k1 , k2 ) is given by (14). In order to gain some intuition on the above result, note that the function F (λ1 , λ2 ) is a homogeneous polynomial of order 4(m − 1). In the Rayleigh case
3
i1 X i2 m−1 X X X m−1 1 π4 F (λ1 , λ2 ) = f (i1 , i2 , k1 , k2 ) (λ1 λ2 ) 2 (i1 +i2 −2(k1 +k2 ))+m−1 (λ1 −λ2 )−i1 −i2 +2(k1 +k2 +m−1) Γ(4m − 2) i1 =0 i2 =0 k1 =0 k2 =0 (13)
i1 k1
i2 k2
m−1 i1
m−1 i2
(−1)−i1 −2i2 +3m+1 2−2(k1 +k2 +1) Γ − i21 − i22 + m − 21 f (i1 , i2 , k1 , k2 ) = Γ − i21 − i22 + m Γ − i21 − i22 + k1 + k2 + m + 12 i1 i2 i1 i2 i1 i2 × Γ − − + k1 + k2 + m Γ − + + k1 − k2 + 2m − 1 Γ − − k1 + k2 + 2m − 1 (14) 2 2 2 2 2 2 0.6 Simulated Theoretical
1
)
0.4
p(
m m=10 0.2
m=1
m=2
0.0 0
2
4
6
1
Fig. 1. The Nakagami eigenvalue distribution function for H (2×2) and m = 1, 2, 10, 100 (Ω = 1).
(m = 1), F is a constant and one recovers the classical JEDF of a Wishart matrix [13]: 1 − (λ1 +λ2 ) Ω p(λ1 , λ2 ) = e (λ1 − λ2 )2 (15) 2Ω4 The effect of the Nakagami-m envelope distribution on the JEDF is therefore expressed by the polynomial F (λ1 , λ2 ). To illustrate the effect of the parameter m, Fig. 1 shows the resulting marginal eigenvalue distribution p(λ) for m = 1, 2, 10, ∞ (see also Corollary 2). It can be observed that as the parameter m increases, the eigenvalues concentrate on the interval [0, 4Ω], with higher probability on the boundary of the interval. It can be also seen in Fig. 1 that the result is in perfect agreement with Monte-Carlo simulations. Proof: In order to get the JEDF of W, the following steps need to be performed: 1) the joint distribution of H can be easily found, since its entries are i.i.d.; 2) the matrix H is decomposed as H = LQ, where L is a
complex lower triangular matrix with real positive diag onals and Q is a complex unitary matrix QQ† = I ; 3) therefore, W = LL† ; 4) finally, performing the eigenvalue decomposition W = SΛS† , one obtains the JEDF of W. Since the entries of the channel matrix H are independent, their joint distribution is given by ! 2 Y m tr HH† |hij |2(m−1) p (H) = K22 exp − Ω i,j=1 (16) Using the LQ decomposition, the matrix H can be written as H = LQ, where the matrix Q is given as [17] ejφ2 sin(θ) ejφ1 cos(θ) (17) Q= −ej(φ3 −φ2 ) sin(θ) ej(φ3 −φ1 ) cos(θ) and the variables are defined in the following range 0 ≤ φ1 , φ2 , φ3 ≤ 2π , 0 ≤ θ ≤ π/2. The matrix L is given by l11 0 (18) L= l21R + j l21I l22 The Jacobian of this transformation is given by J = 3 l l11 22 sin(θ) cos(θ), so the joint probability density function (PDF) of L and Q is given by m tr(L L† ) − K22 Ω 4m−1 p (L, Q) = 2m−1 e l11 l22 sin2m−1 (2θ) 2 m−1 2 × l22 cos2 (θ) + T1L sin2 (θ) − T2L sin (2θ) m−1 2 × T1L cos2 (θ) + l22 sin2 (θ) + T2L sin(2θ) (19)
„
«
where
2 2 T1L = l21I + l21R
and
(20)
T2L = l22 (l21I sin (φ1 + φ2 − φ3 ) − l21R cos (φ1 + φ2 − φ3 )) (21) Since m is integer, it is possible to use the classical
4
m = 2, the JEDF (11) specializes to
binomial expansion in (19). Therefore „
m tr L L†
«
2(λ1 +λ2 )
( ) − 4e− Ω K22 Ω 4m−1 (λ1 − λ2 )2 p(λ1 , λ2 ) = p (L, Q) = 2m−1 e l11 l22 225π 3 Ω8 2 i1 m−1 i2 m−1 × λ41 + λ2 λ31 + 26λ22 λ21 + λ32 λ1 + λ42 (29) X X X X m−1 i1 m−1 × i2 k1 i1 In the same way for m = 3, the following is obtained i =0 k =0 i2 =0 k2 =0 1 1 3(λ1 +λ2 ) i2 k1 +k2 2m−2−i1 −i2 2(i2 −k2 +i1 −k1 ) l22 × (−1)m−1−i2 T1L T2L 6561e− Ω (λ1 − λ2 )2 k2 p(λ1 , λ2 ) = 1254400π 3 Ω12 2(k1 +i2 −k2 ) 2(i1 −k1 +k2 ) 4m−3−i1 −i2 8 7 ×cos (θ) sin (θ) sin(2θ) λ1 + λ2 λ1 + 37λ22 λ61 + 37λ32 λ51 + 478λ42 λ41 (22) +37λ52 λ31 + 37λ62 λ21 + λ72 λ1 + λ82 (30) Integrating now over θ, φ1 , φ2 and φ3 , the distribution The case m → ∞ requires a slightly different analysis, of p(L) is obtained as (23). The next transformation is which leads to the result below. given by Theorem 3: The marginal eigenvalue distribution of w1 w3 − jw4 † W = LL = (24) the 2 × 2 matrix W = HH† , where H (2 × 2) has w3 + jw4 w2 i.i.d. entries with Nakagami-∞ envelope and uniform where the Jacobian of this change is given by J = phase, is given by 3 l . Using this transformation, the distribution of 4l11 22 1 p(λ) = √ 10
