1. Introduction. Let an n m complex Gaussian random matrix A be distributed

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1 COMMUNICATIONS IN INFORMATION AND SYSTEMS c 23 International Press Vol 3, No 2, pp , October 23 3 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY T RATNARAJAH, R VAILLANCOURT, AND M ALVO Abstract The eigenvalue densities of complex central Wishart matrices are investigated with the objective of studying an open problem in channel capacity These densities are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials The connection between the complex Wishart matrix theory and information theory is given This facilitates the evaluation of the most important information-theoretic measure, the so-called channel capacity In particular, the capacity of multiple input, multiple output MIMO) Rayleigh distributed channels are fully investigated We consider both correlated and uncorrelated channels and derive the corresponding channel capacity formulas It is shown how the channel correlation degrades the capacity of the communication system Keywords Complex random matrix, complex Wishart matrix, complex zonal polynomials, complex hypergeometric functions, Rayleigh distributed MIMO channel, channel capacity AMSMOS) subject classification 94A15, 94A5, 6E5, 62H1 1 Introduction Let an n m complex Gaussian random matrix A be distributed as A CN, I n Σ) with mean E{A} = and covariance cov{a} = I n Σ Then the matrix W = A H A is called a complex central Wishart matrix and its distribution is denoted by CW m n, Σ) In this paper, we investigate the densities of the eigenvalues of complex central Wishart matrices and their applications to multiple input, multiple output MIMO) channel capacity We consider that the elements of random matrices are complex Gaussian distributed with zero mean and arbitrary covariance matrices This will enable us to consider the beautiful but difficult theory of complex zonal polynomials also called Schur polynomials 11]), which are symmetric polynomials in the eigenvalues of a complex matrix 14] Complex zonal polynomials enable us to represent the densities of the eigenvalues of these complex Wishart matrices as infinite series The theory of these complex Wishart matrices is used to evaluate the capacity of MIMO wireless communication systems Note that the capacity of a communication channel expresses the maximum rate at which information can be reliably conveyed by the channel 1] In a wireless communication system, data is delivered from a transmitter to a receiver using radio waves or other electromagnetic waves The waves, however, may be reflected off objects in the environment and scattered randomly while Received on April 3, 23; accepted for publication on July 1, 23 This wor was partially supported by the Natural Sciences and Engineering Research Council of Canada Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave, Ottawa ON K1N 6N5 Canada Corresponding author, remi@uottawaca, tel , fax

2 12 T RATNARAJAH, R VAILLANCOURT, AND M ALVO propagating from the transmitter to the receiver Therefore, transmitted signals are attenuated and phase shifted during the transmission This channel effect can be modeled by complex channel coefficients A MIMO channel can be represented by an n r n t complex random matrix H CN, I nr Σ), where n t and n r are the number of inputs or transmitters) and outputs or receivers) of the wireless communication system If Σ = σ 2 I nt then the channel is called an uncorrelated Rayleigh distributed channel, otherwise it is called a correlated Rayleigh distributed channel Recently, industrial researchers have exploited the use of MIMO systems to meet the demand for higher bit rates in wireless communications These studies show that MIMO systems increase capacity significantly over single input, single output SISO) systems For example, when n = min{n t, n r }, a MIMO uncorrelated Rayleigh distributed channel achieves almost n more bits per hertz for every 3-dB increase in signal-to-noise ratio SNR) compared to a SISO system, which achieves only one additional bit per hertz for every 3-dB increase in SNR 18] However, the channel coefficients from different transmitter antennas to a single receiver antenna can be correlated This channel correlation degrades the capacity 4], 17] The channel correlation depends on the physical parameters of a MIMO system and the scatterer characteristics The physical parameters include the antenna arrangement and spacing, the angle spread, the angle of arrival, etc One of the objectives of this paper is to evaluate this capacity degradation by deriving closed form ergodic capacity formulas for correlated channels and their numerical evaluation This paper is organized as follows Section 2 provides the necessary tools for deriving the distribution theory and channel capacity Complex central Wishart matrices are studied in Section 3 The capacity of MIMO channels is formulated in Section 4 and the computational methods are given in Section 5 sequel 2 Preliminary tools In this section we present tools that will be used in the 21 Complex zonal polynomials First, we define the multivariate hypergeometric coefficients a] α) κ which frequently occur in integrals involving zonal polynomials Let κ = 1,, m ) be a partition of the integer with 1 m and = m Then 2] a] α) κ = a 1α ) i 1) i=1 where a) = aa + 1) a + 1) and α = 1 for complex and α = 2 for real multivariate hypergeometric coefficients, respectively In this paper we only consider the complex case; therefore, for notational simplicity we drop the superscript 9], ie, a] κ := a] 1) κ = i a i + 1) i i=1

3 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY 121 The complex zonal polynomial also called Schur polynomial 11]) of a complex matrix X is defined in 8] by 1) C κ X) = χ κ] 1)χ κ] X), where χ κ] 1) is the dimension of the representation κ] of the symmetric group given by 2) χ κ] 1) =! m i<j i j i + j) m i=1 i + m i)! and χ κ] X) is the character of the representation κ] of the linear group given as a symmetric function of the eigenvalues λ 1,, λ m of X by )] det λ j+m j i 3) χ κ] X) = )] det λ m j i Note that both the real and complex zonal polynomials are particular cases of the general α) Jac polynomials C κ α) X) See 2] for details Again α = 1 for complex and α = 2 for real zonal polynomials, respectively For the same reason as before, we shall drop the superscript of Jac polynomials, as was done in 1), ie, C κ X) := C κ 1) X) The following basic properties are given in 8]: tr X) = κ C κ X) and 4) Um) C κ AXBX H )dx) = C κa)c κ B), C κ I m ) where dx) is the invariant measure on the unitary group Um), normalized to mae the total measure unity, and where C κ I m ) = 2 2! ] 1 2 m = κ ] r 1 2 m i<j 2 i 2 j i + j) r κ i=1 2, i + r i)! r i=1 ) 1 m i + 1) 2 i Note that the partition κ of has r nonzero parts 22 Complex hypergeometric functions The probability distributions of random matrices are often derived in terms of hypergeometric functions of matrix arguments The following definitions of hypergeometric functions with a single and double matrix argument are due to Constantine 5] and Baer 2]

4 122 T RATNARAJAH, R VAILLANCOURT, AND M ALVO Definition 1 The hypergeometric function of one complex matrix is defined as 5) pf α) q a 1,, a p ; b 1,, b q ; X) = = κ a 1 ] α) κ a p ] α) κ b 1 ] α) κ b q ] α) κ C α) κ X),! where X C m m and {a i } p i=1 and {b i} q i=1 are arbitrary complex numbers Note that κ denotes summation over all partitions κ of and α = 1 and 2 for complex and real hypergeometric functions, respectively In this paper we consider only the complex case, and hence, we shall drop the superscript, ie, p F q := p F q 1) Note that none of the parameters b i is allowed to be zero or an integer or half-integer m 1)/2 Otherwise some of the terms in the denominator will be zero 14] Remar 1 The convergence of 5) is as follows 14]: i) If p q, then the series converges for all X ii) If p = q + 1, then the series converges for σx) < 1, where the spectral radius σx) of X is the maximum of the absolute values of the eigenvalues of X iii) If p > q +1, then the series diverges for all X, unless it terminates Note that the series terminates when some of the numerators a j ] κ in the series vanish Special cases are F X) = etrx), 1F a; X) = deti X) a, and F 1 n; ZZ H ) = Un) etrze + ZE)dE), where Z is an m n complex matrix with m n, etr denotes the exponential of the trace, etr ) = exptr )) and ZE denotes the complex conjugate of ZE Definition 2 The complex hypergeometric function of two complex matrices is defined by pf q a 1,, a p ; b 1,, b q ; X, Y ) = where X, Y C m m The splitting formula is a 1 ] κ a p ] κ C κ X)C κ Y ), b 1 ] κ b q ] κ! C κ I m ) = pf q AEBE H )de) = p F q A, B) Um) 3 The complex central Wishart matrix In this section, we describe the complex central Wishart distribution and give the joint eigenvalue density of the complex central Wishart matrix eigenvalue density of the complex central Wishart matrix κ From this density we derive a single unordered

5 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY 123 The definition of the complex central Wishart distribution is as follows Definition 3 Let W = A H A, where the n m matrix A is distributed as A CN, I n Σ) Then W is said to have the complex central Wishart distribution with n degrees of freedom and covariance matrix Σ, denoted by W CW m n, Σ) Let W CW m n, Σ) with n m Then the density of W is given by 6) fw ) = 1 CΓ m n)det Σ) n etr Σ 1 W ) det W ) n m, where CΓ m n) denotes the complex multivariate gamma function, CΓ m n) = π mm 1)/2 m Γn + 1) Next, we consider the eigenvalue density of a complex Wishart matrix Proposition 1 Let W be an arbitrary m m positive definite complex random matrix with distribution function fw ) Then the joint density function of the eigenvalues, λ 1 > λ 2 > > λ m >, of W is 7) fλ) = πmm 1) CΓ m m) λ λ l ) 2 <l Um) feλe H )de), where Λ = diagλ 1,, λ m ) and W = EΛE H is the eigendecomposition of W The following proposition gives the joint density of the eigenvalues of a complex Wishart matrix 8] Proposition 2 Suppose that n > m 1 and consider the m m positive definite Hermitian matrix W CW m n, Σ) Then the joint density of the eigenvalues, λ 1 > λ 2 > > λ m >, of W is 8) fλ) = πmm 1) det Σ) n CΓ m m)cγ m n) λ λ l ) 2 λ n m <l Um) etr Σ 1 EΛE H) de), where Λ = diagλ 1,, λ m ) Moreover, etr Σ 1 EΛE H) 9) de) = F Σ 1 EΛE H) de) Um) Um) = F Σ 1, Λ ) = = κ C κ Σ 1 )C κ Λ)! C κ I m ) Proof By substituting the complex Wishart density 6) into 7) and noting that det W = det EΛE H = m λ we obtain 8)

6 124 T RATNARAJAH, R VAILLANCOURT, AND M ALVO Note that the integral in 8) depends on the population covariance matrix Σ only through its eigenvalues υ 1,, υ m This can be seen by writing Σ = F ΥF H, where F Um) and Υ = diagυ 1,, υ m ) Now we have etr Σ 1 EΛE H) de) = etr F Υ 1 F H EΛE H) de) Um) = = Um) Um) Um) etr Υ 1 F H EΛE H F ) de) 1 etr Υ ÊΛÊH) dê) 1 = F Υ ÊΛÊH) dê) Um) = F Υ 1, Λ ) C κ Υ 1 )C κ Λ) =,! C κ I m ) = where Ê = F H E Um) and de) = dê) This was observed in 14] In general, the integral in 8) is not easy to evaluate An infinite series representation for this integral in terms of complex zonal polynomials is shown in 9) Note that the density given in 8) is an ordered eigenvalue density and the unordered eigenvalue density is obtained by dividing 8) by m!, ie, π mm 1) det Σ) n 1) λ n m λ λ l ) 2 F Υ 1, Λ ) m! CΓ m m)cγ m n) Let Υ 1 = diaga 1,, a m ) Then F Υ 1, Λ ) can be written 1] as 11) F Υ 1, Λ ) CΓ m m) det exp a i λ j ))] = π mm 1)/2 m <l λ λ l ) m <l a l a ) A single unordered eigenvalue density is given by the following theorem to be used for computing the correlated MIMO channel capacity in Section 5 <l Theorem 1 Suppose that n > m 1 and consider the m m positive definite Hermitian matrix W CW m n, Σ) Then the single unordered eigenvalue density fλ 1 ) of W is given by 12) π mm 1)/2 m fλ 1 ) = an m! CΓ m n) m <l a l a ) m i 1)peri 1,,i m ) exp a ij λ j j=1 { } m 1)per1,,m) dλ, κ l=1 λ n m+ l l where i denotes summation over all permutations i 1,, i m ) of 1,, m), denotes summation over all permutations 1,, m ) of,, m 1), and per 1,, =2

7 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY 125 m ) is or 1 depending on the permutation being even or odd Similarly for peri 1,, i m ) Proof The single unordered eigenvalue density is obtained by substituting 11) in 1) and integrating with respect to λ 2,, λ m, ie, 13) π mm 1)/2 m fλ 1 ) = an m! CΓ m n) m <l a l a ) det exp a i λ j ))] λ λ l ) The integrand in 13) can be written as det exp a i λ j ))] λ λ l ) <l e a 1λ 1 e a 1λ m e a 2λ 1 e a 2λ m = det <l λ n m det λ n m 1 1 λ 1 λ m λ1 m 1 λ m 1 m m dλ =2 e amλ1 e amλm e a 1λ 1 e a 1λ m λ n m 1 λ n m m = det det e amλ1 e amλm λ n 1 1 λ n 1 m m = i 1)peri 1,,i m ) exp a ij λ j j=1 { },m) 1)per1, λ n m+ l l The result follows l=1 λ n m Note that, if Σ = σ 2 I m, then the joint density of the eigenvalues λ 1,, λ m has a simple form and does not require a zonal polynomial representation Proposition 3 Let W CW m n, σ 2 I m ) with n > m 1 Then the joint density of the eigenvalues, λ 1 > λ 2 > > λ m >, of W is 14) gλ) = πmm 1) σ 2 ) nm λ n m λ λ l ) 2 exp 1 CΓ m m)cγ m n) σ 2 where Λ = diagλ 1,, λ m ) <l Proof Putting Σ = σ 2 I m in Proposition 2 and noting that etr 1σ ) 2 EΛEH de) = etr 1σ ) 2 Λ Um) = exp 1 σ 2 m i=1 de) Um) λ i ) m λ ),

8 126 T RATNARAJAH, R VAILLANCOURT, AND M ALVO completes the proof The single unordered eigenvalue density fλ 1 ) of W is given by the following theorem Theorem 2 Let W CW m n, σ 2 I m ) with n > m 1 Then the single unordered eigenvalue density fλ 1 ) of W is given by 15) fλ 1 ) = πmm 1) σ 2 ) nm m! CΓ m m)cγ m n) m λ n m λ λ l ) 2 exp 1 σ 2 = 1 m m ϕ λ 1 )] 2, <l m ) m λ dλ where ϕ form the orthonormal set which can be obtained by applying the Gram Schmidt procedure to the sequence of functions as 16) λ +n m)/2 e λ/2σ2), =, 1, 2,, m 1 Proof See 19] for similar wor The joint eigenvalue distribution can be written fλ) = K m! m λ λ l ) 2 <l = K det m! 1 1 λ 1 λ nt λ n m exp 1 ) ] σ 2 λ λ m 1 1 λ m 1 m 2 m λ = K n m)/2 1 e λ 1 2σ 2 λ n m)/2 m det m! λ n+m)/2 1 1 e λ 1 2σ 2 λ n+m)/2 1 m 2 ϕ = K 1 λ 1 ) ϕ 1 λ nt ) 1 det m!, ϕ m λ 1 ) ϕ m λ m ) where ϕ is defined in Theorem 2 and satisfies ϕ λ)ϕ l λ) dλ = δ l The last determinant squared in 16) can be expanded as r,s 1)perr 1,,r m ) 1) pers 1,,s m ) =2 λ n m exp 1 ) ] σ 2 λ e λ m 2σ 2 e λ m 2σ 2 ϕ r λ )ϕ s λ ) 2

9 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY 127 where r,s denotes summation over all permutations r 1,, r m ) and s 1,, s m ) of 1,, m) and perr 1,, r m ) is or 1 depending on the permutation being even or odd Similarly for pers 1,, s m ) Hence, fλ 1 ) can be obtained by integrating 16) with respect to λ 2,, λ m, ie, m fλ 1 ) = fλ) dλ =2 = K 1 m! r,s 1)perr1,,rm) 1) pers1,,sm) m ϕ r λ )ϕ s λ ) dλ =2 = K 1 m! r,s 1)perr1,,rm) 1) pers1,,sm) ϕ r1 λ 1 )ϕ s1 λ 1 ) = K 1m 1)! m ϕ λ 1 )] 2 m! = 1 m ϕ λ 1 )] 2 m Since fλ 1 ) dλ 1 = 1, then K 1 = 1 2 δ r s Remar 2 The evaluation of 15) for σ 2 = 1 is given in 3], 12] and 18], where the following formula 17) fλ 1 ) = 1 m m ϕ l λ 1 )] 2 is obtained with ] 1/2 ϕ +1 λ) = λ n m)/2 e λ/2! 1 + n m)!! eλ m n d λ dλ e λ λ n m+ ) ] 1/2 = λ n m)/2 e λ/2! L n m λ), =,, m 1, + n m)! and L n m λ) is the generalized Laguerre polynomial of order l=1 4 The channel capacity A MIMO channel can be represented by an n r n t complex random matrix H, where n t and n r are the number of inputs or transmitters) and outputs or receivers) of the communication system, as shown in Figure 1 The complex signal received at the jth output can be written as n t 18) y j = h ij x i + v j, i=1 where h ij is the complex channel coefficient between input i and output j, x i is the complex signal at the ith input and v j is complex Gaussian noise The signal vector

10 128 T RATNARAJAH, R VAILLANCOURT, AND M ALVO received at the output can be written as y 1 h 11 h nt1 = x 1 + v 1, y nr h 1nr h nt n r x nt v nr or, in vector notation, 19) y = Hx + v, where y, v C n r, H C n r n t, and x C n t constrained to ρ, The total power of the input is E{x H x} ρ or tr E{xx H } ρ We shall deal exclusively with the linear model 19) and derive the capacity of MIMO channel models in this section In this wor, we are particularly interested in Rayleigh v 1 x 1 h 11 + y 1 h 12 h 21 v 2 x 2 h 22 + y 2 h 2nr xn t h n 1 t h n 2 t h n t nr h 1nr + v nr y nr Fig 1 A MIMO communication system distributed channels The following proposition defines this channel model 13] Proposition 4 Let z = re iθ = h ij ) CN, σ 2 ), where r = z and θ = arg z Moreover, we have var{z} = E z 2 = σ 2 and fz) = 1 πσ 2 exp z 2 The density of the magnitude or envelope r is called the Rayleigh density and is given by 2) hr σ 2 ) = { ) 2r r σ exp 2 2 σ 2 r, r < σ 2 )

11 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY 129 The distribution of the phase θ is uniform and its density is given by { 1 21) θ σ 2 2π θ < 2π, ) = otherwise We assume that H is a complex Gaussian random matrix whose realization is nown to the receiver, or equivalently, the channel output consists of the pair y, H) The input power is distributed equally over all transmitting antennas Moreover, if we assume a bloc-fading model and coding over many independent fading intervals, then the Shannon or ergodic capacity of the random MIMO channel is given in 18] by 22) C = E H {log det I nt + ρ )} H H H, n t where the expectation is evaluated using a complex Gaussian density By Proposition 4, if H CN, I nr Σ) then the channel is Rayleigh distributed This is typical of fixed or mobile communication environments In the calculation of capacity we assume n r n t In this case, the distribution of the channel matrix is given by H CN, I nr Σ) Therefore, the distribution of an n t n t complex Wishart matrix is given by W = H H H CW nt n r, Σ) Here the covariance matrix of the rows of H is denoted by Σ, which is an n t n t Hermitian matrix 5 Computation of the capacity In a correlated Rayleigh channel, the distribution of an n r n t channel matrix H is given by H CN, I nr Σ), with n r n t Note that the off-diagonal elements of an n t n t Hermitian matrix Σ are nonzero for correlated channels In other words, the channel coefficient from different transmitter antennas to a single receiver antenna is correlated The following lemma is required in the sequel Lemma 1 If X is an n m n m) full ran matrix and the function fx) depends on X through X H X, then 23) fx H X)dX) = πnm CΓ m n) det A)n m fa) X H X=A Proof Since X H X = A, we have fx H X)dX) = fa) X H X=A X H X=A dx) Let X = ET and A = T H T, where E H E = I m and T is an upper triangular matrix with real and positive diagonal elements Then, from 15, Theorems 4 and 2] we have the following Jacobians for the change of variables dx) = t 2n 2+1 dt )E H de)

12 13 T RATNARAJAH, R VAILLANCOURT, AND M ALVO and Hence Moreover, we have Therefore, da) = 2 m dx) = 2 m m m t 2m 2+1 dt ) t 2n 2m da)e H de) t = det T = det T H T ) 1/2 = det A) 1/2 fa) dx) = 2 m fa)det A) n m E X H X=A CV H de) m,n The last equality follows from 15, Theorem 5] = πnm CΓ m n) det A)n m fa) The channel capacity is given by the following theorem We shall assume that the realization of H is nown to the receiver, or equivalently, the channel output consists of the pair y, H) Theorem 3 Consider the correlated Rayleigh channel, ie, H CN, I nr Σ), with n r n t If the input power is constrained by ρ, then the capacity C is given by 1 log det I CΓ nt n r )det Σ) nr nt + ρ ] W det W ) n r n t etr Σ 1 W ) dw ), n t where W = H H H where W > Proof From formula 22), the capacity C is given by C = log det I nt + ρ ] H H H fh)dh), n t H fh) = π nrnt det Σ) nr etr Σ 1 H H H ) Using Lemma 1, we can write C as C = π nrnt det Σ) nr log det I nt + ρ ] H H H etr Σ 1 H H H ) dh) n t = π n rn t det Σ) n r W > H H H=W 1 = CΓ nt n r )det Σ) nr log det W > H log det I nt + ρ ] H H H etr Σ 1 H H H ) dh)dw ) n t I nt + ρ n t W ] det W ) nr nt etr Σ 1 W ) dw )

13 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY 131 This completes the proof Theorem 3 can also be obtained by using 22) and the complex central Wishart density given in 6) Now, using the eigenvalue density of a complex central Wishart matrix, the correlated Rayleigh channel capacity can be expressed as follows Theorem 4 Consider the correlated Rayleigh channel, ie, H CN, I nr Σ), with n r n t If the input power is constrained by ρ, then using the joint eigenvalue density of the complex central Wishart matrix W = H H H we can write the capacity C as K log Λ> { nt ] 1 } + ρnt n t λ λ n r n t n t <l λ λ l ) 2 F Σ 1, Λ ) n t dλ, where λ 1 > > λ nt > are the eigenvalues of W, Λ = diagλ 1,, λ nt ), and K = πn tn t 1) det Σ) n r CΓ nt n t )CΓ nt n r ) Proof From Theorem 3, the capacity C is given by 24) C = E W {log det I nt + ρ )} { W = E Λ log n t nt 1 + ρ n t λ ] )} The result follows by using the following joint eigenvalue density see Propositions 2), fλ 1,, λ nt ) = πntnt 1) det Σ) nr CΓ nt n t )CΓ nt n r ) n t λ n r n t n t <l λ λ l ) 2 F Σ 1, Λ ) The proof is complete Theorem 5 Consider the correlated Rayleigh channel, ie, H CN, I nr Σ), with n r n t If the input power is constrained by ρ, then using the single unordered eigenvalue density we can write the capacity C as 25) C = n t E λ1 log1 + ρ/n t )λ 1 )] The density fλ 1 ) is given by 26) π n tn t 1)/2 n t fλ 1 ) = anr n t! CΓ nt n r ) n t <l a l a ) n t i 1)peri 1,,i nt ) exp a ij λ j j=1 { } n t nt 1)per 1,, nt ) dλ, l=1 λ nr nt+ l l =2

14 132 T RATNARAJAH, R VAILLANCOURT, AND M ALVO where i denotes summation over all permutations i 1,, i nt ) of 1,, n t ), denotes summation over all permutations 1,, nt ) of,, n t 1) and per 1,, nt ) is or 1 depending on the permutation being even or odd Similarly for peri 1,, i nt ) Note that a 1,, a nt ) are eigenvalues of Σ 1 Proof From 24), C can be written as n t )] )] 27) C = E λ log 1 + ρnt λ = n t E λ1 log 1 + ρnt λ 1, where the expectation is with respect to λ 1 Because we are using the single unordered eigenvalue density 12), the result follows 51 Correlated Rayleigh n r 2 channel matrix In this subsection, a numerical evaluation of a correlated Rayleigh n r 2 channel matrix is given Thus, we assume that we have a two-input n t = 2), n r -output communication system operating over a correlated Rayleigh fading environment typical mobile wireless environment) As mentioned before, the joint eigenvalue density of a central Wishart matrix depends on the population covariance matrix Σ only through its eigenvalues υ 1,, υ nt, ie, F Σ 1, Λ ) = F Υ 1, Λ ), where Υ = diagυ 1,, υ nt ) Let n t = 2 and Υ 1 = diaga 1, a 2 ) Then we have 1] 28) F Υ 1 1, Λ) = a 2 a 1 )λ 1 λ 2 ) exp { a 1 λ 1 + a 2 λ 2 )} exp { a 1 λ 2 + a 2 λ 1 )}] The following theorem gives the correlated Rayleigh channel capacity for an n r 2 matrix Theorem 6 Consider a two-input correlated Rayleigh channel, H CN, I nr Σ), with n r 2 If the input power is constrained by ρ, then the capacity C is 29) a n r 1 C = a 2 a 2 a 1 )Γn r ) a 1 a n r 2 a 2 a 1 )Γn r ) a n r 1 log 1 + ρ ] 2 λ 1 λ nr 1 1 e a 1λ 1 dλ 1 a 2 a 1 )Γn r 1) a nr 2 + a 2 a 1 )Γn r 1) log 1 + ρ ] 2 λ 1 λ n r 1 1 e a 2λ 1 dλ 1 log 1 + ρ ] 2 λ 1 λ n r 2 1 e a1λ1 dλ 1 log 1 + ρ ] 2 λ 1 λ n r 2 1 e a2λ1 dλ 1, where λ 1 is an unordered eigenvalue of W = H H H and a 1, a 2 ) are eigenvalues of Σ 1 Proof By 28), the unordered eigenvalue density of W is given by 3) fλ 1, λ 2 ) = a 1a 2 ) nr λ 1 λ 2 ) nr 2 λ 1 λ 2 ) 2a 2 a 1 )Γn r )Γn r 1) e a 1λ 1 a 2λ 2 e a1λ2 a2λ1]

15 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY 133 Now, integrating with respect to λ 2 and noting that 31) x a 1 e x/b dx = Γa)b a, we obtain the density of λ 1, fλ 1 ) = { 1 a n r 1 a 2λ nr 1 2a 2 a 1 ) Γn r ) 1 e a 1λ 1 an r 1 λn r 2 1 e a 1λ 1 Γn r 1) a 1a nr 2 λnr 1 1 e a 2λ 1 Γn r ) + an r 2 λn r 2 1 e a 2λ 1 Γn r 1) It is easy to see that fλ 1 ) dλ 1 = 1 Finally, evaluating 25) with fλ 1 ) gives 29) Table 1 shows the capacity in nats 1 for an n r 2 correlated Rayleigh fading channel matrix with correlation coefficient 9 } Note that each column represents different levels of input power or signal-to-noise ratio SNR) in db Figure 2 shows the capacity in nats vs n r for the correlation coefficient 9 Figure 3 shows the capacity vs SNR and Figure 4 shows the capacity vs the correlation coefficient From these tables and figures we note the following: i) the capacity is decreasing with increasing channel correlation, ii) the capacity is increasing with increasing n r and SNR ] 1 9 Note that the covariance matrix is Σ = and its eigenvalues are and 1 Hence Υ = diag19, 1), a 1 = 1/19, and a 2 = 1/1 Note also that the off-diagonal element of Σ gives the correlation between the channel coefficient from different transmitter antennas to a single receiver antenna, ie, { E{h ij h 9 i = 1, 2, j = l = 1,, n r, l} = otherwise This off-diagonal element is called a channel correlation coefficient or correlation coefficient 52 Uncorrelated Rayleigh n r 2 channel matrix In this subsection, the numerical evaluation of an uncorrelated Rayleigh n r 2 channel matrix is given In other words, we assume we have a two-input n t = 2), n r -output communication system operating over an uncorrelated Rayleigh fading environment, which is a typical fixed wireless environment The following theorem gives an expression for the capacity C Theorem 7 Consider a two-input uncorrelated Rayleigh channel, ie, H CN, I nr σ 2 I 2 ), with n r 2 If the input power is constrained by ρ, then the 1 In 29), if we use log e then the capacity is measured in nats If we use log 2 then the capacity is measured in bits Thus, one nat is equal to e bits/sec/hz e = 2718 )

16 134 T RATNARAJAH, R VAILLANCOURT, AND M ALVO Table 1 The capacity in nats for a two-input, n r-output communication system operating over a correlated Rayleigh fading channel, where ρ is signal-to-noise ratio in db and the correlation coefficient is equal to 9 ρ in db n r db 5 db 1 db 15 db 2 db 25 db 3 db 35 db db 2 Capacity in nats) db db Number of outputs n r Fig 2 Capacity vs number of outputs for SNR =, 5, 1, 15, 2, 25, 3, 35 db Note that H is an n r 2 correlated Rayleigh fading channel matrix with correlation coefficient equal to 9

17 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY 135 Capacity in nats) n = 6 r n = 2 r n = 8 r n = 1 r n = 4 r Signal-to-noise ratio Fig 3 Capacity vs SNR for correlation coefficient 2, n t = 2, and n r = 2, 4, 6, 8, 1, ie, H is an n r 2 correlated Rayleigh fading channel matrix Capacity in nats) n = 8 r n = 4 r n = 6 r n = 1 r 7 6 n = 2 r Correlation coefficient Fig 4 Capacity vs correlation coefficient for SNR = 2 db, n t = 2, and n r = 2, 4, 6, 8, 1, ie, H is an n r 2 correlated Rayleigh fading channel matrix

18 136 T RATNARAJAH, R VAILLANCOURT, AND M ALVO capacity C is given by 32) C = σ2 ) n r 1 Γn r ) 2σ2 ) n r Γn r 1) + σ2 ) n r+1 Γn r + 1) Γn r )Γn r 1) log 1 + ρ ] 2 λ 1 λ nr 1 e λ 1/σ 2 dλ 1 log 1 + ρ ] 2 λ 1 λ nr 1 1 e λ 1/σ 2 dλ 1 where λ 1 is an unordered eigenvalue of W = H H H log 1 + ρ ] 2 λ 1 λ n r 2 1 e λ 1/σ 2 dλ 1, Proof By 14), the unordered eigenvalue density of W is 33) fλ 1, λ 2 ) = σ2 ) 2n r λ 1 λ 2 ) n r 2 λ 1 λ 2 ) 2 2Γn r )Γn r 1) e λ 1+λ 2 )/σ 2 Integrating with respect to λ 2 and using 31), we obtain the density of λ 1, fλ 1 ) = σ2 ) n r 1 λ nr 1 2Γn r ) e λ 1/σ 2 σ2 ) n r Γn r 1) λnr 1 1 e λ 1/σ 2 + σ2 ) nr+1 Γn r + 1) λ nr 2 1 e λ 1/σ 2 2Γn r )Γn r 1) It is easy to see that fλ 1 ) dλ 1 = 1 Finally, evaluating 25) with fλ 1 ) gives 32) Table 2 shows the capacity in nats for an n r 2 uncorrelated Rayleigh fading channel matrix with different levels of input power Figure 5 shows the capacity in nats vs n r for different signal to noise ratios It is clearly seen from the table and figure that the capacity is increasing with increasing n r and SNR 6 Conclusion In this paper, joint and single unordered eigenvalue densities of complex central Wishart matrices are derived These densities are used to derive formulas for the capacity of correlated and uncorrelated MIMO Rayleigh channels The capacity of n r 2 MIMO Rayleigh channel matrices are computed for both correlated and uncorrelated channels This study shows how the channel correlation degrades the capacity of the communication system

19 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY 137 Table 2 The capacity in nats for a two-input, n r -output communication system operating over an uncorrelated Rayleigh fading channel, where ρ is signal-to-noise ratio in db ρ in db n r db 5 db 1 db 15 db 2 db 25 db 3 db 35 db db 2 Capacity in nats) db db Number of outputs n r Fig 5 Capacity vs number of outputs for SNR =, 5, 1, 15, 2, 25, 3, 35 db Note that H is an n r 2 uncorrelated Rayleigh fading channel matrix

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