INVERSE EIGENVALUE STATISTICS FOR RAYLEIGH AND RICIAN MIMO CHANNELS E. Jorswieck1 , G. Wunder1 , V. Jungnickel1 , T. Haustein1
Abstract Recently reclaimed importance of the empirical distribution function of the eigenvalues of Wishart matrices was attached to the analysis of MIMO systems in [1]. But for the analysis of MIMO systems with signal processing at the transmitter [2] it is necessary to know the empirical distribution function of the inverse eigenvalues and the first and second moment of these distributions. We extend the results on the eigenvalue distribution and develop a method for computing the moments of the inverse eigenvalues. We show the impact of these results on MIMO system design with examples and verify our results with numerical simulations.
1
Introduction
Multiple antennas can be used at the transmitter and receiver forming a multiple input multiple output (MIMO) system to enhance the performance and capacity of a wireless communication system [3]. To reach the promised capacity and performance knowledge of the statistics of typical MIMO channels is useful. For the analysis of the bit error performance in MIMO systems and of the required transmit power in water-filling or channel-inversion transmission schemes, it is mandatory to know at least the first and second moment of the inverse eigenvalue (EV) distribution of the random Rayleigh and Rician MIMO channel matrices. Especially for the capacity evaluation of MIMO channels with channel state information (CSI) at the transmitter or at the receiever or at both [4] the eigenvalue distribution is important. The results regarding the eigenvalue distribution of Wishart matrices and their application to MIMO system design are well known (see for example [5] Theorem 1 and [6]). We develop results for the empirical distribution function of the inverse eigenvalues and present a method for computing the moments in the Stieltjes domain. Further the impact of these results on the MIMO system design especially for systems with signal processing at the transmitter is shown. The average noise gain in zero-forcing systems is evaluated. The average transmit power in channel inversion mode [2] is computed in order to normalize the transmit power for system comparison. The theory yields the same results for Rayleigh and Rician MIMO channels. Finally the differences between these two channel models are discussed. First the signal model is briefly reviewed. For a flat-fading channel the MIMO link can be described by x = Hs + n (1) Heinrich-Hertz-Institut f¨ ur Nachrichtentechnik Berlin GmbH Einsteinufer 37, D-10587 Berlin, Germany, e-mail:
[email protected] 1
where s and x are the transmitter (Tx) signal vector from M transmit antennas and the receiver (Rx) signal vector from N receive antennas, respectively. 2 n ∼ CN (0, σN )
(2)
is the noise vector with independent identically distributed (iid) complex entries and H is the channel matrix. For a Rayleigh channel, the matrix H consists of iid complex Gaussian distributed entries with zero PLOS mean HRay ∼ CN (0, 1). The Rician channel is characterized by the Rician factor K = PRayleigh as the ratio of line-of-sight (LOS) power to non-LOS power and the matrix H consists of iid complex Gaussian entries with mean K. The channel matrix can be decomposed as H = UDVH .
(3)
U and V are N × M and M × M matrices with orthonormal columns, respectively, and D is a M × M diagonal matrix. VH is the conjugate transpose of V. In the case M ≤ N , the diagonal matrix consists of M singular values (SV) D = diag(λ1 . . . λM ). In this paper, the statistics of the inverse SVs λ1i and their application to MIMO transmission systems are investigated.
2
Empirical distribution function for the inverse eigenvalues
2.1
Preliminary results
The following theorem was given in [7] (Theorem 1.1) Theorem 1: Assume that • XN is N ×n matrix with complex independent and identically distributed entries with variance one. • n is a function of N and the ratio
n N
converges as n → c > 0 as N → ∞ N
(4)
• DN is a diagonal matrix with realvalued random entries τ1N , . . . , τnN with empirical distribution function (edf) that converges almost surely in distribution to a probability distribution function (pdf) H(τ ) as N → ∞. • Define where AN
BN = AN + XN DN XH N is Hermitian N × N matrix with nonrandom entries.
(5)
• The matrices XN , DN and AN are independent. Then almost surely the distribution function of the eigenvalues of BN converges weakly, as N → ∞, to a distribution function F , whose Stieltjes transform m(z) (z ∈ C + ) satisfies µ ¶ Z τ dH(τ ) m(z) = mAN z − c . (6) 1 + τ m(z)
The Stieltjes transform of a distribution function FB (λ) is given by Z 1 mB (z) = dFB (λ). λ−z 2.2
(7)
Application of Theorem 1
With the following definitions Theorem 1 is used to compute the edf of the eigenvalues of the channel matrix H. We set XN = HH (M ≤ N ), AN = 0M ×M and DN = IN ×N . The Stieltjes transform R (λ) of the EV distribution of A is given by m(z) = dF with F (λ) = δ(λ). One gets the Stieltjes λ−z transform of the EVs of HH H 1 m(z) = . (8) 1 −z + c 1+m(z) The explicit form of the Stieltjes transformes edf is given for z 6= 0 by m(z) =
1 (c−1−z) + 2 z −
√
c2 −2c−2cz+1−2z+z 2 ) . z
With the Stieltjes inversion formula 1 F ([a, b]) = lim π η→0
Z
b
ImmF (ξ + iη)dξ
the empirical density functions (edf) for the EVs ( 0 f (x) = √ c,x
with a = (1 − 2.3
√
c)2 and b = (1 +
(x−a)(b−x) 2πcx
√
(9)
a
if x < a or x > b, if a ≤ x ≤ b.
(10)
c)2 can be obtained.
Inverse Eigenvalues
Starting from (10) with fc,y (y) =
1 f (1) y 2 c,x y
fc,y (y) =
0 q
if y
a1 ,
≤ y ≤ a1 .
(11)
is obtained as edf for the inverse eigenvalues with a and b defined as above.
3
Mean and variance
In many cases one needs only the first few moments of the edf of the inverse eigenvalues to make statements about the average performance of the MIMO system oder about the ergodic capacity of a MIMO system.
Mean and variance - if they exist - can be calculated either directly from the edfs (10), (11) or from the Stieltjes transformed via the derivatives of m(z) Z ∞ dF (t) (n) m (0) = = mn+1 (12) tn+1 0 with mn as the n-th non-central moment of the inverse EVs. The latter is more simple since all moments can be calculated from the n-th derivative of m at point 0. To obtain the mean of the inverse EVs for HH H for example we must take the implicit formula for the Stieltjes transformed (8) and compute the derivative with respect to z. We start from (8) with m(z) =
1 c −z+ 1+m(z)
↔
(13)
−z · m(z) − z · m2 (z) + c · m(z) = 1 + m(z).
(14)
Now the derivative at the point z = 0 yields c · m(1) (z = 0) − m(1) (z = 0) − 1 = 0 ↔ 1 N m(1) (z = 0) = c−1 = N −M = E[ λ12 ] i
(15) (16)
Analog the variance and moments of higher order can be computed. Mean and variance of the inverse EVs for HH H are 1 N ] = 2 λi N −M 1 1 Var[ 2 ] = . λi (1 − c)3 E[
4
(17) (18)
Rician channels
The theory yields the same edf for the inverse eigenvalues of Rayleigh and Rician channel matrices. Because of the asymptotic view in Theorem 1 this behaviour can be explained: There is one dominant EV in Rician channels that becomes greater than all the other EVs. The probability mass for the one dominant EV reduces to zero, if N → ∞ and the number of EVs increases. Therefore asymptotically, the inverse EVs of Rician and Rayleigh channel matrices have the same empirical probability distribution. For practical systems of order M = 8 . . . 12 mean and variance of the inverse EVs of Rayleigh and Rician channels do differ, of course. The difference depend on the number of receive antennas but not on K as shown in figure (4).
5
Basic observations
From (17) it follows, that for square MIMO systems (M = N ) the mean value of the inverse EVs does not exist. Already with one additional Rx antenna the mean value exists and equals one. The variance of the inverse EVs reduces with increasing c (18).
Figure 1: Mean value of inverse eigenvalues of Rayleigh and Rician channels with 8 Tx antennas
5.1
Impact on noise gain
In a simple zero-forcing MIMO system with channel state information (CSI) only at Rx the data signals are reconstructed by multiplication by the Moore-Penrose pseudo inverse of H denoted by H† y = H† x = H† Hs + H† n = s + H† n. (19) with the pseudo inverse channel matrix H† = (UDVH )† = VD−1 UH .
(20)
The isotropic noise vector is weighted by the inverse SVs (noise gain). At the decision unit the average noise variance for one data stream with (17) is given by 1 σ2 · N H · EH,n [|VD−1 U n|2 ] = n . M N −M
(21)
Obviously, the average noise gain is reduced for N > M . 5.2
Impact on channel inversion schemes
If CSI at Tx is known, linear channel inversion for the uplink instead of (19) can be used H
and at the receiver
˜ s = VD−1 V s
(22)
y = s + VUH n.
(23)
This system approach seems to convert the fading channel to an AWGN channel. Of course this is not true. To allow a fair comparison to other system approaches the Tx power has to be normalized. To normalize the total Tx power, Tx signals ˜ s must be weighted by the square root of (17) in order to obtain an average Tx power of M . Equation (18) is a measure for the dynamics of the Tx power. The dynamics can be reduced by increasing the number of Rx antennas.
6
Monte-Carlo simulation
Figure (6) shows a histogram over 10000 simulated Rayleigh channel matrices and the theoretical results. The theoretical results fit well to simulation data, except of the substructure in the EV distribution which disappears for large N .
Figure 2: Simulated and theoretical distribution of a) eigenvalues of HHH and b) inverse eigenvalues of HHH .
7
Conclusions
We reviewed analytic expressions for the empirical distribution function of the EVs for Rayleigh and Rician MIMO channels and developed new expressions for the empirical distribution function and mean and variance of the inverse EVs for Rayleigh MIMO channels. The difference between the theoretical and simulated result for Rician channels was explained. The practical meaning of the developed results on the zero forcing system approach with CSI only at the Rx and to linear channel inversion with CSI at Tx was shown.
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