Abstract
This paper presents a general expression for the marginal distributions of the ordered eigenvalues of certain important random matrices. The expression, given in terms of matrix determinants, is compacter in representation and more efficient in computational complexity than existing results in the literature. As an illustrative application of the new result, we then analyze the performance of the multiple-input multiple-output singular value decomposition system. Analytical expressions for the average symbol error rate and the outage probability are derived, assuming the general double-scattering fading condition.
) with its Hermitian transposition (given that 
). If 
, the product matrix was termed the pseudo-Wishart matrix [
tensor. After that, a simpler expression for the eigenvalue distribution was presented by Sun et al. [
-dimensional) determinants were involved.
-dimensional) determinants, and importantly, it replaces many functions in Sun's result with constant numbers, greatly improving the computational efficiency. As an illustrative application of the new expression, we analyze the performance of MIMO SVD systems, assuming the (uncorrelated) double-scattering [
denotes the transposition of a matrix; 
denotes the Hermitian transposition of a matrix; 
denotes an 
matrix with only zero elements; 
denotes the 
identity matrix; 
denotes that 
is an 
complex matrix; 
is the 
th element of a matrix 
; 
denotes the determinant of a matrix; 
is the determinant of a matrix whose 
th element is 
; 
is the expectation of a random variable with respect to 
; 
denotes that 
is an 
complex Gaussian matrix with a mean value 
, a row correlation 
, and a column correlation 
.
)], for more details. It is also worth noting that, for non-Gaussian random matrices, obtaining exact expressions on their joint eigenvalue distributions is generally difficult. Very few results can be found in the literature. In this paper, we focus on exact eigenvalue distributions, and thus, we consider mainly Gaussian and Gaussian-related random matrices.) Indeed, this common structure (formulated as the proposition below) was previously reported in [
denote a Hermitian random matrix discussed in Section 1, and let 
, 
, denote the nonzero ordered eigenvalues of 
. Then, the joint PDF of 
can be expressed as
, 
is the joint PDF of 
, 
is a constant coefficient, 
is a generic function, 
and 
are 
and 
matrices (
), respectively, whose elements are given by
being an arbitrary constant, 
's and 
's being a generic function.
and 
denote two mutually independent complex Gaussian matrices).
) Single-user MIMO systems:
with 
, then the joint PDF of the eigenvalues of the Wishart matrix 
is [
, 
, and
, and 
, with 
, 
, and 
being three natural numbers, then the nonzero ordered eigenvalues of 
are jointly distributed as
, 
, and
and 
are defined by
being the modified Bessel function of the second kind [
)].
].
) Multiuser MIMO systems:
with 
, 
with 
, and 
has 
positive and descendingly ordered eigenvalues 
, then the eigenvalues of 
are jointly distributed as [
, 
, and
being the generalized hypergeometric function [
)].
, 
with the matrix 
having 
positive eigenvalues in descendent order 
, then the joint PDF of the nonzero eigenvalues of 
is [
, 
, and
being the upper incomplete Gamma function [
)].
is given by (1), the marginal CDF of the 
th largest eigenvalue 
can be expressed as (
)
, 
is a permutation of 
that satisfies 
and 
. The second summation is over all permutations, that is, 
in total.
)],
is an 
matrix with each element being a function of 
, and 
is identical to 
, except that all elements in the 
th column are replaced by their derivatives with respect to 
.
terms. That form allows closed-form evaluation of moments and characteristic functions of the eigenvalues.) These results, however, needed separate expressions to represent the eigenvalue distributions of Wishart (i.e., 
) and pseudo-Wishart (i.e., 
) matrices. In contrast, Theorem 1 unifies the two cases (
and 
) with only a single expression. It is also worth noting that, although another unified expression could be found in [
tensor 
. (Letting 
be a rank-
tensor, that is, 
for 
, the "determinant" of 
, denoted by 
, is given by [
where 
and 
are permutations of the integers 
, the summation is over all possible permutations, and 
is the sign of the permutation.) which was computationally complex, especially comparing to our new result in a conventional (
-dimensional) determinant form. Perhaps the most related work in the literature is [
] in the following proposition. After comparing the two results, one can clearly see that our expression is much more efficient in computational complexity, since the functions 
in (15) are replaced by constant numbers 
in (13).
can be alteratively expressed as
. Substituting (1) into (17) and invoking the generalized Cauchy-Binet formula [
] the multi-nested integration can be carried out analytically. As such, we get the desired result.
of 
(
) are jointly distributed as (5), the marginal CDF of the 
th largest eigenvalue 
can be expressed as
is given in (7), the second summation is over all combinations of 
and 
with 
being a permutation of 
, and
) and (
)]. The results are given in (20). Substituting (5) into (13) and using (21) completes the proof.
) of the double-scattering channel matrix was reported earlier in [
(
), when 
, 
, and 
. The analytical results are computed by (18), and the Monte Carlo results are based on 
channel realizations. A perfect agreement is observed between the analytical and Monte Carlo curves. 
is modeled as [
, 
, 
, 
, and 
are the numbers of transmit antennas, receive antennas, and the scatterers, respectively. The matrix 
represents the fading channel between the transmitter and the scatterers, while 
represents the channel between the scatterers and the receiver. The introduction of the double-scattering model is due to the fact that [
, 
, 
, and 
as they were defined in (6).
transmit and 
receive antennas. The received vector 
can be expressed as
is the channel matrix, 
is the vector of signals transmitted, and 
is the complex additive white Gaussian noise (AWGN) vector with zero mean and identity covariance matrix. In MIMO SVD, assuming perfect channel state information (CSI) at the transmitter, the transmit vector 
is formed by mapping 
(
) modulated symbols 
(
) onto 
transmit antennas via a linear precoding:
is the spatial pre-coding matrix. Here, the columns of 
are the right singular vectors of 
corresponding to the 
largest singular values. Under the assumption of perfect CSI at the receiver, the decision statistics of MIMO SVD, denoted by 
(
), is obtained by weighting the receive signal 
with a spatial equalizing matrix 
are the left singular vectors of 
corresponding to the 
largest singular values. After such pre-coding and equalization, the MIMO channel is decomposed into a set of equivalent single-input single-output (SISO) channels, whose input-output relation is 
is the 
th largest eigenvalue of 
, and 
is the complex AWGN with zero mean and unit variance (i.e., 
variance per complex dimension). Hereafter, we term these SISO channels as the sub-channels of MIMO SVD. Letting 
denote the power allocated to the 
th subchannel, the instantaneous SNR of the 
th subchannel can be expressed as (
)
s.
is a function of all eigenvalues 
, leading to an intractable SER expression of each subchannel [
. For convenience sake, we assume equal power allocation is employed, that is, 
with 
denoting the total transmit power (normalized by the noise variance). As such, the SNRs of the subchannels are ordered as 
, and the outage probability of the overall system is dominated by the worst subchannel (corresponding to 
). The exact expression on outage probability is then obtained by substituting the CDF (18) into the equation below
-PAM, etc.) [
-PSK [
)])
is the instantaneous SNR, 
is the Gaussian 
-function, 
and 
are modulation-specific constants (e.g., 
, 
for BPSK), the average SER of the 
th subchannel of the MIMO SVD system can be expressed as (after some algebraic manipulations)
transmit antennas, 
receive antennas, and 
scatterers by a three-tuple 
. We also assume that all subchannels are active (i.e., 
), upon which equal power allocation is employed (i.e., 
for all 
).
dB to evaluate the impact of scatterer insufficiency on the outage probability of MIMO SVD. Three channel configurations are considered: 
, 
, and 
. Results from standard Rayleigh channel (i.e., 
) is also provided for the purpose of comparison. The analytical results are computed with (29), and each Monte Carlo result is based on 
channel realizations. From the figure, we observe an exact agreement between the analytical and Monte Carlo curves. Also, we observe that the lack of scattering certainly degrade the performance of the system, which is consistent with our intuition.
,
,
, and
.
double-scattering channel, using uncoded BPSK modulation. It is shown that all analytical results agree with the Monte-Carlo curves perfectly. It is also observed that the first and second strongest subchannels outperform the third subchannel significantly. This indicates that further improvements (in SER) is possible if only a subset of subchannels is used. In-depth analysis along this direction can be found in [
are the nonzero descendingly ordered eigenvalues of 
, with 
, 
, 
, and 
being given by (6). We also define the following new notations 
with 
being its nonzero descendingly ordered eigenvalues. Then, we take three steps to get the joint PDF of 
. First of all, we get the joint PDF of 
, that is, 
. Next, we obtain the joint PDF of 
conditioned on 
, that is, 
. Finally, we average the conditional joint PDF 
over 
to get the unconditional joint PDF 
. Details on this condition-and-average procedure are given below.
of 
(
). Based on the result of [
, conditioned on 
. To this end, we note that if 
is rank deficient, (i.e., 
), 
are the eigenvalues of 
, where 
is a diagonal matrix with 
as its diagonal elements, and 
is a complex Gaussian matrix with statistically independent, zero-mean, unit-variance elements. Knowing this, we get the conditional joint PDF of 
by invoking [
]:
)]
by averaging conditional PDF over 
, and 
. The integration above can be evaluated in a closed form with the generalized Cauchy-Binet formula (see, e.g., [
]). We finally arrive at the expression below
)]
denote the unordered version of 
. Then, by the symmetry of (1), we get the joint PDF of 
is due to the change in function domains when comparing with (1). This joint PDF can be simplified as follows:
is an 
matrix defined by
. The usefulness of this form will become apparent immediately.
by using the fact that 
, with 
and 
being two square matrices of the same size (a similar method was used in [
is a permutation of 
, and the summation is over all permutations. The usefulness of the joint PDF in this form will become apparent immediately.
)], the marginal CDF of the 
th largest variable 
can be expressed as (note that [
)] deals with random variables in ascendent order. However, the result there can be easily rewritten to cover the descending-order cases by appropriate change of variables)
being the CDF of 
. Obviously, the desired marginal CDF 
depends directly on an intermediate CDF 
. As we show below, this intermediate CDF can be obtained by the use of the joint PDF in (B.5)
for 
, that is, 
are the indices of 
in the permutations. Noticing that all integrals above are independent of the order of 
(
and 
, resp.), we can further simplify the summation as
to denote a permutation of 
that satisfies 
and 
. Then, the CDF above is equivalent to
, that is, 
in total. Substituting (B.12) into (B.6) yields the desired result.