4 Application of majorization in wireless communications.

4.1 Spatial Correlation in Multiple Antenna Systems

Recently, there is a transition in communication theory of how fading variations are judged. The time variation and spectral variation of the propagation channel are nowadays welcome. In fact, they are exploited to increase the reliability and spectral efficiency in mobile communications systems. It is well known, that fading variations in time, space, and frequency, increase the diversity of the system (e.g., [159]).

With the introduction of MIMO systems the question about how to model, analyze, and exploit the spatial correlation that is observed at the transmit antenna array and the receive antenna array (see, e.g., [165, ch. 2] for outdoor scenario and [102, 160] for MIMO channels). If the antenna geometry is simple, e.g., a uniform linear array (ULA), the antenna correlation matrix leads to a Toeplitz structure. Since multiple antennas are on both sides of the link, there may or may not be correlation between transmit and receive antenna pairs. In the Kronecker model, the correlation is modeled locally at the transmit and receive side and in between rich multipath scattering is assumed.

4.1.1 The Kronecker Model

Consider the quasi-static block fht-fading MIMO channel H. The correlation of the channel matrices arises in the common downlink transmission scenario in which the base station is un-obstructed [131]. We follow the model in [38] where the subspaces and directions of the paths between the transmit antennas and the receive cluster change more slowly than the actual attenuation of each path.

The most general form of the correlation model consists of a very large correlation matrix of size ([n.sub.T] - [n.sub.R] x [n.sub.T] - [n.sub.R]) which incorporates the transmit and receive correlation, i.e., it is the expectation of the outer product of the vectorized channel matrix

[kappa] = E[vec(H) x vec[(H).sup.H]] (4.1)

The correlation matrix K in (4.1) expresses the correlation between each transmit or receive element to every other transmit or receive element. Often, the transmit and the receive antenna array are spatially divided. Then, the following simplifcation is possible (see, e.g., [30]):

Defnition 4.1 (Kronecker correlation model). If the transmit correlation is independent of the receive antenna and the receive correlation does not depend on the transmit antenna, the correlation matrix in (4.1) is a block-matrix that is given by

[kappa] = [R.sub.R] [cross product] [R.sub.T] (4.2)

and the corresponding correlation model is called Kronecker correlation model.

The channel matrix H for the case in which we have the Kronecker assumption and correlated transmit and correlated receive antennas is modeled as

H = [R.sup.1/2.sub.R] x W x [R.sup.1/2.sub.T] (4.3)

with transmit correlation matrix [R.sub.T] = [U.sub.T][D.sub.T][U.sup.H.sub.T] and receive correlation matrix [R.sub.R] = U [sub.R][D.sub.R][U.sup.H.sub.R]. [U.sub.T] and [U.sub.R] are the matrices with the eigenvectors of [R.sub.T] and [R.sub.R], respectively, and [D.sub.T], [D.sub.R] are diagonal matrices with the eigenvalues of the matrix [R.sub.T] and [R.sub.R], respectively. The random matrix W has zero-mean independent complex Gaussian identically distributed entries, i.e.,W ~ CN(0, I). The matrix W models the rich multipath environment between transmit and receive antenna array.

The constructed matrix H in (4.3) satisfes (4.2) because vec(AXC) = ([C.sup.T] [cross product] A)vec(X) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In Figure 4.1, some of the basic assumptions in MIMO channels are illustrated. Often, it is assumed that the base station antennas are mounted on roof top of high buildings or towers. Therefore, less local scatterer surround the base station antenna array and increased spatial correlation can be observed. In contrast, the mobile moves around surrounded by buildings, cars, trees, and pedestrians. Therefore, it is often assumed that the mobile antennas are spatially uncorrelated. Note that polarization diversity provides an additional degree of freedom [96]. The analysis in [28, 29] is adapted to several special practical scenarios in which so called keyholes occur. For example, in transmission scenarios in which we have long corridors (see Figure 4.1) the channel can be singular. This is not because of correlation at the transmitter or the receiver but because of a keyhole in between.

[FIGURE 4.1 OMITTED]

In the case in which each receive antenna observes the same correlation between the transmit antennas, i.e., the transmit correlation is independent of the receive antenna and vice versa the receive correla tion is independent of the transmit antenna, the correlation model in (4.1) simplifies to the model in (4.3). Note that the Kronecker model arises not only in MIMO communications but also in the modeling of electroencephalography (EEG) data. Methods to estimate the correlation matrices under the Kronecker assumption are described in [154].

Note that the Kronecker model is a limited correlation model that can only be applied successfully under certain conditions on the local scattering at the transmitter and receiver [85, 103]. Therefore, a more generalized model is to allow a sum of Kronecker products [11], i.e.,

[kappa] = [n summation over (k=1)] [R.sup.R.sub.k] [cross product] [R.sup.T.sub.k]. (4.4)

However, it turns out that even the model (4.4) cannot cover the complete set of positive semi-definite correlation matrices. One counter example is explicitly given here for the case [n.sub.T] = [n.sub.R] = 2 (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4.1.2 A Measure of Spatial Correlation

In order to provide a measure of correlation, we take two arbitrarily chosen transmit correlation matrices [R.sup.1.sub.T] and [R.sup.2.sub.T] with the constraint that trace([R.sup.1.sub.T]) = trace([R.sup.2.sub.T]) = [n.sub.T] which is equivalent to

[[n.sub.T] summation over (l=1)] [[lambda].sup.T,1].sub.l] = [[n.sub.T] summation over (l=1)] [[lambda].sup.T,2.sub.l], (4.5)

with [[lambda].sup.T,1].sub.l], 1 [less than or equal to] l [less than or equal to] [n.sub.T], and [[lambda].sup.T,1].sub.l], 1 [less than or equal to] l [less than or equal to] [n.sub.T] are the eigenvalues of the covariance matrix [R.sup.1.sub.T] and [R.sup.2.sub.T], respectively.

This constraint regarding the trace of the correlation matrix [R.sub.T] is necessary because the comparison of two transmission scenarios is only fair if the average path loss is equal. Without receive correlation, the trace of the correlation matrix can be written as

tr([R.sub.T]) = [[n.sub.T]. summation (i=1)] [(E[H[H.sup.H]]).sub.ii] = [[n.sub.T] summation. (i=1)]E[[|[h.sub.i|.sup.2]]. (4.6)

However, the RHS of (4.6) is the sum of the average path loss from the transmit antenna i = 1, ..., [n.sub.T]. In order to study purely the impact of correlation on the achievable capacity separately, the average path loss is kept fixed by applying the trace constraint on the correlation matrices [R.sup.1.sub.T] and [R.sup.2.sub.T].

We will say that a correlation matrix [R.sup.1.sub.T] is more correlated than [R.sup.2.sub.T] with descending ordered eigenvalues [[lambda].sup.T,1].sub.1] [greater than or equal to] [[lambda].sup.T,1].sub.2] [greater than or equal to] ... [greater than or equal to] [[lambda].sup.T,1.sub.[n.sub.T]] [greater than or equal to] 0 and [[lambda].sup.T,2].sub.2] [greater than or equal to] [[lambda].sup.T,2].sub.2] [greater than or equal to] ... [greater than or equal to] [[lambda].sup.T,2.sub.[n.sub.T]] [greater than or equal to] 0 if

[m.summation over (k=1)] [[lambda].sup.T,1.sub.k] [greater than or equal to] [m.summation over (k=1)] [[lambda].sup.T,2.sub.k] 1 [greater than or equal to] m [greater than or equal to] [n.sub.T] - 1. (4.7)

The measure of correlation is defined in a natural way: the larger the fist m eigenvalues of the correlation matrices are (with the trace constraint in (4.6)), the more correlated is the MIMO channel. As a result, the most uncorrelated MIMO channel has equal eigenvalues, whereas the most correlated MIMO channel has only one non-zero eigenvalue which is given by [[lambda].sub.1 = [n.sub.T].

The following definition provides again themeasure for comparison of two correlation matrices.

Definition 4.2 (Measure for spatial correlation). The transmit correlation matrix [R.sup.1.sub.T] is more correlated than [R.sup.2.sub.T] if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.8)

One says that the vector consisting of the ordered eigenvalues [[lambda].sup.T.sub.1] majorizes [[lambda].sup.T.sub.2], and this relationship can be written as [[lambda].sup.T.sub.1] [??][[lambda].sup.T.sub.2] like in Definition 2.1.

Remark 4.1. Note that our defnition of correlation in Definition 4.2 differs from the usual definition in statistics. In statistics a diagonal covariance matrix indicates that the random variables are uncorrelated. This is independent of the auto-covariances on the diagonal. In our definition, we say that the antennas are uncorrelated if in addition to statistical independence, the auto-covariances of all entries are equal. This difference to statistics occurs because the direction, i.e., the unitary matrices of the correlation have no impact on our measure of correlation.