5 Application of matrix-monotone functions in wireless communications.

5.1 Generalized Multiple Antenna Performance Measures

Multiple-antennas can improve the spectral efficiency and reliability in wireless communications systems. In recent years, it was discovered that MIMO systems have the ability to reach higher transmission rates than one-sided array links [137, 158]. First, we review recent results for MISO systems since this case has recently gained much attention. In [157], the potential of multiple antenna systems was pointed out. The capacity of a MISO system with imperfect feedback was fist analyzed in [149] and [99, 100]. In [55, 63], the optimum transmission strategy with covariance knowledge at the transmit array with respect to the ergodic capacity was analyzed. In [21, 111, 121], the problem of downlink beamforming problem in MISO systems was solved. In [54], the ergodic capacity in the non-coherent transmission scenario with only covariance knowledge at the transmitter and the receiver, is studied. Many results regarding the capacity of MISO and MIMO systems under different levels of CSI and the corresponding transmission strategies are recently published [40].

It has been shown that even partial CSI at the transmitter can increase the capacity of a MISO system. Recently, transmission schemes for optimizing capacity in MISO mean-feedback and covariance-feedback systems were derived in [99, 149]. The capacity can be achieved by Gaussian distributed transmit signals with a particular covariance matrix. In a block-fading model, the general signal processing structure which achieves capacity independent of the type of CSI consists of a Gaussian codebook, a number of beamformers and a power allocation entity [9, 149]. Additionally, it was proved that the optimal transmit covariance matrix in the covariance feedback case has the same eigenvectors as the known channel covariance matrix. The complete characterization of the impact of correlation on the ergodic capacity in MISO systems can be found in [68]. In addition to the capacity other performance metrics like the MMSE were analyzed in the literature, e.g., [117, 120]. The multiuser MIMO system optimization is performed with respect to sum MSE in [129], with per-user MMSE requirements in [128].

The analysis and design methodology of single-antenna and beamforming systems was extended and generalized to multiantenna systems. Many novel approaches and techniques were developed and a unmanageable bulk of papers, reports, and books were produced. However, some ideas occurred inherently as persistent concepts in many works. The main goal of this section is to detect these main concepts and express them on a meta-level by constructing a unified framework.

In order not to have different statements for different performance metrics, we present here a unifying framework in which a class of functions serves as the performance metric. The underlying mathematical structure is described by the representation of Lowner from 'Matrix-Monotone Function." Let us start with some motivating and illustrative examples.

5.1.1 Examples in Single-User MIMO Systems

Consider the quasi-static block-fbt fading MIMO system in Figure 5.1.

The transmit signals are complex Gaussian distributed random vectors with zero mean and transmit covariance matrix Q. The transmitter structure that corresponds to this type of signaling is described as follows: The transmit covariance matrix is given by Q = E ([xx.sup.H]).

[FIGURE 5.1 OMITTED]

Using the eigenvalue decomposition of Q = [U.sub.Q][[LAMBDA].sub.Q][U.sup.H.sub.Q], it becomes obvious how one can construct a particular transmit covariance matrix. The input data stream d(k) is split into m parallel data streams [d.sub.l](k), ..., [d.sub.m](k). Each parallel data stream is multiplied by a factor [square root of ([p.sub.1])], ..., [square root of ([p.sub.m])] and then weighted by a beamforming vector [u.sub.l], ..., [u.sub.m], respectively. The number of parallel data streams is less or equal to the number of transmit antennas (m [less than or equal to] [n.sub.T]). The beamforming vectors have size 1 x [n.sub.T] with [n.sub.T] as the number of transmit antennas. The [n.sub.T] signals of each weighted data stream [x.sup.i](k) = [d.sub.i](k) x [square root of ([p.sub.i])] x [u.sub.i] are added up x(k) = [[summation].sup.m.sub.i=1] [x.sup.i](k) and sent. By omitting the time index k for convenience we obtain in front of the transmit antennas

x = [m.summation over (t=1)][d.sub.l] x [square root of ([p.sub.l])] x [u.sub.l]. (5.1)

The transmit signal in x has a covariance matrix Q with eigenvalues [p.sub.1], ..., [p.sub.m],0, ...,0 and eigenvectors [u.sub.1], ..., [u.sub.m]. In order to construct a transmit signal with a given covariance matrix, two signal processing steps are necessary: the power control [p.sub.1], ..., [p.sub.m] and the beamformers [u.sub.1], ..., [u.sub.m]. The sum transmit power [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is constrained, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The first performance measure is the ergodic capacity [10], i.e., the rate that can be transmitted reliable over ergodic (infinite) many channel realizations by codes with very long (infinite) block length. (1) For the system in Figure 5.1 with [n.sub.T] transmit and [n.sub.R] receive antennas and n = min([n.sub.T], [n.sub.R]), m = max([n.sub.T], [n.sub.R]) it is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.2)

with SNR [rho] = 1/[[sigma].sup.2.sub.n], channel matrix H, and expectation with respect to H. The channel matrix is zero-mean iid Rayleigh distributed. A very important property that has been used many times is the invariance property of the channel statistics, i.e., left and right multiplication of H with an unitary matrix U [94]. Furthermore, the function in (5.2) is obviously monotone increasing in [rho], in tr Q for fixed Q/tr Q and also in Q, i.e., the function inside the trace is matrix-monotone with respect to Q. Further on, the function is concave with respect to Q.

Next, consider the slightly modified system in Figure 5.2. In addition to the additive white Gaussian noise, another additive noise with colored covariance matrix is added. That can correspond to either intra- or inter-cell interference or to any other type of jamming signal.

[FIGURE 5.2 OMITTED]

The ergodic capacity for the system in Figure 5.2 is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.3)

In (5.3), [??] is the positive definite noise plus interference matrix. Note that the function in (5.3) is concave in Q and convex in Z.

Next, consider a completely different performance measure, the normalized minimum mean-square error (MMSE). The linear MMSE receiver reduces the computational complexity at the receiver side. The MSE can be evaluated for each fading state and each symbol. The average MSE described the quality of data transmission. If we apply the linear MMSE receiver, the performance metric changes from the average mutual information to the normalized MSE [150]. In general, the Wiener filter for a linear system y = ax + n can be described as w = [R.sub.xy][R.sup.-1.sub.yy]. The reason, why we speak about the normalized MMSE is that there are two cases for deriving the MMSE. In the first case, the actual transmit signal x is considered. In the second case, the source signal before linear precoding is considered. In the first case, the resulting weight w or the resulting MSE expression must be normalized with the transmit covariance matrix. However, both approaches lead to the same result. We consider the first approach: The linear MMSE receiver weights the received signal vector y by the Wiener filter

[??] = [rho]Q[H.sup.H][[[??]+[rho]HQ[H.sup.H]].sup.-1] y. (5.4)

The covariance matrix of the estimation error [R.sub.epsilon] is given by

[R.sub.[epsilon]] = [E.sub.H][([??] - x)[([??] - x).sup.H]] = Q - Q[H.sup.H][[[??] + [rho]HQ[H.sup.H]].sup.-1]HQ. (5.5)

The average normalized sum MSE is defined as the trace error covariance matrix of the estimation error in (5.5) [49, 65]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.6)

and its average over channel realizations is called average sum MSE. Note, that the MSE is convex in Q and concave in Z.

In SISO systems, the relationship between the rate, the SNR, and the MSE is quite simple, i.e., C = log(1 + SNR) = log(1/MSE). In MIMO systems, the connection is more complicated due to the spatial dimension, e.g., the pairwise error probability (PEP) between X and [??] can be upper bounded by P(X [right arrow] [??]) [less than or equal to] exp(-[rho][parallel]H(X - [??])[[parallel].sup.2]). The connection between the performance measure mutual information and MSE is highlighted in the next subsection.

5.1.2 Relationship between MSE and Mutual Information

There are at least three connections between the MSE and the capacity that provide intuitive insights into the meaning of these performance metrics. The first is a function theoretic relationship, the second is a direct relationship by linear algebra, and the third is an analytical relationship following from the second one.

5.1.2.1 Function Theoretic Relationship

Let us compare the capacity and the normalized sum MSE. First, we rewrite the capacity as

C = tr log(I + [rho][Z.sup.-1/2]HQ[H.sup.H][Z.sup.-1/2]).

The capacity expression is the trace of a matrix valued function. Let us denote the matrix valued function as [[PHI].sub.1](X) = log (I + X). Then the capacity can be written as

C = tr[[PHI].sub.1]([rho][Z.sup.-1/2]HQ[H.sup.H][Z.sup.-1/2]). (5.7)