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Description
This chapter has introduced the design of nonlinear DF MIMO transceivers with full CSI based on majorization theory and the generalized triangular decomposition (GTD) algorithm. Two different problem formulations have been considered: one based on a global performance measure subject to the overall input power constraint and another one based on minimizing the input power subject to individual QoS constraints.
The optimal solution is characterized as follows. The optimum DFE is always the MMSE-DFE, which can be easily computed via the QR decomposition of the augmented matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = QR. In a remarkably similar vein to the linear transceiver designs, the precoder P for the nonlinear DF transceiver design also has the form P = [V.sub.H] diag ([square root of (p)])[[OMEGA].sup.[dagger]], where [V.sub.H] contains the right singular vectors of the channel matrix H (whose role is to diagonalize the channel), p denotes a power allocation over the eigen-subchannels, and [OMEGA] is a unitary matrix that can be obtained using the GTD algorithm.
For the design based on some global performance measure [f.sub.0], all the cost functions for which the composite function [f.sub.0] [??] exp is Schurconvex lead to the same solution, i.e., the so-called UCD. The precoder of the UCD scheme applies the waterfilling power allocation p to maximize the channel mutual information and the rotation [OMEGA] such that the DFE yields substreams with identical MSE performance. On the other hand, if [f.sub.0] [??] exp is Schur-concave, then the optimum precoder P = [V.sub.H] diag ([square root of (p)]) and the nonlinear DF transceiver design degenerates to linear diagonal transmission.
For the design with individual QoS constraints, we simplify the transceiver optimization problem to a power allocation problem for which we have a simple multi-level waterfilling algorithm. We have shown through several numerical examples that the DF based nonlinear designs have remarkable performance gain over the linear designs.
Interestingly, the nonlinear MIMO transceiver designs can be implemented in two forms by exploiting the duality between the DFE and DPC as well as the uplink-downlink duality. Finally, the problem of designing CDMA sequences has been shown to be a special case of the MIMO transceiver design problem.
4.A Appendix: Mutual Information and Wiener Filter
Consider Gaussian input x ~ N(0, I). For the MIMO channel given in (4.1), the instantaneous mutual information between the input and output of the channel is [147]
I(x;y) = log |I + HP[P.sup.[dagger]][H.sup.[dagger]]| bps/Hz. (4.101)
There is an interesting link between the mutual information and the MSE matrix when using the Wiener filter (or, equivalently, the linear MMSE receiver). As shown in (3.18), the Wiener filter yields an estimate of x with MSE matrix (see (3.21))
E = [(I + [P.sup.[dagger]][H.sup.[dagger]]HP).sup.-1]. (4.102)
It follows from (4.101) and the equality |I + AB| = |I + BA| that
I(x;y) = -log |E|. (4.103)
With the Wiener filter, the original MIMO channel is converted into L scalar subchannels
[x.sub.i] = [[??].sub.i] + [e.sub.i], 1 [less than or equal to] i [less than or equal to] L, (4.104)
where [x.sub.i] ([[??].sub.i]) is the ith entry of x ([??]) and [e.sub.i] is the estimation error with zero mean and variance [E]ii, the ith diagonal element of E. By the orthogonality principle of the MMSE estimation theory [157], Gaussian random variables [[??].sub.i] and [e.sub.i] are statistically independent. Hence, E[[|[[??].sub.i]|.sup.2]] + E[[|[e.sub.i]|.sup.2]] = E[[|[x.sub.i]|.sup.2]] = 1. Therefore, the... |
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