and DFE. While having the same performance, lazy precoders are more desirable than the ZP- BD-GMD systems in terms of implementation cost. This is due to the fact that lazy precoders can transmit data without CSI at the transmitter and precoding matrix multiplication.
These properties make the proposed ZP-BD-GMD systems more favorable designs in practical implementation than the optimal systems.
The content of this chapter is mainly drawn from [130], and portions of it have been presented in [127].
4.1 Outline
This chapter is structured as follows: In Sec. 4.2, we will introduce the communication model and some preliminaries. Sec. 4.3 describes the proposed ZF-BD-GMD transceiver structure, which uses a block diagonal unitary precoder and a ZF-DFE design based on BD-GMD of the effective channel matrix. Several properties of the ZF-BD-GMD transceiver are discussed. The implementation cost is also analyzed. Sec. 4.4 extends the idea to the MMSE-DFE case. The proposed MMSE-BD- GMD system is discussed. Most of the results will be similar to the ZF case, so this section will be brief. Sec. 4.5 explains that for the two ZF transceivers (ZF-Optimal and ZF-BD-GMD) and the two MMSE transceivers (MMSE-Optimal and MMSE-BD-GMD), there exists a tradeoff between the bandwidth efficiency and the BER performance. Sec. 4.6 discusses the SISO channel case, in which the lazy precoder system is discussed. Sec. 4.7 presents the numerical simulation results related to the topics in the chapter.
received vector. The noise covariance matrix is assumed to beRn=σ2nI.The zero-padded system transmitsNP zero vectors after everyK symbol vectors. That is, inK+NP symbol durations, the following is transmitted: {x1,x2,· · · ,xK,0,· · ·,0}. In order to prevent contamination from previous blocks, one must chooseNP ≥L. The bandwidth efficiency is defined as
= K
K+NP
. (4.2)
Therefore it is desirable to chooseNP = L, so that the BW efficiency is maximized, and equal to K/(K+L). Throughout the chapter, we assumeNP = L. The I/O relation of the zero-padded MIMO frequency selective system can be expressed as an equivalent block channel:
y1
y2 ... yK+L
| {z }
yZP,K
=HZP,K
x1
x2 ... xK
| {z }
xZP,K
+
n1
n2 ... nK+L
| {z }
nZP,K
, (4.3)
where
HZP,K =
H0 0 · · · 0 H1 H0 . .. ...
... ... . .. 0 HL ... . .. H0
0 HL . .. ... ... . .. ... ... 0 · · · 0 HL
, (4.4)
andKin the subscript denotes thatHZP,K hasKNT columns. Note that Eq. (4.3) holds for any NP ≥L. We assume(K+L)NR≥KNT, so that the zero-forcing condition can be satisfied.
We consider the system where the transmitted vector is linear precoded by aNTK×NTKmatrix P:
xZP,K =Ps,
wheres = {sT1,sT2,· · · ,sTK}T, andsiis the NT ×1 transmitted symbol vector. We use the usual
assumption that the transmitted signal is zero-mean, white, and uncorrelated with the noise, i.e., E[sisHj ] =δ(i−j)σ2sIandE[sinj] =0. Here we define a constantζ, which stands for the noise to symbol power ratio:
ζ .
=σn2/σs2. (4.5)
The average power of the transmitted vectorxZP,K is restricted to be less thanKNTσ2s. Note that the power constraint is proportional toKbecauseKis the number of symbol vectors transmitted in one block. SinceE[ssH] =σ2sI, the power constraint can be written as
1
σs2Tr(E[PssHPH]) =Tr(PPH)≤KNT, (4.6) which is a constraint expressed solely in terms of the precoder matrix. We assume each symbol is selected from the same QAM constellation, i.e., no bit allocation is applied. In this case, the BER will be the function of SINR of the input to the decision device (see Eq. (12) in [73]), i.e.,
BER(SINRk) =αQ(βp
SINRk), (4.7)
whereαandβare constants which depend on the constellation, andQ(·)is theQ-function defined asQ(x) = (1/√
2π)R∞
x e−λ2/2dλ. We are interested in the high SNR regime so that the BER function is a convex function of the logarithm of the SINR [90].2
The optimization problem we are interested in is to minimize the average BER by designing a linear precoder and a zero-forcing or MMSE decision feedback equalizer jointly under the power constraint (4.6). We can treat the I/O relation (4.3) as an effective block channel communication sys- tem. For the ZF-DFE case, the optimal solution is suggested by the Theorem 1 in [91]. The optimal precoder is with no loss of generality a unitary matrix. The optimal receiver is the corresponding ZF-DFE solution suggested in Sec. III of [91].
If the receiver is MMSE-DFE instead of ZF-DFE, the optimal precoder will no longer be unitary [90, 37]. Instead, a suitable water-filling power loading to the channel eigenmodes is needed to achieve the optimal performance [37]. However, unitary precoding is usually desired for simplicity
2The property we need for the discussion in this chapter is that the average BER is a Schur-convex function of the logarithm of the effective subchannel gains (see Appendix A.(f) in [90] for details). Therefore, it should be noted that the theorems developed in this chapter (Thm 4.3.4, Thm 4.3.5, Thm 4.5.1, and the corresponding properties of the MMSE-BD- GMD systems) are not restricted to the average BER, but are also true for a broader class of metric.
reasons [142, 56, 99, 141]. Therefore in this chapter, we restrict our interest only to the unitary precoder case. In this case, the optimal system for MMSE-DFE can also be obtained.
Generally, the optimal precoders for both optimal ZF and MMSE systems are full matrices.
Therefore, the implementation of the optimal systems suffers from two disadvantages:
1. In the limited feedback scheme [55, 91], the channel state information is estimated by the receiver, and the optimal precoder information is quantized (or quantized to some prede- termined codebook) and fed back to the transmitter. Since the optimal precoder P is an NTK×NTKunitary matrix, it requires a large codebook for quantizing it to cover the whole space of theNTK×NTKunitary matrices.
2. Computation of the transmitted signalxZP,K =Psis expensive for the full matrix multipli- cation, and takesO(NT2K2)operations.
These two disadvantages are more severe whenKis large, i.e., when the bandwidth efficiency (4.2) approaches unity. To overcome these, we propose using the BD-GMD technique, which was introduced in [47], to design the transceiver. We restrict the precoder to be block-diagonal, i.e., P=diag(P1,P2,· · · ,PK), wherePiis anNT×NT unitary matrix. The block diagonal constraint clearly simplifies the implementation:
1. Since there are onlyNT2K out ofNT2K2 elements that are nonzero, the required size of the codebook is much smaller for covering the precoder matrix space.
2. To form the transmitted vectorxZP,K =Ps, we only need to formxi =Pisifori= 1,· · · , K and concatenate them. The complexity is onlyO(NT2K)instead ofO(NT2K2)as in the optimal precoder case without block diagonal constraint.
Although the block diagonal precoder gives these benefits, it is natural to ask how much the performance degrades due to this constraint. In the following sections, we will discuss several important properties of the proposed ZP-BD-GMD transceivers. It will be shown that the ZP- BD-GMD system has similar performance as the optimal systems when the bandwidth efficiency approaches unity. In addition, the receiver structure of the ZP-BD-GMD systems is computation- ally simple. Also, we will prove that the ZP-BD-GMD systems are optimal within the family of transceivers with DFEs and block diagonal unitary precoders.