3.5.1 Proofs of Lemma 3.2.1
Proof: First note that the zero-forcing constraint is satisfied byGopt:
GoptHF−B= (I+B)(HF)]HF−B=I.
Suppose there is anotherG0 satisfying the zero-forcing constraint with the given Fand B, i.e., G0HF=I+B. Define∆=Gopt−G0. Since bothGoptandG0satisfy the zero-forcing constraint, it follows that
∆G†opt = ∆HF(F†H†HF)−†(I+B)†
= (GoptHF−G0HF)(F†H†HF)−†(I+B)†
= 0.
Therefore
[G0G0†]kk = [(Gopt−∆)(Gopt−∆)†]kk
= [(GoptG†opt+∆∆†]kk
≥ [GoptG†opt]kk,
where we have used∆G†opt=0in these inequalities. Therefore we have smaller sub-channel noise variances if we replaceG0 withGopt, hence with given bit rate and probabilities of error, a lower
transmitted power can be achieved.
3.5.2 Proofs of Theorem 3.2.4
Proof: Part (a) is true because the problem (3.12) discussed in previous sections is a relaxed version of the current problem (3.32). We prove part (b) by constructing a system that achievesPminwhen (3.36) holds. If (3.36) holds then the majorization condition (2.3) can be satisfied by choosing[R]kk
to be positive square roots of
[R]2kk=
M ck2bk(QM
k=1σ2h,k)M1
c2b , fork= 1,2,· · · , M.
σh,k2 , fork=M + 1,· · ·K,
(3.55)
whereK is the rank ofH.Then by the existence of GTD, there exists aK×K upper triangular matrixR, such that the decomposition H = QRP† is true, where Q and Phave orthonormal columns. Now choose the transceiver matricesF,G,and Bas in (3.20), (3.21), and (3.23). Then Ptransis as in (3.25). Substituting from (3.55) we getPtrans=Pminindeed.
Chapter 4
Transceivers Designs for MIMO Frequency Selective Channels
In high rate digital communication systems, multiple-input-multiple-output (MIMO) frequency se- lective (FS) channels complicate the transceiver design process because of the inter-block-interference (IBI) effect. However, by applying the zero-padding precoding technique, we can eliminate the IBI and convert the FS channel into an equivalent MIMO block channel [12, 89]. With MIMO block channels, many researchers have developed transceiver designs to match the channel characteris- tics and to mitigate the noise interference [12, 73, 75, 89, 90]. One of the approaches is to focus on linear precoding and decision feedback equalization (DFE).
If the channel state information (CSI) is available both at the transmitter and the receiver sides, in terms of minimizing the average BER under the transmitted power constraint, the optimal sys- tem with linear precoding and zero-forcing DFE (ZF-DFE) [90], and the optimal system with linear precoding and minimum-mean-square-error DFE (MMSE-DFE) [37], can both be derived from the equivalent block channel matrix. For ZF-DFE, the optimal linear precoder matrix has orthonormal columns; for MMSE-DFE, the optimal linear precoder no longer has orthonormal columns, instead it has suitable power loading on the channel eigenmodes. However, precoding matrix with or- thonormal columns is usually desired for simplicity reasons [142, 56, 99, 141]. Under the unitary precoder constraints, the optimal systems1for both receiver types can be derived. Nevertheless, it is known that the derived optimal systems suffer from two drawbacks. First, they require a large number of bits from the receiver to encode the full precoding matrix and feed it back to the trans- mitter [90]. Second, the full precoding matrix multiplication is computationally complex. For the
1The optimal systems is referred to the optimal designs within the class of systems using unitary precoders and DFEs (ZF or MMSE). These systems will be called the optimal systems through out this chapter.
block channel derived from a zero-padded MIMO FS channel, these disadvantages become more apparent when the block size is large.
The block diagonal GMD (BD-GMD) is proposed in [47] to design memoryless transceivers for MIMO broadcast channels. In this chapter, we consider applying the BD-GMD technique to design thezero-paddedMIMO FS transceiver that solves the two mentioned drawbacks. Two novel systems (which we call ZP-BD-GMD systems) are proposed: the ZF-BD-GMD system, which uses block diagonal unitary precoder and ZF-DFE receiver, and the MMSE-BD-GMD system, which uses block diagonal unitary precoder and MMSE-DFE receiver. We will show the following properties of the proposed ZP-BD-GMD systems:
1. Because of the block diagonal structure of the precoder matrix, the proposed ZP-BD-GMD systems solve the two implementation drawbacks of the optimal systems. It is also shown that the receiver structures are simpler than those of the optimal systems. Therefore, the ZP-BD-GMD systems have a much smaller implementation cost than the optimal systems.
2. For finite block sizes, it can be seen that any block diagonal unitary precoder system solves the above drawbacks. In particular, ZP-BD-GMD systems are minimizers of the average BER within the family of systems that use block diagonal unitary precoder. That is, the ZP-BD- GMD systems have optimality for any block size. As block size gets larger and approaches infinity, the average BER of the ZP-BD-GMD systems also approaches that of the optimal uni- tary precoded systems. In other words, the ZP-BD-GMD systems are asymptotically optimal within the systems that use unitary precoder, as the bandwidth efficiency approaches unity.
3. In all four unitary precoded systems (ZF-Optimal, ZF-BD-GMD, MMSE-Optimal, and MMSE- BD-GMD), there is a tradeoff between the bandwidth efficiency and the average BER perfor- mance. This suggests that one has to carefully design the block length to maintain the target BER for both the ZP-BD-GMD systems and the optimal systems. In [72], a similar tradeoff in single-carrier zero-padded SISO FS channels with linear equalization was reported. Thus, this aspect of our work can be seen as an extension of [72].
4. In the case of the SISO channel, ZP-BD-GMD systems have the same performance as the lazy precoder systems, i.e., systems with identity precoding matrix. Therefore, in SISO channels the lazy precoder systems inherit the benefits of the ZP-BD-GMD systems, making the lazy precoder transceivers asymptotically optimal in the class of systems with unitary precoder
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.