The potential gain from transmitter channel knowledge is quite significant in spatial-multiplex- ing systems. First, we consider a simple theoretical example using singular-value decomposition that shows the potential gain of closed-loop spatial-multiplexing methods. Then we turn our attention to more practical linear-precoding techniques that could be considered in the near to Figure 5.13 (a) D-BLAST detection of the layer 2 of four. (b) V-BLAST encoding. Detection is done dynamically; the layer (symbol stream) with the highest SNR is detected first and then canceled.
2
Antenna Index
Time
1 2
1 3
2 1
4 3 2
4
3 4
Detection Order Interference
Cancelled
3 4
1
Wasted
Wasted Nulled
Antenna Index
Time
2 3 4 1
2 3 4 1
2 3 4 1
2 3 4 1
(a) (b)
Nt
Nt −2 Nt −i
medium term for multiantenna WiMAX systems as a means of raising the data rate relative to the diversity-based methods of Section 5.3.
5.5.3.1 SVD Precoding and Postcoding
A relatively straightforward way to see the gain of transmitter channel knowledge is by consid- ering the singular-value decomposition (SVD, or generalized eigenvalue decomposition) of the channel matrix H, which as noted previously can be written as
(5.62) where U and V are unitary and is a diagonal matrix of singular values. As shown in Figure 5.14, with linear operations at the transmitter and the receiver, that is, multiplying by V andU*, respectively, the channel can be diagonalized. Mathematically, this can be confirmed by considering a decision vector d that should be close to the input symbol vector b. The decision vector can be written systematically as
(5.63)
which has diagonalized the channel and removed all the spatial interference without any matrix inversions or nonlinear processing. Because U is unitary, U*n still has the same variance as n.
Thus, the singular-value approach does not result in noise enhancement, as did the open-loop linear techniques. SVD-MIMO is not particularly practical, since the complexity of finding the SVD of an matrix is on the order of if and requires a substantial amount of feedback. Nevertheless, it shows the promise of closed-loop MIMO as far as high per- formance at much lower complexity than the ML detector in open-loop MIMO.
Figure 5.14 A MIMO system that has been diagonalized through SVD precoding.
H=U VΣ *, Σ
d U y U Hx n U U V Vb n U U V Vb U n
b U n
= ,
= ( ),
= ( ),
= ,
= ,
*
*
* *
* * *
*
+ + + +
Σ Σ Σ
Nt ×Nr O N N( r t2) Nr ≥Nt
V U*
Serial to Parallel
b x = Vb y = Hx+z U*y b^
H=U V*
5.5.3.2 Linear Precoding and Postcoding
The SVD illustratived how linear precoding and postcoding can diagonalize the MIMO channel matrix to provide up to dimensions to communicate data symbols through. More generally, the precoder and the postcoder can be jointly designed based on such criteria as the information capacity [50], the error probability [21], the detection MSE [52], or the received SNR [51]. From Section 5.3.3, recall that the general precoding formulation is
y = G(HFx + n), (5.64) wherex and y are M 1, the postcoder matrix G is M Nr, the channel matrix H is , the precoder matrix F is , and n is . For the SVD example, , G= U*, and F = V.
Regardless of the specific design criteria, the linear precoder and postcoder decompose the MIMO channel into a set of parallel subchannels as illustrated in Figure 5.15. Therefore, the received symbol for the ith subchannel can be expressed as
(5.65) where and are the transmitted and received symbols, respectively, with as usual, are the singular values of H, and αi and βi are the precoder and the postcoder weights, respectively. Through the precoder weights, the precoder can maximize the total capacity by dis- tributing more transmission power to subchannels with larger gains and less to the others—
referred to as waterfilling. The unequal power distribution based on the channel conditions is a principal reason for the capacity gain of linear precoding over the open-loop methods, such as BLAST. As in eigenbeamforming, the number of subchannels is bounded by
(5.66) where corresponds to the maximum diversity order, called diversity precoding in Section 5.3.3) and achieves the maximum number of parallel spatial streams.
Intermediate values of can be chosen to provide an attractive trade-off between raw throughput and link reliability or to suppress interfering signals, as shown in the eigenbeamforming discussion.
5.6 Shortcomings of Classical MIMO Theory
In order to realistically consider the gains that might be achieved by MIMO in a WiMAX sys- tems, we emphasize that most of the well-known results for spatial multiplexing are based on the model in Equation (5.54) of the previous section, which makes the following critical assumptions.
•Because the entries of H are scalar random values, the multipath is assumed negligible, that is, the fading is frequency flat.
•Because the entries are i.i.d., the antennas are all uncorrelated.
•Usually, interference is ignored, and the background noise is assumed to be small.
min(N Nr, t)
× × Nr×Nt
Nt×M Nr ×1 M=min(N Nr, t)
yi =α σ βi i ixi+βini, i= 1,,M,
xi yi E x| i | =2 εx
σi
1≤M ≤min(N Nt, r), M= 1
M=min(N Nt, r) M
Clearly, all these assumptions will be at least somewhat compromised in a cellular MIMO deployment. In many cases, they will be completely wrong. We now discuss how to address these important issues in a real system, such as WiMAX.