As with multiantenna diversity techniques, spatial multiplexing can be performed with or with- out channel knowledge at the transmitter. We first consider the principal open-loop techniques;
we always assume that the channel is known at the receiver, ostensibly through pilot symbols or other channel-estimation techniques. The open-loop techniques for spatial multiplexing attempt
Nr×1 Nr×Nt
Nt ×1 Nr ×1 Nt
εx/Nt
H=
1
11 12 1
21 22 2
h h h
h h h
h h h
N Nt
Nt
Nr N
r N
…
…
…
⎡
⎣
⎢⎢
⎢⎢
⎢⎢
⎤
⎦
⎥⎥
⎥⎥
⎥⎥ ,,
2 r t
σh 2I σz
2I
Nt Nt
min(N Nt, r)log(1+SNR)
min(N Nt, r)
Nr Nt
Nr ≥Nt Nt
to suppress the interference that results from all streams being received by each of the antennas. The techniques discussed in this section are largely analogous to the interference- suppression techniques developed for equalization [48] and multiuser detection [64], as seen in Table 5.1.
5.5.2.1 Optimum Decoding: Maximum-Likelihood Detection
If the channel is unknown at the transmitter, the optimum decoder is the maximum-likelihood decoder, which finds the most likely input vector via a minimum-distance criterion
(5.56) Unfortunately, there is no simple way to compute this, and an exhaustive search must be done over all possible input vectors, where is the order of the modulation (e.g., for QPSK). The computational complexity is prohibitive for even a small number of antennas. Lower-complexity approximations of the ML detector, notably the sphere decoder, can be used to nearly achieve the performance of the ML detector in many cases [34], and these have some potential for high-performance open-loop MIMO systems. When optimum or near- optimum detection is achievable, the gain from transmitter channel knowledge is fairly small and is limited mainly to waterfilling over the channel eigenmodes, which provides significant gain only at low SNR.
5.5.2.2 Linear Detectors
As in other situations in which the optimum decoder is an intolerably complex maximum- likelihood detector, a sensible next step is to consider linear detectors that are capable of recov- ering the transmitted vector x, as shown in Figure 5.12. The most obvious such detector is the zero-forcing detector, which sets the receiver equal to the inverse of the channel Gzf = H–1 when
, or more generally to the pseudoinverse
(5.57) Figure 5.11 A spatial multiplexing MIMO system transmits multiple substreams to increase the data rate.
S/P and Tx
Rx and P/S
x H y
Bits In Bits Out
NtAntennas Nrantennas
Rate = R min(Nt,Nr)
Rate = R min(Nt,Nr)
Rate per stream = R
Nt Nr
xˆ
ˆ arg min ˆ
x= ||y−Hx|| .2
MNt M
M= 4
Nt =Nr
Gzf = (H H* )−1H*.
As the name implies, the zero-forcing detector completely removes the spatial interference from the transmitted signal, giving an estimated received vector
(5.58) BecauseGfz inverts the eigenvalues of H, the bad spatial subchannels can severely amplify the noise in n. This is particularly problematic in interference-limited MIMO systems and results in extremely poor performance. The zero-forcing detector is therefore not practical for WiMAX.
A logical alternative to the zero-forcing receiver is the MMSE receiver, which attempts to strike a balance between spatial-interference suppression and noise enhancement by simply min- imizing the distortion. Therefore,
(5.59) which can be derived using the well-known orthogonality principle as
(5.60) Table 5.1 Similarity of Interference-Suppression Techniques for Various Applications, with Complexity Decreasing from Left to Right
Optimum Interference
Cancellation Linear Equalization (ISI)
Maximum likelihood sequence detection
(MLSD)
Decision feedback equalization (DFE)
Zero forcing minimum mean square error
(MMSE) Multiuser Optimum multiuser
detection (MUD)
Successive/parallel inter- ference cancellation,
iterative MUD
Decorrelating, MMSE Spatial-multiplexing
Receivers
ML detector sphere decoder (near optimum)
Bell Labs Layered
Spaced Time (BLAST) Zero forcing, MMSE
Figure 5.12 Spatial multiplexing with a linear receiver S/P
Linear Receiver
Gzf
or
Gmmse
x H y
Input
Symbols P/S EstimatedSymbols
xˆ=G yzf =G Hxzf +G nzf =x+(H H* )−1H n* .
G Gy x
mmse=arg min EG || − || ,2
Gmmse H H z I H Pt
= ( * ) ,
2
1 *
+σ −
where is the transmitted power. In other words, as the SNR grows large, the MMSE detector converges to the ZF detector, but at low SNR, it prevents the worst eigenvalues from being inverted.
5.5.2.3 Interference cancellation: BLAST
The earliest known spatial-multiplexing receiver was invented and prototyped in Bell Labs and is called Bell Labs layered space/time (BLAST) [24]. Like other spatial-multiplexing MIMO systems, BLAST consists of parallel “layers” supporting multiple simultaneous data streams.
The layers (substreams) in BLAST are separated by interference-cancellation techniques that decouple the overlapping data streams. The two most important techniques are the original diag- onal BLAST (D-BLAST) [24] and its subsequent version, vertical BLAST (V-BLAST) [28].
D-BLAST groups the transmitted symbols into “layers” that are then coded in time inde- pendently of the other layers. These layers are then cycled to the various transmit antennas in a cyclical manner, resulting in each layer’s being transmitted in a diagonal of space and time. In this way, each symbol stream achieves diversity in time via coding and in space by it rotating among all the antennas. Therefore, the transmitted streams will equally share the good and bad spatial channels, as well as their priority in the decoding process now described.
The key to the BLAST techniques lies in the detection of the overlapping and mutually interfering spatial streams. The diagonal layered structure of D-BLAST can be detected by decoding one layer at a time. The decoding process for the second of four layers is shown in Figure 5.13a. Each layer is detected by nulling the layers that have not yet been detected and canceling the layers that have already been detected. In Figure 5.13, the layer to the left of the layer 2 block has already been detected and hence subtracted (canceled) from the received sig- nal; those to the right remain as interference but can be nulled using knowledge of the channel.
The time-domain coding helps compensate for errors or imperfections in the cancellation and nulling process. Two drawbacks of D-BLAST are that the decoding process is iterative and somewhat complex and that the diagonal-layering structure wastes space/time slots at the begin- ning and end of a D-BLAST block.
V-BLAST was subsequently addressed in order to reduce the inefficiency and complexity of D-BLAST. V-BLAST is conceptually somewhat simpler than D-BLAST. In V-BLAST, each antenna simply transmits an independent symbol stream—for example, QAM symbols. A vari- ety of techniques can be used at the receiver to separate the various symbol stream from one another, including several of the techniques discussed elsewhere in this chapter. These tech- niques include linear receivers, such as the ZF and MMSE, which take the form at each receive antenna of a length vector that can be used to null out the contributions from the interfering data streams. In this case, the postdetection SNR for the ith stream is
(5.61) Pt
Nt
Nr Nt −1
γ σ
ε
i
x r i
i Nt
= 2|| || = 1, ,
,
w 2
where wr,i is the ith row of the zero-forcing or MMSE receiver G of Equation (5.57) and Equation (5.60), respectively.
Since this SNR is held hostage by the lower channel eigenvalues, the essence of V-BLAST is to combine a linear receiver with ordered successive interference cancellation. Instead of detecting all streams in parallel, they are detected iteratively. First, the strongest symbol stream is detected, using a ZF or MMSE receiver, as before. After these symbols are detected, they can be subtracted out from the composite received signal. Then, the second-strongest signal is detected, which now sees effectively interfering streams. In general, the ith detected stream experiences interference from only of the transmit antennas, so by the time the weakest symbol stream is detected, the vast majority of spatial interference has been removed.
Using the ordered successive interference cancellation lowers the block error rate by about a fac- tor of ten relative to a purely linear receiver, or equivalently, decreases the required SNR by about 4 dB [28]. Despite its apparent simplicity, V-BLAST prototypes have shown spectral effi- ciencies above 20 bps/Hz.
Despite demonstrating satisfactory performance in controlled laboratory environments, the BLAST techniques have not proved useful in cellular systems. One challenge is their depen- dence on high SNR for the joint decoding of the various streams, which is difficult to achieve in a multicell environment. In both BLAST schemes, these imperfections can quickly lead to cata- strophic error propagation when the layers are detected incorrectly.