The basic problem of MIMO detection is visualized in Figure 3.2 by a simplistic two- dimensional real-valued signal space. The orthogonal signal space of the constellation OMT in Figure 3.2a is transformed by the channel matrix H into the signal space visualized in Figure 3.2b. A noise vector n is drawn from the circular white Gaussian noise indicated by the gray circles in Figure 3.2b. For a transmit symbol s this results in the exemplarily visualized received symbol vector y.
3.2. The MIMO Demapping Problem 47 0 1 ! Tx an te n n a 2 1 0 ! Tx antenna 1 s (a)transmitted s h1,2 h2,2 ! h1,1 h2,1 ! s ˆ s y (b)received y 0 1 ! Tx an te n n a 2 1 0 ! Tx antenna 1 s H+y ˆ s (c)erroneous ZF decision
Figure 3.2: Visualization of the hard decision MIMO demapping problem reduced
to a two-dimensional real-valued signal space. transmitted symbol vector s
received symbol vector y= Hs+n
erroneously estimated transmit symbol vector ˆs noise visualization
optimum hard decision boundaries
zero forcing (ZF) hard decision boundaries optimum hard decision boundary for s ZF decision boundary for ˆs
A straightforward MIMO demapping approach called zero-forcing (ZF) is the re- versal of the channel influence and thus the multiplication of the received symbol vector y with the pseudo-inverse channel matrix H+ given by (3.6). The result of this operation is simply quantized to the nearest constellation vector ˆs as given by (3.7). Since this “search” for the nearest constellation vector can be performed in the con- stellation vector signal space simply by a truncation1 of the least significant bits its complexity is reasonably low.
H+ = HHH−1HH (3.6) ˆ s = arg min s∈OMT n ks−H+yk2o. (3.7) The (hard decision) MIMO detection problem becomes visible when comparing the received signal space in Figure 3.2b and the result of the multiplication with H+ as visualized in Figure 3.2c. In Figure 3.2b the correct hard decision can still be made since the Euclidean distance between s and y is shorter than the one between ˆs and
y. Thus, the received symbol y is still located within the optimum decision bound- aries of s derived from the nearest neighbor criterion (light gray Voronoï diagram in
1Rounding a value x∈Rto nearest integers 2k+1, k∈Zonly requires the operation⌊x⌋ ∨1. There-
fore, QAM constellation grids with Re{x}, Im{x} ∈ {2k+1|k∈Z, 2Q/2−1≤k<2Q/2−1}are partic- ularly hardware friendly.
48 Chapter 3. Demapping Algorithms for Iterative MIMO Reception
Figure 3.2b [191]). However, the ZF decision boundaries indicated by the orthogonal dashed lines in Figure 3.2c do not match the optimum hard-decision boundaries. The regions confined by ZF overlap significantly with the neighbor regions of the opti- mum hard decision. Therefore, the received symbol vector y is located within the ZF boundaries of ˆs instead of s as shown in the example in Figure 3.2c. Thus, erro- neous decisions are likely to happen with the zero-forcing approach since it suffers significantly from the transformation of a spatially white noise distribution (allowing Euclidean distance comparisons) into a spatially correlated distribution.
In order to improve the MIMO detection, various algorithm classes and variants have been developed which cover a wide trade-off between computational effort and algorithmic performance e.g. in terms of error rates. Closed-loop approaches trade- off the detection effort at the receiver side against bandwidth and power on the link back to the transmitter or against computational complexity on the transmitter side. Such approaches like eigenmode signaling [32, 144] and precoding [195] require in- stantaneous channel knowledge at the transmitter side obtained from information fed back from the receiver to the transmitter and sophisticated prediction algorithms at the transmitter side.
This work focuses on the transmission/reception model with an open-loop sce- nario without the need to feed back information to the transmitter. Prominent ap- proaches for the open-loop scenario will be briefly introduced in the following sec- tions. A particular focus is put on sphere-decoding (SD) algorithms in Section 3.5 since this class of algorithms offers a superior algorithmic performance. Among the large set of sphere-decoding algorithms, this work mainly focuses on single tree- search (STS) soft-input soft-output sphere-decoding algorithms published in [172]. Although the worst-case computational effort for these algorithms tends to be high, the average-case and best-case complexity is reasonably low. Therefore, sphere decod- ing offers interesting trade-offs between algorithmic performance and architectural efficiency.
As a primary reference, minimum mean square error (MMSE) detectors are dis- cussed in this work since they are of high importance for recent VLSI implementa- tions. Particularly, a soft-input soft-output MMSE demapper with parallel interference cancellation (MMSE-PIC) has been published recently in [173]. This architecture is, at the time writing this work, the only other known VLSI implementation of a soft-in- put soft-output MIMO demapper and thus the reference of choice for the soft-input soft-output architecture introduced in Chapter 5.
Algorithmic performances in terms of frame error rate (FER) are shown in Fig- ure 3.3 for a 16-QAM scenario and a selected set of MIMO demapping algorithms. It is clearly visible that depending on the degree of sophistication, a wide SNR range can be covered. Though Figure 3.3 might lead to a search for the most algorithmically ef- fective algorithm (lowest FER at lowest SNR), this figure and the following discussion are intended only as an introduction of MIMO demapping basics. The focus of later architecture-related chapters will be put on an approach for reasonable multi-dimen- sional and multi-constraint efficiency and flexibility comparisons for MIMO demap- ping architectures rather than on the identification of an ultimate MIMO demapper
3.3. Optimum and Near-Optimum Demapping 49