−1 0 1 2 3 4 5 x 108 0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
flop count EIGSD
−1 0 1 2 3 4 5
x 108 0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
flop count SD
Figure 2.12: Flop count histograms for SD and EIGSD algorithms, m = 12, SN R = 18 [dB], D={−152,−132, . . . ,132,152}12
the expected flop count of the EIGSD and SD algorithms for different large values of M.
As can be seen, the larger the set of allowed integers the better the EIGSD performance.
60 80 100 120 140 105
106 107 108 109
Flop count, m=7
M
flop count
EIGSD SD
0 1000 2000 3000 4000
103 104 105 106 107 108 109 1010
Flop count, m=3
M
flop count
EIGSD SD
Figure 2.13: Computational complexity of the SD and EIGSD algorithms, m varies, D = {−M2−1,−M2−3, . . . ,M2−3,M2−1}m
algorithm will perform faster than the standard sphere decoding algorithm. This is due to the additional computations required by the modified algorithm to find a lower bound in each node of the search tree. Hence a natural conclusion of our work: a lower bound on one hand has to be as tight as possible in order to prune the search tree as much as possible, and on the other hand it should be efficiently computable. Led by these two main requirements, in this chapter we introduced a general framework, based on theH∞ estimation theory, for computing the desired lower bounds. Several special cases of lower bounds were deduced from this framework. We explicitly studied four such lower bounds, and employed them for sphere decoding. The first two correspond to relaxation of the search space to either a sphere or a polytope, while the third one is a slight generalization of the spherical lower bound. The last special case corresponds to bounding the integer least-squares problem with the smallest eigenvalue and requires smaller computational effort than any of the previously mentioned bounds. In addition toH∞framework for computing lower bound on the integer least-squares problem, we introduced an SDP-based framework for computing desired lower bound relevant in cases when the original problem is binary.
Simulation results show that the modified sphere-decoding algorithm, incorporating the lower bound based on the smallest eigenvalue and on the SDP-duality theory, outperforms in terms of complexity the basic sphere decoding algorithm. This is not always the case with the aforementioned alternative bounds, and is due to their efficient implementation which is effectively only linear in the dimension of the problem.
Effectively all algorithms developed in this chapter can be divided in two groups de- pending on the type of the problem that they were designed for. The first group (SDsdp, GSPHSD, SPHSD, PLTSD) is specifically designed for binary problems, while the second group (EIGSD) is specifically designed for higher-order constellation problems. From the results that we presented, the SDsdp, GSPHSD, and EIGSD algorithms seem to outper- form the standard SD in the simulated regimes in terms of flop-count. Furthermore the distributions of their flop counts have a significantly shorter tail than the distribution of the SD. However, SPHSD and PLTSD don’t perform as well as the standard SD in terms of the flop count and flop count histogram. These results suggest that using a lower-bounding technique is useful, but only if the lower bound can be computed in a fast manner.
We should also point out that, although we derived it in order to improve the speed of the sphere decoding algorithm, the general lower bound on integer least-squares problems is an interesting result in itself. In fact, the proposed H∞ estimation framework for the efficient computation of lower bounds on the difficult integer least-squares problems may find applications beyond the scope of this thesis.
The results we present indicate potentially significant improvements in the speed of the sphere decoding algorithm. However, we should note that the proposed H∞-estimation- based framework for bounding integer least-squares problems is only partially utilized. In fact, there are several degrees of freedom in the generalH∞-based bound that are not fully exploited. It is certainly of interest to extend the current work and use the previously mentioned degrees of freedom to further tighten the lower bound. If, in addition, this can be done efficiently, it might even further improve the speed of the modified sphere decoding algorithm.
Chapter 3
Non-coherent ML Detection in Multi-Antenna Systems
In multi-antenna communication systems, channel information is often not known at the receiver. To ensure the practical feasibility of the receiver, the channel parameters are often estimated via the transmission of training symbols and then employed in the design of signal-detection algorithms. Such a scenario is possible when the environmental conditions are not changing rapidly and was considered in the previous chapter. However, in some applications, due to limited systems resources and/or rapid time variation of the channel parameters, explicitly learning the channel coefficients becomes infeasible. In this chapter we consider the problem of maximum-likelihood (ML) detection in single-input multiple-output (SIMO) systems (see Figure 3.1) when the channel information is completely unavailable at the receiver and when the signalling at the transmitter is q-PSK. It is well known that finding the solution to this optimization requires solving the integer maximization of a quadratic form which is, in general, an NP-hard problem.
In this chapter we consider solving this problem exactly and approximately. To solve it exactly we introduce the so-called out-sphere decoder algorithm, which we consider as a counterpart to the standard sphere decoder used in coherent detection and discussed in the previous chapter. In addition to developing the out-sphere decoder, we analyzed its complexity as well. Since the problem has a natural statistical setup, we considered the expected value of its complexity. We provided an explicit analytical upper bound on the expected complexity of the out-sphere decoder.
Figure 3.1: Single-input multiple-output (SIMO) system
Besides developing the exact out-sphere decoder, we propose an approximate algorithm which is based on a certain modification of a standard semi-definite program (SDP) relax- ation. We derive a bound on the pairwise error probability (PEP) of the proposed algorithm and show that the algorithm achieves the same diversity as the exact maximum-likelihood (ML) detector. Furthermore, we prove that in the limit of large system dimension this bound differs from the corresponding one in the exact ML case by at most 3.92 dB if the transmitted symbols are from a 2- or 4-PSK constellation, and by at most 2.55 dB if the transmitted symbols are from an 8-PSK constellation. This suggests that the proposed algorithm requires moderate increase in the signal-to-noise ratio (SNR) in order to achieve performance comparable to that of the ML detector but with often significantly lower com- putational effort.
3.1 Introduction
Multi-antenna wireless communication systems are capable of providing reliable data trans- mission at very high rates. The channel in such systems is, in principle, unknown to the
receiver and needs to be estimated either prior to, or concurrently with, the detection of the transmitted signal. However learning channel coefficients requires time and energy, which in environments with rapidly changing conditions and limited system resources can be impractical. In this chapter we study the problem of ML detection when the channel information is unavailable at the receiver. The system that we study has a single transmit antenna and multiple receive antennas.
We assume a standard flat-fading channel model for multi-antenna systems similar to the one used in the previous section (see Figure 3.2),
X = rρT
Msh+W. (3.1)
Here T denotes the number of time intervals during which the channel remains constant, M = 1 is the number of the transmit antennas; N is the number of the receive anten- nas; ρ is the signal-to-noise ratio (SNR); X is a T ×N matrix of received symbols; s is a T ×1 transmitted symbol vector comprised of components si, for which it holds that si = √1
Tej2rπq , r = 1, . . . , q, and q is an integer power of 2; h is an 1×N channel matrix whose components are independent, identically distributed (i.i.d.) zero-mean, unit-variance complex Gaussian random variables; and W is an N ×T noise matrix whose components are i.i.d. zero-mean, unit-variance complex Gaussian random variables. Furthermore, we assume that the components of h and W are uncorrelated and thatT ≥N, which is often the case in practice.
The rest of this chapter is organized as follows. In Section 3.2 we recall what the criterion for non-coherent ML-signal detection is. In Section 3.3 we propose an algorithm (which we call out-sphere decoder) for solving the problem of non-coherent ML detection exactlyand analyze its expected complexity. In Section 3.4 we consider solving non-coherent ML detection problem approximately. First in section 3.4.1 we propose a trivial polynomial time algorithm that achieves full diversity. In Section 3.4.2 we introduce an SDP-based approximate algorithm for solving the ML-detection problem. In Section 3.4.3 we compute
Figure 3.2: Mathematical model of SIMO system
its pairwise error probability (PEP). In Section 3.4.4 we asymptotically analyze the PEP performance in the case of large system dimensions. In Section 3.4.5 we briefly comment on the complexity of the proposed algorithms. In Section 3.5 we summarize the obtained results and suggest several possible directions for a future work.