Contribution of This Thesis

We briefly outline the contribution of this work.

1. We propose iterative alternating optimization to improve the capacity of distributed AF MIMO relay network. The proposed algorithm is based on matrix low-rank updating and thus could efficiently generate optimized results.

2. Noise-dominant models is proposed to simplify system design on distributed AF MIMO relay network. We show the models serve as upper bounds of system capacity and lead to several efficient design algorithms.

3. Based on noise-dominant models and relay selection, we develop two corresponding relay gains designs. We also discover the performance of proposed algorithm could be modeled as equivalent MIMO antenna selection systems. Thus we could reply on the well-studied characteristics of MIMO antenna selection to approximate and analyze the proposed algorithms.

4. To circumvent the potential expensive computation burden of selection-based schemes, we further transform and simplify the design problem in order to consider and uti-lize all the relays simultaneously. For the relay noise dominant model, we develop algorithms based on low-leakage beamforming designs. As to destination noise dom-inant model, we propose to simplify the problem by matrix triangularization. An

algorithms with specifically constrained solution space is developed to iteratively improve the results.

5. To perform OFDM channel estimation with highly dispersive delay profile but with few pilot or training symbols, we apply subspace-based algorithm to efficiently utilize limited pilot resource. Since the unknown subspace is critical to this approach, we apply OMP and invent GMP to iterative reconstruct the subspace. The proposed MP based algorithms could bypass some limitations of other algorithms and utilize the information brought by consecutive OFDM symbols. Also we discover that AF distributed relay network with OFDM transmission could be modeled as single-hop OFDM with equivalent highly dispersive channel. Thus the proposed subspace-based algorithms is suitable to handle the channel estimation for the relay network of interest.

Chapter 2

Capacity of Relay Network and Alternating Optimization

In this chapter, we first briefly describe the system model of amplify-forwarding dis-tributed relay network used in this work, then present capacity (i.e. mutual information) of this model and the associated optimization problem. To simplify the optimization and gain some insights into system design, we discuss the effects of relay transmission power scaling which ultimately leads to an upper bound of capacity. As our first approach to capacity optimization, in the third part of this chapter we present the alternating op-timization. Specifically, this approach applies successive greedy optimization, low-rank updating of matrix computation, and quadratic approximation. Finally some numerical results are shown to confirm the performance of proposed approach.

2.1 System Model and Capacity of Relay Networks

2.1.1 System Model and Power Limits

We consider a distributed MIMO relay system consisting of one source terminal, one destination terminal and L single-antenna relay terminals. Both source and destination are equipped with M antennas. To preserve the degree of freedom in the end-to-end transmission, it is reasonable to assume L ≥ M . Fig. 2.1 demonstrates a conceptual representation for the system of interest.

Let x and y (x, y ∈ C^M) denote the signal vector transmitted from the source and received at the destination, respectively. Let r = [r(1) r(2) · · · r(L)]^T represent the for-warding gain vector of relay network, wherein the ith relay performs amplify-forfor-warding by multiplying the received signal with the complex gain r(i) and then transmitting the result. Signals in the system pass through two MIMO channels, F and G^H, to go from the source to the destination, where F ∈ C^{M ×L} and G^H ∈ C^L×M denote channel between

des-Fig. 2.1: MIMO system with distributed relays.

tination and relays and channel between relays and source, respectively. The superscript ()^H defines matrix Hermitian transpose. The received signals at the relays are assumed to be subject to additive complex circular white Gaussian noise n_R ∼ CN (0, σ_R²I_L), where I_L denotes an identity matrix of size L × L. Likewise, the received signals at the des-tination are assumed to be subject to additive complex circular white Gaussian noise n_D ∼ CN (0, σ_D²I_M). Assume that there is no direct propagation path from the source to the destination. The received signal vector y at the destination can be described as

y = F diag(r)G^Hx + F diag(r)n_R+ n_D

, F RG^Hx + F Rn_R+ n_D (2.1)

where R is a diagonal matrix defined from r, and diag(r) denotes a diagonal matrix whose diagonal terms are equal to r.

To optimize the system capacity fairly, transmission power limits are imposed on the source and the relays. Specifically, we assume that the source transmits independent signal streams over its M antennas with equal power σ_x², subject to a total power limit of P_S. Further, the relays are subject to a total power limit of P_R. Mathematically, these constraints can be expressed as

PS ≥ tr(E{xx^H}) = M σ²_x,

P_R ≥ tr(E{(RG^Hx + Rn_R)(RG^Hx + Rn_R)^H})

=

L

X

i=1

(σ_R² + σ_x²|G⁽ⁱ⁾|²)|r(i)|² ,

L

X

i=1

p(i)|r(i)|²,

where |G⁽ⁱ⁾|² = (G⁽ⁱ⁾)^HG⁽ⁱ⁾, with G⁽ⁱ⁾ representing the ith column of G and being asso-ciated with the channel vector between the source and the ith relay.

2.1.2 Noise Whitening and Capacity

Note that the noise vector (F Rn_R+n_D) in (2.1) as received by the destination is spatially correlated. To find the system capacity, a noise whitening filter W^−1/2 at the destination is used, where W is given by

W = E{(F Rn_R+ n_D)(F Rn_R+ n_D)^H}

= σ_D²(I + σ²(F R)(F R)^H), (2.2)

with σ , σ^R/σD. Let H denote the noiseless end-to-end equivalent MIMO channel, i.e., H ,pσ²_xF RG^H. Then we may derive the system capacity as a function of R as [41]

log₂C(R) , log2det(I_N + H^HW⁻¹H)

= log₂det(W + HH^H) − log₂det(W ) (2.3) where det(·) denotes matrix determinant. The optimization problem can be stated as

R_opt = arg max

R C(R) (2.4)

subject to

P_S ≥ M σ_x², P_R ≥

L

X

i=1

p(i)|r(i)|². (2.5)

By observing (2.3) it is clear increasing source transmitting power or σ_x is always beneficial to capacity regardless of channel condition or the design of R. Thus we may readily set σ_x² = M P_S. Further, the combination of σ²_x and G matrix in (2.3) could be replaced by a scaled version of G without lose any generality, so we ignore σ_x hereafter in this work.