the presented results will be valid.
The rest of this chapter is organized as follows. In Section 3.2, the signal and system models are introduced. In Section 3.3, the E2E performance analysis of the proposed system is demonstrated by deriving the output SINRs for the first and second hops along with their corresponding PDF. Furthermore, the outage probabilities and ASER are de- rived. In Section 3.5, the E2E upper bound capacity of this system is derived. Section 3.6 presents simulation results and discussion, and finally, the chapter’s conclusions are drawn in Section 3.7.
3.2
Signal and System Model
In this chapter, a wireless communications system is considered in which the source com- municates with destination via relay as shown in Fig. 3.2, in addition to the description and definition mentioned in Table 5.1. The EF relay operates as a FD transceiver with Ntxtransmit and Nrxreceive antennas. Furthermore, the source hasNs antennas used to send the signal, while, Nd antennas at the destination are used for receiving. The chan- nels between the source and the relay, the relay-to-destination, and the relay output to its input are considered in this chapter as flat Rayleigh fading channels and they are de- fined as Hsr ∼ CN (0, INs×Nrx), Hrd ∼ CN (0, INtx×Nd), and Hrr ∼ CN (0, INtx×Nrx), respectively. The assumption that theHrr links are flat comes from the fact that passive suppression of SI has been implemented in the analogue domain via antenna separation and shielding to suppress the LoS path [9]. In addition, AWGN is defined generally as n ∼ CN (0, σ2
nI). In this chapter, it is assumed that there is no directly available source- to-destination path, i.e. Hsd = 0, and all E2E communications occurs via the relay.
As highlighted in Section 3.1, the FD-MRC-MIMO system applies transmit and re- ceive beamforming at the source terminal and relay, respectively. The weight vectors at both transmitter and receiver are designed to maximize the SNR of the desired path by exploiting the eigen-transmissions and providing full diversity gain [51, 52]. This con- sequently leads to improved SINR for fixed SI and noise power levels [68]. This can be obtained by using wsr
tx = usrmax as MIMO transmit beamforming and wsrrx = Hsrusrmax at the receiver as MRC reception, whereusr
maxis a unit norm eigenvector corresponding to the largest eigenvalue λsr
max of the Wishart matrix HHsrHsr, where unit norm implies that the Euclidean norm of usr
max is unity, i.e. kusrmaxk 2
= 1. This is due to the fact that maximizing SNR is subject to determining the squared-spectrum norm of the matrixHsr,
3.2 Signal and System Model
Table 3.1: Model parameters.
Notation Description/Definition
Source-Relay Parameters
so Modulated symbol
wsr
tx MRC weighting coefficients for transmitting with respect toHsr
Hsr source-relay channel coefficients
Ξsr The channel estimation error ofHsr Relay Parameters
r Sum of the received signal from the source coupled withsioverHrr Grx Spatial filter determined by the first unitary matrix of SVD(Hrr)
z Output of the filterGrx
(wsr
rx)H MRC weighting coefficients for detection with respect toHsr ˜
z Weight received symbol
F MMSE coefficient of the EF-relay
sef Output ofF
wrd
tx MRC weighting coefficients for transmitting with respect toHrd Gtx Spatial filter determined by the second unitary matrix of SVD(Hrr)
si Transmitted symbols from relay to destination
Hrr Self-interference Channel
Ξrr The channel estimation error ofHrr Relay-Destination Parameters yD The Received signal at the destination (wrd
rx)H MRC weighting coefficients for detection with respect toHrd ˆ
so The received symbol at the destination after applying MRC toyD
Hrd Relay to destination Channel
Ξrd The channel estimation error ofHrd
which suggests that the signal is transmitted from source-to-relay over the strongest path of Hsr [51, 52]. The same procedure can be used in the path of relay-to-destination of Hrdto obtainwrdtx andwrdrx.
3.2 Signal and System Model
Source
RELAY
Destination
Demod. Mod. (w r d r x ) H Nr x Ntx Nd Ns r z bso ez so F sef Hs r Gr x Gtx Hr r [N tx × Nr x ] Hr d yD si w s r tx (w s r)rx H w r d tx Figure 3.2: FD-MRC-MIMO system with SIC using NSP .3.2 Signal and System Model
This research focuses on FD-MIMO based relay systems, which offer additional de- grees of freedom in the spatial domain [9]. Spatial suppression schemes have been pro- posed and applied extensively for this issue. This is achieved by adding a receive filter, Grx ∈ CNrx×Nrx, at the input of the FD relay, and a transmit filter, Gtx ∈ CNtx×Ntx, at the output of relay, as illustrated in Fig. 3.2. Both of these filters are designed based as eigen-beamformers using the SVD of the SI channel of the relay, Hrr, with Hrr = UrrΣrrVrrH, where Urr ∈ CNrx×Nrx and Vrr ∈ CNtx×Ntx are unitary matrices, i.e. UrrUHrr = UHrrUrr = I and VrrVHrr = VrrHVrr = I. Here, Urr and Vrr are con- structed using orthogonal column vectors of Hrr. In addition, Σrr ∈ RNrx×Ntx is a diagonal matrix containing in descending order the singular values, σrr[i] ≥ 0, for i = 1, 2, ..., min{Nrx, Ntx} of Hrr[8] [29].
The target in designing the filters from the SVD of the SI channel is to remove loop- back interference. This can be satisfied asGrxHrrGtx= 0, which is referred to as NSP. This method can be used when the SI signal is not perfectly known due to linear and non- linear distortion induced in the transmit/receive chains. However, the channel estimation error, which will be discussed in more details later in this chapter, will cause residual SI, which impacts negatively on the the overall performance of the system. In order to design GrxandGtx, there are several approaches that can be utilized as in [2, 8, 9, 20, 29, 69–71]. As the emphasis of this chapter is on the performance analysis of FD-MIMO relays in the presence of SIC, the approach outlined in [20] and [70] have been employed, which is suitable for MIMO systems with the same number of transmit and receive antennas, i.e. Nrx = Ntx. In this method, the two spatial filters are designed by selecting one of the two options in (3.1) in order to satisfyminkGtxHrrGrxk2F [9], i.e.
Grx = [u(0)rr u(0)rr]H, if Gtx= [v(1)rr v(1)rr], (3.1a) Grx = [u(1)rr u(1)rr]H, if Gtx= [v(0)rr v(0)rr], (3.1b)
whereu(0)rr andv(0)rr represent the first half columns of the matricesUrr andVrr, respec- tively, while, u(1)rr andv(1)rr represent the second half columns of matrices Urr and Vrr respectively. In addition, for non-square matrices and/or for the case of rank deficiency ofHrrwhen the channels are not totally independent, (3.2) can be used to design the two