Open access peer-reviewed chapter

Beamforming in Wireless Networks

By Mohammad-Hossein Golbon-Haghighi

Submitted: May 2nd 2016Reviewed: October 19th 2016Published: December 14th 2016

DOI: 10.5772/66399

Downloaded: 929

Abstract

This chapter is about the beamforming approach in wireless 5G networks, which involves communication between multiple source-destination pairs. The relays can be multiple-input multiple-output (MIMO) and/or distributed single-input single-output (SISO), and full channel state information of source-relays and relay-destinations are assumed to be available. Our design consists of a two-step amplify-and-forward (AF) protocol. The first step includes signal transmission from the sources to the relays, and the second step contains transmitting a version of the linear precoded signal to the destinations. Beamforming is investigated only in relay nodes to reduce end user’s hardware complexity. Accordingly, the optimization problem is defined to find the relay beamforming coefficients that minimize the total relay transmit power by keeping the signal-to-interference-plus-noise ratio (SINR) of all destinations above a certain threshold value. It is shown that this optimization problem is a non-convex, and can be solved efficiently.

Keywords

  • beamforming
  • 5G wireless networks
  • MIMO
  • optimization

1. Introduction

Recently, cooperative communication has become one of the appealing techniques that can be used in 5G wireless relay networks to achieve spatial diversity and multiplexing, which overcomes the channel impairments caused by several fading effects and destructive interference. Though various cooperative communication schemes exist [1, 2], the AF scheme is more attractive due to its simplicity since the relays simply forward the amplitude phase-adjusted version of received signals to destinations. In Ref. [2], a distributed beamforming relay system with a single transmitter-receiver pair, and several relaying nodes have been proposed. The authors assumed that perfect channel state information (CSI) is available at all relay nodes. Although the same scenario is investigated in Ref. [3], the second-order statistics of all channel coefficients are assumed to be available at the relays. Furthermore, the beamforming weights are obtained in order to maximize the signal-to-noise ratio (SNR) at destination subject to holding the relay power above a certain threshold value.

In the past three decades, code-division-multiple-access (CDMA) systems have been extensively investigated as the one of the important candidates for transmitting data over single channels while sharing a fixed bandwidth among a large number of users [4]. The design of receivers to increase the number of supported users, in these systems, has been explored in Ref. [5, 6]. In Ref. [6], joint channel estimation and data detection based on an expectation-maximization (EM) algorithm [7] is proposed. The authors have shown that the proposed receiver achieves a near-optimum performance with modest complexity. Furthermore, the authors in Ref. [5] designed a double stage linear-detection receiver to increase the number of supported users on the system. This design requires complex processing at the receiver’s side instead of using a precoding scheme at the transmitter where more hardware complexity is tolerable. Therefore, the authors in Ref. [8] studied a MIMO CDMA system implementing zero-forcing beamforming (ZFBF) as an efficient precoding technique.

Though various complex multiuser detection techniques that can be used in CDMA systems [9], the unconventional matched filter receiver is chosen at destination nodes due to the intractability of the precoding design when other forms of detectors are used. In this article, we have focused on the optimization of the beamforming weights applied to the outputs of matched filter banks to minimize the total relay transmit power subject to a target SINR of all destinations. Our proposed distributed CDMA-relay network can easily overcome the other multiplexing schemes such as space division-multiple access (SDMA), time division-multiple access (TDMA) or frequency division-multiple access (FDMA). The SDMA schemes [10] in which sources, destinations and relays are distributed in the space, have two disadvantages. First, these schemes should have a significant number of relays in proportion to their users to be able to overcome channel impairments at destinations. Although the SDMA scheme with the limited number of relays cannot compensate the interference power, our CDMA schemes can easily satisfy the network QoS due to their ability to decrease the interference effect at destinations. So, the second disadvantage of SDMA is the inefficient use of hardware communication resources. In the SDMA scheme, if the number of users increases, the network data rate can significantly decrease. Therefore, the number of relays should be considerably increased to be able to satisfy the QoS constraints, which is costly for the network operator.

Notation: We denote the complex conjugate, transpose, Hermitian (conjugate transpose) and inner product operators by (⋅)*, (⋅)T, (⋅)H and ⟨ ⋅ ⟩, respectively. We use E{⋅} to denote statistical expectation. trace{⋅} and Rank{⋅} represent the trace and rank of the matrix, respectively. Vec(⋅) is the vectorization operator stacking all columns of a matrix on top of each other; ⊗ represent the Kronecker product of two matrices and A>0stands for semi-definite conic inequality that means A is a non-negative semi-definite matrix.

2. 5G wireless system and equations

Consider a wireless relay network with d pairs of source-destination (peers) communicating without a direct link through R MIMO or SISO relay antennas. In this chapter, a two-step AF protocol is used. In the first step, each source user broadcasts its spread symbol toward the relays. A matched filter is applied in each relay in order to retrieve the source’s signals. In the second step, the adjusted and spread signals by the relays are transmitted to destinations.

3. MIMO relay networks

In this section, a peer-to-peer MIMO-relay network with d pairs of source-destination nodes is considered, as shown in Figure 1. It is assumed that all source and destination nodes are equipped with one SISO antenna and each source attempts to maintain communication with its corresponding destination. It is assumed that there is no direct link between source and destination pairs due to path loss and deep shadowing and all nodes are working in a half-duplex mode. We use a two-step AF protocol. During the first step, each source broadcasts its signals to MIMO-relay. Then, after applying the beamforming weights at MIMO-relay, the adjusted signals transmit to all destinations.

Figure 1.

A MIMO-relay network (from M.H. Golbon et al. [11]).

Let sk stands for the kth source symbol that is assumed to be independent of the other sources, that is, Esksl*=Pkδkl. Denote the channel coefficient from the kth source to the rth relay as frk and the channel coefficient from rth relay to kth destination as grk. Then, the received signal at the rth relay is given by:

χr=l=1dfrlsl+ωr,r1R,E1

where ωr is the noise at the rth relay. For simplicity, Eq. (1) can be rewritten as:

χ=l=1dflsl+ω,E2

where χ ≜ [χ1, χ2, … , χR]T, ω ≜ [ω1ω2, … , ωR]T, fl ≜ [f1l, f2l , … , fRl]T.

The received signal in MIMO relay has been processed by the beamforming weights, that is, W ∈ R × R, which should be designed appropriately. Finally, each MIMO-relay antenna transmits the following signal to destinations:

γ=WχR×1E3

The rth entry of γ is the signal transmitted by rth MIMO antennas. Finally, the received signal at the kth destination is given by

yk=gkTγ+ζkE4

where ζk(t) is the noise at the kth receiver. We can easily rewrite Eq. (4) as:

yk=gkTWχ+ζk=gkTW(l=1dflsl+ω)+ζk=gkTWl=1dflsl+gkTWω+ζk=gkTWfkskdesired received signal+gkTWl=1,lkdflslinterferencepart+gkTWω+ζknoisepartE5

The three last terms of Eq. (5) are the desired received signal, interference and noise at the kth destination, respectively. The object of the network beamforming is to minimize the total relay transmit power subject to maintaining every destination SINR above a pre-defined threshold value γth (as a QoS parameter of the network). In this case, the instantaneous SINR for kth destination simply becomes the desired signal power of the desired signal to the power of interference plus noise. So, the optimization problem can now be written as

MinimizewPRSubjecttoSINRkγthkk{1,2,...,d}E6

where PR is the total relay transmit power, w stands for beamforming weights, SINRk and γk denote the received SINR and the target SINR (threshold value) at the kth destination node, respectively.

First, using Eq. (3), the total relay transmit power can be calculated as

PR=EγHγ=EχHWWHχ=traceWHRxWE7

where Rx ≜ E{χχH} and it can be calculated as:

Rx=l=1dPlEflflH+σω2IR×RE8

For any conforming matrices M, N and Z, the following equation holds

traceMZHNZ=vecZHMTNvecZE9

Therefore, Eq. (7) can be rewritten as the following quadratic form:

PR=vecWHIR×RRxTvecW=wHTwE10

where w ≜ vec(W) and T ≜ (IR × R ⊗ Rx).

Using Eq. (5), the desired signal power at the kth destination can be obtained as:

PSk=PkEfkHWHgk*gkTWfk=PkvecWHRfkTRgkRkvecW=wHRkwE11

where RfkEfkfkH,RgkEgk*gkTand RkPkRfkTRgk.

Also, using Eq. (5), the received noise power at kth destination can be calculated as:

PNk=EωHWHgk*gkTWω+σςk2=σω2traceWHRgkW+σςk2=vecWHIRgkΝkvecW=wHΝkw+σςk2E12

where ΝkIRgk.

Finally, the power of the received interference at the kth destination can be computed as

PIk=El=1,lkdflslHWHgk*gkTWl=1,lkdflsl=tracePkEl,m=1l,mkdflfmHFkWHRgkW=vecWHFkTRgkvecW=wHIkwE13

where FkPkEl,m=1l,mkdflfmHand IkFkTRgk.

In this case, the instantaneous SINR for kth destination simply becomes the desired signal power of the desired signal to the power of interference plus noise. So, the optimization problem can now be written as

MinimizewwHTwSubjecttoSINRk=wHRkwwHΝk+Ikw+σςk2γthkk12dE14

Since wHΝk+Ikw+σςk20, the constraints of the optimization problem can be formulated as

wHRkγthkΝk+Ikwγthkσςk2E15

In this problem, if all the matrices RkγthkΝk+Ikare negative semi-definite for all k, the problem is convex and can be solved uniquely. However, the feasible set of our optimization problem is empty since wHRkγthkΝk+Ikw0for all k and w. Therefore, RkγthkΝk+Ikis non-negative definite matrix which results in non-convex inequality constraints, hence the quadratically constrained quadratic programming (QCQP) problem is non-convex and NP-hard in general. However, we will show that a simple near optimal solution can be found in our problem. First, we replaced our QCQP problem with a semi-definite programming (SDP) problem. Let us define DkRkγthkΝk+Ik,XwwHand using the fact that trace(AB) = trace(BA) (when A is an m×n and B is an n×m matrix), the optimization problem Eq. (14), can recast to

MinimizeXtraceTXSubjecttotraceDkXγthkσςk2,k1dRankX=1,X0E16

This optimization problem is non-convex, because the Rank(X) = 1 constraint is non-convex. We relax the problem by ignoring this non-convex constraint and convert it to a convex SDP problem. The following semi definite representation (SDR) form is the relaxed version of the problem Eq. (16).

MinimizeXtraceTXSubjecttotraceDkXγthkσςk2,k1dX0E17

The optimal value of the relaxed problem is a lower bound of the optimal value of SDP problem (Eq. 16).Well-known semi-definite problem solvers such as SeDuMi or CVX can solve the above problem in polynomial time using interior point methods. If the optimal value of Eq. (17), that is, Xopt, is rank one, then its principal eigenvector is exactly the optimal solution of the original optimization problem. Since the solution of Eq. (17) is not always rank one, one can use randomization techniques [10] to obtain an approximate solution of the original problem from the solution of the relaxed problem. The randomization technique is finding the best solution from the candidate sets of beamforming vectors generated from Xopt [12]. Luo et al. [13] and Chang et al. [14] analyzed the accuracy of these techniques for different semidefinite problems, and it has been found that the randomization technique has acceptable performance in practical scenarios [15]. Therefore, the eigenvalue decomposition of Xopt can be calculated as Xopt = VDVH. Then the candidate sets of beamforming vectors is generating as xc = VD(1/2)pc, where pc is a circularly symmetric complex, and zero mean, unit variance white Gaussian vector, that is, pc ∈ R × 1 ∼ ℂΝ(0, 1). Hence, it can be easily recognized that the vector xc satisfies E{xcxcH} = Xopt. This candidate vector generation should perform several times and in each iteration, any vector (or scaled version) that satisfies SINR constraints of problem Eq. (17) is saved as a candidate vector (xc) along with corresponding objective values. The vector generation should be repeated for a predefined number of times. The final minimum solution can be achieved by a simple minimization over the obtained objective values as an approximate solution of the problem.

Then, solving problem Eq. (16) from xc becomes finding a proper scaling factor of β0. Applying β to Eq. (17), the following problem will be attained

MinimizeXβtraceTXSubjecttoβtraceDkXγthkσςk2,k1dRankX=1,X0E18

In the above algorithm, the acceptable scaling factors are those that satisfy β trace(TkX) ≥ 0. Thus, the maximum scaling factor should be selected as

β=maxk=1,,dγthkσςk2traceDkXE19

Consequently, the approximate solution of problem (Eq. 16) is βxc. In our case, after an acceptable number of iterations (around 100 iterations), the solution of the randomization problem approached to its lower bound (the optimal value of relaxed problem). Therefore, Xopt is an acceptable and a near optimal solution to the original non-convex problem. Another optimal solution of Eq. (16) can be found using a penalty function in the objective part of the problem and converting the objective function into the difference of two convex functions subject to current convex constraints [16], and applying an effective non-smooth optimization algorithm based on the sub-gradient of rank one constraint.

For examination, we assumed that channel state information is known at a processing center and the beamforming weights are optimized and spreaded to the nodes from this processing Center [17]. In each simulation snapshot, the channel coefficients frk, grk are generated as i.i.d circularly symmetric complex Gaussian random variables with variances of σf2=σg2=10dB. Also, it is assumed that we have the same output power at sources, that is, Pkk=1d=10dBand we set γthkk=1d=γth,σωi2i=1R=σςk2k=1d=0dB.

Figures 2 and 3 show the minimum MIMO-relay transmit power PTminversus destination SINR threshold value γth, for different values of σf2,σg2. It can be seen from Figures 2 and 3 that the better quality of uplink and/or downlink channels can decrease the minimum MIMO-relay transmit power for a certain threshold value.

In Figures 4 and 5, we examine the network performance by changing the number of MIMO-relay antennas and number of source-destination pairs. As expected, more power saving will be obtained by increasing the number of MIMO antennas and/or decreasing the number of user nodes.

Figure 2.

Minimum MIMO-relay transmit power PTmin versus destination SINR threshold value γth, for different values of σf2 and σg2=10dB.

Figure 3.

Minimum MIMO-relay transmit power PTmin versus destination SINR threshold value γth, for different values of σg2 and σf2=10dB.

Figure 4.

Minimum MIMO relay transmit power PTmin versus destination SINR threshold value γth, for different number of antennas.

Figure 5.

Minimum MIMO relay transmit power PTmin versus destination SINR threshold value γth, for different number of source-destination pairs.

4. MIMO-CDMA relay networks

In the last section, we obtained the optimal beamforming weights for a MIMO relay network. Here, in addition to the multiple antenna technique, CDMA is applied to the network to increase the order of multiuser multiplexing. CDMA systems can share a fixed bandwidth among a large number of users without the need of frequency division or time division between nodes. CDMA introduces a diverse range of trade-off between receiver complexity and system performance.

As shown in Figure 6, a two-step AF protocol is used for this MIMO-relay network. In the first step, each source user broadcasts its precoded signal (i.e. slul(t)) at its maximum power Pl toward the MIMO-relay. At the MIMO-relay, a matched filter is applied to retrieve the source’s signals. In the second step, the adjusted and spreaded signals are transmitted from MIMO-relay to all destinations. Another matched filter is used at each destination to extract its corresponding symbols.

Figure 6.

MIMO-relay multiuser network (from M.H. Golbon et al. [18]).

Let uk(t) denotes a signature waveform that is assigned to the kth source. Then, the received signal at the rth antennas of MIMO-relay is given by

χrt=l=1dfrlslult+ωrtE20

The vector form of Eq. (20) can be written as:

χt=l=1dflslult+ωtR×1E21

where

χtχ1t,χ2t,,χRtT,vtv1t,v2t,,vRtT,flf1l,f2l,,fRlTE22

By denoting the cross correlation between kth user’s codeword to the lth user’s codeword as ρk,l=uktulT0tt=T0, the output signal of the matched filter at the MIMO-relay can be expressed as

γn=χtun*T0tt=T0=l=1dflslultun*T0tt=T0+ωtun*T0tt=T0=l=1dflslρl,n+ςn=γn,k+γn,k+εnE23

where ρl,n is the cross correlation between lth user’s code-word and nth user’s code-word [19]:

ρl,n=ultun*T0tt=T0=ult,untE24

where γn,kγn,− k, and εn are defined as

γn,kl=1,lkdflslρl,nγn,kfkskρk,nεnωtun*T0tt=T0E25

The output of the matched filter in each relay has been processed by the beamforming weights Wl ∈ R×Rd, which should be designed appropriately. We define the output of the matched filter bank as Γ = [γ1Tγ2T, … , γdT]T ∈ Rd×1, the adjusted MIMO-relay signals can be written as

ξl=WlΓR×1,l1dE26

Another filter bank is applied to the output of each MIMO antenna, which generates R×d filtered data. This data are processed in a processing center in the MIMO relay to achieve the proper symbol vector, which can be transmitted in each user’s subspace. After beamforming by the above linear operation, the MIMO-relay transmits the following modulated and precoded signal to destination nodes:

ψt=l=1dξlultR×1E27

The rth entry of ψ(t) is the signal transmitted by rth relay antenna. Then, the received signal at the kth destination is given by

ykt=gkTψt+ζktE28

where ζk(t) is the noise at the kth receiver, which is also assumed to be N(0, 1). Finally, each destination node convolves the received signals by its code-word to retrieve its corresponding data. So, the retrieved signal will be

λk=yk(t)uk*(T0t)|t=T0=gkTl=1dξlul(t)uk*(T0t)|t=T0ρl,k+ζk(t)uk*(T0t)|t=T0ςk=gkTl=1dξlρl,k+ςk=gkTl=1dWlΓρl,k+ςk=gkT(l=1dρl,kIR×RWl)Γ+ςk=gkT(ρkW)Γ+ςk=gkTρkW(Γk+Γk+Γεn)+ςk=gkTρkWΓkdesired received signal+gkTρkWΓkinterferencepart+gkTρkWΓεn+ςknoisepartE29

where ςk is the noise at the kth receiver, and the following notations are defined for simplicity:

rkρ1,kρ2,kρd,k1×dρkrkIR×RR×RdWW1TW2TWdTTRd×RdΓkγ1,kTγ2,kTγd,kTTΓkγ1,kTγ2,kTγd,kTTΓεnε1Tε2TεdTTΓ=Γk+Γk+ΓεnRd×1E30

The object of the network beamforming is to minimize the total relay transmit power subject to maintaining every destination SINR above a pre-defined threshold value γth(as a QoS parameter of the network).

First, using Eq. (27), the total MIMO-relay transmit power can be obtained as:

PR=Eψt,ψt=El=1dξlulT0tH*n=1dξnuntt=T0,=El=1dWlulT0tΓH*n=1dWnuntΓt=T0,=EΓHl=1dWlHulT0t*n=1dWnuntΓt=T0,=EΓHl=1dn=1dWlHulT0t*untρl,nt=T0WnQΓ=EΓHQΓE31

where Ql=1dj=1dWlHρl,jWjand the inner product of vectors x(t), y(t) is defined as

xt,ytxHtytdt=xHT0t*ytt=T0E32

For simplicity, Q can be represented by the following quadratic formml:

Q=W1W2WdHρ1,1IR×Rρ1,2IR×Rρ1,dIR×Rρ2,1IR×Rρd1,dIR×Rρd,1IR×Rρd,d1IR×Rρd,dIR×RW1W2WdE33

The kernel of the above form can be expressed as a Kronecker products as follows:

Q=WHϒIR×RRd×RdWE34

where ϒρ1,1ρ1,2ρ1,dρ2,1ρd,1ρd,d. Thus, Eq. (31) can be rewritten as:

PR=E(ΓH(WH(ϒIR×R)W)Γ)=trace(WH(ϒIR×R)WE(ΓΓH))=vec(W)H(E(ΓΓH)T(ϒIR×R))Tvec(W)=wHTwE35

where w ≜ vec(W) and T ≜ E(ΓΓH)T ⊗ (ϒ ⊗ IR × R).

Also, the instantaneous desired signal power at the kth destination is calculated as:

Psk=EgkTρkWΓkΓkHWHρkTgk*E36

By defining Γk ≜ μkSk and μkrkTfk, Eq. (36) can be rewritten as

Psk=PkE[gkTρkWμkμkHWHρkTgk*]=Pktrace(WHρkTE(gk*gkT)RgkρkWE(μkμkH)Rμk)=vec(W)H(RμkTPk(ρkTRgkρk))vec(W)=vec(W)H(RμkT(Pkτk))Rkvec(W)=wHRkwE37

where Rμk, Rgk, τk and Rk are defined as

RμkEμkμkH, RgkEgk*gkT, τkρkTRgkρkand RkRμkTPkτk

Also, the received noise power at kth destination is given by:

PNk=EgkTρkWΓεnΓεnHWHρkTgk*+σςk2=traceWHρkTRgkρkWEΓεnΓεnH+σςk2PNk=vecWHEΓεnΓεnHTτkvecW+σςk2=wHΝkw+σςk2E38

where ΝkEΓεnΓεnHTτk. Also, it can be easily proved that:

EΓεnΓεnH=u*tuTtdtσω2IR×RRd×RdE39

Finally, the power of the received interference at the kth destination can be computed as

PIk=EgkTρkWΓkΓkHWHρkTgk*=traceWHρkTEgk*gkTρkWEΓkΓkH=traceWHρkTRgkρkWEΓkΓkH=vecWHEΓkΓkHTτkIkvecW=wHIkwE40

The instantaneous SINR for kth destination simply becomes the desired signal power of the desired signal to the power of interference plus noise. So, the optimization problem can now be written as

MinimizewwHTwSubjecttoSINRk=wHRkwwHΝk+Ikw+σςk2γthkk12dE41

By defining DkRkγthkΝk+Ik,XwwH, the optimization problem can recast to

MinimizeXtraceTXSubjecttotraceDkXγthkσςk2,k1dRankX=1,X0E42

We solve this optimization problem in a same way as the previous section. The first simulation scenario was carried out to consider the total MIMO-relay transmit power versus destination SINR threshold value, for different values of users’ correlation factors. The averaged results are shown in Figure 7. The network consists of two source-destination pairs and four MIMO-relay antennas. Figure 7 shows that the total MIMO-relay transmit power in all cases increases by raising γth. Furthermore, Figure 7 indicates that when the signature sequence correlation ρk,l increases, more total transmit power is needed to ensure SINR constraints at destination nodes.

Figure 7.

Minimum MIMO-relay transmit power PTmin versus γth, for R=4, u=2.

When ρk,l approaching one, the problem downgrades to the SDMA network and the system loses the benefits of CDMA technique. Also, increasing the signal dependency by increasing the correlation factor results in the more infeasibility rate of the constraints. Therefore, when the correlation factor increases from 0 to 0.75, there is little difference between the curves, but when ρk,l increases beyond 0.75, it can be seen that the difference becomes considerably larger. As a result, a large power gain can be achieved when moving from ρk,l = 1, by a small reduction of ρk,l. To study the effect of the number of relay nodes and the number of source-destination pairs in terms of quality of matched filter output, we have examined a network with ρk,l = 0.75.

Figures 8 and 9 display the minimum relay transmit power versus γth, for different number of MIMO-relay antenna and different number of user pairs. As normally expected, more power saving can be achieved by increasing the number of relays or decreasing the number of users. Comparing Figures 8 and 9 with Figure 7 reveals that decreasing the correlation factor will be much more efficient for saving network power than increasing the number of relays.

Figure 8.

Minimum relay transmit power PTmin versus γth, for R=2 and ρl,m = 0.75.

Figure 9.

Minimum relay transmit power PTmin versus γth, for u=2 and ρl,m = 0.75.

5. Distributed relay networks

In this section, we considered a distributed relays network, instead of MIMO-relay. The optimization problem is defined to find the relay beamforming coefficients that minimize the total relay transmit power by keeping the SINR of all destinations above a certain threshold value.

Consider a wireless relay network with d pairs of source-destination (peers) communicating without a direct link through R single relay antennas, as shown in Figure 10. A two-step AF protocol is used. In the first step, each source user broadcasts its spread symbol toward the relays. A matched filter is applied in each relay in order to retrieve the source’s signals. In the second step, the adjusted and spread signals by the relays are transmitted to destinations. Another matched filter is used at each destination to extract its corresponding symbols. Let sk stands for the kth source symbol that is assumed to be independent of the other sources, that is, Esksl*=Pkδkland uk(t) denotes a signature waveform that is assigned to the kth source. Then, the received signal at the rth relay is given by:

χrt=l=1dfrlslult+ωrt,r1RE43

Figure 10.

Distributed relay network.

where ωr(t) is the noise at the rth relay. By denoting the cross correlation between kth user’s codeword to the lth user’s codeword as ρk,l=uktulT0tt=T0, the output signal of the matched filter at the rth relay can be expressed as

νr=χrtuT0tt=T0=k=1dfrkskρk+ςr=νr,k+νr,k+nr,r1RE44

where the following definitions have been used:

utu1t,,udtTnrωrtuT0tt=T0ρkuktuT0tt=T0=ρk,1ρk,dT,νr,kfrkskρk,νr,kl=1,lkdfrlslρlE45

The output of the matched filter in each relay has been processed by the beamforming weights Wr ∈ d × d, which should be designed appropriately. So, it can be expressed as

γr=Wrνrd×1,r1RE46

Another filter bank is applied to the output of each relay, which generates d filtered data. These data are processed in the relay in order to achieve the proper symbol vector, which can be transmitted in each user’s signal subspace. After beamforming by the above linear operation, the rth relay transmits the following modulated and precoded signal by a CDMA technique

ψrt=γrTut,r1RE47

The vector forms of Eq. (47) can be written as

ψt=ψ1t,ψ2t,,ψRtT=γ1γRTut=W1ν1,,WRνRTut=WHTutE48

The rth entry of ψ(t) is the signal transmitted by rth relay and W ≜ [W1, …, WR] ∈ d × Rd, H ≜ BD(ν1, …, νR) ∈ Rd × R, where BD(⋅) denotes the block diagonalization of matrices. Thus, the total received signal at the kth destination is given by

ykt=gkTψt+ζktE49

where ζk(t) is the noise at the kth receiver and gk ≜ [g1kg2k  … gRk]T is the vector of downlink channel coefficients. Finally, each destination node convolves the received signals by its code-word to retrieve its corresponding data. So, the retrieved signal will be

λk=yk(t)uk*(T0t)|t=T0=gkTl=1dξlul(t)uk*(T0t)|t=T0ρl,k+ζk(t)uk*(T0t)|t=T0ςk=gkTl=1dξlρl,k+ςk=gkTl=1dWlΓρl,k+ςk=gkT(l=1dρl,kIR×RWl)Γ+ςk=gkT(ρkW)Γ+ςk=gkTρkW(Γk+Γk+Γεn)+ςk=gkTρkWΓkdesired received signal+gkTρkWΓkinterferencepart+gkTρkWΓεn+ςknoisepartE50

where:

ςkζktukT0tt=T0HkBDν1,kνR,kHkBDν1,kνR,kHςnBDς1,ς2,,ςRH=Hk+Hk+HςnE51

The three last terms of Eq. (50) are the desired received signal, interference and noise at the kth destination, respectively.

The object of the network beamforming is to minimize the total relay transmit power subject to maintaining every destination SINR above a pre-defined threshold value γth (as a QoS parameter of the network). First, using Eq. (48) the total relay transmit power can be obtained as

PR=Eψt,ψt=EWHTuT0tT*WHTutt=T0=EutTW*H*HTWTutdt=ETrW*H*HTWTututTdt=traceEH*HTWTμW*=vecWTEH*HTTμvecW*=wTTw*E52

where

TEH*HTTμ,μut*uT0tTt=T0=utuTtdt=ρ1,1ρ1,dρd,1ρd,dE53

Note that using Eqs. (48) and (44), E(H*HT) can be obtained as

EH*HT=BDEν1*ν1T,,EνR*νRT,Eνr*νrT=Ek=1dfrk*sk*ρk*+nr*k=1dfrkskρkT+nrT=PkEk=1dfrk*frkρk*ρkTEnr*nrT,Enr*nrT=Eωr*tutdtωr*tuTtdt=Eωr*tωrtutuTtdt,Eωr*tωrt=σω2,utuTtdt=μE54

Using Eq. (50), the desired signal power at the kth destination can be obtained as

PSk=EρkTW*Hk*gk*gkTHkTWTρk=traceEHk*gk*gkTHkTWTρkρkTW*=vecWTτkTρkρkTvecW*=wTRkw*E55

where τkEHk*gk*gkTHkT=PkFkGkρkρkT, Fkfk*fkT,Gkgk*gkTand fk ≜ [f1kfRk]T, Rk ≜ τkT ⊗ ρkρkT. Also, the received noise power at kth destination is given by

PNk=EρkTW*Hnk*gk*gkTHnkTWTρk+σςk2=traceEHnk*gk*gkTHnkTWTρkρkTW*+σςk2=wTΝkw*+σςk2E56

where

ΝkϒkTρkρkT,ϒkEHnk*gk*gkTHnkT=EHnk*GkHnkT=σω2GkμE57

The relay noises are assumed to be zero-mean and independent with the equal noise power. So, we have

EωtωHtdt=σω2IR×R,ωtω1t,,ωRtTE58

Finally, the power of the received interference at the kth destination can be computed as

PIk=EρkTW*Hk*gk*gkTHkTWTρk=traceEHk*gk*gkTHkTWTρkρkTW*=vecWTθkTρkρkTvecW*=wTIkw*E59

where Ik ≜ θkT ⊗ ρkρkT and

θkEHk*gk*gkTHkT=l=1,lkdFlGlρlρlTPlE60

In this case, the instantaneous SINR for kth destination simply becomes the desired signal power of the desired signal to the power of interference plus noise. So, the optimization problem can now be written as

MinimizewwTTw*SubjecttoSINRk=wTRkw*wTΝk+Ikw*+σςk2γthkk12dE61

Since wTΝk+Ikw*+σςk20, the constraints of the optimization problem can be formulated as

wTRkγthkΝk+Ikw*γthkσςk2E62

In this problem, if all the matrices RkγthkΝk+Ikare negative semi-definite for all k, the problem is convex and can be solved uniquely. However, the feasible set of our optimization problem is empty since wTRkγthkΝk+Ikw*0for all K and W. Therefore, RkγthkΝk+Ikis non-negative definite matrix which results in non-convex inequality constraints, hence the QCQP problem is non-convex and NP-hard in general. However, we will show that a simple near optimal solution can be found in our problem. First, we replaced our QCQP problem with a semi-definite programming (SDP) problem. Let us define DkRkγthkΝk+Ik,Xw*wT, the optimization problem can recast to

MinimizeXtraceTXSubjecttotraceDkXγthkσςk2,k1dRankX=1,X0.E63

The problem is non-convex, because the Rank(X) = 1 constraint is non-convex. We relax the problem by ignoring this non-convex constraint and convert it to a convex SDP problem. The following semi definite representation (SDR) form is the relaxed version of the problem (Eq. 63).

MinimizeXtraceTXSubjecttotraceDkXγthkσςk2,k1dX0E64

This optimization problem has been solved in a same way as the previous sections. Figure 11 shows the total relay transmit power versus destination SINR threshold value, for different values of users’ correlation factors. The network consists of two source-destination pairs and four relays. Figure 11 shows that the total relay transmit power in all cases increases by raising γth. Furthermore, Figure 11 indicates that when the signature sequence correlation ρk,l increases, more total transmit power is needed to ensure SINR constraints at destination nodes. When ρk,l approaching one, the problem downgrades to the SDMA network and the system loses the benefits of CDMA technique. Also, increasing the signal dependency by increasing the correlation factor, results in the more infeasibility rate of the constraints.

Figure 11.

Minimum relay transmit power PTmin versus γth, for R=4, u=2.

Figure 12 displays the minimum relay transmit power versus γth for different number of relays and users. As normally expected, more power saving can be achieved by increasing the number of relays or decreasing the number of users. Comparing Figure 2 with Figure 3 reveals that decreasing the correlation factor will be much more efficient for saving network power than increasing the number of relays.

Figure 12.

Minimum relay transmit power PTmin versus γth.

Figure 13 shows the minimum relay transmit power versus the network data rate (D) for distributed CDMA, SDMA and TDMA schemes. In Figure 13, we consider a network with four relays and four source-destination pairs. For the sake of comparison fairness, we need to ensure that different schemes are compared with the same average source powers. So, we assume that the source power of CDMA and SDMA are one fourth of those in TDMA scheme. For Figure 13, the network data rate has the following relation to the SINR threshold value, D = w log2(1 + SINRth). Signature sequences of the user are randomly generated for 100 trials and the best code in term of least maximum correlation is chosen for performance comparison.

Figure 13.

Minimum relay transmits power versus D, for R=4, u=4.

Also, it can be seen from Figure 13 that the minimum relay transmitted power increases with the increase of D. For the SDMA scheme, the problem quickly becomes infeasible due to the power of interference at destinations. So, for establishing connections between four users, SDMA-based networks should use at least 40 relays to overcome the TDMA scheme. Since the QoS constraints are less stringent in CDMA scheme, the network can establish the communication between source-destination pairs for a larger range of D. Consequently, it can be observed from Figure 13 that the CDMA-based network can establish the source-destination connections with a significantly lower relay transmit power as compared to other schemes.

6. Computational complexity

Since the CDMA relay systems have a heavy computational complexity, the aim of this section is to analyze the computational form of related algorithms used in practice [20]. Here, the computational complexity of a standard SDP is introduced and extended to our case. The standard SDP problem with equality constraint is given as:

MinimizeXtraceCXSubjecttotraceAiX=bii1dX0E65

where C and Ai are symmetric n × n matrices, and b ∈ d.So for such a problem the complexity with large-update (or long-step) algorithm [21] based on the primal dual SDP algorithm is

Οnlognlogn/εE66

where ε denotes the accuracy parameter of the algorithm, while this algorithm with small-update (or short-step) still has Οnlogn/εiterations bound [22].

It is shown in Ref. [22] that small update interior point methods (IPMs) are restricted to unacceptably slow progress, while large-update IPMs are more efficient for faster. Also, large update IPMs perform much more efficiently in practice, however, they often have somewhat worse complexity bounds. The complexity order of solving standard SDP problem is polynomial time.

For evaluating the complexity of our SDP problem with inequality constraints, we have to calculate the dimension parameter n. Therefore, we should determine the dimensions of the matrices used in the objective and constraints of the problem Eq. (63). In the Kronecker product of two matrices, if A ∈ n × n and B ∈ m × m, then A ⊗ B will be a nm × nm matrix. According to the new vectors definite in Eq. (65) and sizes of μ ∈ d × d and E(H*HT) ∈ Rd × Rd, dimension of T will be TRd2×Rd2.

Similarly, we can obtain the above conclusion for Dk and X, that is, T,Dk,XRd2×Rd2. It is notable that the constraints of our problem are not the same as the standard SDP form. Therefore, we have to equalize them so that they alter to a type similar to the standard format. In order to achieve this goal, first we have to eliminate the inequality constraints of Eq. (64) by defining yi as:

traceDiX=γiσςk2+yi,X0,yi0fori=1,,dE67

Next, a new variable X^should be defined in order to standardize the problemml:

X^[X0R2d2×d0d×R2d2[y100yd]]E68

As a result, the following standard form will be attained.

MinimizeXtraceT^X^SubjecttotraceD^iX^=bi,X^0fori=1,,dE69

where

D^[D0Rd2×d0d×Rd20d×d],T^i[Ti0Rd2×d0d×Rd20d×d]E70

As a result of the above representation form, n for Eq. (63) would be:

nDistributed_Relay=Rd2+dRd2E71

Also, we can use the same procedure to calculate n for Eqs. (16) and (42):

nMIMO=R2+dR2nMIMO_CDMA=R2d2+dR2d2E72

Therefore, the complexity for problems (16), (42) and (63) for MIMO, MIMO-CDMA, and distributed-relay networks are as follows:

ΟRlogR2logR2/ε,ΟRdlogR2d2logR2d2/ε,ΟRd2logRd2logRd2/εE73

while a SDMA relay network has the complexity order of ΟRlogRlogR/ε.

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Mohammad-Hossein Golbon-Haghighi (December 14th 2016). Beamforming in Wireless Networks, Towards 5G Wireless Networks - A Physical Layer Perspective, Hossein Khaleghi Bizaki, IntechOpen, DOI: 10.5772/66399. Available from:

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