In this paper , in order to attain the maximum ergodic capacity and signifi cantly increase the spectral efficiency of wireless communication systems, novel linear beamforming is proposed for dual-hop amplify-and-forward (AF) multi-relay networks. The linear beamforming is designed based on the maximization of the signal-to-interference-plus-noise ratio (SINR) and signal-to- leakage-and-noise ratio (SLNR). The channel state information (CSI) is used in applying this new design to multi-relay (MR) nodes between the source and relays as well as relays and destination. The beamforming optimization problem is solved by using the Fukunaga-Koontz Transform (FKT). The scheme can achieve intra-node array and distributed gains by using multiple antennas and multi-relays (MRs). The performance of the proposed scheme demonstrates that by considering interference mitigation criteria the ergodic capacity at a significant level is improved as compared to the conventional techniques. Therefore, the proposed techniques based on the maximization of the signal-to-interference-plus-noise ratio (SINR) and signal-to-leakage-and-noise ratio (SLNR) relay processing outperform other conventional techniques in case of a multi-relay dual-hop network in terms of ergodic.
Due to high data rate, delivery dream, and interference, free demand of wireless services, the existing and future mobile communication technologies such as 4G and 5G, and cognitive radio are employing advanced technologies like MIMO and multi-relay (MR) systems. The technologies are being pursued to attend reliable communications with high data rate and low energy consumption providing high service quality. In wireless networks, multiple MIMO relays are used to achieve diversity in signal transmission and to improve overall network performance confrontations. Due to path loss, shadowing, multipath fading and signal interference effects, the channel capacity, reliability, and quality are debased in mobile communication. This is assumed as common transmission impairments of wireless transmission. To tackles these issues, micro-base stations are deployed which require substantial running and capital costs. For the future as well as existing wireless communication systems, a reliable and cost-effective technique is used called the relay-assisted transmission technique. This technique can enhance the quality of the wireless network between the source and destination and can improve base station coverage at the cell edge [
The amplify-and-forward (AF) relay technology is more attractive because of low complexity and simplicity of application and decode-and-forward (DF) relay in which processing delay occurs due to the decoding and encoding process [
In noise-limited scenarios, the MF beamforming technique is important to deliver better performance. However, the MF performance in interference-limited scenarios is poor [
However, this paper contributes to the design of relay receive beamforming based on SINR-Max using the FKT. It is worth mentioning that the scheme handles the noise and interference in both, the MR receives beamforming design and downlink precoding design. This novel relay processing design can achieve the distributed and intra-node array gain by controlling the interference at the reception of the relay nodes and intra-user interference because of the leakage at the destination. Intra-node array gain is achieved when multiple antennas are deployed at individual node of the network, whereas, the distributed array gain is obtained through various relay nodes connected in parallel [
In this research, a dual-hop single user AF MR-based MIMO (DH-AF-MIMO) network is studied as presented in
establish the communication between the source and destination. The source node first sends the information signal, processes it, and then forwards it to the destination by using MR nodes, which possesses multiple antennas at the individual relay. The relays with multiple antennas give better performance [
The information signal transmission is distributed into two phases. In the first phase, the modulated signal vector “s” is transmitted to MR nodes by the source as N s × 1 and it satisfies the source power constraint as:
E { s s H } = P s N s I N s
where;
P s = source transmit power
I N s = identity matrix of N s × N s .
The information signal at the kth relay node is as follows:
s r k = ∑ k = 1 K H k s + n r k (1)
where,
H k = [ H 1 T , H 2 T , ⋯ , H K T ] T , which is the channel matrix of dimension “ N r K × N s ”;
k = 1 , 2 , ⋯ , K .
E { n r k n r k H } = σ r 2 I N r
where;
σ r 2 = noise variance at the relay node.
In the second stage, linear processing is performed through the AF relay nodes by the multiplication of a beamforming matrix “ F k ” of “ N r × N r ” dimension with the received signal vector “ s r , k ” and then forwarded to the destination. The network performance is improved by the beamforming matrix and the relay transmit power constraint is satisfied as:
E { r k H r k } ≤ P k
where;
P k = relay transmit power
The transmitted signal by kth relays to the destination is shown as:
r k = ∑ k = 1 K F k H k s + n r k (2)
where;
r k = kth relay node transmit power
Therefore, the condition of a power constraint for the precoded signal that is transmitted by AF relay terminals is given as:
δ ( r k ) = t r { F k ( P s N s H k H k H + σ r k 2 I N r ) F k H } ≤ P k (3)
where;
ρ k = power normalizing coefficient [
This can also regulate the relay transmit power which is given as:
ρ k = P k δ ( r k ) . (4)
After relay linear processing, the received signal is shown as:
y = ∑ k = 1 K ρ ( r k ) G k F k H k s + ∑ k = 1 K G k F k n r k + n d (5)
where;
G k = complex channel matrix of “ N s × N r K ” dimension between the kth relay nodes k = 1 , 2 , ⋯ , K and the destination i.e. G k ∈ ℂ N d × N r channels where G k = [ G 1 , G 2 , ⋯ , G k ] .
The elements of g j , i symbolize the channel gain from the relay transmit antenna “i” to the destination receive antenna “j” i.e. i = 1 , ⋯ , N r and j = 1 , ⋯ , N d . Since, n d is n d ∈ ℂ N d CGN with zero-mean and covariance, which meets the noise variance as:
E { n d n d H } = σ d 2 I N d
where,
σ d 2 = noise power at the destination.
In relay receive beamforming, the cancellation of interference by the ZF technique will cost noise enhancement. Nevertheless, the SNR is maximized by minimizing the noise with the help of the MF beamforming technique but it does not take into consideration the interference cancellation. To deal with this issue, the SINR-Max criterion based on the relay receive beamforming is designed using the FKT. Therefore, the criterion like SINR-Max can provide a reconciliation between the noise and interference by eliminating the interference and minimizing the noise [
In this section, SINR-Max based relay receive beamforming for multiple AF relays is designed and discussed using the FKT. For relay receivers, the beamforming vectors can be used in such a way that at k relay nodes, the beamforming can maximize the SINR. The interference is controlled by the beamforming between multiple antennas of MRs for the network as seen in
H c = [ H 1 H , H 2 H , ⋯ , H K H ] (6)
where; H c represents the combined matrix.
For each relay receive antenna, the channel matrix is given as:
H ^ k l = [ H ^ k l H , H k − 1 H , H k + 1 H , ⋯ , H K H ] H (7)
In an MR system, the SINR, represented by “ γ k l ”, is obtained when the received signal power is divided by the indesired signal and noise power at k relays. It is expressed as:
γ k l = ‖ h k l b k l ‖ F 2 σ r l 2 + ∑ l = 1 l ≠ k L ‖ H i b k l ‖ F 2 + ∑ j = 1 j ≠ k N k ‖ h i b k l ‖ F 2 (8)
It can also be written as:
γ k l = b k l H ( h k l H h k l ) b k l B k l H H k l H H k l B k l + σ r k l 2 I N r (9)
where,
b k l = beamforming vector
h k l = complex row vector of the lth receive antennas of k relay nodes i.e. l = 1 , ⋯ , N k
σ r k l 2 = noise variance at lth antenna elements for kth relays i.e. σ r k l 2 = σ 1 l 2 , ⋯ , σ K l 2 and b k l = b 1 , ⋯ , b N k , whichis vector-by-vector relay beamforming for an individual antenna element of the kth relays.
The numerator term in (9) is representing the desired signal, while the term in the denominator is representing the combination of interference and noise power at the lth receive antennas of kth relays. The vectors are developed independently, based on the vector-by-vector relay beamforming. Therefore, the antenna-wise SINR-Max beamforming at “k” relays is set as:
B ^ k l = arg max B k l ∈ C N r × 1 S I N R k l ( B k l ) . (10)
Or
b k l = arg max b k l ∈ C N r × 1 b k l H ( h k l H h k l ) b k l B k l H H k l H H k j B k l + σ r k l 2 I N r . (11)
B k = [ b k 1 , ⋯ , b k N k ] .
where;
B k = beamforming matrix for k relays
k = 1 , ⋯ , K
l = 1 , ⋯ , N k
As the interference caused by the signal leakage cannot be canceled by using conventional techniques like the RZF technique, which uses an adjustable parameter to control the interference. However, the interference is not canceled out completely by the non-zero value of the regularization parameter. Although, the leakage-based downlink precoding can show robustness against the interference between multiple antennas at the destination. In this design, the interference due to leakage and noise at the destination side is taken into account for vector by vector downlink precoder design as discussed in [
The channel matrix between the destination and MR nodes at the downlink of the given system is represented by the notation “ G k ”.
G k = [ G 1 H , G 2 H , ⋯ , G K H ] H (12)
For jth antennas of k relay nodes to the destination, the channel matrix is given as:
G k j = [ G k − 1 H , G k + 1 H , ⋯ , G K H ] H (13)
The relay precoder design is developed with the relay power constraint as:
F ^ k j = arg max F k j ∈ C N r × 1 S L N R k j ( F k j ) .
s.t δ ( r k ) ≤ P r . (14)
where, “ F k j ” is the kth RTP for jth receiving antennas.
The SLNR per antenna is obtained when the desired signal power is divided by the leakage signal and noise power and is given by:
P A - S L N R k j = ‖ g k j f k j ‖ F 2 ∑ i = 1 j ≠ k L ‖ G k f k j ‖ F 2 + ∑ i = 1 j ≠ k N d ‖ g k f k j ‖ F 2 + σ d j 2 (15)
where;
g k j = complex signal of jth destination antennas where, j = 1 , ⋯ , N d
σ d j 2 = noise variance at the destination i.e. σ d j 2 = σ 1 j 2 , ⋯ , σ d j 2
f k j = vector-by-vector kth RTP for jth antenna elements i.e. f k j = f 1 , ⋯ , f N d
When (15) is maximized, the precoding vectors per antenna are generated as
f k j = arg max F k j ∈ C N r × 1 f k j H ( g k j H g k j ) f k j f k j H ( G k j H G k j + σ r k j 2 I N r ) f k j (16)
F k j = [ f 1 , ⋯ , f N d ]
Here “ F k j ” represents the kth RTP matrix for jth receive antennas at the destination, which supports maximizing the SLNR.
For this objective function, different precoders are selected to maximize the effectiveness of the wanted signal analogous to the interference and noise due to the leakage. The multiplication of relay receive beamforming matrix from (11) and (16) gives:
F = F k j × B k . (17)
For the selection of feature, Koontz and Fukunaga suggested a statistical method for high-dimensional data [
The U and D matrices are used to build the transformation operator P , which is also termed as the FKT [
P = U D − 1 / 2
D = diagonal matrix with eigenvalues, which are non-zero and positive singular values i.e. D = d i a g { λ 1 , ⋯ , λ r } , λ 1 ≥ ⋯ ≥ λ r > 0 .
U = identity matrix.
The matrices S 1 and S 2 are transformed into subspace by using the P operator with lesser dimension. After the transformation, two new matrices i.e. S ˜ 1 and S ˜ 2 are obtained which are given as:
S ˜ 1 = P T S P and S ˜ 2 = P T S P
where S ˜ 1 and S ˜ 2 are the transformed target and background matrices and their sum should be equal to the identity matrix i.e.
S ˜ 1 , and S ˜ 1 = I − S ˜ 2 . (18)
The same eigenvectors and eigenvalues are shared by S ˜ 1 and S ˜ 2 as these are transformed by using the same operator. If v is the eigenvector of S ˜ 1 with the eigenvalue λ 1 then, S ˜ 1 v = λ 1 v [
( I − S ˜ 2 ) v = λ 1 v . (19)
S ˜ 2 v = ( 1 − λ 1 ) v . (20)
In (18), S ˜ 1 holds important information for the targeted class whereas, S ˜ 2 has more significant information for the background class. The training stage is completed after obtaining these two matrices.
The optimization problem of the relay receive beamforming for k number of relays is formulated to FKT as follows:
S ˜ 1 = h k l H h k l . (21)
S ˜ 2 = H k H H k + σ r k 2 I N r . (22)
And their sum is given as:
S = S ˜ 1 + S ˜ 2 = H c H H c + σ r k 2 I N r . (23)
The FKT factor can be calculated by using the facts of (21) and (22) for individual relay receive antenna, which generates the shared eigenspace matrices S ˜ 1 and S ˜ 2 . They complement one another in a manner that the best-principled eigenvectors of the matrix S ˜ 1 are the least principled eigenvectors for the matrix S ˜ 2 and vice versa. The FKT factor, after multiplying with the eigenvectors of unitary matrix U , gives the eigenvalue of matrix D of transformed antenna covariance matrix [
The summary of the algorithm steps to compute the FKT factor for MR nodes is given in
The received signal is given as:
y d = ∑ j = 1 k ≠ j N d ρ G F H s + ∑ j = 1 k ≠ j N d G F s i + G F n r k + n d (24)
where;
G s d = ρ G F H
I = G F s i (Interference due to signal leakage)
z = G F n r k + n d (Noise at the destination).
According to [
R = [ r 11 r 12 r 13 0 r 22 r 23 0 0 r 33 ] . (25)
The destination filter based on QRD is given as:
W = ( Q s d ) H . (26)
The SINR is obtained when the transmitted signal is divided by the interfering signals and noise [
SINR = ( P s N d ) r j , j 2 ‖ ( ρ ( Q s d ) H G F ) j ‖ 2 σ r l 2 + σ d j 2 . (27)
where r j , j 2 are jth entries in the diagonal of the matrix “ R ”. Finally, the network ergodic capacity is given as:
| Steps | Activities |
|---|---|
| Input | SR channel matrices and noise variance at the relay terminal ( H k and σ r l 2 ) |
| 1. | Calculation ofthe summation of covariance matrices S ˜ 1 and S ˜ 2 . |
| 2. | Computation of the transform factor P from singular value decomposition of S |
| 3. | For l = 1 , ⋯ , N k relay receive antennas |
| 4. | For k = 1 , ⋯ , K number of relays |
| Using the FKT factor, transform the lth receive antennas covariance matrix S ˜ 1 from (21) | |
| Transformation of P to S ˜ 1 and selection of the first eigenvector v l of matrix S ˜ 1 | |
| b k l = P v l is the relay receive beamforming vector for lth antennasand k relay receive antennas | |
| 5. | Multiplication of B k l with F j |
| 6. | End |
| Output | F = B k l × F j the relay precoding matrix |
C e r g = E { H , G } { 1 2 ∑ j = 1 k ≠ j N d log 2 ( 1 + SINR ) } (28)
For the dual-hop multi-relay assisted network, the upper-bound capacity is given as:
C u p p e r = E { H k } k = 1 K { 1 2 log 2 det ( I N s + P s N s σ r 2 ∑ k = 1 K H H H ) } (30)
where,
P s = total source transmit power.
1/2 = time-sharing factor (for signals of the receiving and forwarding process.
In this section, the MCS is used to present the theoretical results of the proposed scheme. In all simulations, dimensionally uncorrelated MIMO channel matrices are developed between the source to the relay nodes and the relay nodes to the destination without loss of generality. These matrices are identically distributed, independent, and their entries are CG random variables with zero-mean components and unit-variance i.e. C N ( 0 , 1 ) . At the relay and destination terminal, the noise variances are considered equal i.e.
σ r 2 = σ d 2
For the validation of the performance of the proposed scheme, the performance results of the existing techniques are compared.
The performance of a DH-MR-MIMO network is evaluated in terms of the ergodic capacity. From the source node, independent data streams are transmitted by using separate antennas having equal distribution power. The SNR of the S-R and R-D channels are given as S N R s r = P s / σ r 2 and S N R r d = P k / σ d 2 . The channel knowledge is considered at the relay terminals for S-R and R-D channels [
The results from the MCS of the recommended beamforming technique for the DH-MR-network are balanced for 5000 independent channels to evaluate the performance of the suggested scheme. The performance of other available precoding methods that are discussed in [
This section discusses the evaluation of the suggested and compared schemes. The results are obtained by using different number of antennas at each node by taking various number of relay nodes. The ergodic capacity curves of the proposed scheme and other beamforming techniques are compared and shown in
The graph shows the results of the suggested and additional beamforming techniques. It is clear from the figure that at each SNR value, the recommended technique outperforms other techniques.
The regularization factor is used for the suggested scheme, which clearly shows that the MF-ZF technique is outperformed by the MF-RZF technique. When the value of the regularization parameter is reduced, then the scheme approaches the MF-ZF beamforming. Since the MF technique performs well at low SNR values, the MF-ZF technique outperforms the ZF-ZF technique only when the SNR values are low, although the interference is eliminated by the ZF beamforming that will cost the noise enhancement. The results clearly show that at low SNR values, the ZF-ZF scheme gives slow performance but it regains its performance at a high SNR value.
The plots of Figures 3-5 represent the efficiency of the suggested beamforming scheme at an increased number of relay nodes from one to six by keeping the
| Parameters | Value |
|---|---|
| InputRelay protocol | Amplify-and-forward |
| Network configuration | DH-MR-MIMO system as shown in |
| Antenna configurations at each node | N s = N r = N d = 3 , and N s = N r = N d = 6 |
| Noise variance | σ r 2 = σ d 2 |
| Channel between S-R and R-D is assumed | Rayleigh flat fading |
| S N R s r | S N R s r = P s / σ r 2 |
| S N R r d | S N R r d = P k / σ d 2 |
| Noise | AWGN |
| CSI | Considered perfect at the relay terminals |
same number of antennas i.e. three antennas at each node of the network. The distributed array gain is achieved from MR nodes, which helps in increasing the ergodic capacity, whereas, diversity gain is achieved by the number of antennas at each node. The efficiency of the proposed technique is validated by its performance gain against the interference in contrast with other techniques that control the interference between several antennas of MR nodes and destination nodes.
The results achieved by the suggested technique and other beamforming techniques are represented in the graph. The network is equipped with
N s = N r = N d = 3 and K = 3 (K is the number of relay nodes with three antennas at each node).
The results achieved by the suggested technique and other beamforming techniques are represented in the graph. The network is equipped with N s = N r = N d = 3 and K = 6 .
The graph represents the performance comparisons of the suggested scheme with other precoding schemes having perfect CSI at the relay node, under system configuration of N s = N r = N d = 6 and K = 6 .
The simulation results show that when SINR-Max is considered for beamforming, the design gives better performance alongside the inter-stream interference at
| No. | Beamforming at the AF relay nodes | Capacity gain for 3 antennas at each node and different number of relay nodes | ||
|---|---|---|---|---|
| Number of relay nodes | ||||
| 1 | 2 | 3 | ||
| bit/s/Hz | bit/s/Hz | bit/s/Hz | ||
| 1 | Proposed technique | 16.15 | 20.67 | 23.55 |
| 2 | MF-PA-SLNR | 14.26 | 18.44 | 21.17 |
| 3 | MF-RZF | 13.26 | 17.31 | 19.80 |
MR nodes. Whereas, the relay downlink provides unwanted signal reduction due to the signal leakage at the destination. Moreover, when the FKT is used in the proposed technique, it effectively solves the issue of optimization. It is clear from the FKT algorithm that the computational burden of beamforming is reduced because the transformation factor needs to be computed only one time for MRs.
However, when the number of antennas is increased at each terminal from three to four, the measured sum capacity using the proposed scheme is increased as compared to the MF beamforming at the relay receive side. Similarly, as the number of antennas rises to eight, the sum capacity also increases linearly. Based on the results and discussions mentioned above, it is concluded that the proposed technique for AF relay is more efficient as compared to other techniques. The performance of the whole system is enhanced in terms of the ergodic capacity by adding more relays due to the diversity order achieved by an additional number of relays. Moreover, the increasing number of antennas at individual nodes increases the capacity and decreases the number of relay nodes, which may result in complexity reduction.
A linear beamforming scheme has been proposed using the Fukunaga-Koontz Transform (FKT) to control the interference for amplify and forward multiple relay MIMO assisted dual-hop network, where the co-channel interference (CCI) was considered at relay receivers and interference at the destination node due to leakage signal. The performance of the proposed scheme demonstrates that by considering interference mitigation criteria the ergodic capacity at a significant level is improved as compared to the conventional techniques. Therefore, the proposed techniques based on the maximization of the signal-to-interference-plus-noise ratio (SINR) and signal-to-leakage-and-noise ratio (SLNR) relay processing outperform other conventional techniques in terms of ergodic capacity.
The results in this work provide a strong foundation for future work in the performance of wireless network. One area of the future work is to optimize the Performance of Massive MIMO systems, to limit inter-cell interference and to improve the use of the available frequency bands by covering cognitive radio and mobile to mobile communications(D2D), to guarantee a level of end-to-end quality of service in different mobile wireless environments. Another area is maturing enough beamforming algorithms in order to exploit Millimeter wave to its full potential. It requires improvement in the networks when personal base stations (BSs) and satellite connectivity can get merged into cellular communication.
This work would not have been possible without the financial support of Doctoral School of Science and Technology of the University of N’Djamena’s, EcoleNationaleSuperieure des Technologies de l’Informationet de la Communication (ENASTIC) and Islamic Developement Bank (IsDB).
The authors declare no conflicts of interest regarding the publication of this paper.
Ali, A.M., Adoum, B.A., Bachar, I.S. and Khayal, M.Y. (2021) Beamforming Scheme for MIMO Relay Based 5G and beyond Wireless Network. Open Journal of Applied Sciences, 11, 1157-1176. https://doi.org/10.4236/ojapps.2021.1110087