Robust Beamforming Techniques for Non-Orthogonal Multiple Access Systems with Bounded Channel Uncertainties
Faezeh Alavi Kanapathippillai Cumanan Zhiguo Ding Alister G. Burr











Robust Beamforming Techniques for Non-Orthogonal Multiple Access Systems with Bounded Channel Uncertainties
    Faezeh Alavi, Kanapathippillai Cumanan, Zhiguo Ding and Alister G. Burr


    Received: date / Accepted: date
===========================================================================================================




In this letter, we propose a robust  
for non-orthogonal multiple access (NOMA) based multiple-input single-output (MISO)
downlink systems.
In , the robust power minimization problem is studied with imperfect channel state information (CSI),
where the beamformers are designed by incorporating norm-bounded channel uncertainties to provide the required quality
of service at each user. This robust scheme is developed based on the worst-case performance optimization framework.
In terms of beamforming vectors, the original robust design is not convex and therefore, the robust beamformers cannot
be obtained directly. To circumvent this non-convex issue, the original intractable problem is reformulated into a
convex problem, where the non-convex constraint is converted into a linear matrix inequality (LMI) by exploiting S-Procedure.
Finally, simulation results are
provided to demonstrate the effectiveness of the proposed robust
design.



Non-orthogonal multiple access (NOMA), multiple-input single-output
(MISO), robust beamforming, worst-case performance optimization






§ INTRODUCTION



Non-orthogonal multiple access (NOMA) is a promising multiple access technique for 5G networks

which has the potential to address the 
issues associated with the exponential growth of data traffic
such as spectrum scarcity and massive connectivity <cit.>.
In contrast to conventional multiple access schemes,
NOMA allows different users to efficiently share the same resources (i.e., time, frequency and code)
at different power levels so that the user with lower channel gain is served with a higher power and vice versa.
In this technique, a successive interference cancellation (SIC) approach is employed at receivers to separate multi-user
signals, which significantly enhances
the overall spectral efficiency.
In other words,
NOMA has the capability to control the
interference by sharing resources
while increasing system throughput
with a reasonable additional complexity <cit.>.

Recently, a significant amount of research has focused in studying
several practical issues in NOMA scheme.
In particular, beamforming designs for multiple antenna NOMA networks have received  
a great deal of interest in the research community due to their additional degrees of freedom and diversity gains <cit.>. 
A general framework for a multiple-input multiple-output (MIMO) NOMA system has been developed for both the downlink
and the uplink in <cit.> whereas 
the throughput maximization problem was studied for a two-user MIMO NOMA system in <cit.>.
The sum  rate maximization problem for a multiple-input single-output (MISO) NOMA has been investigated in <cit.>
through the minorization maximization algorithm. 
In most of the existing work, beamforming designs have been proposed  
for NOMA schemes with the assumption of perfect channel state information (CSI) at the transmitter <cit.>.
However, 
this assumption might not be always valid for practical scenarios 
due to channel estimation and quantization errors <cit.>. On the other hand, channel uncertainties significantly influence the performance of the

SIC based receivers as the 


decoding order of the received multi-user signals is determined with respect to the users' effective channel gains.


Therefore, it is important to take into account the channel
uncertainties especially in the beamforming design for NOMA networks.
Motivated by this practical constraint, we
focus on robust beamforming design based on the worst-case performance optimization framework
to tackle the norm-bounded channel uncertainties <cit.>.
In <cit.>, the robust beamforming design has been developed
for providing secure communication in wireless networks with   
imperfect CSI.
 
By incorporating the 
bounded channel uncertainties, the robust sum power minimization
problem 
is investigated in <cit.>
for a downlink multicell network with the worst-case signal-to-interference-plus-noise-ratio (SINR) constraints
whereas the robust weighted sum-rate maximization was studied for multicell downlink MISO systems in <cit.>.
In <cit.>, a robust minimum mean square error based beamforming technique is
proposed for multi-antenna relay channels with imperfect CSI between the relay and the users.
In the literature, there are two types of NOMA schemes considered:
I) clustering NOMA <cit.>, II) non-clustering NOMA <cit.>.
In the clustering NOMA scheme, all the users in a cell are grouped into N clusters with two users
in each cluster, for which a transmit beamforming vector is designed to support those two users
through conventional multiuser beamforming designs. The users in each cluster are supported by a NOMA
beamforming scheme. However, in the non-clustering NOMA scheme, there is no clustering and each
user is supported by its own NOMA based beamforming vector.


In <cit.>, the authors studied a robust NOMA scheme for the MISO channel
to maximize the worst-case achievable sum rate with a
total transmit power constraint.

In this letter, we follow the second class of research where NOMA scheme applied between all users and there is the spectrum
sharing between all users in cell. Then, we propose a robust beamforming design for NOMA-based MISO
downlink systems.
In particular, the robust power minimization problem is solved
based on worst-case optimization framework to provide the required quality of service at each user
regardless of the associated channel uncertainties.
By exploiting  the original  problem is
converted into a convex one by recasting the non-convex constraints into linear matrix inequality (LMI) forms.
Simulation results are provided
to validate the  
of the robust design by comparing the performance of the robust scheme with that of the non-robust approach.
The work in <cit.> also studied the worst-case based robust scheme
for MISO NOMA system, however, there are main differences between our proposed scheme and the work in <cit.>.
A clustering NOMA scheme is developed in <cit.> by grouping
users in each cluster.
In this scheme, a single beamformer is designed to transmit the signals for all users
in the same cluster whereas, in this letter, the signal for each user is transmitted with a dedicated beamformer.
In addition, both beamforming designs are completely different as the work in <cit.>
proposes robust sum-rate maximization based design whereas this letter solves robust power minimization problem
with rate constraint on each user. In terms of solutions, the work in <cit.>
exploits the relationship between MSE and achievable rate and derives an equivalent non-convex problem, which
is decoupled into four sub-problems and those problems are iteratively solved to realize the solution of the original problem.
In this letter, the robust power minimization problem is formulated by deriving the worst-case achievable rate.
The original problem formulation turns out to be non-convex and we exploit S-Procedure and semidefinite
relaxation to convert it to a convex one. Hence, the work in <cit.> and the proposed work in
this letter are different including problem formulation and the solution approaches.





§ SYSTEM MODEL AND PROBLEM FORMULATION

We consider NOMA-based downlink transmission where a
base station (BS) sends information to K users 
U_1,U_2, …, U_K.

It is assumed that the BS is equipped with M antennas whereas each user consists of a single antenna. 
The channel coefficient vector between the BS and the k^th user U_k is denoted by 𝐡_k ∈ℂ^M × 1  (k = 1,…,K)
and 𝐰_k∈ℂ^M × 1  represents the corresponding beamforming vector of the k^th user U_k.
The received signal at U_k is given by

    y_k = 𝐡_k^H 𝐰_k s_k + ∑_m≠ k𝐡_k^H 𝐰_m  s_m + n_k,     ∀ k,




where s_k denotes the symbol intended for U_k

and  represents a zero-mean additive white Gaussian noise with variance σ_k^2.
The power of the symbol s_k is assumed to be unity, i.e., 𝔼(|s_k|^2)=1. In practical scenarios, it is difficult to provide perfect CSI at the transmitter due to  
channel estimation and quantization errors. Therefore, we consider a robust beamforming design to overcome these channel uncertainties.


In particular, we incorporate norm-bounded channel uncertainties in the design as




    𝐡_k=ĥ_k +Δĥ_k,  Δĥ_k_2=𝐡_k-ĥ_k_2 ≤ϵ,




where ĥ_k, Δĥ_k and
ϵ≥ 0 denote the estimate of 𝐡_k, the norm-bounded channel estimation error and the channel estimation error bound, respectively.

In the NOMA scheme, user multiplexing is performed in the power domain
and the SIC approach is employed at receivers to separate signals between different users.
In this scheme, users are sorted

based on the norm of their channels,

i.e., 𝐡_1_2≤𝐡_2_2≤…≤𝐡_K_2.
For example, the k^th user decodes the signals intended for the users from U_1 to U_k-1
using the SIC approach 
whereas the signals intended for the rest of the users (i.e., U_k+1,…,U_K) are treated as interference at the k^th user.
Based on this SIC approach, the l^th user can detect and remove
the k^th user's signals for 1 ≤ k < l <cit.>.


Hence, the signal at the l^th user after removing the first k-1 users' signals to detect the k^th user is represented as




    y_l^k=𝐡_l^H 𝐰_k s_k+∑_m=1^k-1Δĥ_l 𝐰_m s_m+∑_m=k+1^K𝐡_l^H 𝐰_m s_m+n_l,





    ∀ k,   l∈{k,k+1,…,K},




where the first term is the desired signal to detect s_k and 
the second term is due to imperfect CSI at the receivers during 
the SIC process. 
Due to the channel uncertainties,  the signals intended for the users U_1,…,U_k-1 cannot be completely removed by the l^th user. The third term is the interference
introduced by the signals intended to the users U_k+1,…,U_K. 


According to the SIC based NOMA scheme, the l^th user should be able to detect
all k^th (k<l) user signals. Thus,
the achievable rate of U_k can be defined as follows:




    R_k=log_2 (1+min_l ∈{k, k+1, …, K} SINR^k_l),




where SINR^k_l 
denotes the SINR 
of the k^th user's signal at the l^th user which can be written as

    SINR_l^k= 𝐡_l^H 𝐰_k𝐰_k^H 𝐡_l∑_m=1^k-1Δĥ_l^H 𝐰_m𝐰_m^H Δĥ_l+∑_m=k+1^K𝐡_l^H 𝐰_m𝐰_m^H𝐡_l+σ_l^2.




For this network setup, we study robust power minimization by incorporating channel uncertainties to satisfy the required SINR at each user. This robust beamforming design is developed by considering the worst-case SINR of each user, which can be formulated as





    min_𝐰_k∈ℂ^M×1∑_k=1^K𝐰_k^2_2, 
       𝐬.𝐭.min_Δĥ_l_2 ≤ϵ(min_l ∈{k, k+1, …, K}SINR^k_l)≥γ_k^min,  ∀ k,





where γ_k^min=(2^R_k^min-1) is the minimum required SINR to achieve a target rate R_k^min at U_k.





§ ROBUST BEAMFORMING DESIGN







The problem formulation in (<ref>) is not convex and the optimal robust beamformers cannot be obtained directly. To tackle this issue, we introduce a new matrix variable 𝐖_k=𝐰_k 𝐰_k^H
and reformulate the original robust problem in (<ref>) into the following optimization framework without loss of generality:






    min_𝐖_k∈ℂ^M× M∑_k=1^K Tr(𝐖_k), 
       𝐬.𝐭.  ϖ_kl,   ∀ k,  l=k,…,K,
           𝐖_k ≽0, rank(𝐖_k)=1,     ∀ k,


where ϖ_kl is defined in Appendix <ref>.










However, the reformulated problem in (<ref>) is still not convex
for two reasons; the rank-one constraint and unknown channel uncertainties, i.e., Δĥ_k,
which lead to an intractable problem.
The rank-one constraint in (<ref>)  
can be relaxed by exploiting semi-definite relaxation (SDR).
To remove the unknown channel uncertainties and solve the original problem with available knowledge of imperfect CSI (error bound), we employ  to recast the non-convex constraints into LMIs. 




Lemma 1:
By relaxing the rank-one constraints on 𝐖_k, the original problem in (<ref>) can be recast into
the following convex problem: 





    min_[ 𝐖_k∈ℂ^M× M,;     λ_kl≥ 0 ]∑_k=1^K Tr(𝐖_k), 
       𝐬.𝐭. 
    𝐖_k ≽0, 𝐂_kl≽0,   ∀ k,  l=k,…,K





where 𝐂_kl is defined in Appendix <ref>.











 Please refer to Appendix <ref>.





The problem in (<ref>) 
is a standard semidefinite  (SDP) and can be efficiently solved using interior-point 
The optimal solution for the original problem in (<ref>) can be
obtained through extracting the eigenvector corresponding to the maximum
eigenvalue of the rank-one solution of (<ref>).
Thus, the following lemma
holds to show that the optimal solution to (<ref>) is rank one.




Lemma 2:
Provided the problem in (<ref>) is feasible, there always exists a rank-one optimal solution {𝐖^*_k}.








 Please refer to Appendix <ref>.

































































§ SIMULATION RESULTS

To assess the performance of the proposed robust beamforming approach, we consider a single cell downlink transmission, where a multi-antenna BS serves single-antenna users which are uniformly distributed over the circle with a radius of 1000 meters around the BS, but no closer than d_0=100 meters. 




The small-scale fading of the channels is Rayleigh which represents an isotropic scattering environment. We model the large-scale fading effects as the product of path loss and shadowing fading. The log-normal shadowing is considered with standard deviation σ_0=8 dB, scaled by (d_k/d_0)^-β to incorporate the path-loss effects where d_k is the distance between U_k and the BS, measured in meters and β=3.8 is the path-loss exponent. Throughout the simulations, it is assumed that the BS is equipped with eight antennas (M = 8) and it serves three users (K = 3). The noise variance at each user is assumed to be 0.01 (i.e., σ_k^2 = 0.01)  
and the target rates for all users are the same. 
The term “Non-robust scheme" refers to the scheme where
the BS has imperfect CSI without any information on the channel uncertainties and the beamforming
vectors are designed based on imperfect CSI without incorporating channel uncertainty information.




First, we study the impact of channel uncertainties on the required total transmit power.
Fig. <ref> depicts the required total transmit power against
different SINR thresholds for the robust and the non-robust NOMA schemes as well as OMA scheme with different error bounds.

As seen in Fig. <ref>,
the robust scheme requires more transmit power
than that of the non-robust scheme.
This is because the robust scheme satisfies the required
SINR all the time, at the price of more transmit power at the BS
whereas the non-robust scheme does not.


The difference between the required transmit power for the robust and the non-robust schemes increases with error bounds.
This is because incorporating all possible sets of errors in the beamforming design to satisfy high SINR thresholds
requires more transmit power in the robust scheme.
Moreover, as seen in , the conventional framework, orthogonal multiple access (OMA), requires
more transmit power to achieve the same rate in comparison with NOMA scheme. This demonstrates that the NOMA scheme
yields a better performance in terms of spectral and energy efficiencies.

Next, we evaluate the performance of the proposed robust and non-robust schemes in terms of the minimum achieved SINR between users.


Fig.  <ref> provides cumulative  function (CDF) and probability density function (PDF)  from 1000 random sets of channels with error bounds of 0.06  (ϵ = 0.06)
where the SINR threshold has
been set to 10 dB at each user.
As evidenced by the results, the robust scheme outperforms the non-robust scheme
in terms of minimum achieved SINRs. In addition, the robust scheme satisfies the SINR thresholds all the time
regardless of the channel uncertainties whereas
 the non-robust design fails to
satisfy the minimum SINR requirements. 






















§ CONCLUSION




In this letter, we propose a robust beamforming
design for the downlink of a NOMA based MISO network
by taking into account the norm-bounded channel uncertainties.
However, the original robust problem formulation is not convex due to the imperfect CSI.




To cope with this challenge, we exploited 
to reformulate the original non-convex problem into a convex optimization
framework by recasting the original non-convex constraints into an LMI form.






Simulation results demonstrate that the
proposed robust scheme offers a better performance than the non-robust approach
by satisfying the SINR requirement at each user all the time regardless of associated channel uncertainties.



equationsection





§ DERIVATION OF Π_KL
 
















The equivalent transformations of (<ref>) can
be obtained as ϖ_kl as follows:




    {[ min_Δĥ_k_2 ≤ϵ(𝐡^H_k 𝐖_k 𝐡_k/∑_m=1^k-1Δĥ_k^H 𝐖_m Δĥ_k +∑_m=k+1^Kh^H_k 𝐖_m 𝐡_k +σ_k^2)
    ≤γ^min_k,;   min_Δĥ_k+1_2 ≤ϵ(𝐡^H_k+1𝐖_k𝐡_k+1/∑_m=1^k-1Δĥ_k+1𝐖_m Δĥ_k+1+∑_m=k+1^K𝐡^H_k+1𝐖_m 𝐡_k+1+σ_k+1^2);                                                                                      ≤γ^min_k,;                                                                                              ⋮;   min_Δĥ_K_2 ≤ϵ(𝐡^H_K 𝐖_k 𝐡_K/∑_m=1^k-1Δĥ_K 𝐖_m Δĥ_K +∑_m=k+1^K𝐡^H_K 𝐖_m 𝐡_K +σ_K^2)
    ≤γ^min_k, ].
    
       ⇔min_Δĥ_l_2 ≤ϵ(𝐡^H_l 𝐖_k 𝐡_l/∑_m=1^k-1Δĥ_l^H 𝐖_m Δĥ_l +∑_m=k+1^K𝐡^H_l 𝐖_m 𝐡_l+σ_l^2)
    ≤γ^min_k 
       ≜ϖ_kl,  ∀ k,  l=k,…,K.







§ PROOF OF LEMMA 1 
 



To incorporate the channel uncertainties in the robust
optimization framework, we exploit S-procedure 
to convert the non-convex constraint into LMI form.





























By applying  <cit.>, the constraint (<ref>) is 

    Δĥ_l^H 𝐈Δĥ_l -ϵ^2 ≤ 0 ⇒Δĥ_l^H (∑_m≠ k𝐖_m - 𝐖_k/γ^min_k )Δĥ_l





    +2Re{ĥ_l^H(∑_m=k+1^K𝐖_m -𝐖_k/ γ^min_k)Δĥ_l }+ĥ_l^H
    (∑_m=k+1^K𝐖_m
        - 𝐖_k/ γ^min_k )ĥ_l + σ_l^2 ≤ 0,




Then, the constraint (<ref>) can be reformulated with  as the following semidefinite constraint

    𝐂_kl=  [[               λ_kl𝐈+ϕ_k+ν_k                     ϕ_k ĥ_l;                   ĥ_l^H ϕ_k ĥ_l^H ϕ_k ĥ_l-σ_k^2-λ_klϵ^2 ]] ≽0,

where 


This completes the proof of Lemma 1.             ▪




§ PROOF OF LEMMA 2 
 



To prove Lemma 2, we examine the Karush-Kuhn-Tucker
(KKT) conditions of (<ref>). First, let 

𝐘_k∈ℂ^M× M,  and μ_kl∈ℝ_+
denote the dual variable of the
constraints in (<ref>), 
respectively. Then, the Lagrangian
dual function of (<ref>) can be written as




    ℒ(𝐖_k,λ_kl,𝐓_kl,μ_kl,𝐘_k)=∑_k  Tr(𝐖_k)-∑_k  Tr(𝐘_k𝐖_k)-
       ∑_k,l Tr(𝐓_kl𝐀_1)-∑_k,l Tr[𝐓_kl𝐇^H_lϕ_k 𝐇_l]-∑_k,l Tr(𝐓_kl𝐀_2),







where 𝐇_l=[𝐈  𝐡_l] and




    𝐀_1=([           λ_kl𝐈  0; 0   -σ_k^2-λ_klϵ^2 ]),   𝐀_2 = ([ ν_k   0; 0     0 ]).




The following KKT conditions hold for (<ref>) 




    ∂ℒ/∂𝐖_k=0⇒𝐘_k+𝐇_l𝐓_kl𝐇^H_l/γ^min_k= 𝐈
                   + ∑_j=1^k-1𝐇_l𝐓_jl𝐇^H_l
    +∑_j=k+1^K𝐓_jl,
       𝐖_k𝐘_k=0,
       (𝐀_1+ 𝐇^H_lϕ_k 𝐇_l+ 𝐀_2)𝐓_kl=0.


We premultiply (<ref>) by 𝐖_k, i.e.,




    𝐖_k 𝐇_l𝐓_kl𝐇^H_l/γ^min_k=𝐖_k (𝐈 + ∑_j=1^k-1𝐇_l𝐓_jl𝐇^H_l
    + ∑_j=k+1^K𝐓_jl),




Then, we can write the following rank relation




    rank(𝐖_k)   =rank[𝐖_k (𝐈+∑_j=1^k-1𝐇_l𝐓_jl𝐇^H_l
    +∑_j=k+1^K𝐓_jl)]
       =rank(𝐖_k 𝐇_l𝐓_kl𝐇^H_l)
       ≤min{rank(𝐇_l𝐓_kl𝐇^H_l), rank(𝐖_k)}.

Based on (<ref>), it is required to show
rank(𝐇_l𝐓_kl𝐇^H_l)≤1 if we claim rank(𝐖_k)≤1.

First, we consider the following equations and Lemma 3:


    [𝐈  0]𝐇^H_l=𝐈,   [𝐈  0]𝐀_1=λ_kl(𝐇^H_l-[0_M   𝐡_l]),
       [𝐈  0]𝐀_2=ν_k(𝐇^H_l-[0_M   𝐡_l]),




Lemma 3: If a block Hermitian matrix 
then the main diagonal matrices 𝐁_1 and 𝐁_4 must be positive
definite (PSD) matrices <cit.>. 










We pre-multiply [𝐈  0] and post-multiply 𝐇^H_l
by (<ref>), respectively, and applying the equalities in (<ref>):




    λ_kl(𝐇^H_l-[0_M  𝐡_l])𝐓_kl𝐇^H_l+ν_k(𝐇^H_l-[0_M  𝐡_l])𝐓_kl𝐇^H_l
       +ϕ_k𝐇_l𝐓_kl𝐇^H_l=0⇒
       (λ_kl𝐈+ϕ_k+ν_k)𝐇_l𝐓_kl𝐇^H_l=(λ_kl𝐈+ν_k)[0_M  𝐡_l]𝐓_kl𝐇^H_l


By applying Lemma 3 to (<ref>), we can claim (λ_kl𝐈+ϕ_k+ν_k)≽0
and is nonsingular; thus, multiplying by a nonsingular
matrix will not change the matrix rank. Thus, the following
rank relation holds:




    rank(𝐇_l𝐓_kl𝐇^H_l)=   rank((λ_kl𝐈+ν_k)[0  𝐡_l]
     𝐓_kl𝐇^H_l)
       ≤rank([0  𝐡_l])≤ 1.




This completes the proof of Lemma 2.             ▪


















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