Novel Channel Aware Power Control for a Multi-User Downlink NOMA Network

In this letter, we consider a downlink multi-user (MU) non-orthogonal multiple access (NOMA) network. We demonstrate that utilizing knowledge of the channel gains to determine the NOMA power allocation coefficients can dramatically improve throughput performance. Considering practical imperfect successive interference cancellation, expressions are derived for the optimum power allocation that ensures the minimum outage probability for the signalling scheme. The level of successive interference cancellation at each user and the decoding order are specified. It is shown that the proposed decoding order and the power allocations result in the maximum throughput. Expressions are derived for the throughput with these power allocations. Channel knowledge is exploited to determine the minimum power required for non-outage, and an expression is derived for the average value of this minimum power requirement. Computer simulations validate the derived expressions.

Novel Channel Aware Power Control for a Multi-User Downlink NOMA Network Anand Jee , Graduate Student Member, IEEE, and Shankar Prakriya , Senior Member, IEEE Abstract-In this letter, we consider a downlink multi-user (MU) non-orthogonal multiple access (NOMA) network.We demonstrate that utilizing knowledge of the channel gains to determine the NOMA power allocation coefficients can dramatically improve throughput performance.Considering practical imperfect successive interference cancellation, expressions are derived for the optimum power allocation that ensures the minimum outage probability for the signalling scheme.The level of successive interference cancellation at each user and the decoding order are specified.It is shown that the proposed decoding order and the power allocations result in the maximum throughput.Expressions are derived for the throughput with these power allocations.Channel knowledge is exploited to determine the minimum power required for non-outage, and an expression is derived for the average value of this minimum power requirement.Computer simulations validate the derived expressions.
Index Terms-Non-orthogonal multiple access, channel state information, outage probability, throughput, energy efficiency.

I. INTRODUCTION
T HE EMERGENCE of Internet of Things (IoT), massive machine-type communication (mMTC), and wireless services requiring low latency, high spectral efficiency (SE), and better energy efficiency (EE) [1] has increased the need for efficient utilization of the limited radio resources as well as the available energy, causing a fundamental change in the realm of wireless communication [2].In this context, Non-Orthogonal Multiple Access (NOMA) has surfaced to fulfill the demands/requirements of mMTC [3] by allowing multiple users to share the same time/frequency resources while assigning different power levels to each user [4].This is in contrast to traditional orthogonal multiple access schemes (OMA), where orthogonal resources are assigned to users to avoid interference [5].
In NOMA, users are distinguished by their power levels, and their symbols are superimposed and transmitted concurrently [6].At the receiver, the symbols are decoded by successive interference cancellation (SIC) [5].However, the performance of NOMA depends heavily on the power allocation scheme used [7].The conventional power allocation methods for NOMA [6], [7], [8], [9], such as Proportional Fairness (PF) [6], [7], [8], and Max-Min Fairness (MMF) [9], do not fully exploit the channel state information (CSI) to allocate power to users.Further, the optimal power allocation for max-min fairness in NOMA systems while considering joint beamforming is presented in [10].Joint user pairing and power The authors are with the Department of Electrical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India (e-mail: anandjee7@ ee.iitd.ac.in; shankar@ee.iitd.ac.in).
Digital Object Identifier 10.1109/LWC.2023.3330150allocation is considered in [11], [12] to maximize the sum rate, and it is demonstrated that performance of the NOMA system with optimal user pairing is significantly better than with random pairing or OMA.In mMTC applications [13], where the number of devices can be very large, these conventional methods may not be sufficient to meet the high throughput and low latency requirements.A power allocation scheme for multi-user downlink NOMA has been suggested in [14] to maximize the minimum sum rate of users.To address the limitations of conventional power allocation [15], we propose a novel CSI-assisted power control for a multi-user (MU) downlink NOMA network.The proposed CSI-based power control adjusts the power allocation to users dynamically based on their channel gains, which can vary widely due to the fading nature of wireless links.Therefore, this dynamic power allocation can significantly improve the overall throughput (and SE) of the system, which is crucial for mMTC applications.Additionally, the proposed framework has the potential to be a reliable solution for supporting mMTC applications in IoT systems.Due to its importance in facilitating transmission to a large number of devices concurrently, this problem of practical interest is studied in this letter.The novel primary contributions in this letter are as follows: • We investigate (for the first time) the performance of a MU downlink NOMA network.K carefully selected subset out of N users are served concurrently.The proposed CSI-assisted power control scheme operates at the source, where the power levels are adjusted to achieve high throughput.• The generalized throughput-optimal power allocation coefficients for users being served are obtained in closed-form while considering the effect of imperfect SIC (which is essential from the practical point of view).• The level of the SIC at each of the K selected user and the decoding order to be used are specified.Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.U n and the source by h n ∼ CN (0, 1/λ).Note that λ = d L SU , where L is the path-loss exponent, and d SU is the Euclidean distance between the source and user.The channel coefficients are assumed to be of quasi-static Rayleigh fading type, and change independently from one signalling frame to another.The additive zero-mean complex Gaussian noise at all the receiving nodes is assumed to be of variance N o .Denote by P S the transmit power used by the source.After a pilot is transmitted by the source, a timer-based mechanism [18] can be used to identify users with ordered channel gains.The source therefore has knowledge of |h n | 2 for users scheduled to transmit.Let g n = |h n | 2 .Denote the ordered channel gains by g (n) with g (1) ≤ g (2) ≤ . . ., ≤ g (N ) and the corresponding ordered users by U (n) .Denote the symbols transmitted to the user U (n) by s (n) .In this letter, it is assumed that K out of these N users are picked for downlink transmission using NOMA in any signalling frame in an effort to increase access to the spectrum for users being served.In a traditional NOMA scheme, strong users are paired with weak users.Here, we assume that the user U (N ) with the strongest channel gain g (N ) is selected.We then look for K − 1 other users to receive data concurrently.We assume that the users U (N −l k ) are picked, with k = 1, 2, . . ., K and l 1 = 0.The source uses superposition coding (SC) to transmit the symbols of all the users simultaneously.The superimposed signal transmitted by the source is received at all the users where α (N −l k ) denotes the fraction of overall transmit power allocated to s (N −l k ) .The received signal sample at U (N −l k ) is expressed as (1) where k = 1, 2, . . ., K, and w (N −l k ) ∼ CN (0, N o ) is the additive Gaussian noise sample at U (N −l k ) .The users use SIC to decode the symbols.In a typical NOMA setup, the user U (N −l K ) with the weakest channel gain only decodes its own symbols (does not perform SIC), while the one with the next weakest channel gain U (N −l K −1 ) decodes s (N −l K ) first, cancels it, and then decodes s (N −l K −1 ) .The user U (N ) = U (N −l 1 ) with the strongest channel gain performs N − 1 SICs to decode its own symbol s (N ) .In the presence of channel knowledge at the source, it is not immediately clear if the user selection scheme and the traditional decoding order are optimal.Denote the signal-to-interference-plus-noise ratio (SINR) to decode the (N − l ) th symbol at the (N − l m ) th user by Γ ) .It is clear that with SIC, the SINR can be expressed as where No and denotes the fraction of the residual interference power caused by imperfect SIC ( ∈ [0, 1)).The above is valid provided m ≤ .In the above, it is understood that when + 1 > K , the summation is taken to be zero.

III. PERFORMANCE ANALYSIS
In the following, the NOMA power allocations that result in minimum outage probability are determined, and the performance with this optimal power allocation is analyzed.The outage probability p o is defined as where γ th = 2 R − 1 is the threshold signal-to-noise ratio (SNR) with R denoting the target rate.Note that the above ensures the success of each stage of the SIC for all the users.It is clear from (2) that Γ The NOMA power allocations that result in minimum outage is Proof: Noting that l 1 = 0 and equating the Γ Using this α (N ) in Γ , and then equating it to γ th , we have Substituting α (N ) , α (N −l 2 ) from ( 5) and ( 6) into Γ ) and then equating to γ th , α (N −l 3 ) is obtained as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Similarly, substituting successively in Γ (N −l ) (N −lm ) and performing mathematical manipulations, the expression for the remaining power allocation coefficients can be obtained, and the generalized expression for the power allocation coefficient can be expressed as in (4).
The outage condition in (3) is rewritten in terms of the sum of NOMA power allocations of K users being served, and is mathematically expressed as in the following Lemma.
Lemma 2: The outage condition can be written in terms of the NOMA power allocations as (7) Proof: Successful decoding of the respective symbols at all the K selected users is possible if Therefore, the outage probability can be expressed as Substituting α (N −l k ) from ( 4), p o is obtained as in (7).Corollary 1: A highly accurate approximate closed-form expression for the outage probability is given by where Ψ 1 = 1+γ th 1+ γ th and Ψ 4 = 1+ γ th γ th − (1 + Ψ 1 ).Proof: For a detailed proof, please refer to Appendix A. Lemma 3: A high SNR expression for p o is given by , (10) Proof: By neglecting the higher order terms of 1 ρ S in (9) and using e θ ≈ 1 − θ, (10) can be readily obtained.
Remark 1: It is clear from Lemmas 3 that the SIC error effects the performance even in the high SNR region and an increase in ρ S improves the outage performance.
Corollary 2: The user selection that minimizes the outage is Proof: It becomes evident from the expression of α (N −l k ) in (4) that the power allocation is inversely proportional to the channel gains, and hence selecting a subset of users with the maximum channel gains results in minimum α (N −l k ) .Thus, the outage can be minimized by maximizing the values of g (N −l ) and g (N −l k ) .Consequently, choosing where k takes on values from 1 to K, is optimal.
Remark 2: Picking K users with the largest channel gains reduces the value of the power allocation coefficients.This helps in power saving, and helps in accommodating a large number of users, thereby ensuring larger throughput.
Lemma 4: When K users are selected as per Corollary 2, the minimum power required is Proof: By substituting ρ S = P S No in (7), it can be clearly seen that the minimum value of P S to be used when the signalling is not in outage is obtained as in (11).
The above is valid provided P min S < P max , where P max is the peak power constraint at the source.
Corollary 3: When NOMA power apportioning is as per Lemma 1 whenever the signalling is not in outage, a highly accurate approximate closed-form expression for the minimum average power ( P min S ) is given by ( 12), shown at the bottom of the page, where No γ th .Proof: For a detailed proof, please see Appendix B. Lemma 5: Change of decoding order (interchange of decoding of different user symbols) at any user, or interchange of SIC levels of two users increase the value of K k =1 α (N −l k ) in ( 4).This degrades the outage performance and thus reduces the throughput and increases energy consumption.
Proof: By changing the decoding order of the p th and p + 1 th symbols, the new power allocation α(N −l k ) is obtained as (13) It is clear from ( 4) and ( 13) that α(N −l k ) > α (N −l k ) , and therefore the sum of power allocations will be larger with the changed decoding order.This leads to higher outage probability, reduced throughput, and increased energy consumption.

IV. THROUGHPUT AND ENERGY EFFICIENCY
Using the expressions derived for p o in (7), throughput in bits per channel (bpcu) is: τ = KR(1 − p o ).Efficient utilization of energy is crucial for modern communication systems that are limited in power, and support a vast number Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
(a) (b) (c) of interconnected IoT devices.In this context, EE (η E ) is a critical performance indicator that represents the relationship between the throughput (τ ) and energy consumed (E ).
Utilizing the derived expressions for τ , the proposed system's EE [7] can be expressed as η E = τ E , where E is the energy consumption in a signalling interval of duration T, i.e., E = P S T .
V. SIMULATION RESULTS This section presents Monte simulations to verify the derived analytical expressions, and demonstrate the superiority of the proposed scheme.It also draws some valuable insights into the performance characteristics.We assume d SU = 1.5, and L = 3 [7].Performance of the proposed CSI-assisted MU NOMA network is compared with its OMA counterpart (special case with K = 1).
Fig. 2(a) depicts the variation of p o w.r.t.R while considering various combinations of N, K, and ρ S .The precision of the derived analytical expressions in equation ( 9) can be clearly confirmed through simulations.It can be clearly seen that increasing the targeted rate degrades the outage performance.The figure also demonstrates that for a fixed number of selected users, increasing the number of users as well as increasing the transmit SNR leads to a significant improvement in the outage performance.
Fig. 2(b) depicts the variation of τ w.r.t.R for the different values of N, K, and ρ S .The accuracy of the derived analytical expressions is validated through simulations.It is evident that the throughput initially increases as the value of the target rate increases, reaching a maximum value before declining to zero.There exists an optimal value of the target rate, which varies depending on N and K.In addition, it is observed that increasing K while keeping N constant enhances throughput performance at lower target rates but degrades it at higher target rates.On the other hand, increasing N while maintaining a fixed K improves throughput performance across the entire range of the target rates.Performance of the proposed framework is compared with the OMA scheme, which allocates the entire power to the user with the best channel gain (K = 1).It is also depicted that the SIC error significantly effect the throughput performance for the higher values of target rate whereas its effect almost negligible for the lower value of target rate.It is depicted that the proposed schemes provide a huge gain compared to its OMA counterpart.

VI. CONCLUSION
In this letter, we analyzed the performance of a multiuser (MU) downlink non-orthogonal multiple access (NOMA) network for the first time, assuming that limited channel state information (CSI) is available at the source.Channeldepended generalized throughput-optimal power allocation coefficients were derived.The user selection scheme and the decoding orders were specified.Assuming fixed-rate transmission, closed-form expressions were derived for important performance metrics such as outage probability, throughput, and EE while consider the scenario of imperfect successive interference cancellation (SIC) which is a crucial aspect for the practical implementation.Further, it was found that selecting an optimal target rate maximizes system performance.Furthermore, performance of the proposed MU downlink NOMA system was compared to its OMA counterpart, and it was demonstrated that the proposed system offers higher throughput.

APPENDIX A PROOF OF COROLLARY 1
It is extremely difficult to solve and get a closed-form expression for p o in (7) due to the involvement of the sum of the inverse of K ordered RVs.However, the large-valued order statistics dominate, and this can be approximated as . Performing mathematical rearrangements and noting that U > V, the above is expressed as where the joint PDF f (V )(U ) (v , u) can be expressed using [19, 2.2.1], and the PDF and CDF of exponentially distributed RVs U and V, as follows Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
On solving the inner integral in (15) after using the binomial expansion, we get Substituting for λ(u− Ψ 1 ρ S Ψ 4 ) = t in the above and then solving using [17, eq. ( 5)], p o is obtained as in (9).

APPENDIX B PROOF OF COROLLARY 3
It is extremely difficult to find a closed-form expression for (11).This is because of the involvement of the sum of the inverse of K ordered RVs.However, since large-valued order statistics are dominating, this can be approximated as To obtain the average value of P min S (denoted by P min S ), the above is averaged over the joint PDF of RVs U and V.By doing so while considering U > V, P min S < P max and performing mathematical manipulation, P min S is obtained as Substituting, f (V )(U ) (v , u) form ( 16) and using the binomial expansion, the above is expressed as follows ) where χ 2 = Ψ 1 χ 1 (i+1)λ .Using [17, eq. ( 4)], I 2 becomes Using [16, 3.352.2] in (20), I 1 becomes  Pmax Ψ 2 ) = t only in the first term and then using relation [4], [17] and [16, 3.352.2] in the first and second term respectively, I 1 is obtained as (24) Substituting I 2 and I 1 from ( 22) and (24), into (20), P min S can be obtained as in (12).

Fig. 2 (
c) illustrates the variation of η E w.r.t.R for the different values of N, K and P S = 1.5W.The figure clearly demonstrates that for a fixed K, increasing the number of users leads to a significant improvement in EE (21.29% if K = 8 and N increases from 15 to 20).It can be clearly seen that an optimal value of R exists that maximizes the EE.It is observed that by optimally choosing R, EE can be maximized.Further, it can be clearly seen from Fig.2(b) and Fig.2(c) that increasing K narrows down the valid range of R.
(23) Using relation [20, 5.1.19],substituting λ(u − Ψ 1 Manuscript received 15 October 2023; accepted 1 November 2023.Date of publication 6 November 2023; date of current version 9 February 2024.This work was supported by DST-SERB funding under Grant CRG/2021/000578.The associate editor coordinating the review of this article and approving it for publication was G. Chen.(Corresponding author: Shankar Prakriya.)