Toward UL-DL Rate Balancing: Joint Resource Allocation and Hybrid-Mode Multiple Access for UAV-BS Assisted Communication Systems

A number of studies on UAV-BS assisted communications have been conducted [18, 19, 20, 21]. In [18], UAV serves the ground users through time division multiple access (TDMA) scheme where at most one ground user can be served by UAV at each time slot. In [19], the user scheduling and UAV trajectory design were jointly optimized to further enhance the minimum average rate among users. In [20], a UAV assisted orthogonal frequency division multiple access (OFDMA) system was investigated to meet the heterogeneous traffic demands of users, and the minimum average rate among all users was maximized by joint bandwidth assignment, power allocation and UAV trajectory design. Nevertheless, the minimum average rate among users may decrease significantly as the number of users increases and it is difficult to guarantee all users’ heterogeneous rate requirements, especially in some crowed hotspots.

In order to provide enhanced system spectrum efficiency (SE) performance and make full use of the underutilized wireless resources in orthogonality based approaches [18, 19, 20, 21], power-domain non-orthogonal multiple access (NOMA) has been applied to UAV assisted systems, where a group of users share the same frequency band simultaneously with different power levels [23, 22, 24, 25]. In [22], Liu et. al. introduced the UAV assisted NOMA system and summarized their superiorities in terms of SE, multiple access, communication coverage, outage probability and energy efficiency (EE), over the UAV assisted orthogonal multiple access (OMA) methods. Although UAV assisted NOMA can provide a higher average rate, the instantaneous rate demands of users in each NOMA group may not be guaranteed due to the effects of 1) UAV movement; 2) the inter-user interference from other users within the same NOMA group; and 3) error propagation introduced by the imperfect successive interference cancellation (SIC) decoding process [26, 27, 28]. In this case, to guarantee the rate requirements of users, a hybrid OMA/NOMA terrestrial communication system was investigated in [29] and [30], where dynamic mode switching between OMA and NOMA is utilized based on the instantaneous channel conditions. Nevertheless, the work in [29] and [30] for terrestrial communications cannot be applied to a UAV-BS assisted wireless system because of UAV’s mobility and the high correlations between air-ground channels. Hence, to meet the heterogeneous traffic demands of ground users and consider fairness among users, it is significant to explore the UAV assisted hybrid non-orthogonal/orthogonal multiple access systems and its corresponding resource allocation.

In addition, as we have explained, UL plays an increasingly significant role in UAV assisted communications. It is therefore important to balance the UL/DL rates to meet the users’ demands [11, 12, 13]. While most aforementioned studies regarding UAV assisted communications [14, 15, 16, 17] have focused on DL design only and lack of a consideration of joint DL and UL optimization. As a consequence, the achievable UL rate is generally much lower than DL rate, which makes it difficult to guarantee the increasing rate requirements of UL communication. In addition, in UL NOMA transmission, the power back-off strategy was adopted [31] [32], where the users with weaker channel conditions were assigned less transmission power, as opposite to the power allocation principle of DL NOMA transmission[26]. Hence, the work on DL NOMA or UL NOMA is not applicable to joint UL-DL optimization, and the joint UL-DL resource allocation for UAV assisted NOMA systems with UL and DL rate balancing remains an open challenge.

Motivated by the above open issues, we propose a UAV assisted communication system with hybrid mode multiple access (HMMA) and joint UL-DL optimization, where both high rate-oriented and instantaneous rate-sensitive services are demanded simultaneously and the rates for UL and DL are required to be quasi-balanced. Our contributions are summarized as follows:

1. To the best of our knowledge, this is the first explicit investigation of joint UL-DL optimization of bandwidth assignment, power allocation and trajectory design for the UAV assisted systems to achieve comparable UL-DL average rates and accommodate heterogeneous rate demands at UL and DL. Our work fills up the gap in current literature of UAV assisted communications [14, 15, 16, 17] which have focused on DL design only, leaving the UL rate much lower than DL rate and failing to guarantee the increasing rate requirements of UL communication.

2. We propose HMMA in which NOMA is used to achieve high average data rate, and OMA to meet users’ instantaneous rate demands by compensating for the rates of users with poor channel conditions or strong inter-user interference. The proposed HMMA enables a higher degree of freedom in multiple access and allows dynamic resource allocation for each access mode. As a result, it is able to meet a wide range of heterogeneous traffic demands of users, and provide a superior minimum average rate performance and hence a better fairness among users, compared to the UAV assisted OMA [20] or NOMA [26] approaches.

3. Making use of the reciprocity of the air-ground channel, we develop a joint UL-DL resource allocation (J-ULDL-RA) algorithm to conduct UL-DL bandwidth assignment, power allocation and UAV trajectory design in an alternative manner. In addition, by theoretically proving that the sum UL/DL transmission power is a convex function of UL/DL rates, the non-convex power allocation problem can be written as an equivalent convex form with respect to the rate of users, thus leading to the closed-form optimal power allocation solution.

4. To mitigate the impact of the SIC error propagation in HMMA transmission, we propose an enhanced-HMMA (E-HMMA) scheme. The enhancement lies in that, in addition to the users suffering from poor channel conditions or strong inter-user interference, the users with severe error propagation caused by imperfect SIC are also taken into consideration for rate compensation by E-HMMA. The proposed E-HMMA scheme demonstrates higher robustness against the SIC error propagation and enables an enhanced minimum average rate than the HMMA scheme. Based on the proposed E-HMMA scheme, a joint UL-DL resource allocation algorithm with error propagation (J-ULDL-RAEP) is further developed for UAV assisted systems. Last but not least, it is also confirmed that both the proposed joint UL-DL optimization algorithms, namely J-ULDL-RA and J-ULDL-RAEP, can lead to fast convergence.

It is noteworthy that this paper is a substantial extension of our previous work in [1], as we propose a joint UL-DL resource allocation algorithm supported by a closed-form optimal solution to power allocation, whereas no power allocation solution was considered in [1]; also, a more generic system model with more than two users per NOMA group can be supported, as opposed to only two users per NOMA group in [1].

The rest of this paper is organized as follows. Section II introduces the UAV assisted HMMA system model and problem formulation. In Section III, the joint UL-DL resource allocation algorithm is proposed. The E-HMMA scheme is presented in Section VI. Section V presents the numerical results, followed by conclusion in Section IV.

Similar to [23, 22, 24], to make a balance between complexity and system performance, we assume the same time and frequency resource element can be shared by $L$ $(L\geq 2)$ users. For convenience, we assume the users are divided into $M$ groups with $L$ users in each group and keep the group fixed during the flight of UAV. For DL NOMA transmission, the UAV transmits the signals to $L$ users simultaneously over the same frequency resource [26], the received superposed signal ${D_{m,n,k}^{\text{DL}}}$ of user $k$ in NOMA group $m$ ($k\in\mathcal{K}_{m}$) at time slot $n$ can be expressed as

where ${P_{m,n,k}^{\text{NO,DL}}}$ and ${s_{m,n,k}^{\text{DL}}}$ denote the transmission power and the signal of user $k$ at DL, respectively. $z^{\text{DL}}$ is the Gaussian noise with spectral power density $N_{0}$.

At the user end, SIC is adopted to decode the superposed signals. For DL NOMA transmission, the SIC decoding is generally in the increasing order of channel gains, i.e., the signal of the user with a weak channel gain (e.g., user $i$) is first decoded. Subsequently, the user with a higher channel gain (user $j$) removes the signal of user $i$ from the superposed signals and continues to decode its own signal [23][28]. Denote the bandwidth allocated to NOMA group $m$ for DL transmission as ${B_{m,n}^{\text{{NO},DL}}}$. As mentioned above, there are $L$ multiplexed users in each NOMA group, based on [26], to fairly compare the performance between NOMA and OMA under the same resources, a time slot in NOMA can be regarded as the combination of $L$ time slots in OMA. Hence, the rate of user $k$ for DL NOMA transmission at time slot $n$ can be obtained as

where ${\mathcal{K}^{\text{DL}}_{m,k}}$ denotes the set of users in NOMA group $m$ that have a higher channel gain than user $k$, i.e., $\mathcal{K}_{m,k}^{\text{DL}}=\left\{{i}|{{H}_{{i},n}}\geq{{H}_{k,n}}\cap{i}\in\mathcal{K}_{m}^{\text{DL}}\right\}$.

As mentioned above, though NOMA can provide a superior system performance via multi-user multiplexing, the instantaneous DL rate requirements of some users in NOMA group cannot always be guaranteed due to UAV movement and the inter-user interference introduced in NOMA transmission [23, 22, 24]. To assure the DL instantaneous rate requirements of users and enhance user fairness, in our proposed HMMA, we allocate a fraction of bandwidth and transmission power apart from NOMA to serve these users with OMA to boost up their instantaneous rates. Denote the OMA bandwidth and transmission power for user $k$ in DL HMMA as ${B_{k,n}^{\text{{O},DL}}}$ and ${P_{k,n}^{\text{{O},DL}}}$, respectively. Specifically, ${B_{k,n}^{\text{{O},DL}}}=0$ indicates that user $k$ is allocated with no OMA bandwidth and requires no rate compensation in DL. The compensated rate $R_{k,n}^{\text{{{O}},DL}}$ of user $k$ can be obtained as

The total DL rate of user $k$ at time slot $n$ is

For UL transmission, the same user grouping is used as DL. Note that due to the difference between UL and DL in NOMA, the HMMA scheme is operated differently in UL and DL. The differences between DL/UL NOMA lie in the following [23] [24] [28]. 1) Unlike the DL NOMA where UAV transmits the superposed signals to multiple users, in UL NOMA, the multiplexed users non-orthogonally transmit signals to the UAV over the same frequency resource element. 2) In DL NOMA, SIC is performed at the user side in the increasing order of users’ channel gains [23] [28]. Whereas, in UL NOMA, SIC is conducted at the UAV side and the signal of a user with a higher channel gain is first decoded and removed from the superposed signals, so that the inter-user interference of weak users could be smaller. After that, the UAV continues to decode the signals of other users with weak channel gains [24].

Based on the above, the received UL superposed signal $D_{m,n}^{\text{UL}}$ at UAV from the users in NOMA group $m$ can be expressed as

where ${P_{m,n,k}^{\text{NO,UL}}}$ and ${s_{m,n,k}^{\text{UL}}}$ denote the UL transmission power and the signal of user $k$ at time slot $n$, respectively [24]. Then the UAV decodes the superposed signals by the SIC approach. Denote the bandwidth allocated to group $m$ for UL NOMA transmission as ${B_{m,n}^{\text{NO},\text{UL}}}$, after SIC, the rate of user $k$ at UL NOMA transmission is given by

where ${\mathcal{K}^{\text{UL}}_{m,k}}$ denotes the set of users in ${\mathcal{K}^{\text{UL}}_{m}}$ that have lower channel gains than user $k$, i.e., $\mathcal{K}_{m,k}^{\text{UL}}=\left\{{i}|{{H}_{{i},n}}\leq{{H}_{k,n}}\cap{i}\in\mathcal{K}^{\text{UL}}_{m}\right\}$.

Based on (6), the UL sum rate of users in NOMA group $m$ $R_{m,n}^{\text{sum,UL}}$ can be obtained as

It can be observed from (7) that the UL inter-user interference ${\sum\limits_{j\in\mathcal{K}_{{m,k}}^{\text{UL}}}{P_{m,n,j}^{\text{NO,UL}}H_{j,n}}}$ is naturally eliminated while deriving the UL sum rate of users, which can be utilized to help optimize UAV trajectory as detailed in Subsection III-B.

Similarly, to guarantee the instantaneous UL rate demands of users at each time slot, we allocate a fraction of bandwidth and transmission power to serve those users with poor channel conditions or severe inter-user interference using OMA. Then, the compensated rate of user $k$ for UL transmission at time slot $n$ can be expressed as

where ${B_{k,n}^{\text{{O},{UL}}}}$ and ${P_{k,n}^{\text{{O},{UL}}}}$ denote the compensated UL bandwidth and transmission power of user $k$, respectively. Based on (6) and (8), the UL rate of user $k$ after rate compensation is

The total UL sum rate of users in NOMA group $m$ $R_{m,n}^{\text{group,UL}}$ can then be written as

We aim to maximize the minimum average rate among all users at both UL and DL transmissions by jointly optimizing UL-DL bandwidth assignment, power allocation and UAV trajectory design, while guaranteeing users’ heterogeneous traffic demands. Specifically, we apply the concept of minimum-rate ratio (MRR) $\alpha$ for each user to indicate the ratio of the minimum data rate required (for instantaneous rate-sensitive traffic) over the total average rate (for both high rate-oriented and instantaneous rate-sensitive traffic) in $N$ time slots [20], which specifies the percentage of the required instantaneous rate-sensitive traffic versus that of the high rate-oriented traffic, according to its applications, i.e., $\alpha$ fraction of its total average rate is instantaneous rate-sensitive and the remaining ($1-\alpha$) fraction is high average rate-oriented.

Denote the minimum average rate among users at UL and DL transmissions as $\eta\!=\!\mathop{\min}\limits_{i,k}\frac{1}{N}\!\!\sum\limits_{n=1}^{N}\!\!{R_{k,n}^{i}}$, $k\in\left\{1,2,...,K\right\}$, with $i\in\{\text{UL,DL}\}$ denoting as the indicator for UL or DL. For bandwidth assignment and power allocation, let $\mathbf{B}^{\text{DL}}\!=\!\left\{[{B}_{m,n}^{\text{NO,DL}}]_{M\times N},[{B}_{k,n}^{\text{O,DL}}]_{K\times N}\right\}$ and $\mathbf{B}^{\text{UL}}\!=\!\left\{[{B}_{m,n}^{\text{NO,UL}}]_{M\times N},[{B}_{k,n}^{\text{O,UL}}]_{K\times N}\right\}$ denote the bandwidth assignment matrix, $\mathbf{P}^{\text{DL}}\!=\!\left\{[{P}_{m,n,l}^{\text{NO,DL}}]_{M\times N\times L},\right.$ $\left.[{P}_{k,n}^{\text{O,DL}}]_{K\times N}\right\}$, $\mathbf{P}^{\text{UL}}\!=\!\left\{[{P}_{m,n,l}^{\text{NO,UL}}]_{M\times N\times L},[{P}_{k,n}^{\text{O,UL}}]_{K\times N}\right\}$ denote the power allocation matrix for DL and UL, respectively. Then, the overall bandwidth for DL and UL transmissions can be expressed as $B_{n}^{\text{DL}}=\sum\limits_{m=1}^{M}{B_{m,n}^{\text{NO,DL}}}+\sum\limits_{k=1}^{K}{{B_{k,n}^{\text{O,DL}}}}$ and $B_{n}^{\text{UL}}=\sum\limits_{m=1}^{M}{B_{m,n}^{\text{NO,UL}}}+\sum\limits_{k=1}^{K}{{B_{k,n}^{\text{O,UL}}}}$, respectively. Denote $\mathbf{Q}$ as the UAV trajectory matrix. For UAV energy consumption, we consider two main components, namely, the communication related power ($P_{t}^{\text{DL}}$ and $P_{t}^{\text{UL}}$) and propulsion power $P(S_{n})$. In particular, the propulsion power is consumed to keep UAV aloft and support its movement and can be approximated as a convex function with respect to UAV’s flying speed $S_{n}$ [8].

Then, with the consideration of heterogeneous rate requirements and UL-DL rate balancing, the optimization problem for the UAV assisted HMMA systems can be formulated as

where (11a) guarantees the average rate of users at UL/DL transmission higher than a lower bound $\eta$, which is the objective to be optimized. (11b) denotes the UL/DL instantaneous rate requirements of users at each time slot. (11c) constrains the transmission power budget for DL and UL, respectively. (11d) indicates that the total utilizable bandwidth for UL and DL transmissions is limited to $B$. (11e) guarantees that the propulsion power of UAV should be not higher than the maximum propulsion power $P^{\text{Pro}}_{\max}$ in each time slot. Trajectory related constraints are confined by (11f) and (11g).

In its current form, problem $\mathbf{P0}$ is difficult to be solved directly. First, the rates ${R_{k,n}^{{\text{DL}}}}$ and ${R_{k,n}^{{\text{UL}}}}$ in constraints (11a)-(11b) are non-convex with respect to bandwidth assignment, power allocation and trajectory optimization variables ($\mathbf{B}^{\text{DL}}$, $\mathbf{B}^{\text{UL}}$, $\mathbf{P}^{\text{DL}}$, $\mathbf{P}^{\text{UL}}$ and ${\bf{Q}}$). Second, with a fixed UAV trajectory ${\mathbf{Q}}$, ${R_{k,n}^{{\text{DL}}}}$ and ${R_{k,n}^{{\text{UL}}}}$ are still non-convex due to the inter-user interference in the denominator of (3) and (6). Third, with a given bandwidth assignment and power allocation solution $\mathbf{B}^{\text{DL}}$, $\mathbf{B}^{\text{UL}}$, $\mathbf{P}^{\text{DL}}$, $\mathbf{P}^{\text{UL}}$, neither ${R_{k,n}^{{\text{DL}}}}$ nor ${R_{k,n}^{{\text{UL}}}}$ are concave or convex functions of ${\bf{Q}}$. To address these challenges, we transform $\mathbf{P0}$ into the following

Comparing (12c) and (12d) with (11b), it can be observed that the feasible solution of $\mathbf{P1}$ is generally a subset of that of $\mathbf{P0}$, and problems $\mathbf{P1}$ and $\mathbf{P0}$ are equivalent if all of the ground users can achieve equivalent average rate [20]. Now, problem $\mathbf{P1}$ is still non-convex. To this end, $\mathbf{P1}$ can be decomposed into several sub-problems, namely (1) bandwidth assignment and power allocation, and (2) UAV trajectory design, which can be alternatively solved by the proposed joint UL-DL resource allocation algorithm to obtain effective and low-complexity solutions, as detailed in Section III.

Now, bandwidth assignment and power allocation have been obtained in Subsection III-A, the UAV trajectory problem for the UAV assisted HMMA systems can be expressed as

where successive convex optimization method is utilized to solve the non-convex problem $\mathbf{P1.5}$.

Due to the difference between UL and DL HMMA transmission, the UL/DL inter-user interferences in (2) and (6) are different, which leads to different operations in UL/DL for the UAV trajectory design. First, for DL transmission, though the NOMA/OMA rates of users ($R_{k,n}^{\text{NO,DL}}$, $R_{k,n}^{\text{O},\text{DL}}$ in (2) and (3)) are non-convex functions of $\mathbf{q}[n]$, they are evidently convex with respect to $||\mathbf{q}[n]-{{\mathbf{w}}_{k}}|{{|}^{2}}$. We can use the first-order Taylor’s series expansion of $||\mathbf{q}[n]-{{\mathbf{w}}_{k}}|{{|}^{2}}$ as the lower bound of users’ rates [9]. The lower bound of the rate $R_{k,n}^{\text{DL,lb}}$ of user $k$ is given in (20), with $\phi_{k,n}^{r}=||{{\mathbf{q}}^{r}}[n]-{{\mathbf{w}}_{k}}|{{|}^{2}}$.

On the other hand, for UL transmission, due to the inter-user interference of the user $k$ in the denominator of (6), the UL NOMA rate of user $k$ is non-convex with respect to the term $||\mathbf{q}[n]-{{\mathbf{w}}_{k}}|{{|}^{2}}$, which is different from the case at DL. As a result, the method utilized in DL NOMA is not applicable for UL NOMA. It is worth noting that the inter-user interference can be naturally eliminated when calculating $R_{m,n}^{\text{sum,UL}}$ in (7), the UL sum rate of users in each NOMA group, which makes $R_{m,n}^{\text{sum,UL}}$ convex with respect to $||\mathbf{q}[n]-{{\mathbf{w}}_{k}}|{{|}^{2}}$. As a result, we transform constraint (12b) to $({\text{24b}}):\frac{1}{NL}\sum\limits_{n=1}^{N}{(R_{m,n}^{\text{sum,UL}}+\sum\limits_{k\in{{\mathcal{K}}_{m}}}{R_{k,n}^{\text{O,UL}}})}\geq{{\eta}},\forall k,m$, where both $R_{m,n}^{\text{sum,UL}}$ and $\sum\limits_{k\in{{\mathcal{K}}_{m}}}{R_{k,n}^{\text{O,UL}}}$ are convex with respect to ${{\Phi}_{k,n}}=||\mathbf{q}[n]-{{\mathbf{w}}_{k}}|{{|}^{2}}$. In this way, the lower bound of the overall UL rate of users $R_{m,n}^{\text{group,UL,lb}}$ in NOMA group $m$ can be derived, which is given in (22).

Based on the above transformations, the original trajectory problem can be approximated as

which is now a convex quadratic problem and can be effectively solved by CVX.

The J-DLUL-RA algorithm is given in Algorithm 1. The bandwidth assignment and power allocation sub-problem in $\mathbf{P1.2}$ and $\mathbf{P1.4}$, and the trajectory design sub-problem in $\mathbf{P1.6}$ are alternately solved until convergence.

In the following, we analyze the computational complexity of the proposed J-ULDL-RA algorithm. Since the bandwidth assignment problem $\mathbf{P1.2}$ is a linear program with respect to ${\mathbf{B}^{\text{DL}},\mathbf{B}^{\text{UL}}}$, the complexity of solving $\mathbf{P1.2}$ is in the order of $O(2K^{3}N^{3})$. The complexity of obtaining the closed-form power allocation solution in $\mathbf{P1.4}$ is $O(KN)$. Note that in CVX, the interior-point method is generally employed to solve the optimization problems, the computational complexity of trajectory design in $\mathbf{P1.6}$ is on the basis of the complexity analysis of interior-point method to solve the corresponding convex conic problem [20]. Since $\mathbf{P1.6}$ mainly contains $K(1+L)/L$ second-order cone constraints with size $2N+1$, $K(1+L)/LN$ second-order cone constraints with size $3$, $2N-1$ second-order cone constraints of size $4$, and one linear equality constraint with size $4$, referring to the analytical expression in [20][38][39], the total computational complexity of solving $\mathbf{P1.6}$ is $O(K^{\frac{3}{2}}N^{\frac{7}{2}})$. Denote $i_{\max}$ as the number of iterations required in Algorithm 1 for convergence. Since all problems $\mathbf{P1.2}$, $\mathbf{P1.4}$ and $\mathbf{P1.6}$ require to be solved in each iteration, whose complexities are $O(2K^{3}N^{3})$, $O(KN)$ and $O(K^{\frac{3}{2}}N^{\frac{7}{2}})$, respectively. Given $i_{\max}$ to record the maximum allowable number of iterations, the whole computational complexity of Algorithm 1 can be obtained as $O(i_{\max}(2K^{3}N^{3}+KN+K^{\frac{3}{2}}N^{\frac{7}{2}}))$.

As mentioned in Section II, for DL NOMA SIC decoding, the superposed signals are generally decoded by users in the increasing order of channel gains [23][28]. Nevertheless, due to limited computational capacity, there exists common impairment among NOMA users in SIC process, which may cause error propagation to the next-level decoding [24][28]. Let $\omega$ denote the proportion of SIC residual power. Based on (2), the rate of user $k$ at DL NOMA transmission with imperfect SIC decoding can be obtained as

where $\omega\sum\limits_{j\in\{\mathcal{K}^{\text{DL}}_{m}/{\mathcal{K}^{\text{DL}}_{m,k}}\}}{{{P}_{m,n,j}^{\text{NO,DL}}}H_{k,n}}$ denotes the SIC residual power at user $k$.

After that, E-HMMA scheme is conducted for rate compensation to achieve enhanced minimum average rate among users and a higher user fairness. E-HMMA differs from HMMA by also taking into account the error propagation effect for rate compensation in addition to the poor channel conditions or strong inter-user interference in HMMA. With the E-HMMA scheme, the compensated rate $\hat{R}_{k,n}^{\text{{{O}},DL}}$ of user $k$ can be obtained as

where ${{B}_{k,n}^{\text{{{O}},DL}}}$, ${{P}_{k,n}^{\text{{{O}},DL}}}$ and ${{B}_{k,n}^{\text{{{OE}},DL}}}$, ${{P}_{k,n}^{\text{{{OE}},DL}}}$ denote the compensated bandwidth and transmission power of user $k$ in the E-HMMA scheme to combat the inter-user interference and SIC error propagation, respectively. For example, when $L=2$ with ${H_{1,n}}>{H_{2,n}}$, according to (25), the strong user $1$ may suffer from severe SIC error propagation when decoding the signal of user $2$, which deteriorates the rate of user $1$. Therefore, the compensated bandwidths of user $1$ in the E-HMMA scheme are ${{B}_{1,n}^{\text{{{OE}},DL}}}\geq 0$ and ${{B}_{1,n}^{\text{{{O}},DL}}}=0$. On the other hand, since the weak user $2$ undergoes inter-user interference from user $1$, its compensated bandwidths can be ${{B}_{2,n}^{\text{{{O}},DL}}}\geq 0$ and ${{B}_{2,n}^{\text{{{OE}},DL}}}=0$.