Quicker solution for interference reduction in wireless networks
2018; Institution of Engineering and Technology; Volume: 12; Issue: 14 Linguagem: Inglês
10.1049/iet-com.2017.0684
ISSN1751-8636
AutoresKalpana Naidu, Ramesh Babu Battula,
Tópico(s)Advanced MIMO Systems Optimization
ResumoIET CommunicationsVolume 12, Issue 14 p. 1661-1670 Research ArticleFree Access Quicker solution for interference reduction in wireless networks Kalpana Naidu, Corresponding Author Kalpana Naidu kalpana@nitw.ac.in Department of ECE, NIT-W, Warangal, IndiaSearch for more papers by this authorRamesh Babu Battula, Ramesh Babu Battula Department of CSE, MNIT, Jaipur, IndiaSearch for more papers by this author Kalpana Naidu, Corresponding Author Kalpana Naidu kalpana@nitw.ac.in Department of ECE, NIT-W, Warangal, IndiaSearch for more papers by this authorRamesh Babu Battula, Ramesh Babu Battula Department of CSE, MNIT, Jaipur, IndiaSearch for more papers by this author First published: 18 July 2018 https://doi.org/10.1049/iet-com.2017.0684Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, a closed-form solution is presented for the constrained water-filling problem (CWFP) where in powers allocated to the user resources maximise the user's capacity under an interference constraint and a total power constraint. Total power constraint in CWFP is to save the energy in wireless network. To solve CWFP, erstwhile algorithms compute powers for all the resources iteratively. Unlike existing algorithms, the proposed method calculates the number of resources that gets positive (or non-zero) powers using the concepts of traditional water-filling problem. Later, powers are allocated to the resolved resources (that gets positive powers) alone using the closed-form solution, which reduces the number of computations. It can be discerned that the computational complexity of the given solution is of the order of , where is the total number of resources; which is remarkably lower than that of the prior algorithms specified by an order of . 1 Introduction In any wireless network, user's data transmission rate (or capacity) is to be maximised to the possible extent. Maximum powers can be allotted to the user's resources so as to increase the capacity of the user. However, assigning more powers to the resources of the user may create more interference to the resources of the other user (or interfered user). Besides, battery used at the user's transmitter runs out of its charge quickly if more than the essential power is allotted to user's resources [1, 2]. So, we have to conserve the total power that is used for all the resources of the transmitting user. At the same time, the total interference created to the resources of the interfered user is to be within the tolerable interference limit. Traditional water-filling problem (TWFP) allocates powers to the resources of the user such that the user's total power budget constraint is fulfilled while maximising the user's capacity [3–6]. Along with this power budget limit, the interference from the transmitting user to the interfered user is also to be within the fixed tolerable limit in the constrained water-filling problem (CWFP). User's resources are the sub-carriers in orthogonal frequency division multiplexing (OFDM)-based cognitive radio (CR) network or the discrete multi-tone modulation(DMT) tones in asymmetric digital subscriber line (ADSL) system. Similarly, in wireless sensor networks, the user's resources are the normal frequency bands. Likewise, channels in heterogeneous networks are also considered as the user's resources. Iterative water-filling algorithm (IWA) used in [7–10] allocates the powers to the resources under the total power budget constraint. Likewise, iterative algorithm of [11] is used for allocating the powers to the user's resources with the aim to minimise the interference to the interfered users. In the same manner, various iterative algorithms that solve the CWFPs for the resources mentioned above are given in [12, 13]. All the above-mentioned iterative algorithms find powers for all the resources iteratively. However, huge number of iterations involved gives rise to increased computational complexity in these iterative algorithms [14]. In addition, such iterative algorithms may ensue in loss of capacity in practical applications as they take long time to converge to the optimum solution. Unlike the aforementioned iterative algorithms, IWA of [15] allots powers to multiple users of uplink with lesser number of iterations. However, IWA of [15] is not applicable to CWFP since IWA of [15] does not deal with the downlink scenario wherein, base station performs power allocation to multiple users while making sure that powers allocated to the resources of kth user produce only the tolerable interference to mth user (with ). However, implementation time becomes very small for closed-form solutions. Consequently, closed-form solution for TWFP is obtained in [16] by initially finding the number of positive powers. In the same way, the proposed solution also allocates powers for CWFP using the closed-form solution if the number of resources getting positive powers (= L) is known a priori in CWFP. Since the previous algorithms are not computing ‘L’, powers are to be evaluated for all resources iteratively for many times in the previous algorithms. The proposed algorithm overcomes this drawback by computing ‘L’ for CWFP using the precise range reckoned for all Lagrange multipliers. Once ‘L’ is evaluated, the proposed algorithm finds at most ‘L’ powers in a single step (as all the remaining resources are to be allotted with zero powers). As far as the authors know, this is the first time that ‘L’ is found for CWFP. In this way, the proposed algorithm requires finite number of operations to solve CWFP. The rest of this paper is organised as follows. Section 2 gives the system model. The closed-form equations for performing the power allocation are proposed in Section 3 along with finding the number of non-zero powers. In Section 4, computational complexity of the proposed algorithm is compared with that of the IWA given in [7, 8, 12]. Simulation results are presented in Section 5. Section 6 concludes the paper. 2 System model for the CWFP In this section, six scenarios are discussed in Sections 2.1–2.6, wherein the CWFP arises. Based on these scenarios, a unified model is presented in Section 2.7. 2.1 Device-to-device (D2D) communication D2D communication helps in upgrading the throughput of the cellular network with lesser energy consumption [17]. As revealed in Fig. 1, device1 sends data directly to device2. Owing to this data transmission in between the devices, cellular network gets interference, since cellular communication also happens on the same channels as revealed in Fig. 1. This interference to the cellular network is to be kept under control. Figure 1Open in figure viewerPowerPoint Interference from D2D to cellular network 2.2 Heterogeneous communication In heterogeneous communication, cell phone gets connected to the available radio access technology (RAT) in that location. When available radio spectrum is distributed among the co-deployed RATs, interference is created in between the co-located RATs [18] as indicated in Fig. 2. Figure 2Open in figure viewerPowerPoint Interference in heterogeneous communication In Fig. 2, RAT1 is using CDMA and RAT2 is using the combination of FDMA and TDMA. When a cell phone comes to the vicinity of RAT1, cell phone should start using the CDMA technology to interact with RAT1 as RAT1 is operated on CDMA. Similarly, cellphones that arrive to the surroundings of RAT2 have to follow the blend of FDMA and TDMA technologies. As CDMA cell phone uses all the available frequency channels, it creates interference to RAT2 cell phone as revealed in Fig. 2. 2.3 Long-term evolution (LTE) network In LTE network, macro cell sends data to its users on sub-carriers. As Femto cells also use the same sub-carriers as that of macro cell, Femto cells get interference from macro cell [6]. Total interference created to all the femto cells is controlled in this paper. 2.4 OFDM-based CR network In OFDM-based CR network, secondary user (SU) utilises the unoccupied sub-carriers of primary user (PU) for transmitting its data. However, because of timing and frequency mismatch, SU sub-carriers may not have orthogonality with the PU sub-carriers. Consecutively, SU sub-carriers produce interference to PU's data [19]. 2.5 DMT-based ADSL system When DMT systems (or modems of ADSL service) belonging to different service providers do not provide orthogonality to the sub-channels (or tones) of other DMT system users, then sub-channels of different DMT systems interfere with each other [9, 20]. 2.6 Multi-band wireless sensor network In wireless sensor networks, if there are two sensors within the communication range of each other, then data transmission of one sensor will create interference to other sensor's data [21]. In the multi-hop wireless sensor network shown in Fig. 3, sensor1 sends the sensed data to relay1 (or the intended destination). Figure 3Open in figure viewerPowerPoint Wireless sensor network At the same time, sensor2 also sends the data to relay2. From relay1 and relay2, data finally goes to the coordinator which analyses the received data to infer about the sensors' environment. Since sensor1 and relay2 are in close proximity to each other geometrically and both the sensors are using the same frequencies, interference comes from sensor1 to relay2. 2.7 Generalised system model For all the above scenarios, system model is generalized here. For that, consider a transmitter–receiver pair that transmits its data on resources as shown in Fig. 4. Owing to the data transmission from the transmitting user, there is interference to resources of another transmitter–receiver pair called the interfered user. Figure 4Open in figure viewerPowerPoint Transmitting users and interfered users in different networks It is presumed here that channel state information is available with the transmitting user. Let , to be the channel gain from the transmitting user to the intended receiver over the kth resource as given in Fig. 4. Moreover, is the total power budget of the transmitting user. In addition, maximum amount of interference that the interfered user can bear is the interference threshold . For instance, PU should not get more than amount of interference from CR transmissions [6, 22]. Power allocated to the kth resource of the transmitting user is . Further, variance of the combined additive white Gaussian noise and interference is . If the weight of the kth resource is ; then maximum sum capacity of the transmitting user is given by [23] (1) In (1), . Weight has the following interpretations: (a) can represent the bandwidth of the kth resource. If the resource is sub-carrier, then is equal for all k [4, 24, 25]. (b) Moreover, priorities of the resources can be considered in [3]. Defining as the parameter that describes the channel gain from the kth resource of the transmitting user to (all the resources of) the interfered user (as revealed in Fig. 4), the total interference from all the resources of the transmitting user to the interfered user is given as [14] (2) Note that , to values depend on the type of network and are given in [19, 26, 27]. The problem is to allocate powers to the resources of the transmitting user which will maximise the data transmission rate (C) of transmitting user. To be able to conserve the transmitting user's battery life, sum of powers allotted to resources does not exceed the power budget . Also, in order not to disturb the interfered user's original data reception; interference from the transmitting user to the interfered user should not go beyond . Accordingly, optimisation problem is specified as (3) (4) (5) and (6) in (4) refers to the following: (a) in (4) makes the transmitting user to follow the power budget. (b) For CR network, sensing probability for kth resource is incorporated in as in [28]. For instance, if probability of missed detection is more for kth resource, then power allotted for that resource should be less in order to avoid interference to PU. Using Karush–Kuhn–Tucker conditions, the optimal power allocation obtained for (3)–(6) is given by [3, 4] (7) where and are Lagrange multipliers; ; and . Equation (7) defines the CWFP. However in (7), and are not known and hence are determined using IWA given in [7, 8, 12] as the number of non-zero 's are not known a priori. Unlike the prior algorithms, the proposed solution identifies the resources that get positive powers by making use of the attained bounds for and . Subsequently, powers are discerned only for the ascertained resources alone using the closed-form solution, in order to reduce the computational complexity. 3 Closed-form solution for the CWFP Here, we propose closed-form equations for calculating the powers to be assigned to resources. 3.1 Equal 's and equal 's Consider the case when and . In this case, (7) can be rewritten as where . Then the solution for this case, assuming 's are arranged in ascending order, is given by Theorem 1.Solution of the constrained water-filling optimisation problem for equal 's and equal 's is given by [23, Theorem 2] where L is calculated from with , and . In effect, if all 's and 's are same; then there is only one constraint that is actually effective and so the problem reduces to the TWFP. 3.2 Different 's and different 's In order to determine the closed-form solution for the general case, we need to solve for , and the number of non-zero (positive) powered resources (L). This process is carried out in three steps: (a) Rewriting CWFP [or (3)–(6)] in the form of TWFP. (b) Ordering of the 's. (c) Simplifying and solving by applying Theorem 1. As per [3, 4, 6], optimal capacity is obtained when equality is followed in (4) and (5). Then, we have the following equations: (8) (9) In (8) and (9), is replaced with L. This is because of the fact that, the first L resources only have non-zero (positive) powers once the resources are shuffled in a particular order. Substituting (7) in (8) and rearranging, we have (10) where (11) Similarly, substituting (7) in (9) and rearranging, we get (12) where (13) Multiplying (12) with and (10) by and adding, we have (14) While getting the above equation, and are taken into consideration. Rewriting (7), we obtain (15) where (16) Substituting (14) into (16) and rearranging, we procure (17) From (16), . Besides, the authors in [19, 26, 27] show that involves square terms as represents channel power gain. So, are positive. Moreover, is positive, which induces to be positive. Besides, all Lagrange multipliers are positive. Hence, all the above compels to be positive (i.e. ) always. Note that (15) differs from the TWFP in two aspects: (i) Instead of the ‘water-level’ variable in TWFP, we have the polynomial here. So, it is difficult to assign a water level here in the CWFP. (ii) is a function of the resource index ‘k’. In order to overcome these differences, (15) can be rewritten as (18) where . Equation (18) allows to be regarded as ‘water-level’ variable. Without loss of generality, we presume that , are sorted in the ascending order. Additionally, , , and are also sorted in the sorting order of . Further, assuming that L powers are non-zero, it follows from (18) that 0; and (of course, we have to find L). Substituting (18) in (8), we obtain (19) Replacing in (19) gives out the equation (20) Substituting (17) in (20) and rearranging, we have (21) where . Define (22) Using this in (19), we get (23) In effect, we have reduced CWFP to TWFP. We have Theorem 2.Solution of the constrained water-filling optimisation problem is given by the solution of (24) Proof.Substituting in (23), we have the desired result. Substituting (22) in (24), we have (25) Note that Theorem 2 reduces the problem of constrained water-filling optimisation to that of traditional water filling with a variable scaling constant which is a function of the water level ; while in TWFP [5], the scaling constant is not a function of . Owing to this, there are similarities and differences in the solution. Similar to TWFP [5], the value of is obtained by solving (24). However; in TWFP, solving for with a constant scaling factor results in a single unique solution that achieves the optimal capacity. Solving , with as a function of (or) solving (21) results in L different solutions. Of these, only one solution is optimal. We have Theorem 3.The optimal solution of the water-filling algorithm with variable scaling factor is the smallest positive root of (21) denoted as . Proof.It is shown in Appendix 2 of Section 8.2 that the value that satisfies (8) and (9) lies within the range . Appendix 2 of Section 8.2 further shows that, (21) has only one root in between 0 and (including ). Hence, the smallest positive root of (21) (denoted with ) becomes the optimal solution for (24). value that corresponds to is calculated from (14) as (26) We are now in a position to give the main theorem of this paper. We have Theorem 4.The solution of the general constrained water-filling optimisation problem is specified by (27) where L is found from with (28) Proof.For a given L, Theorem 3 gives as the smallest positive root of (21). The values of , and can be calculated from . We can now apply results of Theorem 1 to obtain the powers in CWFP.Substituting (25) in (22) gives out the powers () as in Theorem 1 as (29)From (18) and (22), we can tell that (30)Substituting (29) in (30), we have (27) that gives us the powers of CWFP ().In order to find L (the number of positive powered resources), take from (27) 0 to get (31)If RHS of (31) is taken as , then becomes (28). (31) and (28) further indicate that .Likewise from , ; we get , . This is obtained in the same way as that of (31) and (28).All the above points out that L value can be obtained from . The example for finding L is shown in Fig. 5. Figure 5Open in figure viewerPowerPoint Finding the number of non-zero (positive) powered resources Using Theorem 4, , calculated for all the resources satisfy the condition indicating that resources get positive powers. If , are calculated again for Q resources with taken as , then it should satisfy the same condition confirming that all of resources indeed get positive powers. However, this may not happen because of the following reasons: (i) calculation needs . However, calculated for resources is not same as that of computed for Q resources. Consequently, values evaluated for resources are different from those of values computed for Q resources. (ii) Once is calculated, are sorted in ascending order of . Owing to this, sorted for resources are different from those of the sorted for Q resources. Due to the aforementioned reasons, L value obtained for resources may be different from that of L value procured for Q resources. As a result, we have to find how many number of resources actually gets positive powers. 3.3 Positive powers convergence method Here, we propose the ‘positive powers convergence method’ to find the actual number of non-zero powers. (i) Initially, we start with resources. (ii) Find , for L resources with . These 's satisfy the condition . (iii) If L is equal to Q, then the starting number of resources (= L) is converging (i.e. same) with the ending number of resources [= Q in ]. So, it can be concluded that L number of resources get positive powers definitely. The remaining is to calculate ‘L’ powers using (27). (iv) If L Q, it means that we still did not get the correct number of positive powers. So, take . For this new L value, repeat steps (ii)–(iv) until becomes valid. It is observed here that the proposed algorithm obtains L directly, whereas IWA does not have this provision to get L explicitly. 3.4 Power allocation Algorithm 1 gets L. Once L is obtained, powers are allotted to the first L resources using (27). The remaining resources are allocated with zero powers. As proposed algorithm procures powers using the closed-form solution, computational complexity is reduced profoundly in the proposed solution. Algorithm 1.(Algorithm to procure the number of positive powers () in CWFP)Require: Inputs required are , , , , , ; and .Ensure: Output is number of positive powers (). 1: Initialise . 2: Find and using (11) and (13). 3: Find as the smallest positive value of the solution of equation (21). 4: Find from (14) as . 5: Evaluate from (16) and ; . 6: Sort , , , , , in the ascending order of . 7: while ( and ) do 8: (a) Calculate and ; . . 9: (b) Find Q that satisfies the condition . The number of resources that get positive powers is possibly Q. 10: If [(Q is equal to L) and ()] then 11: (i) If Q is equal to L; then finally, we got the number of resources that get positive powers which is equal to L. 12: (ii) Interference to the interfered user is within the bounds for L resources if . The proof for this is given in Appendix 8.1. 13: (iii) Exit the algorithm. 14: end If 15: if [(Q is equal to L) and ()] then 16: Since the interference to the interfered user exceeds the limit for Q resources, reduce the number of resources by 1. i.e. . 17: end if 18: if (Q is not equal to L) then 19: Take . For the first L values of , , and ; do the steps 2–7 and 19 until L becomes equal to Q. 20: end if 21: end while 4 Computational complexities for CWFP It is showed here that the computational complexity of the proposed algorithm (described in Section 3) is substantially smaller than that of IWA given in [7, 8, 12] for implementing CWFP. Comparisons between the two are given below: (A) In IWA, initialised and values decide the number of iterations. However, in the proposed algorithm, and values are solved directly with the closed-form solution without doing any initialisation for and values. For some initialised values of and , the number of iterations required in IWA is specified in Table 1. It is interpreted from Table 1 that the number of iterations is more in IWA. Table 1. IWA results for , , , and value value No. of assigned assigned in in iterations initially initially W W required in [7, 8, 12] 0.5 1 670,887 0.4 1 376,316 0.4 1 835,431 0.5 1 1,386,774 0.8 2 589,183 0.9 2 262,923 2 2 4 2,249,174 2 2 2 1,543,022 1 4 3 2,583,729 0.5 2 76,350 0.5 3 77,700 0.5 4 91,976 0.5 3 80,296 0.5 0.5 1 6,517,481 0.1 0.1 1 1,452,811 0.8 0.8 3 67,907 1.5 1.5 3 70,580 The number of operations involved in IWA per one iteration are: (i) Calculation of requires 2 multiplications, additions, subtractions and 2 divisions. (ii) To find , it requires multiplications and additions. (iii) To calculate , multiplications and additions are required. (iv) For calculation, it requires 1 subtraction. (v) For calculation, it requires 1 subtraction. (vi) If 0, then change the value as: new + . To do this, 1 addition and 1 division are required. (vii) If , then change the value as: new . To do this, 1 addition and 1 division are required. The above implies that IWA necessitates number of FLOating-Point operationS (flops) per one iteration. However, more than one iteration is required for performing the power allocation in IWA. Thus, IWA demands a total of ‘(number of iterations) ’ flops. (B) Number of flops entailed for the proposed algorithm of Section 3 are given below: (flops given in the following steps are for L resources) (i) Calculation of using (11) and using (13) requires 2L multiplications and 2L additions. (ii) To find the number of flops required to find , (21) is rearranged as the polynomial equation in ‘ ’. Coefficients of the polynomial equation are specified in a row vector . This row vector is converted into another matrix [] of size L L. Eigenvalues of this matrix [] are the roots of the polynomial equation. Finding the eigenvalues of a square matrix requires ) flops. i.e. finding requires ) flops. (iii) To calculate , ; 1 division, L multiplications and L additions are required. To calculate , to L; it requires L multiplications. (iv) Doing ; requires L divisions. To calculate , using (28) recursively; additions, 2L multiplications and L subtractions are required. (v) and were already calculated while calculating . Taking r and d values in (27) gives So, to calculate ; ; it requires 1 addition, L subtractions and divisions. (vi) If all sums and multiplications are done recursively, then number of flops could be reduced further. If the sequence of resources that comes by following Algorithm 1 is ; then implementation of the proposed algorithm of Section 3 needs flops. (C) Table 2 provides the computational complexities of various algorithms to implement CWFP. In Table 2, IWA initialises and to 0.5 and , respectively. It can be discerned from Table 2 that approximately the number of flops needed in IWA of [7, 8, 12] the number of flops necessary in the proposed solution of Section 3. It can also be observed from Table 2 that the proposed algorithm takes three or four iterations at the most in contrast to the IWAs. This is because of the fact that proposed algorithm evaluates Lagrange multipliers ( and ) using the closed-form solution, whereas IWA computes them iteratively and requires more iterations before producing the converged solution [29]. Besides, the proposed algorithm determines L exclusive powers alone in one shot using the closed-form solution (Remaining L powers are zero in the proposed algorithm.) On the contrary, IWA calculates all of powers iteratively, that too with more number of iterations. In addition, the proposed algorithm computes sums recursively. Owing to all these, the proposed algorithm gets least computational complexity. Table 2. Number of flops necessary to implement CWFP for , , , , and 4.1 Optimal powers are achieved from the proposed method The following proposition validates about the proposed method ensuring optimal powers. Propositon 1.Proposed algorithm produces optimal number of powers (=L) and hence yields optimal capacity. Proof. (a) Algorithm 1 checks ‘ ’ to ensure that number of positive powers does not exceed L. (b) Algorithm 1 makes sure that interference to the interfered user does not go beyond . (c) Total power distributed among L resources is exactly . This gives optimal power allocation for L resources. Consequently, L resources yield optimal powers which in turn produce optimal capacity. Thus, the proposition is proved. 5 Numerical examples and simulation results To elucidate the proposed algorithm, numerical example is furnished now. 5.1 Analogy for CWFP is given below (32) (33) For simplicity purpose, = 1 is taken. and = [0.003, 0.04, 0.5]. Now, Algorithm 1 is used for finding the optimal powers. For L = 3, evaluated and are 1.9 and 0.0735, respectively. The solution of (21) for L = 3 is [1.5789, 0.7803, 32.2721]. Then, becomes 0.7803, which is the smallest positive value of the solution. that corresponds to this is 20.6452. here. Now, values evaluated using are [1.0794, 2.0583, 14.2290]. For these values, comes out to be [0.5397, 0.6175, 1.4229]. As 's are already there in the ascending order, there is no need to sort any of the parameters now. 's are [0, 0.0721, 1.2096, ]. Since , Q becomes 2. Here, L = 3 Q = 2. Hence, make = 2. For the first two values of , and ; newly found and are 1.8 and 0.0235. Equation (21) solved for the first two values of and gives values as [0.659, 1.1111]. Smallest positive value of these values becomes = 0.659. For this , we get as 34.6301. New values calculated are [1.1576, 3.102]. For these new values, 's become [0.5788, 0.9306]. Since, 's are already in ascending order, no parameter is sorted now. New 's are [0, 0.3039, ], which implies that . This inturn gives Q = 2. Now, L became equal to Q. Also, implying that interference to the interfered user is . Powers are calculated for the first two resources using (15). Then, powers become [0.8108, 0.1892, 0]. Finally, we can observe here that and . This concludes that optimal powers are obtained. 5.2 Simulation results Simulations are performed in MATLAB R2010b. Channel parameters and , , are calculated using the Rayleigh fading model with 0.1 variance. Parameters applied in simulations are specified in Table 3. Table 3. Parameters used for the simulations The simulation results of the powers allotted to SU sub-carriers in OFDM-based CR network (or to the transmitting user's tones in DMT-based ADSL system or to the macro cell sub-carriers in LTE network) for = 30 and = 29 are shown in Fig. 6. Fig. 6 depicts that the proposed algorithm allots same powers as that of IWA to the resources of the transmitting user. We can further observe that the proposed algorithm does not allot negative powers. Figure 6Open in figure viewerPowerPoint Powers allotted to SU sub-carriers from both IWA (shown in red colour) and the proposed algorithm (shown in blue colour) for and 6 Conclusion After finding the number of non-zero powered resources, powers are allotted to those non-zero powered resources alone using the closed-form solution. It is discerned that the proposed algorithm achieves optimal powers. Moreover, the number of flops essential in the proposed algorithm O () the number of flops necessary in the IWA of [7, 8, 12]. 8 Appendix 8.1 Appendix 1: Condition for the interference to be with in the interference limit Here, we derive the condition for the interference to be with in the interference limit . Substituting (15) in (5), we get 8.2 Appendix 2: Proof that minimum of the positive roots of (21) produces maximum powers This proof is carried out in two steps: (a) First, we show that value that produces maximum powers lies in the range , (b) Later, we prove that there is at most one root for (21) that is there in between 0 and . This root becomes the minimum of the positive roots of (21) and also brings optimal powers. 8.2.1 Why should be with in the range to get optimal powers. (i) Equation (7) indicates that minimum value brings out the maximum possible power . (ii) Equation (26) implies that, if , then becomes negative. However, Lagrange multipliers ( and ) cannot be negative. Hence, cannot exceed . (iii) Similarly, if , then becomes negative. Therefore, we cannot take values that are more than . (iv) If , then is zero [from (26)]. Subsequently, powers [from (7)] are . These powers produce and . (v) Likewise, = 0 gives . Accordingly, powers are . These 's bring in and . (vi) Consequently, (iv) andw(v) point out that when a Lagrange multiplier is zero, corresponding constraint does not bind. (vii) Thus, we can conclude that, in order to get and ; and should follow the range and 0 . As a result, it can be deduced from (i) and (vii) that, out of all the roots of (21), should be the minimum, lying in to allot optimal powers. 8.2.2 Show that there is at most one root for (21) in the range . Equation (21) is taken alternatively as To show that there is at most one root for ‘ ) = 0’ in the range , it is enough to show that derivative of ) is either positive or negative [30]. Derivative of ) is (34) Expanding , we obtain (35) Using (7)–(9) in (35), we procure (36) Applying (36) in the denominator of (34), we obtain (37) Substituting (37) in (34), we obtain (38) Expanding we get (39) Substituting (39) in (38), we get (40) is taken as (41) Applying Cauchy–Schwarz inequality in (41) to get (42) Using (42) in (40), we procure (43) Since 0, is positive. 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