Skip to main content

Advertisement

A Novel Method for Improving Fairness over Multiaccess Channels

Article metrics

Abstract

It is known that the orthogonal multiple access (OMA) guarantees for homogeneous networks, where all users have almost the same received power, a higher degree of fairness (in rate) than that provided by successive interference cancellation (SIC). The situation changes in heterogeneous networks, where the received powers are very disparate, and SIC becomes superior to OMA. In this paper, we propose to partition the network into (almost) homogeneous subnetworks such that the users within each subnetwork employ OMA, and SIC is utilized across subnetworks. The newly proposed scheme is equivalent to partition the users into ordered groups. The main contribution is a practical algorithm for finding the ordered partition that maximizes the minimum rate. We also give a geometrical interpretation for the rate-vector yield by our algorithm. Experimental results show that the proposed strategy leads to a good tradeoff between fairness and the asymptotic multiuser efficiency.

1. Introduction and Preliminaries

Rate allocation in multiuser communication systems is an important task which should consider simultaneously the fairness and the spectral efficiency. This paper is focused on fairness of multiple-access (MA) schemes working under maximum spectral efficiency evaluated in terms of sum rate. The state of art is the method recently introduced in [1]. However, the main drawback of this algorithm is a significant decrease of the asymptotic multiuser efficiency (AME) [24]. We propose a new strategy that combines the strengths of two different MA schemes such that to guarantee a good tradeoff between fairness and AME.

1.1. System Model

Consider a single-antenna Gaussian MA channel with users transmitting to the base station (BS). The system model can be written as [1, Example  1]

(1)

where is the received signal, models the fading channel from the th user to the BS, and is the symbol transmitted by the th user. The additive noise is assumed to be white circular Gaussian with variance for each real and imaginary component. Under the hypothesis that the transmitting powers of the users are constrained such that , , we have

(2)
(3)

where is the rate of the th user and . The interested reader can find in [5, Chapter 6] a comprehensive discussion on the significance of (2) and (3). The following two methods can be applied to achieve equality in (2):

  1. (1)

    OMA: orthogonal multiple-access with degrees of freedom (DOF) allocated proportional to users' received powers;

  2. (2)

    SIC: successive interference cancellation.

We refer to [5, Chapter 6] for more details on OMA, SIC, and the definition of DOF.

It is also pointed out in [5] that, whenever the received power is almost the same for all users, that is, the network is homogeneous, OMA guarantees a higher degree of fairness (in rate) than that provided by SIC. The situation changes in heterogeneous networks, where the received powers are very disparate: if the decoding is performed in the decreasing order of the received powers, then SIC becomes superior to OMA. However, the SIC systems have drawbacks which do not exist for OMA. Because the signals received from the users are estimated and subtracted from the composite signal one after the other, the inaccurate estimation for the current user makes the next users decoded unreliably. This deficiency becomes more severe when the number of users increases. In fact, it is known that SIC works well only when a specific disparity of the powers is enforced (see, e.g., [6, 7] and Chapter 5 in [8]).

To measure the fairness and the performance, we employ two criteria that have been used frequently in the past. For instance, it is customary to evaluate the fairness with the following max-min criterion: a rate vector is called max-min fair (MMF) if and only if an increase in the rate of one user results in the decrease in the rate of one or more users who have smaller or equal rates [1, 9]. Additionally, we consider the AME. Note that AME quantifies the loss of performance when the interferer users are present and the background noise vanishes [24]. More precisely, AME is a measure of degradation in bit error rate because of the presence of multiple-access interference in a white Gaussian channel.

1.2. Basics of the New Method

Our approach exploits the beneficial aspects of both OMA and SIC. Because we do not aim to improve fairness by sacrificing the throughput, we assume that (2) is satisfied with equality.

The key idea is to partition the network into (almost) homogeneous subnetworks such that the users within each subnetwork employ OMA, and SIC is utilized across subnetworks. Since OMA is applied to (almost) homogeneous subnetworks, it is likely that the degree of fairness is not deteriorated. The application of SIC to subnetworks and not directly to users allows to decrease the number of decoding stages, which potentially improves the performance.

Given that the number of users is , we assume that the number of subnetworks is . The newly proposed scheme is equivalent to partition the users into ordered groups. Note that the order matters because it corresponds to the order in which the groups are decoded. Remark for that the grouping method is the same with OMA. Moreover, the grouping method is identical with SIC for . Similarly to conventional SIC, the max-min rate achieved in this case depends on the order in which the groups are decoded. We consider the family of all ordered partitions of the users into nonempty groups. Then we pick up the ordered partition for which the minimum rate is maximized, and we name it (basic ordered grouping ofusers intogroups). Conventionally, coincides with OMA, and we write . Obviously, .

Furthermore, one can select again from the ordered partition which maximizes the minimum rate. The new selection is dubbed . Remark that the rate vector which corresponds to is not necessarily the same with the max-min fair rate vector that was defined in Section 1.1. However, is guaranteed to be max-min fair among all possible user groupings for which the sum capacity is achieved.

We investigate how the fairness can be evaluated for OMA, SIC, and BORG. In this context we demonstrate for a fundamental property, which allows us to introduce a low-complexity search method for choosing from all ordered partitions of users into groups.

We give also a geometrical interpretation for the rate-vector yield by our algorithm. More exactly, we point out the connections between the outcome of the proposed method and the polymatroid structure of the capacity region as it is used in multiuser information theory [1, 10, 11].

During recent years, several works have exploited the polymatroid structure for optimizing the fairness in multiaccess systems [1, 12]. The main idea is based on the fact that particular points within the sum capacity facet of the polymatroid can be achieved by successive decoding and time sharing. Then, the effort is focused on finding the time-sharing coefficients which give the fairest point. For the sake of comparison, we pick up the method from [1], which is based on time sharing, and we name it TS. It is clear that TS cannot be inferior to our method if the criterion is the fairness in the multiaccess system. But since TS is a linear combination of successive decoders with different decoding orders, it suffers from the same deficiencies like the ones mentioned earlier for SIC.

The rest of the paper is organized as follows. Section 2 contains the main contribution, where we show how a low-complexity search algorithm can be devised to find the ordered partition which maximizes the minimum rate. The geometrical interpretation of the result is included. In Section 3, the newly proposed method is compared with OMA, SIC, and TS in a simulation study which comprises four different network models. In all cases, the new strategy provides the best tradeoff between fairness and AME.

2. Fairness

2.1. Formulas for OMA and SIC

It is well known for OMA method that the degree of fairness among users is lowered when their received powers are very disparate [5]. This drawback can be easily understood from the formula which gives the rate of the th user [5, Chapter 6]:

(4)

where . From the identity above, we have for all , where . Hence, the rates are as disparate as the received powers are, which leads to unfair rates in heterogeneous networks. For example, if , then the minimum rate tends also to zero.

When SIC is applied, the fairest rate vector is obtained by decoding the users in the decreasing order of their received powers [1]. Consequently, the rate of the th user has the expression [5]

(5)

With the convention that denotes the number of users whose received power is smaller than , the following inequality is readily obtained: . It shows that, as long as is not much smaller than , then does not tend to zero when . Hence, it is likely that SIC has a higher degree of fairness than OMA when the received powers are very disparate.

2.2. BORG and Its Low-Complexity Implementation

Consider the following scenario: users are divided into nonempty groups . For an arbitrary , we use to denote the sum of the received powers for the users that belong to the group . It is clear that and .

To be in line with the previous literature, we adopt the convention that the order of the groups in the successive decoding is , where is a permutation of the set . For simplicity, we denote by ORG the ordered partition which is given by the sequence of subsets . By combining the results from (4) and (5), we get the rate of the th user

(6)

For writing the equation above more compactly, we have assumed that the th user belongs to the group .

The naive approach for finding when is to search among all ordered partitions of the users into groups, then to compute the minimum rate in each case, and eventually to pick up the partition which maximizes the minimum rate. This leads to a huge computational burden and makes the method unpractical. We show below how the number of ordered partitions to be considered can be reduced significantly.

We need some more definitions. Let be a vector of strictly positive integers whose sum is equal to . The ordered partition is of type if for all the cardinality of is . Given and , we denote by the family of all ordered partitions of type . It is important to remark that for all partitions within this family we have that (i) the order of the subsets is the same and is given by the reverse order of the permutation ; (ii) the cardinality of the th subset is the same, namely, .

Additionally, for two arbitrary subsets and , we write if the received powers of all users within are greater than those of the users within . When the condition is not satisfied, we write .

Theorem 1.

Let . For fixed and , consider all ordered partitions that belong to . In this class, the ordered partition that maximizes the minimum rate for a given set is the one which satisfies the condition

(7)

The proof is deferred to the appendix.

Now we are prepared to formalize the result which shows the decrease in computational complexity.

Corollary 2.

For , we have the following.

  1. (i)

    To select by brute-force search amounts to compute the minimum rate for

    (8)

    different ordered partitions.

  2. (ii)

    Theorem 1 allows to reduce to

    (9)

    the number of ordered partitions that are considered in the evaluation process.

Proof.

  1. (i)

    In the case of brute-force search, it is easy to note that the rate vector must be computed for all ordered partitions of the users into nonempty subsets. Hence, the number of partitions to be considered equals , where is the Stirling number of the second kind, and its closed-form expression is given by [13]. This proves the result in (8).

  2. (ii)

    From Theorem 1, we know that for all permutations there exists a single ordered partition of type that must be considered, namely, the one which satisfies (7). For finding , we must evaluate a single rate vector for each vector type. This implies that the number of partitions which are investigated equals the number of ways that the integer can be written as a sum of strictly positive integers. According to [13], this number is given by (9).

To gain more insight, let us suppose that users and groups. Corollary 2 points out that the number of competing partitions for the selection of can be reduced from 55980 to 36, which implies a significant decrease of the computational complexity. However, by using the result from (9), it is easy to verify that the number of rate vectors which must be evaluated for selecting is .

2.3. Geometrical Interpretation

We resort to the polymatroid structure of the capacity region [1, 10, 11], to give a new interpretation of BORG. The object of interest is the polyhedron defined by

(10)

where the set function is a mapping for all subsets of to the positive real numbers. For the problem that we study, it is convenient to choose

(11)

Then, it is a simple exercise to verify that the set function defined in (11) satisfies (i) (normalized); (ii) if (increasing); (iii) (submodular). Thus, according to the definition from [14], is a polymatroid. Moreover, the hyperplane given by is the sum-capacity facet of [1]. We analyze next the points within the sum-capacity facet that correspond to OMA, SIC, and ORG. To get the point which corresponds to OMA, we rewrite the identity in (4) as

(12)

For SIC, we take to be an arbitrary permutation of the set , and we assume that the users are decoded in the reverse order of . Therefore, the rate of the th user is

(13)

The formula in (5) is easily obtained from (13) for the particular case when is chosen such that . More importantly, (13) shows that, for each permutation , is a corner point of the polymatroid (see [10, 14] for more details).

With the convention that the th user belongs to , where is a permutation of , the expression in (6) is equivalent to

(14)

Observe that, in general, the rate vector given by (14) does not correspond to a corner of .

To enhance intuition, we depict in Figure 1(a) the polymatroid for when the two users have equal received powers (). Similarly, in Figure 1(b), it is shown for the case when . In both cases, the sum-capacity facet is the segment whose endpoints are the corners and .

Figure 1
figure1

The polymatroid when users. The points marked on the sum-capacity facet correspond to various MA methods. Here, means that user is decoded before user . Two different cases are considered: (a) homogeneous network (), (b) heterogeneous network ().

Because , the number of groups for the BORG method can be either or . Thus, we are interested in the points within the sum-capacity facet that correspond to and , respectively. As we already know, is the same with OMA, and coincides with SIC, where with . Consequently, is chosen by selecting between OMA and the one which maximizes the minimum rate. For completeness, we consider also the point TS that corresponds to the degree of fairness provided by the method from [1], which finds optimum weights for the time sharing between and .

Note in Figure 1(a) that the OMA point is the fairest on the sum-capacity facet. In this case, it is obvious that also corresponds to the fairest point. Moreover, the method from [1] gives the same weight to both and , which makes the TS point coincide with the OMA point. The situation changes in Figure 1(b), where and . This leads to . Remark also in Figure 1(b) that, even if is the best among OMA and SIC, the minimum of its rate vector is slightly smaller than the minimum rate for TS.

Next, we demonstrate by simulations the capabilities of various MA schemes.

3. Simulation Results

3.1. Evaluation Criteria

As it was already mentioned, the fairest rate vector is obtained by applying TS or, equivalently, by time sharing between the corner points of the sum-capacity facet. To find the fairest rate vector and also the optimal time-sharing coefficients, we have implemented in Matlab the algorithms III and IV from [1].

Let us assume that the number of runs for a specified set of experimental conditions is . An arbitrary method, say MET, is compared with TS by computing the Normalized Min-Rate with formula

(15)

where is the minimum of the rate-vector yield by MET in the th run. Similarly, is the minimum of the TS rate vector in the th run.

The second figure of merit that we consider for evaluating the MA schemes is the AME, which is generally denoted by . AME takes values in the interval and attains its maximum when OMA is utilized. Therefore, we have for all (see Chapter 5 in [3]).

To keep SIC inline with what we have in the corner points of the sum-capacity facet, we assume that all cancellations are perfect, and what is forwarded to the next decoder has no residual error from the already decoded users [15]. Furthermore, suppose that in each step a matched filter is used as decoder such that for computing the AME of the th user we can apply the formula (3.123) from [3]:

(16)

Note that the expression above takes into consideration the system model from (1). Additionally, it is assumed that the users are decoded in the decreasing order of the received powers. We emphasize that we do not use formula (7.31) from [3] because it was derived for SIC with residual errors propagated from previous steps.

It is clear that, for , we do not need to compute AME for all ordered partitions of the users into subgroups but only for . With slight abuse of notation, we assume that is the ordered partition , where .

More importantly, the BORG method combines the features of both OMA and SIC such that (i) the users within each group are orthogonal one to each other; (ii) the groups share the entire channel. It is evident that only the second characteristic determines the degradation of the AME. Hence, the expression of AME can be derived straightforwardly from (16) by taking into account that, when decoding the group , the role of interferer is played by the groups . If the th user belongs to the group , then we have

(17)

Remark in the expression above that AME is the same for all the users within the -group.

It is worth mentioning here that does not necessarily coincide with the grouping that maximizes the AME. For example, if the received powers are , , , , , , , , and , then the optimum AME is produced by the ordered partition , where , , and . The inequality implies , which shows clearly that cannot be the ordered partition .

We conclude the short discussion on the second figure of merit, by noticing that, whenever an experiment is repeated times, we calculate for each method MET the Average AME

(18)

where is the AME for the th user in the th run.

In the examples outlined below, the Normalized Min-Rate and the Average AME are employed to compare the performance of the following MA schemes: TS, , , , , and . In our settings, the number of users is , and the number of runs for each set of experimental conditions is . Additionally, the power of the Gaussian noise is taken to be one (). Four different network models are considered.

3.2. Examples

Model I

To quantify the degree of network heterogeneity, we consider the ratio between the power of the strongest user and the power of the weakest user: . The larger is , the more heterogeneous is the network. For a fixed value , we take and . The powers are chosen to be outcomes from a uniform distribution on , and the experiment is repeated times. This selection guarantees that the mean power is equal to 100. When , a single realization is considered, namely, .

We plot in Figure 2(a) the Normalized Min-Rate obtained for various MA schemes when increases from 0 dB to 30 dB. Due to the definition in (15), the graph for TS is a straight line parallel to -axis. Note in the same figure that the degree of fairness is very high for OMA when is close to 0 dB, but it decreases rapidly when the heterogeneity of the network increases. By contrast, SIC has a very low degree of fairness in homogeneous networks, but it improves with the increase of the network heterogeneity such that for  dB, SIC is clearly superior to OMA.

The beneficial effects of the newly proposed strategy can be observed for , which performs as well as OMA for small , but surpasses both OMA and SIC for large . More interestingly, good results are obtained not only when searching for the optimum , but also when the number of groups is kept fixed. Remark for the heterogeneous networks that performs very similarly with .

In Figure 2(b), we show the Average AME for the six methods which are compared. As it was already pointed out previously, OMA achieves always the maximum possible AME. We can notice from Figure 2(b) that SIC and TS yield the poorest Average AME. This drawback appears for the two schemes because the firstly decoded users receive high interference, which makes their AME close to zero. It is remarkable that and have AME superior to that of SIC. Moreover, the performance of approaches the Average AME of OMA when increases.

Figure 2
figure2

Experimental results for Model I: (a) Normalized Min-Rate versus ; (b) Average AME versus . The number of users is . Note that is expressed in dB, and for each the reported results are obtained from runs. The following MA methods are compared (for each method we indicate the color and the marker symbol used in plots): TS (black left-pointing triangle), (blue asterisk), (magenta right-pointing triangle), (green diamond), (red square), and (brown circle).

Model II

Let , , , and . We simulate a network such that are uniformly distributed on , and are uniformly distributed on . The parameter controls the degree of heterogeneity of the network. When , all the users belong to a single cluster, and the increase of makes the network to consist of two disjoint clusters.

The results plotted in Figure 3 are obtained by varying the value of from 0 to 180. For each value of , different realizations of are generated. In terms of fairness and AME, the performance of is the same with that of for . Remark in Figure 3(a) that the graph of coincides with that of when is larger than 100. A similar fact can be also observed in Figure 3(b). So, automatically adapts to the topology of the network.

Figure 3
figure3

Experimental results for Model II: (a) Normalized Min-Rate versus ; (b) Average AME versus . Note that is not expressed in dB. The number of users (), the number of runs (), and all graphical conventions are the same like in Figure 2.

Model III

We consider again a network including two clusters. This time, the received powers of the users are generated as suggested in [16]. Let such that . We take , where if and if . The distribution of the random variable is Chi-Square with two degrees of freedom.

Remark that the heterogeneity of the network is measured by the difference , which we increase from 0 to 90. The Normalized Min-Rate and the Average AME calculated for each based on runs are shown in Figure 4. It is easy to observe the following outcome of the experiment. Because the Chi-Square distribution has infinite support, OMA does not provide fairness in rate allocation when . Due to the same reason, for all values of , the degree of fairness yield by is inferior to that of even if, for , the number of groups equals the "true" number of clusters. However, when comparing the Average AME, is ranked the second after OMA which achieves the maximum possible value.

Figure 4
figure4

Experimental results for Model III: (a) Normalized Min-Rate versus ; (b) Average AME versus . Note that is not expressed in dB. The number of users (), the number of runs (), and all graphical conventions are the same like in Figure 2.

Model IV

Following the suggestion of one of the reviewers, we briefly investigate the case of users uniformly distributed over a two-dimensional area. For the sake of concreteness, let be the distances from the BS to the users. According to the large-scale model, we have , where is the received power from the th user, is the received power from a transmitter located at distance one from the BS, is the distance from the th user to the BS, and is the path loss exponent [17, 18].

It is widely accepted that for urban area cellular radio [18, Table  3.2]. In our settings, we choose the path loss exponent to be and . For two arbitrary bounds and with property , the squared distances are selected to be uniformly distributed on . Hence, the mean power has the expression . Let us consider various values of between 1.2 and 3.0, and for each we choose such that . Conventionally we take . It is easy to verify that implies , or equivalently all the users are located on a circle whose center coincides with the BS. Obviously, this corresponds to the case of a homogeneous network. In fact, for all , the quantity can be used to measure the heterogeneity of the network: the bigger is , the larger is the difference , which makes the values of , , more disparate.

In Figure 5, we plot the Normalized Min-Rate and the Average AME. They are computed for each based on runs, while for one single realization is considered. By comparing the results within Figure 5 with those from Figure 2, we can observe that the multiaccess schemes have a similar behavior for Model IV and Model I.

As a final remark, we note that for all four network models, finding the partition which corresponds to is faster than applying the TS optimization strategy. In all runs, the execution time for was at about of the execution time for TS. When the number of users is very large, the computational complexity can be decreased by searching for with a fixed , instead of finding the partition . We observe from the numerical examples that and have an acceptable level of performance.

Figure 5
figure5

Experimental results for Model IV: (a) Normalized Min-Rate versus ; (b) Average AME versus . Note that is not expressed in dB. The number of users (), the number of runs (), and all graphical conventions are the same like in Figure 2.

4. Conclusion

In this paper, we investigated how OMA and SIC can be combined to improve fairness in Gaussian wireless networks. The newly proposed method divides the network into (almost) homogeneous subnetworks such that the users within each subnetwork employ OMA, and SIC is utilized across subnetworks. Equivalently, the users are partitioned into ordered groups. The main theoretical result which we proved for any shows that the ordered partition which maximizes the minimum rate can be found with a low-complexity algorithm. Moreover, it was demonstrated experimentally that the user grouping strategy guarantees a good tradeoff between fairness and the asymptotic multiuser efficiency.

References

  1. 1.

    Maddah-Ali MA, Mobasher A, Khandani AK: Fairness in multiuser systems with polymatroid capacity region. IEEE Transactions on Information Theory 2009, 55(5):2128-2138.

  2. 2.

    Verdu S: Optimum multiuser asymptotic efficiency. IEEE Transactions on Signal Processing 1986, 34: 890-897.

  3. 3.

    Verdu S: Multiuser Detection. Cambridge University Press, Cambridge, UK; 1998.

  4. 4.

    Yang B, Danilo-Lemoine F: Asymptotic multiuser efficiency of a decorrelator based successive interference cancellation DS-CDMA multiuser receiver. Proceedings of Military Communications Conference (MILCOM '06), 2006-7.

  5. 5.

    Tse D, Viswanath P: Fundamentals of Wireless Communications. Cambridge University Press, Cambridge, UK; 2005.

  6. 6.

    Viterbi AJ: Very low rate convolutional codes for maximum theoretical performance of spread-spectrum multiple-access channels. IEEE Journal on Selected Areas in Communications 1990, 8(4):641-649. 10.1109/49.54460

  7. 7.

    Warrier D, Madhow U: On the capacity of cellular CDMA with successive decoding and controlled power disparities. Proceedings of the 48th IEEE Vehicular Technology Conference (VTC '98), May 1998, Ottawa, Canada 3: 1873-1877.

  8. 8.

    Buehrer RM: Code Division Multiple Access(CDMA). Synthesis Lectures on Communications 2006, 2: 1-192.

  9. 9.

    Bertsekas DP, Gallager RG: Data Networks. Prentice-Hall, Upper Saddle River, NJ, USA; 1987.

  10. 10.

    Tse DNC, Hanly SV: Multiaccess fading channels-part I: polymatroid structure, optimal resource allocation and throughput capacities. IEEE Transactions on Information Theory 1998, 44(7):2796-2815. 10.1109/18.737513

  11. 11.

    Zhang X, Chen J, Wicker SB, Berger T: Successive coding in multiuser information theory. IEEE Transactions on Information Theory 2007, 53(6):2246-2254.

  12. 12.

    Shum KW, Sung CW: Fair rate allocation in some Gaussian multiaccess channels. Proceedings of IEEE International Symposium on Information Theory (ISIT '06), July 2006, Seattle, Wash, USA 163-167.

  13. 13.

    Flajolet P, Sedgewick R: Analytic Combinatorics. Cambridge University Press, Cambridge, UK; 2009.

  14. 14.

    Edmonds J: Submodular functions, matroids, and certain polyhedra. Proceedings of Calgary International Conference on Combinatorial Structures and Applications, 1970, Calgary, Canada 69-87.

  15. 15.

    Cover T, Thomas J: Elements of Information Theory. John Wiley & sons, New York, NY, USA; 2006.

  16. 16.

    Jagannathan K, Borst S, Whiting P, Modiano E: Scheduling of multi-antenna broadcast systems with heterogeneous users. IEEE Journal on Selected Areas in Communications 2007, 25(7):1424-1434.

  17. 17.

    Pahlavan K, Krishnamurthy P: Principles of Wireless Networks: A Unified Approach. Prentice-Hall, Upper Saddle River, NJ, USA; 2002.

  18. 18.

    Rappaport TS: Wireless Communications: Principles and Practice. Prentice-Hall, Upper Saddle River, NJ, USA; 2002.

  19. 19.

    Cloud M, Drachman B: Inequalities: With Applications to Engineering. Springer, Berlin, Germany; 1998.

Download references

Acknowledgment

This work was supported by the Academy of Finland, Project nos. 113572, 118355, 134767, and 213462.

Author information

Correspondence to Seyed Alireza Razavi.

Appendix

Appendix

Proof of Theorem 1

First we demonstrate two auxiliary results that are instrumental in proving Theorem 1.

Lemma 3.

  1. (i)

    Let with . The following inequality holds:

    (A1)
  2. (ii)

    For that satisfy simultaneously the conditions and , we have

    (A2)

Proof.

For an arbitrary , the following inequality is well known [19]:

(A3)

where denotes the natural logarithm. By using this result together with some elementary calculations, the inequalities in (A.1) and (A.2) are readily obtained.

  1. (i)

    Since , there exists such that . So, the left-hand side of (A.1) can be expressed as . It is easy to check that the first derivative of is strictly negative for all :

    (A4)

    Note that the inequality in (A.4) is a straightforward consequence of (A.3). The fact that is a monotonically decreasing function proves the inequality in (A.1).

  2. (ii)

    The inequality guarantees that there exists such that . Moreover, we have because . For all , we define . In (A.2), the left-hand side is equal to with , whereas the right-hand side coincides with . To prove the inequality, it is enough to show that is a monotonically decreasing function or, equivalently, to verify that the first derivative of is strictly negative:

    (A5)

    The inequality in (A.5) was obtained by applying (A.3).

Lemma 4.

For a given set , let and be two ordered partitions from . If there exists such that

(C1) for all , 

(C2) , 

(C3) , 

then the minimum rate corresponding to is not larger than the minimum rate corresponding to .

Proof.

Without loss of generality, we assume for all . Additionally, we make the assumption that do not exist two different users for which the received powers are the same. With the exception of the symbol , all the notations employed in connection with the ordered partition are the same with those utilized for .

The formula in (6) leads to the following expression of the minimum rate for the users within :

(A6)

where is the received power of the weakest user within and . The equation above together with condition () lead to the identity for all . Hence, for proving Lemma 4, it is enough to show that

(A7)

This inequality can be obtained from the three results which are outlined below.  

Result 1.

.

To verify the inequality, we note that

(A8)
(A9)
(A10)
(A11)

In (A.9), we use the fact that , which is a consequence of ()–(). The inequality in (A.10) is derived by applying (A.2) with , , , and .

Result 2.

If , then .

First we consider the case for which we get:

(A12)
(A13)
(A14)
(A15)
(A16)

Because contains the weakest users of , condition implies , which leads to the identity in (A.13). The inequality in (A.14) can be verified by operating in (A.1) the following substitutions: , , and . It is easy to check that . Hence, , which proves the inequality in (A.15).

Under the hypothesis , we have

(A17)
(A18)
(A19)
(A20)

The identity in (A.17) is the same with the one from (A.13). Then we write the inequality in (A.2) for the particular case when , , , , and we get (A.18). Observe that . Additionally, implies , which yields the inequality in (A.19).

Result 3.

If , then . Note that

(A21)
(A22)
(A23)
(A24)
(A25)

The identity in (A.22) is obtained by applying the same type of reasoning like the one used to demonstrate (A.13). Then we focus on (A.23), which is proved by choosing , , in (A.1). Furthermore, we get (A.24) from .

The inequalities in (A.11), (A.16), (A.20), and (A.25) lead to (A.7), which concludes the proof of Lemma 4.

Now we give the proof of the theorem.

Proof of Theorem 1.

Given , we consider the ordered partition that belongs to . For an arbitrary , we define the following elementary transformation, which is denoted by :

  1. (i)

    if , then is transformed to another ordered partition from , say , such that the conditions () and () from Lemma 4 are satisfied;

  2. (ii)

    otherwise, the ordered partition remains unchanged.

Assume that we apply to . The newly obtained ordered partition is then transformed by , and the process continues until is applied. Due to the definition of , the resulting ordered partition is guaranteed to contain the strongest user in the group which is decoded first. If the type-vector has the property that and the second strongest user was not yet included in the group which is decoded first, then we move it to this group by iterating again from to . When , the second strongest user can be moved to the second decoded group by using similar transformations.

Based on the observations above, we remark that each ordered partition from can be transformed, in a finite number of steps, to the unique partition from which satisfies the condition in (7). Each step consists in transforming the current ordered partition to a new one by applying , where . We conclude the proof by mentioning that, at each step, the minimum rate increases or remains constant (see Lemma 4).

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Keywords

  • Corner Point
  • Minimum Rate
  • Receive Power
  • Rate Vector
  • Successive Interference Cancellation