A distributed power level selection method and system for cellular wireless networks under joint constraints

The present invention relates to power level selection in cellular wireless networks. In particular, the present invention relates to the selection of power levels in a plurality of smaller wireless cells, taking into account the presence of larger wireless cells. More specifically, the present invention relates to the selection of power levels in a plurality of femtocells or picocells servicing a small area such as a private home or a larger condominium within a wider area served by the macrocell. ..

Cross-reference of related applications This application claims the benefits of US Patent Provisional Application No. 61 / 380,730, filed September 8, 2010, which is incorporated herein by reference in its entirety.

Today's cellular wireless networks provide services using large base stations, called macrocells, that transmit and receive communication channels over a relatively large area. While wireless technology is improving rapidly, service providers have introduced a number of smaller cellular wireless cells in private homes, shopping centers and other complexes to free up macrocell capacity, within smaller cells. We are exploring the value of providing better service quality. The present invention provides a method for selecting distributed generation levels in areas including many small wireless cells (eg, femtocells that cover private homes or even apartments, or picocells that cover larger complexes such as shopping centers). Present. Also, the power level selection method must take into account the presence of larger wireless cells (eg, macrocells) that serve the entire area not covered by the smaller cells.

The advent of small cells such as femtocells or picocells presents new opportunities and challenges for cellular carriers. Because these small cells use a cellular licensed spectrum, they can interfere with the use of macrocells, the infrastructure used by cellular carriers. Therefore, the power level selected for the femtocell must be adjusted to prevent unacceptable interference between femtocells and to macrocells. Although the method covers small wireless cells introduced within the area serviced by large wireless cells, the terms femtocell and macrocell will be used below. Non-Patent Document 1 and Non-Patent Document 2 present a summary paper on a femtocell network integrated within the area serviced by macrocells.

Consider an area with multiple femtocells and macrocells where the power level of the macrocell is fixed (hereinafter, small cells are referred to as femtocells and large cells are referred to as macrocells). The purpose is to determine the power level of each femtocell to meet the following constraints: (i) Each femtocell can provide an appropriate signal-to-interference noise ratio (SINR) over the entire area covered by the femtocell. (ii) The macrocell can provide the appropriate SINR over the entire area serviced by the macrocell.

Suppose a set of cell-dependent critical locations is given as input for each femtocell and each macrocell. The critical position is expected to be the worst expected position in the region covered by SINR. Practical power level selection methods should consider the suitability of SINR at the critical position associated with each femtocell and each macrocell.

Several papers have been published on determining minimum-power solutions. In the context of the present invention, these papers select the minimum power level of each femtocell so that its SINR is at least as large as the specified parameter. Although these treatises focus on different applications, the challenges and underlying mathematical problems are similar. Examples of these references include Non-Patent Document 3, Non-Patent Document 4, and Non-Patent Document 5. These references present a distributed algorithm for minimum power, where all calculations are performed locally. These references focus on single-layer cells such as femtocells and do not take into account the constraints imposed by existing, larger cells such as macrocells.

Non-Patent Document 6 and KJ Kerpez, T. Lan, K. Sinkar, and L. Kant, Patent Document 1 "System and Method for Resource Allocation of a LTE Network Integrated with Femtocells" extend previous research and both. It presents an algorithm for power allocation in an environment with two types of cells, where the type of power level is the determinant. The former reference seeks the minimum power solution, while the latter imposes constraints on the selected power level while maximizing the data rate. These references address the key challenge of managing two types of cells (femtocells and macrocells), where the open challenge is bounded by areas where the power levels of both femtocells and macrocells change. It means that it does not affect other macro cells that are not included in the area. It may be more appropriate to keep the power level of the macrocell fixed at a particular value while determining the power level of the femtocell in the bounded region.

The present invention provides a distributed algorithm that determines the maximum SINR that can be met by all femtocells, while also satisfying the specified SINR parameters of the macrocell. The power level selected for the femtocell is the minimum power solution with the maximum SINR that can be met by all femtocells. The power level of the macro cell is specified as an input and cannot be changed. This algorithm performs all intensive computations locally on each femtocell independently. The timing of distributed computations is synchronized. The calculation is performed simultaneously in all cells, and after each iteration, information on provisional power level selections in multiple cells is exchanged between femtocells. Finally, the variance calculation converges to the maximum SINR value and the corresponding minimum power solution.

<p><patcit num="1"><text>U.S. Patent Application Publication No. 2011/0183678</text></patcit></p>

<p><nplcit num="1"><text>V. Chandrasekhar, JG Andrews, and A. Gatherer, "Femtocell Networks: A Survey", IEEE Communications Magazine, 46, 59-67, September 2008</text></nplcit><nplcit num="2"><text>H. Claussen, LT W Ho, and LG Samuel, "An Overview of the Femtocell Concept", Bell Labs Technical Journal, 13, No. 1, 221-246, 2008</text></nplcit><nplcit num="3"><text>SV Hanly, "An Algorithm for Combined Cell-Site Selection and Power Control to Maximize Cellular Spread Spectrum Capacity", IEEE Journal on Selected Areas in Communications, 13, 1332-1340, 1995</text></nplcit><nplcit num="4"><text>RD Yates, "A Framework for Uplink Power Control in Cellular Radio Systems", IEEE Journal on Selected Areas in Communications, 13, 1341-1347, 1995</text></nplcit><nplcit num="5"><text>E. Altman and Z, Altman, "S-Modular Games and Power Control in Wireless Networks", IEEE Transactions on Automatic Control, 48, 839-842, 2003</text></nplcit><nplcit num="6"><text>T. Thanabalasingham, SV Hanley, LLH Andrew, and J. Papandriopoulos, "Joint Allocation of Subcarriers and Transmit Powers in a Multiuser OFDM Cellular Network", IEEE ICC Proceedings, 269-274, 2006</text></nplcit><nplcit num="7"><text>P. Lancaster, Theory of Matrices, Academic Press, New York, 1969, p. 285</text></nplcit><nplcit num="8"><text>Lancaster 1969, exercise 4, p. 287</text></nplcit></p>

<p> The emergence of a new architecture, a cellular wireless network with a large number of small cells called femtocells and picocells, integrated within an existing large cell network called macrocells, presents new opportunities and challenges. Femtocells and picocells are used in larger complexes such as private homes and shopping malls, thus freeing up the scarce bandwidth capacity of macrocell networks. However, these small cells use spectra licensed to cellular carriers, creating new challenges. Therefore, these small cells can interfere with the use of macrocells, which is the infrastructure used between small cells and by cellular carriers.</p><p> Consider an area with multiple femtocells and macrocells where the macrocell power level is fixed. The purpose is to determine the power level of each femtocell to meet the following constraints: (i) Each femtocell can provide an appropriate signal-to-interference noise ratio (SINR) over the entire area covered by the femtocell. (ii) The macrocell can provide the appropriate SINR over the entire area serviced by the macrocell.</p>

<p> The present invention provides a new distributed algorithm that determines the maximum SINR that can be met by all femtocells, while also satisfying the specified SINR parameters of the macrocell. The power level selected for the femtocell is the minimum power level at which the femtocell's maximum SINR can be achieved. The power level of the macro cell is specified as an input and cannot be changed. This new algorithm extends the previous algorithm for finding the minimum power solution while satisfying the specified femtocell SINR parameter (or some similar variation), but the previous algorithm is achieved by the femtocell. Does not try to maximize the SINR that can be.</p><p> This new distributed algorithm performs all centralized calculations independently and locally on each femtocell. The calculations are synchronized over time and are performed simultaneously in all cells, after each iteration, information on provisional power level selection in multiple femtocells is exchanged between femtocells. Finally, the calculation converges on the maximum SINR value and the corresponding minimum power solution.</p><p> The present invention will be understood more clearly by reading the following description in conjunction with the accompanying drawings.</p>

<figref num="1">FIG. 6 is a schematic representation of the region serviced by six femtocells and one macrocell and a single critical position of femtocell 101.</figref><figref num="2">FIG. 6 is a schematic representation of the area serviced by six femtocells and one macrocell and a single critical position of the macrocell.</figref><figref num="3">It is a flowchart of a distributed algorithm that calculates the maximum feasible SINR of a femtocell and the corresponding minimum power solution while satisfying the specified SINR parameters of the macrocell.</figref>

Next, with reference to the figure, in particular FIG. 1, example 100 of the area serviced by the six femtocells 101-106 is shown, each of which is represented by a circle surrounding each femtocell. Provide services in a small area. The femtocell can be, for example, a private home or a larger complex, in which case the cell is often referred to as a picocell. In general, these cells are just small cells that provide wireless services in narrow, bounded areas. This entire area is serviced by macrocell 107. Today's technology uses only macrocells in cellular wireless networks. In the future, these services can be provided by mixing large cells (eg, macrocells) and large numbers of small cells (eg, femtocells and picocells).

FIG. 1 shows the critical position 108 associated with femtocell 101. Suppose a set of cell-dependent critical positions is given as the input for each femtocell. The critical position is expected to be the worst position in the area where SINR is serviced by femtocells. In addition, FIG. 1 shows the signal from the femtocell 101 received at the critical position 108 and the interference signals from the other femtocells 102 to 106 and the macrocell 107. It makes sense to assume that the critical position of the femtocell is on the cell boundary, as indicated by position 108 on the border of the femtocell 101.

Next, with reference to FIG. 2, an example 200 of the same region as FIG. 1 serviced by the six femtocells 201-206 and macrocell 207 is shown. FIG. 2 shows the critical position 208 associated with macrocell 207. Again, it makes sense to assume that the critical position of the macrocell is on the border of the femtocell, as indicated by position 208 on the border of the femtocell 204. In any case, the input to the method in the present invention comprises a set of specified critical positions without any particular assumptions. In addition, FIG. 2 shows the signal received from the macro cell 207 at the critical position 208 and the interference signal from the femtocells 201 to 206. For each critical position in the macrocell, the specified macrocell SINR must be met. Note that if the region is serviced by more than one macrocell, it is easy to determine which macrocell will service each critical point (ie, the macrocell that provides the strongest signal). Therefore, for the sake of brevity, this description is limited to a single macrocell without any loss of generality.

The following notation is used. j, k = femtocell index, J is the set of all considered femtocells. m = Macrocell index. s = Index of selected critical position. S_jLet be the set of selected critical positions of the femtocell j, and let Q be the set of the selected critical positions of the macrocell. g_j(s) = Signal loss factor from the center of femtocell j to position s. g_m(s) = Signal loss factor from macrocell to position s. P_m= Transmission power level (also known as signal strength) of the macrocell (input). P_j= Transmission power level of femtocell j (also known as signal strength). P_j, j J is the coefficient of determination. P = {P_j, j J} is P_jRepresents a vector of. N = Noise level (independent of position). T_m= Minimum SINR required for macro cell connection. T_f= Minimum SINR required for femtocell connections (independent of j). It is easy to use different noise levels at each position instead of the same noise level.

Any femtocell power allocation scheme must meet the following constraints:

<maths num="1"><img file="JP2013538020A_D0001.tif" /></maths>

For all j J s S_j (1)

<maths num="2"><img file="JP2013538020A_D0002.tif" /></maths>

s Q (2) For all j J Pj 0 (3)

Constraint (1) ensures that the SINR of the femtocell meets the required threshold at the selected critical position. Constraint (2) ensures that the SINR of the macrocell meets the required threshold at the selected critical position. Note that if multiple macrocells serve the region, constraint (1) includes a term of interference (which is just a constant) for each of these macrocells. Constraint (2) has a single constraint for each critical position s Q associated with the macrocell that provides the strongest signal at that position.

P¹Satisfies constraints (1)-(3), and P for any vector P that also satisfies constraints (1)-(3)¹Assume P. Then P¹Is called the minimum power solution.

The minimum power model (MINP model) is formulated by reorganizing constraints (1) to (3) as follows.

MINP model

<maths num="3"><img file="JP2013538020A_D0003.tif" /></maths>

For all j J s S_j (4.2)

<maths num="4"><img file="JP2013538020A_D0004.tif" /></maths>

s Q (4.3)

Pj 0 (4.4) for all j J

Minimum power solution P = {P_j, j J} (4.1).

The constraint (4.2) is the same as the constraint (1), and the set S_jGuarantees proper SINR of femtocells at critical positions in. To be able to service the position s, s S_jAbout g_jNote that (s)&gt; 0. Constraint (4.2) is any P_jHas a special structure below, bounded by a linear combination of all other femtocell powers with constants added (having non-negative coefficients). Therefore, for all j J

<maths num="5"><img file="JP2013538020A_D0005.tif" /></maths>

Satisfies the constraint (4.2) for all j J

<maths num="6"><img file="JP2013538020A_D0006.tif" /></maths>

Also satisfies constraint (4.2) for any γ 1. Constraint (4.3) is the same as constraint (2). Each of these constraints indicates that the linear combination of femtocell power (with a non-negative coefficient) cannot exceed the specified parameter.

MINP model is a feasible solution P^(feas)Have some j₁About all constraints (4.2)

<maths num="7"><img file="JP2013538020A_D0007.tif" /></maths>

Is satisfied as an inequality in the narrow sense. Then, so that at least one of these constraints is satisfied as an equation, while all others are still satisfied as an inequality.

<maths num="8"><img file="JP2013538020A_D0008.tif" /></maths>

Can be reduced. You can repeat the decrement in one variable at a time. P^(feas)Is feasible (for all j J)

<maths num="9"><img file="JP2013538020A_D0009.tif" /></maths>

) And all s S_jAnd about j J [T_f(P_mg_m(s) + N)] / g_jSince (s) 0, this variable reduction scheme maintains P 0 on all iterations, converges to a feasible solution, and for each j J, at least one constraint that is satisfied as an equation (4.2). ). The solution at that point satisfies all the constraints (4.2) to (4.4) and is the minimum power solution. Furthermore, the solution of the constraint (4.2) that is satisfied as an equation is the minimum power solution. Under the conditions described below, this solution is unique.

Since the minimum power solution is unique, the minimum power solution is Σ_{j J}P_jAlso minimize. Therefore, (Purpose minΣ_{j J}P_jThe MINP model is a linear programming optimization problem that can be solved using commercial software. However, it should be noted that the linear programming method can only be used as a centralized method in which the calculations are centralized for all femtocells.

The references cited above present a variant of the distributed generation level selection algorithm, where each femtocell uses locally available interference measurements to perform all required calculations. Execute locally. In other words, each femtocell performs its power level calculation independently of all other cells until the power levels calculated in the various cells converge to the minimum power solution. Note that these algorithms solve the MINP model without considering constraint (4.3). However, these algorithms can be easily modified to take into account the latter constraint. A version of such an algorithm will be described later.

The present invention has a threshold T in constraint (4.2)._fPresents a new minimum power algorithm that is treated as the coefficient of determination to be maximized. Since constraint (4.2) is non-linear, the resulting problem here is the problem of non-linear programming optimization. The purpose is to meet all constraints (4.2)-(4.4) while maximizing the feasible threshold T._fAnd to determine the corresponding minimum power solution.

The formulation of a new model solved by the present invention, called the maximum threshold model (MAXT model), is as follows.

MAXT model

<maths num="10"><img file="JP2013538020A_D0010.tif" /></maths>

For all j J s S_j (5.2)

<maths num="11"><img file="JP2013538020A_D0011.tif" /></maths>

s Q (5.3) Pj 0 (5.4) for all j J τ_f T_f (5.5) Maximum τ_fAnd the corresponding minimum power solution P = {P_j, j J} (5.1).

Next, referring to FIG. 3, a flowchart 300 of a distributed algorithm called the DSMAXT algorithm is shown, which solves the MAXT model in a distributed manner, that is, the centralized calculation is independent of all other femtocells. , Performed locally in each femtocell. This calculation is iterative and synchronizes across all femtocells. After each iteration, each femtocell shares the provisional power level calculated for that cell with all other femtocells. The iterative process of calculation converges on the optimal solution of the MAXT model, resulting in the femtocell's maximum feasible SINR threshold and the corresponding minimum power solution.

For the time being, for the sake of brevity, each set S_jIs assumed to consist of a single position. Therefore, since the position is specified by the index j, the index s can be deleted. Then the constraints (4.2) and (5.2)

<maths num="12"><img file="JP2013538020A_D0012.tif" /></maths>

For all j J (6)

Can be written, in the formula, g_jkIs the signal loss factor from cell k to the critical position of cell j, g_jmIs the signal loss coefficient from the macro cell to the critical position of cell j. Index j is s S_jNote that it replaces dependence on.

The outline of the algorithm presented in Fig. 3 is shown below.

DSMAXT algorithm Step 301 Solve the MINP model using the distributed synchronous minimum power algorithm (DSMINP algorithm) described below.

av = 0 and

<maths num="13"><img file="JP2013538020A_D0013.tif" /></maths>

To initialize.

b.

<maths num="14"><img file="JP2013538020A_D0014.tif" /></maths>

For all j J (7) To calculate.

c. Check constraint (4.3). If one or more of these constraints are violated, the process proceeds to step 302.

d. For any small ε&gt; 0, for all j J

<maths num="15"><img file="JP2013538020A_D0015.tif" /></maths>

If it is,

<maths num="16"><img file="JP2013538020A_D0016.tif" /></maths>

The minimum power solution of the MINP model called is obtained, and the process proceeds to step 303. If not, return to step b above as v v + 1.

Step 302. Finish. There is no feasible solution for the MAXT model.

Step 303.P⁽⁰⁾= P⁰, Tau⁽⁰⁾= T_fAnd start with v = 0.

Step 304.P^(v)Find the critical position c Q with the smallest macrocell SINR ratio of (this SINR is for all k J)

<maths num="17"><img file="JP2013538020A_D0017.tif" /></maths>

Is the left side of (2)).

Step 305.

<maths num="18"><img file="JP2013538020A_D0018.tif" /></maths>

To calculate.

Step 306.

<maths num="19"><img file="JP2013538020A_D0019.tif" /></maths>

To calculate. During the ceremony

<maths num="20"><img file="JP2013538020A_D0020.tif" /></maths>

U = [u₁, ..., u_J]^T、

<maths num="21"><img file="JP2013538020A_D0021.tif" /></maths>

Is z = [z₁, ..., z_J]^TAnd

<maths num="22"><img file="JP2013538020A_D0022.tif" /></maths>

, J, k J, j k Matrix F = (f_jk) And f_jj= 0, j J.

<maths num="23"><img file="JP2013538020A_D0023.tif" /></maths>

To set. In the equation, TMAX is τ, which guarantees convergence._fThe maximum possible value (which will be derived later).

Step 307. Calculate the power. For all j J

<maths num="24"><img file="JP2013538020A_D0024.tif" /></maths>

Step 308. For any small δ&gt; 0 and ε&gt; 0, for all j J

<maths num="25"><img file="JP2013538020A_D0025.tif" /></maths>

and

<maths num="26"><img file="JP2013538020A_D0026.tif" /></maths>

If, the process proceeds to step 309. If not, update v v + 1 and return to step 304.

Step 309. Finish. The best solution is

<maths num="27"><img file="JP2013538020A_D0027.tif" /></maths>

And P^opt= P^{(v + 1)}Is.

A further description of the steps of the algorithm is provided below.

The DSMAXT algorithm starts in step 301 by solving the MINP model.

MINP model is a feasible solution P^(feas)Have some j₁About all constraints (4.2)

<maths num="28"><img file="JP2013538020A_D0028.tif" /></maths>

Is satisfied as an inequality in the narrow sense. Then, so that at least one of these constraints is satisfied as an equation, while all others are still satisfied as an inequality.

<maths num="29"><img file="JP2013538020A_D0029.tif" /></maths>

Can be reduced. You can repeat the decrement in one variable at a time. P^(feas)Is feasible (for all j J)

<maths num="30"><img file="JP2013538020A_D0030.tif" /></maths>

), And for all j J [T_f(P_mg_m(s) + N)] / g_jSince (s) 0, this variable reduction scheme maintains P 0 on all iterations, converges to a feasible solution, and for each j J, at least one constraint that is satisfied as an equation (4.2). ). The solution at that point satisfies all the constraints (4.2) to (4.4) and is the minimum power solution. This solution is unique under the conditions also described later in Non-Patent Document 4.

Since the minimum power solution is unique, the minimum power solution is Σ_{j J}P_jAlso minimize. Therefore, Σ subject to constraints (4.1) to (4.3)_{j J}P_jMinimizing is a linear programming optimization problem that can be solved in the center using commercial software (eg, IBM's ILOG CPLEX software). It is emphasized that the linear programming method can only be used as a centralized method in which the calculations are performed in the central position, rather than the distributed calculations performed locally in each femtocell. The DSMINP algorithm described in step 301 is a distributed algorithm in which the power level calculation (7) is performed locally in each femtocell. Similarly, the convergence check is calculated locally. The term synchronization refers to the simultaneous computation performed at each iteration in each femtocell.

The DSMINP algorithm is, for example, a minor modification of the previously published algorithm in Non-Patent Document 4 described above, and does not constitute the present invention by itself. As described in Non-Patent Document 4, the DSMINP algorithm does not need to be synchronized across all femtocells. However, the DSMAXT algorithm requires synchronization.

For all j J

<maths num="31"><img file="JP2013538020A_D0031.tif" /></maths>

Please note that. Here, for all j J

<maths num="32"><img file="JP2013538020A_D0032.tif" /></maths>

Is assumed to be. Then recursion (7) is for all j J

<maths num="33"><img file="JP2013538020A_D0033.tif" /></maths>

Will be. Note that the calculation (7) performed on each femtocell requires only the interfering information available via the measurements, without exchanging any information between the femtocells. Constraint (4.3) is used only to identify the feasibility and does not complicate the problem. For example, as evidenced in Non-Patent Document 4, if a feasible solution exists, the DSMINP algorithm is guaranteed to converge to a unique minimum power solution.

The necessary and sufficient conditions for the existence of a feasible solution of MINP MODEL without constraint (4.3) are presented below.

Constraint (6) can be written as P AP + b in matrix notation. Matrix A = (a_jk) Is

<maths num="34"><img file="JP2013538020A_D0034.tif" /></maths>

j, k J, j k a_jj= 0, j J

Is. Vector b = (b_j) Is

<maths num="35"><img file="JP2013538020A_D0035.tif" /></maths>

j J

Given by.

Recursive equation (7) is P^{(v + 1)}= (I + A + A²+ ... A^v) B. For any v (I + A)^-1= I + A + A²+ ... A^v+ (I + A)^-1A^{v + 1}Please note that. Therefore, the necessary and sufficient conditions for the algorithm to converge to the solution (ignoring the constraint (4.3)) are the following set of matrices (I + A).^-1= I + A + A²+ ... A^v+ ... is convergence. This requires that the maximum eigenvalues of matrix A must be less than exactly 1. Violation of one of the constraints (4.3) will detect an infeasible solution, which is due to the very strict macrocell constraints (eg P)._mIs too small, or T_mCan occur (because is too large). Also, the feasibility is the vector P even without the constraint (4.3).^(v)Does not converge (P^{(v + 1)} P^(v)(Please note that) can occur. The matrix A can be rewritten as follows. A = T_fF

{λ₁, ..., λ_n) Is the eigenvalue of F. Then the eigenvalues of A are {T_fλ₁, ..., T_fλ_n). Therefore, a sufficiently small T_fFor, the DSMINP algorithm converges to a solution, which may or may not be feasible for constraint (4.3). Note that the matrix F is not negative and is irreducible. Then, according to the Perron-Frobenius theorem (Non-Patent Document 7), the eigenvalue of F having the largest magnitude, that is, λ₁There is, which is a real number and positive. The necessary and sufficient conditions for the DSMINP algorithm to converge are: T_fλ₁&lt;1 (11) Is.

further,

<maths num="36"><img file="JP2013538020A_D0036.tif" /></maths>

It also means that (Non-Patent Document 8). Therefore, T_fλ₁Sufficient conditions for &lt;1 to hold

<maths num="37"><img file="JP2013538020A_D0037.tif" /></maths>

Is. Of course, even if the conditions are not sufficient, the maximum eigenvalue λ still (for the submatrix F) using the numerical method for determining convergence.₁Can be calculated.

From the above explanation, the minimum power solution P⁰Note that is only a unique solution for the set of linear equations P = AP + b. T_fλ₁When &lt;1, a unique solution exists with P 0. Recursion (7) only facilitates the distributed algorithm to reach that solution.

Here, the MINP model needs to consider for each femtocell j J, the set S._jRecall that it can have multiple critical positions in. Therefore, equation (7) in the DSMINP algorithm

<maths num="38"><img file="JP2013538020A_D0038.tif" /></maths>

j J (13) Replaced by, sufficient condition (12)

<maths num="39"><img file="JP2013538020A_D0039.tif" /></maths>

Replaced by. The selected critical position to be considered may change from one iteration to the next. Nevertheless, as described in Non-Patent Document 4 (when receiving from multiple connections), the proof of convergence still holds in this case. Intuitively, when the algorithm performs several iterations, the set of critical positions does not change, resulting in a proof of a single critical position per femtocell.

Then M in steps 305 and 306_cand

<maths num="40"><img file="JP2013538020A_D0040.tif" /></maths>

Consider the calculation of. When constraint (5.3) is satisfied as an equation at critical position c

<maths num="41"><img file="JP2013538020A_D0041.tif" /></maths>

Is. M_cIs τ_fUsed to calculate a new value for. M_cMust be positive, so macrocells

<maths num="42"><img file="JP2013538020A_D0042.tif" /></maths>

Must be met, which is the required condition τ_fλ₁Should be considered in conjunction with &lt;1.

Condition TMAX τ_fλ₁&lt;1 is satisfied τ_fThe maximum value of. T_fIs τ_fThe condition (12) replaced by is τ_fλ₁Note that this is a sufficient condition for &lt;1 to hold.

Constraints (6) and (15) P τ_f(FP + u) (17) z^TP = M_c (18) Can be written as. In preparation for the iterative algorithm, the inequality in (17) is replaced by the equation. Then, by substituting (18) for (17), the following equation M_c= z^TP = τ_f(z^TFP + z^Tu) (19) Must be met, this is

<maths num="43"><img file="JP2013538020A_D0043.tif" /></maths>

Bring. In step 306 of the DSMAXT algorithm using (20)

<maths num="44"><img file="JP2013538020A_D0044.tif" /></maths>

Is calculated.

Sequence in (9.2)

<maths num="45"><img file="JP2013538020A_D0045.tif" /></maths>

Is monotonously non-decreasing. Then it was calculated by the DSMAXT algorithm

<maths num="46"><img file="JP2013538020A_D0046.tif" /></maths>

Is the optimal solution for the MAXT model. This statement is shown to hold using the following arguments. Assuming the sequence in (9.2)

<maths num="47"><img file="JP2013538020A_D0047.tif" /></maths>

Is non-decreasing, so the sequence {P^(v)} Is also non-decreasing. (Previously, recursion (7) in the DSMINP algorithm was for all j J

<maths num="48"><img file="JP2013538020A_D0048.tif" /></maths>

It was shown to meet. Sequence

<maths num="49"><img file="JP2013538020A_D0049.tif" /></maths>

The same argument holds when is non-decreasing. ) Sequence

<maths num="50"><img file="JP2013538020A_D0050.tif" /></maths>

Is

<maths num="51"><img file="JP2013538020A_D0051.tif" /></maths>

(In the formula, M is the largest M_cIs), so it is bounded above. Therefore, the sequence

<maths num="52"><img file="JP2013538020A_D0052.tif" /></maths>

Converges. In addition, (TMAX) λ₁Since &lt;1, finally the sequence {P. calculated by recursion (10) in step 307.^(v)} Converges.

The above explanation guarantees convergence by ensuring that the parameter τ is monotonically non-decreasing, but experience has shown that the algorithm converges even if no steps are taken to guarantee monotonicity. I know.

When it converges,

<maths num="53"><img file="JP2013538020A_D0053.tif" /></maths>

Is assumed to be. Then

<maths num="54"><img file="JP2013538020A_D0054.tif" /></maths>

Satisfies equation (20) and P^optSatisfies (17) with the equation. Therefore,

<maths num="55"><img file="JP2013538020A_D0055.tif" /></maths>

Satisfies all constraints (5.2)-(5.5) of the MAXT model, where at least one of the constraints (5.3) is satisfied as an equation. P^optIs the minimum power solution that satisfies constraint (5.2) (all satisfied as equations). Therefore,

<maths num="56"><img file="JP2013538020A_D0056.tif" /></maths>

Is the maximum possible τ that can meet all the constraints of the MAXT model_fThe value. Then, when it converges,

<maths num="57"><img file="JP2013538020A_D0057.tif" /></maths>

Is assumed to be. Again, all constraints of the MAXT model are met, where at least one of the constraints (5.3) is met as an equation. Minimum power vector P is τ_fIs a continuous increasing function of τ_fThe latter holds because these power levels approach infinity as they approach that boundary. Sequence {P^(v)} Does not converge and the MAXT model is not feasible, so the threshold τ_fCannot exceed TMAX. In addition, P^optIs

<maths num="58"><img file="JP2013538020A_D0058.tif" /></maths>

Is the minimum power solution that satisfies the constraint (5.2). Sequence {τ^(v)Calculated by the DSMAXT algorithm even if the} is not monotonically non-decreasing and the DSMAXT algorithm converges

<maths num="59"><img file="JP2013538020A_D0059.tif" /></maths>

Note that is the optimal solution for the MAXT model.

The DSMAXT algorithm is a sequence

<maths num="60"><img file="JP2013538020A_D0060.tif" /></maths>

Is expected to converge in most cases, even if is not monotonously non-decreasing. Nevertheless, to ensure convergence in all cases, the sequence

<maths num="61"><img file="JP2013538020A_D0061.tif" /></maths>

The calculation of can be modified so that this sequence is monotonically non-decreasing. For example, (9.1)

<maths num="62"><img file="JP2013538020A_D0062.tif" /></maths>

This can be achieved by modifying to. However, this amendment, when converged, can result in one or more violations of constraint (5.3). Various methods can be implemented to reduce the step size in (21). For example, whenever the left term of the maximum operator in (21) is negative, the algorithm is Δ^{(v + 1)}Backtrack to the latest iteration that brought&gt; 0 and (21)

<maths num="63"><img file="JP2013538020A_D0063.tif" /></maths>

A smaller step size can be used by modifying (in the equation, α is a parameter and 0 &lt;α &lt;1).

The alternative method is T_fIs to use the DSMINP algorithm to solve the MINP model for different values of, and the maximum value that provides a feasible solution is sought for the MAXT model.

<maths num="64"><img file="JP2013538020A_D0064.tif" /></maths>

Is. However, this technique, for example, T_fIt requires solving many such problems using the binary search of.

Constraint (5.2) needs to be considered for each femtocell j J, set S_jRecall that it can have multiple critical positions in. When multiple critical positions are considered for each femtocell, the recursion (10) in step 307 is corrected as follows.

<maths num="65"><img file="JP2013538020A_D0065.tif" /></maths>

j J (23)

In steps 305 and 306, for the most critical position c of all the positions in Q, M_cAnd vector z = [z₁, ..., z_J]^TNote that is calculated. Also, the vector u = [u₁, ..., u_J]^TIn each u_jAnd each row j of the matrix F in equation (9.1) is the most critical position s S_jConsists of. Fixed τ_fSufficient condition for there to be a feasible solution to T_fIs τ_fGiven by the constraint (14) replaced by.

The DSMAXT algorithm is a distributed synchronization method in which each femtocell j J performs power calculations locally independently. To this end, each femtocell has computational and information exchange capabilities via a local computing system. The computing system may be a computer or any type of known or becoming known system, typically a processor, memory device, storage device, input / output device, internal bus, and / Or may include a communication interface for communicating with other computers, such as communication hardware and software. Modules may be systems that perform any "function" that can be embodied as device components, software, programs, or software, hardware, firmware, electronic circuits, and the like. The computing systems must be synchronized to facilitate simultaneous power level calculations on all femtocells.

As will be appreciated by those skilled in the art, the present invention may be embodied as a system, method or computer program product. Therefore, the present invention takes the form of a completely hardware embodiment, a completely software embodiment (including firmware, resident software, microcode, etc.) or an embodiment that combines a software aspect and a hardware aspect. All of these embodiments may be referred to herein as "circuits," "modules," or "systems."

The terms used herein are for the purpose of describing particular embodiments only and are not intended to limit the invention. As used herein, the singular forms "a", "an" and "the" are intended to include the plural form unless the context explicitly indicates otherwise. The terms "comprises" and / or "comprising", as used herein, identify the presence of the features, integers, steps, operations, elements, and / or components mentioned, but one or more. It should be further understood that it does not exclude the existence or addition of multiple other features, integers, steps, operations, elements, components and / or their groups.

If present, the corresponding structures, materials, actions, and equivalents of all means or step plus function elements within the claims are specifically claimed. It is intended to include any structure, material, or act for performing a function in combination with other claimed elements. The description of the invention is presented for purposes of illustration and illustration, but is not intended to be exhaustive or limited to the invention in disclosed form. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and gist of the present invention. Embodiments best describe the principles and practical applications of the invention, and others skilled in the art will understand the invention for various embodiments with various modifications suitable for the particular intended use. Selected and explained to make it possible.

Various aspects of the disclosure are computer / machine usable storage media or computer / machine readable storage media or computer / machine readable storage media that, when performed on a computer, processor, and / or machine, cause the computer or machine to perform steps of the method. It can be embodied as a program, software, or computer instruction stored on the device. A computer-readable storage medium or device may include any tangible device capable of storing computer code or instructions that can be read and executed by a computer or machine. Examples of computer-readable storage media or devices include memory devices such as hard disks, diskettes, random access memory (RAM), read-only memory (ROM), optical storage devices, and other recording media or storage media. However, it is not limited to these.

The systems and methods of the present disclosure may be implemented and implemented on a general purpose computer or a dedicated computer system. The computer system may be any type of known or becoming known system, typically a processor, memory device, storage device, input / output device, internal bus, and / or communication hardware. It may include a communication interface for communicating with other computer systems along with hardware and software.

The terms "computer system" and "computer network" that may be used in this application may include various combinations of fixed and / or portable computer hardware, software, peripherals, and storage devices. A computer system can contain multiple individual components that are networked and otherwise linked to run cooperatively, or can contain one or more stand-alone components. .. The hardware and software components of the computer system of the present application can include fixed and portable devices such as desktops, laptops, and servers, and can be included within these devices. Modules may be systems that perform any "function" that can be embodied as device components, software, programs, or software, hardware, firmware, electronic circuits, and the like.

Distributing methods for power level selection in cellular wireless networks, such as networks with multiple femtocells operating within the area serviced by at least one macrocell, have been described and illustrated, but are attached herein. It will be apparent to those skilled in the art that modified and modified forms are possible without departing from the principles and broad teachings of the invention, which are limited solely by the scope of the claims.