**Chapter 6**

**THE MULTICELL BLOCK-DIAGONALIZATION PRECODING – COMPETITIVE DESIGNAND COORDINATION DESIGN**

**6.1 Competitive Design**

**6.1.1Problem Formulation**

In this chapter, the multicell BD precoding under the competition manner, i.e. , the IA manner is examined, where each BS will selfishly plan its BD precoders without any coordination between the cells. This competitory design is formulated utilizing the game-theory model. See a strategic non concerted game ( SNG ) , here participants are the BSs and final payment maps are the sum-rates of the cells. At each cell, BS will strategically accommodate its BD precoder on a per-cell footing, this avariciously maximizes the amount rate to its connected MSs, capable to constraint on its transmit power. Here presuming all the channels change sufficiently slow, so they are considered to be fixed during the game being played.

Let set of Q participants be. Specifyingas the final payment functionof the player-q. Then,is the scheme profile given from the other participants, player-q will selfishly maximise its final payment map by work outing the undermentioned optimisation job

( 6.1 )

hereis the budget of power at BS-q. To accomplish maximal sum data-rate at cell-q, here it is assumed that the IPN matrixis measured absolutely at the corresponding MS-i and it is fed back to its connected BS. The precoding schemeis captured by IPN matrixat MS-i, as indicated in Equation ( 5.2 ) . Therefore, when a BS other than the BS-q changes its attack, the IPN matrixis besides changed. At this case, MS-i needs to feed back the updatedto its matching BS-q. Clearly, the optimisation job ( 6.1 ) shows that the optimum scheme of BS-q dependance on the schemes of the other BSs. Noting that the optimisation job ( 6.1 ) is conceded with merely local informations such as intracell CSI and the IPN matrices between MSs and its associated BS. Thus, the BD precoding game here is implemented in a to the full circulated manner without any exchanges of signals among the BSs. Sing the restraints, each column of the precoder matrixmust be in void infinite created by.Suppose that one performs the remarkable value decompositionof thematrixas

( 6.2 )

Whereis diagonal,andare unitary matrices, andis formed by the lastcolumns of. Then, any precoding covariance matrixis formulated as, where& A ; gt ; = 0 is an arbitrarymatrix, would do,.

Therefore, the set of all allowable attacks for the participant ‘q’ is defined as the followers:

( 6.3 )

And mathematically, the game has the subsequent construction

( 6.4 )

Pure Nash equilibrium ( NE ) of a game G is explained when

( 6.5 )

At NE, provided the precoding scheme from the other cells, a BS doesn’t have the inducement to independently alter its precoding attack, i.e. , it will accomplish a lesser amount rate with same power restraint.

**6.1.2 Problem Solution: Nash Game Theory And Nash Equilibrium**

**6.1.2.1 Game Theory**

Game theory is the survey of strategic synergistic decision-making, where the result for each participant or “ participant ” depends on the actions of all. Specifically specifying it as “ the survey of mathematical theoretical accounts of struggle and cooperation between intelligent rational decision-makers” . In such a game, each participant before taking a “ scheme ” or taking action, participant must see the picks of others. It is a mindset game believing about each other’s schemes and planing our ain maintaining in head that others think about our scheme.

This scientific discipline is applied today in broad scope of applications. Game theory is chiefly used in, political scientific discipline, economic sciences, psychological science, computing machine scientific discipline, and biological science.

**6.1.2.2 Definition Of ‘Nash Equilibrium ‘**

“NASH EQUILIBRIUM” is explained as a subject of game theory where optimum result of a game is on in which where no participant has got an inducement to go from his/her chosen scheme after sing the pick of an opposition. On the whole, a individual can hold no incremental addition from altering hisactions, believing other participants remain fixed in their schemes. A game may hold zero or multiple Nash equilibrium.

**6.1.2.3Formal Definition**

See a game ( S, degree Fahrenheit ) with N participants,is a set of the scheme profiles andis payoff map foris the scheme set for participant, wherebe the scheme profile of the participantandbe the scheme profile of all the participants except for the participant I. When each participantchooses schemeputing scheme profile asso participantas a final payment.This final payment depends on the scheme profile which has been chosen, i.e. on the scheme attack preferred by the participantevery bit good as schemes that is chosen by all other participants. Strategy profileis an NE if no one-sided divergence in the scheme by any single participant is paid for that participant, i.e. given by

If the inequality purely holds for all the participants and for all operable alternate schemes, so that equilibrium is said to be a rigorous NE. Alternatively of this, for some of the participants, there is an exact equality affectingand some other schemes in the given set S, so that equilibrium is classified as a weak NE. A game might hold either pure scheme or assorted Nash Equilibrium.

**6.1.3 Word picture of the BD Precoding Game’s Nash Equilibrium**

For analysing a strategic non-cooperative game ( SNG ) the two most cardinal questions are studied in this subdivision: the being and the singularity of theNEin game. Using the word picture ( description ) of NE enables one to foretell a stable result of non-cooperative BD precoding design ingame G.

For analysing being of pure NE in a game G, foremost we easy verify the scheme set Sq for player-q as defined in the equation ( 6.3 ) , which is compact and convex, ?q.As the public-service corporation map, is farther written as

) =|( 6.6 )

This public-service corporation map is now the combination summing up and minus as it is uninterrupted map in the profile schemesand the first term is concave in Qq this is because it is a composing of concave log deciding map that preserves its concave shape. The first term consists of both the scheme profiles of participant –q ( Qq ) and scheme profile of other participant ( Q-q ) and the 2nd summing up is independent of ( Qq ) . Therefore this showis a map in Qq which is concave. Consequently there prevails at least a individual pure NE in the game G.

Now to analyze the singularity of NE in game G, the best response scheme at each participant is foremost investigated. In an admissible set of schemes Sq for participant –q the best response scheme must be of this signifierAt BS-q the best response schemecanthen be obtained from the following optimisation job.

( 6.7 )

,

Whereis defined as

( 6.8 )

By eigen-decomposing, the optimum solution to job ( 6.7 ) can be obtained easy from the WF process.

( 6.9 )

wherethe water-level adjusted to run into the powerconstraint.

merely depends on the channels inside the cell at cell-q, so BS-q merely needs to strategically accommodate its precoding matricesas in ( 6.8 ) .From equation ( 6.8 ) for each participant the best response scheme ( Dq ) can be obtained in a closed-form. Due to the nonlinear construction of WF operator it is debatable to analyze the singularity of the game’s NE. From [ 21 ] , [ 24 ] the WF operator is interpreted as a projection onto closed and bulging set. As per [ 21 ] for the SISO multi-carrier system the WF operator is shown as a projection onto a simplex set. Using this description in [ 24 ] it is generalized for the single-user MIMO WF operation. Using this reading in [ 39 ] it is farther generalized to the instance of multiuser MIMO system.

Using the construct prescribed in [ 39 ] the block-diagonal WF solution, can be interpreted as a projection.From this Dq is said to be a closed and bulging set. Projection onto a closed and bulging set holds the non-expansive belongings. This belongings is used to analyse the NE’s singularity of the multicell BD precoding game[ 39 ] .

Specifying the multicell function.

Letandfor any givenand, so utilizing the set of inequalities,

( 6.10a )

( 6.10b )

( 6.10c )

( 6.10d )

( 6.10e )

( 6.10f )

This prompts

whereandis defined as

( 6.11 )

Using the definitions of matrix norms and vector norms in paper [ 39 ] the contraction belongings of the game G.

Using these the status is setup ifWF ( D ) function is said to be in contraction and this implies the singularity in NE in game G. this status besides guarantees convergence of NE from any get downing precoding scheme. S is a nonnegative matrix, there will ever be a positive vector**tungsten**

( 6.12 )

The constructed matrix S consists of all the channels of the system in this instance it becomes really ambitious to fulfill the status C.To get the better of this disputing undertaking a centralised unit is required such as base station accountant ( BSC ) as this helps in roll uping all the channels from the BS and continue farther.

The matrix S is constructed utilizing ( 17 ) comprises of all channels in the system. Because of this, the status ( C ) is really ambitious. So to get the better of this a centralised unit, e.g. , a base-station accountant ( BSC ) , is necessary as it can roll up all the channels from the BSs, to take the farther action. To physically construe and to formalize the status ( C ) . The way loss melting theoretical account in this multicell system, is considered. In the intra-cell, when the BS-MS becomes closer so the distance between them is smaller when compared to the distance between the BSs, the inter cell intervention ( ICI ) becomes less dominant. Due to this, the positive off-diagonal elements of S becomes smaller. This shows a smaller spectral radius of S. Therefore, as MSs are becomes closer to its connected BS, the chance of run intoing the status ( C ) is higher, which so guarantees the singularity of the NE.