**3.4. Computational effort**

10 Will-be-set-by-IN-TECH

*n* ) 

It has to be considered that the sum of concave functions is also concave and that the arguments of the logarithms are linear functions of the powers. Furthermore, the domain of the feasible set is a closed convex set and, therefore, only one global maximum of the problem exists. This maximum can be explicitly calculated by using the method of Lagrange multipliers which is equivalent to the water-filling power allocation result [3]. The result is

subject to

0, *<sup>λ</sup> <sup>β</sup><sup>n</sup>* − 1 

*K*

*n* ) *αC n*

> *N*˜ ∑ *n*=1 *βn*

*N* ∑ *n*=1 *PC <sup>n</sup>* + *<sup>P</sup><sup>R</sup>*

*<sup>n</sup>* ≤ *P*tot. (24)

, <sup>∀</sup>*<sup>n</sup>* <sup>∈</sup> <sup>F</sup>*N*, (25)

*<sup>K</sup>*−<sup>1</sup> , <sup>∀</sup>*<sup>n</sup>* <sup>∈</sup> <sup>F</sup>*N*. (26)

, (27)

, <sup>∀</sup>*<sup>n</sup>* <sup>∈</sup> <sup>F</sup>*N*. (29)

*<sup>P</sup>*noise (28)

*<sup>n</sup>* (|*rK*|−|*r*1|)<sup>2</sup> 4*P*noise*g*<sup>2</sup>(2*dR*

optimization problem is given by

*N* ∑ *n*=1 1 <sup>2</sup> log <sup>1</sup> <sup>+</sup> *<sup>P</sup><sup>R</sup> <sup>n</sup> α<sup>R</sup>*

*PR*

*PC*

*<sup>n</sup>* <sup>=</sup> *<sup>P</sup>*noise *<sup>g</sup>*<sup>2</sup>(*d<sup>C</sup>*

where the factor *β<sup>n</sup>* is defined by

*<sup>n</sup>* <sup>=</sup> *<sup>P</sup>*noise *<sup>g</sup>*<sup>2</sup>(2*d<sup>R</sup>*

*<sup>β</sup><sup>n</sup>* :<sup>=</sup> *<sup>g</sup>*<sup>2</sup>(2*d<sup>R</sup>*

*n* ) *αR n*

*n* ) *αR n*

manner. Then the water-filling level *λ* is a value specified by the inequality

*<sup>β</sup>N*˜ <sup>&</sup>lt; *<sup>λ</sup>* <sup>≤</sup> <sup>1</sup>

*N*˜ ∑ *n*=1

*n* ) *αC n*

4 (|*rK*|−|*r*1|)<sup>2</sup> · max

4 (|*rK*|−|*r*1|)<sup>2</sup> <sup>+</sup> *<sup>g</sup>*<sup>2</sup>(*d<sup>C</sup>*

For the following equations, we assume that the factors *β<sup>n</sup>* are ordered in an increasing

where the number *<sup>N</sup>*˜ with 1 <sup>≤</sup> *<sup>N</sup>*˜ <sup>≤</sup> *<sup>N</sup>* is a suitably chosen integer value for which the

(*βN*˜ <sup>−</sup> *<sup>β</sup>n*) <sup>&</sup>lt; *<sup>P</sup>*tot

This allocation has the following interpretation. The sensor node *Sn* with the lowest *β<sup>n</sup>* gets the largest part of the total power because its communication channels are possibly the best due to the low distances. Therefore the observation of the target object is less interfered by noise and consequently results in better data communication. Sensor nodes with higher distances get smaller parts of the total power and some of them do not get any power at all. The last ones participate neither in the data communication nor in the classification of the target object. Their information reliability is too poor to be considered for data fusion. More and more sensor nodes will become active by increasing the total power. Then the overall classification probability increases because more correct information is provided by

Note that we have used the approximation (22) in order to simplify the maximization problem and to find analytical solutions for all equations. Without any approximation the maximization problem yields the *Lambert's trinomial equation*, which still does not have any analytical solutions. Although the above approximation is only valid for low transmission

0, *<sup>λ</sup> <sup>β</sup><sup>n</sup>* − 1 

*N*˜ *P*tot *<sup>P</sup>*noise +

holds. From (23) and (25) the allocated power for the second channel is determined as

*K <sup>K</sup>*−<sup>1</sup> · max

maximize *PR* <sup>1</sup> ,...,*P<sup>R</sup> N*

given by

inequality

the observations.

In order to calculate the transmission powers (25) and (29) the computation of *βn*, *λ*, and *N*˜ is necessary. The parameters *K*, *N*, *P*tot, *P*noise, *rk*, *α<sup>R</sup> <sup>n</sup>* , and *α<sup>C</sup> <sup>n</sup>* are fixed system parameters which are known to the computation unit. The distances *d<sup>R</sup> <sup>n</sup>* and *d<sup>C</sup> <sup>n</sup>* depend on the position of the target object and are therefore unknown. They can be estimated for example by a tracking algorithm. If these values are also determined, then the equations (25) to (29) can be calculated with little effort, because of simple mathematical operations such as summation and multiplication. The only difficulty is the evaluation of the path loss function *g*, which can include complex mathematical operations. Its complexity depends on the given multipath channel.

However, the computation effort of the equations (25) to (29) is less complex than the evaluation of the classification algorithm such as (6). If one can find simpler algorithms than (6) (see, for example [12]), then the assessment of the calculation effort becomes important and should be considered in detail.
