**4. Numerical results and conclusions**

In this section we present some numerical results obtained by applying the proposed optimization method from Section 3. We simulate target objects with equal prior probabilities *π<sup>k</sup>* = <sup>1</sup> *<sup>K</sup>* <sup>∀</sup>*<sup>k</sup>* <sup>∈</sup> <sup>F</sup>*<sup>K</sup>* in sensor networks with different settings as described in Section 2. In all results, we consider three different kinds of target objects with reflection coefficients chosen as <sup>|</sup>*r*1<sup>|</sup> <sup>=</sup> 0, <sup>|</sup>*r*2<sup>|</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> , and |*r*3| = 1. Furthermore, the path loss function is modeled as line-of-sight propagation. The ratio SNR = 10*dB* log *<sup>P</sup>*tot *P*noise , instead of *received* SNRs, is depicted on the abscissa of all figures.

The verification of the proposed power allocation between both communication links of a single sensor node is shown in Figure 4. The overall error probability of the classification increases for higher SNR values for the case where the allocated power of one link is reduced by 10% and at the same time the power of the other link is stepped up by this 10%. When we reallocate a power amount of 10% − 30% to both links in an inverse manner, then the classification probability remains almost valid. This result shows that the proposed method allocates the given total power nearly optimal to both communication links, especially for higher SNR values.

In Figure 5 another verification of the proposed power allocation is shown, where a network of two sensor nodes is considered. The overall error probability of the classification decreases if we decrease the allocated power of the sensor node, which has the smallest part of the total power, by 10% and allocate this amount of power to the other sensor node. This result shows that the proposed method assigns the given total power suboptimal to the sensor nodes. The curves disperse, because of the approximation (22) which has been used for the equation (23).

As shown in Figure 6 the proposed method yields a better classification probability in comparison to a uniform power allocation where a network of ten sensor nodes is considered.

12 Will-be-set-by-IN-TECH 176 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Power Allocation Procedure for Wireless Sensor Networks with Integrated Ultra-Wide Bandwidth Communications and Radar Capabilities <sup>13</sup>

**Figure 4.** Verification of proposed power allocation between the two communication links of a single sensor node network.

12.7 16.7 20.7 24.7 28.7

**Figure 6.** Comparison of proposed power allocation to a uniform power allocation in a network of ten

*Institute for Theoretical Information Technology, RWTH Aachen University, D-52056 Aachen,*

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aspects for future sensor networks, *IEEE Signal Process. Mag.* Vol. 22: 70–84.

Proposed power allocation

Symbol error-rate Uniform power allocation

Gholamreza Alirezaei, Rudolf Mathar and Daniel Bielefeld

*Conference on Ultra-Wideband*, Syracuse, NY, USA.

*Cogn. Radio Advanced Spectr. Management CogART*, pp. 1–5.

SNR/dB

177

Power Allocation Procedure for Wireless Sensor Networks

with Integrated Ultra-Wide Bandwidth Communications and Radar Capabilities

10−5

sensor nodes.

*Germany*

**Author details**

**5. References**

& Sons, Inc.

11): 5572–5583.

& Sons, Inc.

pp. 23–26.

10−4

10−3

Probability

 of error

10−2

10−1

100

**Figure 5.** Verification of proposed power allocation between two sensor nodes.

In particular, it is shown that the same overall classification probability can be achieved with much lower transmission power, especially for low SNR values, by using an efficient power allocation method. Furthermore, the symbol-error probability of the sensor node with the highest part of the total power is also shown. The classification accuracy is better than the best symbol-error probability for higher SNR values, which affirms the gain of data fusion and illustrates the feasibility of object classification in this kind of distributed sensor networks.

**Figure 6.** Comparison of proposed power allocation to a uniform power allocation in a network of ten sensor nodes.
