# Distributed Power Allocation for Parallel Broadcast Channels with Only Common Information in Cognitive Tactical Radio Networks

- Vincent Le Nir
^{1}Email author and - Bart Scheers
^{1}

**2010**:172013

https://doi.org/10.1155/2010/172013

© V. Le Nir and B. Scheers. 2010

**Received: **31 May 2010

**Accepted: **20 December 2010

**Published: **12 January 2011

## Abstract

A tactical radio network is a radio network in which a transmitter broadcasts the same information to its receivers. In this paper, dynamic spectrum management is studied for multiple cognitive tactical radio networks coexisting in the same area. First, we consider the problem of common rate maximization subject to a total power constraint for a single tactical radio network having multiple receivers and using parallel subchannels (parallel multicast channels). Mathematical derivations show that the optimal power allocation can be found in closed form under multiple hypothesis testing. An outer loop can be used to minimize the power subject to a common rate constraint. Then, we extend the iterative water-filling algorithm to the coexistence of multiple cognitive tactical radio networks without requiring any cooperation between the different networks. The power allocation is performed autonomously at the transmit side assuming knowledge of the noise variances and channel variations of the network. Simulation results show that the proposed algorithm is very robust in satisfying these constraints while minimizing the overall power in various scenarios.

## Keywords

## 1. Introduction

Tactical radio networks are networks in which information (voice and packet based data) is conveyed from one transmitter to multiple receivers. When several coalition nations coexist in the same area, current technologies do not permit reconfigurability, interoperability, nor coexistence of the radio terminals. Software defined radio has been developed for reconfigurability of the terminals with software upgrades and for portability of the waveforms. Cognitive radio has been introduced by Mitola in 1999 as an extension to software-defined radio [1]. Cognitive radio has been developed for spectrum availability recognition, reconfigurability, interoperability, and coexistence between terminals by means of software defined radio technology, intelligence, awareness, and learning [1, 2]. The fundamental principles of cognitive radio are on one hand to identify other radios in the environment that might use the same spectral resources by means of spectrum sensing and on the other hand to design a transmission strategy that minimizes interference to and from these radios by means of dynamic spectrum management. The major goals of cognitive radio are to provide a high utilization of the radio spectrum and reliable communications whenever and wherever needed [2]. Applications of cognitive radio include, but are not limited to, tactical radio networks, emergency networks, and wireless local area networks with high throughput and range.

The broadcast channel has been introduced by Cover in 1972 as a communication channel in which there are one transmitter and two or more receivers [3]. The broadcast channel in which independent messages are sent to the receivers (unicast channel) belongs to the class of degraded channels in which one user's signal is a degraded version of the other signals. Its capacity region is fully characterized and can be achieved by superposition coding [3, 4]. Contrary to a single unicast channel, the sum of unicast channels as well as MIMO broadcast channels is nondegraded [5, 6].

Previous studies on parallel broadcast channels have focused on scenarios in which independent messages are sent to the receivers (parallel unicast channels) [7–10]. The optimal power allocation can be achieved by a multilevel water-filling over the parallel channels, which is an extension of Gallager's 1968 water-filling strategy for single-user parallel Gaussian channels [11]. Some other studies have considered parallel broadcast channels in which simultaneous common and independent messages are sent to the receivers [5, 12, 13], or simultaneous common and confidential messages are sent to the receivers [14].

Contrary to a unicast channel, a tactical radio network can be thought as a multicast channel with only common information. The capacity of a single multicast channel is limited by the capacity of the *worst receiver* [4, 15]. However, less work has been done on parallel multicast channels [16].

In the first part of the paper (Section 2), we extend the water-filling strategy [11] to multiple receivers considering parallel multicast channels with perfect channel state information (CSI) at the transmit side. In this case, the extended water-filling strategy maximizes the common rate subject to a power constraint (inner loop) or minimizes the power subject to a common rate constraint (outer loop). Mathematical derivations show that the optimal power allocation can be found in closed form under multiple hypothesis testing [14, 17, 18].

Distributed multiuser power control has been studied for parallel interference channels, leading to a common strategy known as iterative water-filling [19–21]. Distributed algorithms, although suboptimal, are preferred to centralized algorithms in practical scenarios because of their scalability. In the iterative water-filling algorithm, each network considers the interference of all other networks as noise and iteratively performs a water-filling strategy. At each iteration, the power spectrum of each network modifies the interference caused to all other networks. This process is performed iteratively until the power spectra of all networks converge.

In the second part of the paper (Section 3), capitalizing on the previous results, we introduce an autonomous dynamic spectrum management algorithm based on iterative waterfilling [19] for multiple cognitive tactical radio networks coexisting in a given area and willing to broadcast a common information (voice, data, etc.) to their group. The problem can be modeled as
networks, each network
with a single transmitter willing to send a common message to its corresponding
receivers over
parallel scalar Gaussian subchannels. It is assumed that each transmitter has the knowledge of the channel variations and noise variances in its own network and iteratively updates its power spectrum until a common rate constraint for all receivers is satisfied. Although this paper focuses on multiple cognitive radio networks for tactical communications, the proposed algorithm can be applied to any application requiring spectrum management between multiple cognitive radio networks for parallel multicast channels with only common information. In Section 4, simulation results are given for multiple scenarios and compare the proposed algorithm with the *worst subchannel* strategy. Finally, Section 5 concludes this paper.

## 2. Single Tactical Radio Network

The expression in (2) is the maximization of the minimum of a set of sums of concave functions of . Since the sum and the minimum operations preserve concavity, the objective is concave, and maximizing a concave function yields a convex optimization problem. This max-min optimization problem can be efficiently solved by the approach based on minimax hypothesis testing given in [14, 17, 18]. For two receivers, the optimal power allocation algorithm is given by three steps

Step 1.

If , then the optimal power allocation is and finish.

Step 2.

If then the optimal power allocation is and finish.

Step 3.

Search over all to find that satisfies , then the optimal power allocation is and finish.

The optimal power allocation corresponds to Gallager's water-filling strategy for single-user parallel Gaussian channels [11].

Step 1.

Step 2.

In this formula, the optimal power allocation takes into account the difference between the water-fill functions and the weights of the different receivers.

Figure 2 shows the power control for a single tactical radio network. An inner loop determines the power allocation maximizing the common rate subject to a total power constraint. Then, an outer loop minimizes the power such that a common rate constraint is achieved. Algorithm 1 provides the proposed power allocation for power minimization subject to a common rate constraint. The inner loop and the outer loop correspond to lines 13–21 and 6–30, respectively. Note that if all the steps in the multiple hypothesis testing are needed, the complexity of the algorithm increases exponentially with the number of receivers .

**Algorithm 1:** Minimization of the power subject to a common rate constraint.

(6) for all steps

(13) Calculate according to the roots of(18)

(18) end if

(20) end while

(21) If condition satisfied on exit step loop

(22) end for

(27) end if

(29) end while

## 3. Multiple Cognitive Tactical Radio Networks

Considering jointly the maximization of the aggregate common rate subject to a total power constraint per network in a centralized algorithm is an extensive task, since it would require the knowledge of the channel variations of all the interference terms for all , , , . This knowledge can be acquired through a feedback channel from the receivers to the transmitter of each network assuming that the acquisition time is much lower than the coherence time of the channel fading. To this end, each terminal must be equipped with a spectrum sensing function to estimate the noise variances and a channel estimation function to estimate its channel variations. This information can be further transmitted to a centralized unit. Moreover, even if a centralized cognitive manager was able to collect all the channel state information (CSI) within and between the different networks, solving (22) would require an exhaustive search over all possible 's or a more efficient genetic algorithm.

Therefore, after collecting the noise variances and the channel variations of its network, each transmitter has to apply Algorithm 1 autonomously and to update its power allocation regularly to reach an equilibrium between the different networks. As shown on Figure 4, within each network, an inner loop determines the power allocation maximizing the common rate subject to a total power constraint. This process is updated regularly between all the different networks until they reach a Nash equilibrium. Finally, an outer loop minimizes the power such that a common rate constraint is achieved for each network. The algorithm for the coexistence of multiple tactical radio networks is presented in Algorithm 2.

**Algorithm 2:** Distributed power allocation for minimization of the power subject to a common rate constraint.

(6) for iteration = 1 to 20

(8) for all steps

(15) Calculate according to the roots of(18)

(20) end if

(22) end while

(23) If condition satisfied on exit step loop

(24) end for

(25) end for

(26) end for

(32) end if

(34) end for

(35) end while

## 4. Simulation Results

^{2}. The carrier frequency is chosen to be in the very high frequency (VHF) band ( MHz). The SNR gap for an uncoded quadrature amplitude modulation (QAM) to operate at a symbol error rate 10

^{−7}) is dB. The subchannel bandwidth is kHz, the path loss exponent is , reference distance meters and thermal noise with the following expression:

which gives a noise variance per subchannel of approximately .

### 4.1. Single Tactical Radio Network

*worst subchannel*strategy for the minimization of the power subject to a common rate constraint. The

*worst subchannel*strategy corresponds to the strategy in which the common message codebook is broken into different codebooks for each subchannel; therefore, the common rate is limited by the weakest receiver in each subchannel [23]. This transmission scheme is referred to as "multiple codebook, variable power" transmission [24]. In the

*worst subchannel*strategy, the water-filling is performed on the worst subchannel conditions considering both receivers. More precisely, the values and are compared for each subchannel and the greatest value is selected for the water-filling. For the

*worst subchannel*strategy, the inner loop maximizes the rate of the superposition of the receiver's worst subchannels given by

^{3}Monte Carlo trials for both scenarios. Algorithm 1 provides the minimum power for all scenarios compared to the

*worst subchannel*strategy as this is the optimal strategy. Moreover, for the scenario 2, the

*worst subchannel*strategy uses the maximum power for all common rate constraints.

### 4.2. Multiple Cognitive Tactical Radio Networks

*worst subchannel*strategy for the minimization of the power subject to a common rate constraint with networks whose transmitters and receivers are in a same square area of 1 km

^{2}. To highlight the robustness of our algorithm, we take an extreme scenario in which the receivers see a different noise on their subchannels (as shown in Figure 7). In the first network, a very strong noise ( ) is seen on the 4th subchannel by the first receiver and the 1st subchannel by the second receiver. In the second network, a very strong noise ( ) is seen on the 3th subchannel by the first receiver and the 2nd subchannel by the second receiver. Figure 8 shows the results of the power minimization subject to a common rate constraint ranging from kbps to kbps over 10

^{3}Monte Carlo trials. The results are averaged for both networks. In this scenario, it can be seen that Algorithm 2 outperforms the

*worst subchannel*strategy. Therefore, in practical scenarios in which the interference temperature varies along the subchannel and the receiver locations, Algorithm 2 provides a novel distributed strategy to find the power allocation minimizing the power subject to a common rate constraint. Since it is based on closed-form expressions, the algorithm has reasonable complexity for a low number of receivers. However, the search for the best set of weights requires an exhaustive search over all possible weights. Therefore, to reduce the complexity in a practical algorithm, the weights are taken from a given data set in interval .

## 5. Conclusion

In this paper, dynamic spectrum management was studied for multiple cognitive tactical radio networks coexisting in the same area. First, we have considered the problem of power minimization subject to a common rate constraint for a single tactical radio network with multiple receivers over parallel channels (parallel multicast channels). Then, we have extended the iterative water-filling algorithm to multiple receivers for the coexistence of multiple cognitive tactical radio networks assuming knowledge of the noise variances and channel variations of the network. Simulation results have shown that the proposed algorithm is very robust in satisfying these constraints while minimizing the overall power in various scenarios.

## Funding

This work was carried out in the frame of the Belgian Defense Scientific Research and Technology Study C4/19 funded by the Ministry of Defense (MoD). The scientific responsibility is assumed by its authors.

## Authors’ Affiliations

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