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# Computationally Efficient Power Allocation Algorithm in Multicarrier-Based Cognitive Radio Networks: OFDM and FBMC Systems

*EURASIP Journal on Advances in Signal Processing*
**volume 2010**, Article number: 528378 (2010)

## Abstract

Cognitive Radio (CR) systems have been proposed to increase the spectrum utilization by opportunistically access the unused spectrum. Multicarrier communication systems are promising candidates for CR systems. Due to its high spectral efficiency, filter bank multicarrier (FBMC) can be considered as an alternative to conventional orthogonal frequency division multiplexing (OFDM) for transmission over the CR networks. This paper addresses the problem of resource allocation in multicarrier-based CR networks. The objective is to maximize the downlink capacity of the network under both total power and interference introduced to the primary users (PUs) constraints. The optimal solution has high computational complexity which makes it unsuitable for practical applications and hence a low complexity suboptimal solution is proposed. The proposed algorithm utilizes the spectrum holes in PUs bands as well as active PU bands. The performance of the proposed algorithm is investigated for OFDM and FBMC based CR systems. Simulation results illustrate that the proposed resource allocation algorithm with low computational complexity achieves near optimal performance and proves the efficiency of using FBMC in CR context.

## 1. Introduction

Federal Communications Commission (FCC) has reported that many licensed frequency bands are severely underutilized in both time and spatial domain [1]. Assigning frequency bands to specific users or service providers exclusively does not guarantee that the bands are being used efficiently all the time. Cognitive radio (CR) [2–4], which is an intelligent wireless communication system capable of learning from its radio environment and dynamically adjusting its transmission characteristics accordingly, is considered to be one of the possible solutions to solve the spectrum efficiency problem. By CR, a group of unlicensed users (referred to as secondary users (SUs)) can use the licensed frequency channels (spectrum holes) without causing a harmful interference to the licensed users (referred to as primary users (PUs)) and thus implement efficient reuse of the licensed channels.

Multicarrier communication systems have been suggested as a candidate for CR systems due to its flexibility to allocate resources between the different SUs. As the SU and PU bands may exist side by side and their access technologies may be different, the mutual interference between the two systems is considered as a limiting factor affects the performance of both networks. In [5], the mutual interference between PU and SU was studied. The mutual interference depends on the transmitted power as well as the spectral distance between PU and SU. Orthogonal frequency division multiplexing- (OFDM-) based CR system suffers from high interference to the PUs due to large sidelobes of its filter frequency response. The insertion of the cyclic prefix (CP) in each OFDM symbol decreases the system capacity. The leakage among the frequency subbands has a serious impact on the performance of FFT-based spectrum sensing, and in order to combat the leakage problem of OFDM, a very tight and hard synchronization implementation has to be imposed among the network nodes [6].

The filter bank multicarrier system (FBMC) does not require any CP extension and can overcome the spectral leakage problem by minimizing the sidelobes of each subcarrier and therefore lead to high efficiency (in terms of spectrum and interference) [6, 7]. Moreover, efficient use of filter banks for spectrum sensing when compared with the FFT-based preiodogram and the Thomson's multitaper (MT) spectrum sensing methods have been recently discussed in [6, 8].

The problem of resource allocation for conventional (noncognitive) multiuser multicarrier systems has been widely studied [9–12]. The maximum aggregated data rate in downlink can be obtained by assigning each subcarrier to the user with the highest signal-to-noise ratio (SNR) and then the optimal power allocation that maximizes the channel capacity is waterfilling on the subcarriers with a given total power constraint [9]. In cognitive radio systems, two types of users (SU and PU) and the mutual interference between them should be considered. The use of the power allocation based on conventional waterfilling algorithm is not always efficient. An additional constraint should be introduced due to the interference caused by the sidelobes in different subcarriers. The transmit power of each subcarrier should be adjusted according to the channel status and the location of the subcarrier with respect to the PU spectrum.

Wang et al. in [13] proposed an iterative partitioned single user waterfilling algorithm. The algorithm aims to maximize the capacity of the CR system under the total power constraint with the consideration of the per subcarrier power constraint caused by the PUs interference limit. The per subcarrier power constraint is evaluated based on the pathloss factor between the CR transmitter and the PU protection area. The mutual interference between the SU and PU was not considered. In [14, 15], the authors proposed an optimal and two suboptimal power loading schemes using the Lagrange formulation. These loading schemes maximize the downlink transmission capacity of the CR system while keeping the interference induced to only one PU below a prespecified interference threshold without the consideration of the total power constraint. In [16], an algorithm called *RC algorithm* was presented for multiuser resource allocation in OFDM-based CR systems. This algorithm uses a greedy approach for subcarrier and power allocations by successively assigning bits, one at a time, based on minimum SU power and minimum interference to PUs. The algorithm has a high computational complexity and a limited performance in comparison with the optimal solution. In [17], a low complexity suboptimal solution is proposed. The algorithm initially assumes that the maximum power that can be allocated to each subcarrier is equal to the power found by the conventional waterfilling and then modifies these values by applying a power reduction algorithm in order to satisfy the interference constraints. Experimental results like [18] emphasize the need of low interference constraints where this algorithm has a limited performance. Moreover, the nontransmission of the data over the subcarriers below the waterfilling level or the deactivated subcarriers due to the power reduction algorithm decreases the overall capacity of the CR system. In [19], we give some preliminary research results for resource allocation in OFDM-based CR systems. This preliminary work considers a simple model with one PU. The performance of the algorithm was not compared with neither the optimal nor the existed suboptimal algorithms.

In this paper, considering more realistic scenario with several primary user interference constraints, a computationally efficient resource allocation algorithm in multicarrier-based CR systems is proposed. The proposed algorithm maximizes the downlink capacity of the CR system under both total power and interference induced to the PUs constraints. The CR system can use the nonactive and active PU bands as long as the total power and the different interference constraints are satisfied. The simulation results demonstrate that the proposed solution is very close to the optimal solution with a good reduction in the computational complexity. Moreover, the proposed algorithm outperforms the previously presented algorithms in the literature. The efficiency of using FBMC in CR systems is investigated and compared to OFDM-based CR systems. The rest of this paper is organized as follows. Section 2 gives the system model while Section 3 formulates the problem. The proposed algorithm is presented in Section 4. Selected numerical results are presented in Section 5. Finally, Section 6 concludes the paper.

## 2. System Model

In this paper, the downlink scenario will be considered. As shown in Figure 1, the CR system coexists with the PUs radio in the same geographical location. The cognitive base station (CBS) transmits to its SUs and causes interference to the PUs. Moreover, the PUs base station interferes with the SUs. The CR system's frequency spectrum is divided into subcarriers each having a bandwidth. The side by side frequency distribution of the PUs and SUs will be assumed (see Figure 2). The frequency bands ,,, have been occupied by the PUs (active PU bands) while the other bands represent the nonactive PU bands. Its assumed that the CR system can use the nonactive and active PU bands provided that the total interference introduced to the PU band does not exceed where denotes that the maximum interference power that can be tolerated by the and is the interference temperature limit for .

The interference introduced by the subcarrier to PU, , is the integration of the power spectrum density (PSD), , of the subcarrier across the PU band, , and can be expressed as [5]

where is the total transmit power emitted by the subcarrier and is the spectral distance between the subcarrier and the PU band. denotes the channel gain between the subcarrier and the PU. denotes the interference factor of the subcarrier.

The interference power introduced by the PU signal into the band of the subcarrier is [5]

where is the power spectrum density of the signal and is the channel gain between the subcarrier and PU signal. The PSD expression, , depends on the used multicarrier technique. The OFDM and FBMC PSDs are described in the following subsections.

### 2.1. OFDM System and Its PSD

The OFDM symbol is formed by taking the inverse discrete Fourier transform (IDFT) to a set of complex input symbols and adding a cyclic prefix. This can be written mathematically as

where is the set of data subcarrier indices and is a subset of the set , is the IDFT size, is the length of the cyclic prefix in number of samples, and is the length of the OFDM symbol in number of samples. denotes the pulse shape, while denotes the OFDM symbol.

Following the derivation of the PSD for general baseband signal given in [20], it can be shown that the OFDM PSD is

where is the Fourier transform of , and is the variance of the zero mean (symmetrical constellation) and uncorrelated input symbols. The assumption of the uncorrelated input symbols can be justified because of coding and interleaving in practical symbols [21].

can be chosen as

and hence its Fourier transform is

### 2.2. FBMC System and Its PSD

Each subcarrier in FBMC system is modulated with a staggered QAM (offset QAM) [22]. The basic idea is to transmit real-valued symbols instead of transmitting complex-valued ones. Due to this time staggering of the in-phase and quadrature components of the symbols, orthogonality is achieved between adjacent subcarriers. The modulator and the demodulator are implemented using the synthesis and analysis filter banks. The filters in the synthesis and analysis filter bank are obtained by frequency shifts of a single prototype filter. Figure 3 depicts the structure of the synthesis and analysis filter bank at the transmitter and receiver in FBMC-based multicarrier systems.

The FBMC, also called OQAM/OFDM, signal can be written mathematically as [23]

where is the set of subcarrier indices, is the pulse shape, is an additional phase term, and is FBMC symbol duration. are the real symbols obtained from the complex QAM symbols having a zero mean and variance . Hence, the FBMC symbols have a zero mean and finite variance . The PSD of the FBMC can be expressed by [23]

where is the frequency response of the prototype filter with coefficients with , where and is the length of each polyphase components (overlapping factor) while is the number of the subcarriers. Assuming that the prototype coefficients have even symmetry around the coefficient, and the first coefficient is zero [21], we get

To make a parallel between OFDM and FBMC, we place ourselves in the situation where both systems transmit the same quantity of information. This is the case if they have the same number of subcarriers together with duration of samples for FBMC real data and for the complex QAM ones [21, 23].

From the relations above we can notice that the PSDs of OFDM and FBMC are the summation of the spectra of the individual subcarriers. Using the PHYDYAS prototype filter [24], Figure 4 plots a single subcarrier power spectral densities of the OFDM and FBMC systems. It can be noted that the FBMC system has very small side lobes in comparison with that of the OFDM system. Note that in order to solve the large sidelobes problem in OFDM system, many methods have already been employed, such as the insertion of guard subcarriers [25] or cancelation subcarriers [26], windowing (in time domain) [27, 28], and filtering before transmitting [29]. It is known that the guard subcarriers decrease the spectral efficiency, while windowing reduces the delay spread tolerance and filtering is more complex and introduces distortion in the desired signals [30].

## 3. Problem Formulation

The transmission rate of the subcarrier, , with the transmit power can be evaluated using the Shannon capacity formula and is given by

where is the subcarrier fading gain from the CBS to the user. where is the mean variance of the additive white Gaussian noise (AWGN) and is the interference introduced by the PUs band into the subcarrier. The interference from PUs to the subcarrier is assumed to be the superposition of large number of independent components, that is, . Hence, we can model the interference as AWGN. This assumption may not be valid for low number of PU bands but can be considered as a good approximation for large number of PU bands. The same model can be found in [6, 15, 17]. Remark that the nature of the PUs interference on SUs band is the same on both OFDM and FBMC systems. The difference is only in the SUs interference to the PU bands, which is in that case FBMC has significantly lower interference, because of its significantly smaller sidelobes as compared to those of OFDM.

Assuming that each subcarrier band is narrow, subcarriers can be approximated as channel with flat fading gains [31, 32]. It will be assumed that the channel changes slowly so that the channel gains will be constant during transmission. The total achievable rate for OFDM and FBMC systems is evaluated by summing the transmission rate across the different subcarriers [7, 33]. All the instantaneous fading gains are assumed perfectly known at the CR system and there is no intercarrier interference (ICI). Let to be a subcarrier allocation indicator, that is, if and only if the subcarrier is allocated to the user. It is assumed that each subcarrier can be used for transmission to at most one user at any given time. Our objective is to maximize the total capacity of the CR system subject to the instantaneous interference introduced to the PUs and total transmit power constraints. Therefore, the optimization problem can be formulated as follows:

subject to

where denotes the total number of subcarriers, is the number of users, denotes the interference threshold prescribed by the PU, and is the total SUs power budget. is the number of the active PU bands. Inequality (13) ensures that any given subcarrier can be allocated to at most one user.

The optimization problem is a combinatorial optimization problem and its complexity grows exponentially with the input size. In order to reduce the computational complexity, the problem is solved in two steps by many of the suboptimal algorithms [9–12]. In the first step, the subcarriers are assigned to the users and then the power is allocated for these subcarriers in the second step. Once the subcarriers are allocated to the users, the multiuser system can be viewed virtually as a single user multicarrier system. As proved in [9], the maximum data rate in downlink can be obtained if the subcarriers are assigned to the user who has the best channel gain for that subcarrier as described in Algorithm 1.

**Algorithm 1:** Subcarriers to user allocation

**Initialization:**

Set

**Subcarrier Allocation:**

** for** to **do**

;

**end for**

By applying Algorithm 1, the values of the channel indicators are determined and hence for notation simplicity, single user notation can be used. The different channel gains can be determined from the subcarrier allocation step as follows:

Therefore, problem in (11) can be reformulated as follows:

subject to

The problem is a convex optimization problem. The Lagrangian can be written as [17]

where , , and are the Lagrange multipliers. The Karush-Kuhn-Tucker (KKT) conditions can be written as follows:

and also the solution should satisfy the total power and interference constraints given by (20) and (19). Rearranging the last condition in (23) we get

Since , we get

If , then and hence

Moreover, if , from (24) we get

and since and , we get that .

Therefore, the optimal solution can be written as follows:

or more simply, (28) can be written as the follows:

where . Solving for Lagrangian multipliers is computational complex. These multipliers can be found numerically using ellipsoid or interior point method with a complexity [17, 34]. In what follows we will propose a low complexity algorithm that achieves near optimal performance.

## 4. Proposed Algorithm

The optimal solution for the optimization problem has a high computational complexity which makes it unsuitable for the practical applications. A low complexity algorithm is proposed in [17]. The subcarriers nulling and deactivating throughout this algorithm degrade the system capacity and causing the algorithm to have a limited performance in low interference constraints. To overcome the drawbacks of this algorithm, a low complexity power allocation algorithm will be presented.

As described in [5, 17], most of the interference introduced to the PU bands is induced by the cognitive transmission in the subcarriers where the PU is active as well as the subcarriers that are directly adjacent to the PU bands. Considering this fact, it can be assumed that each subcarrier is belonging to the closest PU band and only introducing interference to it, then the optimization problem can be reformulated as follows:

subject to

where denotes the set of the subcarriers belong to the PU band. Using the same derivation leading to (29), we get

where and are the non-negative dual variables corresponding to the interference and power constraints respectively. The solution of the problem still has high computational complexity which encourages us to find a faster and efficient power allocation algorithm.

If the interference constraints are ignored in , the solution of the problem will follow the well-known waterfilling interpretation [35]

where is the waterfilling level. On the other side, if the total power constraint is ignored, the Lagrangian of the problem can be written as [15]

where is the Lagrange multiplier. Equating to zero, we get

where the value of can be calculated by substituting (35) into to get

It is obvious that if the summation of the allocated power under only the interference constraints is lower than or equal the available total power budget, that is, for all , then (35)-(36) will be the optimal solution for the optimization problem . In most of the cases, the total power budget is quite lower than this summation, and hence the Power Interference (PI) constrained algorithm, referred to as *PI-Algorithm*, is proposed to allocate the power under both total power and interference constraints.

In order to solve the optimization problem , we can start by assuming that the maximum power that can be allocated for a given subcarrier is determined according to the interference constraints only by using (35)-(36) for every set of subcarriers for all . By such an assumption, we can guarantee that the interference introduced to PU bands will be under the prespecified thresholds. Once the maximum power is determined, the total power constraint is tested. If the total power constraint is satisfied, then the solution has been found and is equal to the maximum power that can be allocated to each subcarrier, that is, . Otherwise, the available power budget should be distributed among the subcarriers giving that the power allocated to each subcarrier is lower than or equal to the maximum power that can be allocated to each subcarrier , and hence the following problem should be solved:

subject to

The problem is called *"cap-limited"* waterfilling [36]. The problem can be solved efficiently using the concept of the conventional waterfilling. Given the initial waterfilling solution, the channels that violate the maximum power are determined and upper bounded with . The total power budget is reduced by subtracting the power assigned so far. At the next step, the algorithm proceeds to successive waterfilling over the subcarriers that did not violate the maximum power in the last step. This procedure is repeated until the allocated power does not violate the maximum power in any of the subcarriers in the new iteration. The *"cap-limited"* waterfilling algorithm implementation is described in Algorithm 2.

**Algorithm 2:** Cap-limited waterfilling

() **Initialize** , , and .

() Sort in decreasing order with being the sorted index. Find the waterfilling as follows:

(a) , , .

(b) **While** **do**

, , ,

**end while**

(c) Set

() **repeat**

**if**

Let , , , , and go to step 2;

**end** **if**

**until**

The solution of the problem is satisfying the total power constraint of the problem with equality which is not the case for the different interference constraints . Since it is assumed that , some of the powers allocated to subcarriers will not reach the maximum allowable values. This will make the interference introduced to the PU bands below the thresholds . In order to take advantage of all the allowable interference, the values of the maximum power that can be allocated to each subcarrier should be updated depending on the left interference. The left interference can be determined as follows:

Assuming that is the set of the subcarriers that reach its maximum, that is, for all , then, for all can be updated by applying (35)-(36) on the subcarriers in the set with the following interference constraints:

After determining the updated values of , the *"cap-limited"* waterfilling is performed again to find the final solution . Now, the solution is satisfying approximately the interference constraints with equality as well as guaranteing that the total power used is equal to . A graphical description of the *PI-Algorithm* is given in Figure 5 while the implementation procedures are described in Algorithm 3.

**Algorithm 3:** PI-Algorithm.

() **Initialize** , , , and .

() , sort in decreasing order with being the sorted index.

Find the as follows:

(a) , , .

(b) **while** **do**

, , ,

**end** **while**

(c) Set

() **if**

Let and stop the algorithm.

**end** **if**

() Execute the *"cap-limited"* waterfilling (Algorithm 2) and find the set where .

() Evaluate and set , and apply again only step to update .

() Execute the *"cap-limited"* waterfilling (Algorithm 2) and set .

The computational complexity of Step in the proposed PI-Algorithm (Algorithm 3) is . Steps and of the algorithm execute the *"cap-limited"* waterfilling which has a complexity of , where is the number of the iterations. Step has a complexity of . Therefore, The overall complexity of the algorithm is lower than . The value of is estimated via simulation to be lower than five, that is, . Comparing to the computational complexity of the optimal solution, , the proposed algorithm has much lower computational complexity specially when the number of the subcarriers increased.

## 5. Simulation Results

The simulations are performed under the scenario given in Figure 1. A multicarrier system of cognitive users and subcarriers is assumed. The values of and are assumed to be MHz and watt, respectively. AWGN of variance is assumed. Without loss of generality, the interference induced by PUs to the SUs band is assumed to be negligible. The channel gains and are outcomes of independent, identically distributed (i.i.d) Rayleigh distributed random variables (rv's) with mean equal to "" and assumed to be perfectly known at the (CBS). OFDM and FBMC-based cognitive radio systems are evaluated. The OFDM system is assumed to have a of its symbol time as cyclic prefix (CP). For FBMC system, the prototype coefficients are assumed to be equal to PHYDYAS coefficients with overlapping factor and are defined by [24, 37]

The optimal solution is implemented using the interior point method. We refer to the method proposed in [17] by Zhang algorithm. All the results have been averaged over iterations.

Two interference constraints belonging to two active PU bands, that is, , is assumed as given in Figure 6. Each active PU band is assumed to have six subcarriers where . The achieved capacity using optimal, PI and Zhang algorithms for different interference constraints where is plotted in Figure 7. It can be noted that the proposed PI-algorithm approaches the optimal solution and outperforms Zhang algorithm. The effect of assuming that every subcarrier is belonging to the closest PU band and introducing interference to it only on the net interference introduced to the active PU bands is studied in Figures 8 and 9 for and respectively. It can be observed that the net interference induced using the PI-algorithm is approximately satisfying the prespecified interference constraints which makes the assumption reasonable. Unlike the OFDM-based CR system, the interference induced by the FBMC-based system does not reach the pre-specified thresholds. This is because the FBMC-based CR system reaches to the maximum interference that can be introduced to the PU using the given power budget. Moreover, the interference induced by the proposed algorithm is less than that using Zhang algorithm. Returning to Figure 7, one can notice that the interference constraints after Watt start to have no effect on the achieved capacity of the FBMC system. This indicates also that the FBMC system reaches the maximum interference for the given power budget. The small difference between the net interference values after Watt is due to averaging over different channel realizations. The achieved capacity of the different algorithms is plotted in Figure 10 with lower values of the interference constraints. It can be noticed that Zhang algorithm has a limited performance with low interference constraints because the algorithm turns off the subcarriers that have a noise level more than the initial waterfilling level and never uses these subcarriers again even if the new waterfilling level exceeds its noise level. Moreover, the algorithm deactivates some subcarriers, that is, transmit zero power, in order to ensure that the interference introduced to PU bands is below the prespecified thresholds. The lower the interference constraints, the more the deactivated subcarriers which justifies the limited performance of this algorithm in low interference constraints.

To show the efficiency of transmitting over the active PU bands as well as the nonactive bands, Figures 11 and 12 plot the achieved capacity using the PI algorithm with and without allowing the SUs to transmit over the PU active bands. The capacity of the CR system transmitting on both the active and nonactive bands is more than that one transmitting only on the nonactive band. Since the cognitive transmission in the active PU band introduces more interference to the PUs than the other subcarriers, low power levels can be used in these bands with low interferences constraints. This justifies why the difference between the two systems decreases when the interference constraints decrease.

RC algorithm can be used if there is only one active PU band, that is, . The RC algorithm allocates the subcarriers and bits considering the relative importance between the power needed to transmit and the interference induced to the PU band. In order to compare the proposed PI-algorithm with RC algorithm, One active PU band with "12" subcarriers will be assumed as given in Figure 13. For fair comparison, the same bit mapping used in [16] is considered as follows:

where denotes the maximum number of bits in the symbol transmitted in the subcarrier and denotes the floor function. Figures 14 and 15 show that the proposed algorithm performs better than the RC and Zhang algorithms. In low interference constraints, RC algorithm performs better than Zhang algorithm because of the limited performance of Zhang algorithm with low interference constraints.

For all the so far presented results, the capacity of FBMC-based CR system is higher than that of OFDM-based one because the sidelobes in FBMC's PSD is smaller than that in OFDM which introduces less interference to the PUs. Moreover, the inserted CP in OFDM-based CR systems reduces the total capacity of the system. It can be noticed also that the interference condition introduces a small restriction on the capacity of FBMC-based CR systems which is not the case in OFDM-based CR systems. The significant increase in the capacity of FBMC-based CR systems over the OFDM-based ones recommends the FBMC as a candidate for the CR network applications.

## 6. Conclusion

In this paper, a low complexity suboptimal resource allocation algorithm for multicarrier-based CR networks is presented. Our objective was to maximize the total downlink capacity of the CR network while respecting the available power budget and guaranteeing that no excessive interference is caused to the PUs. With a significant reduction in the computational complexity from to , It is shown that the proposed PI-algorithm achieves a near optimal performance and outperforms the suboptimal algorithms proposed so far. It is found that the net total interference introduced to the PUs band is relatively not affected by assuming that each subcarrier is belonging to the closest PU band and only introducing interference to it. Its demonstrated also that capacity of the CR system uses the nonactive as well as the active bands is more than that only uses the nonactive bands. Simulation results prove that the FBMC-based CR systems have more capacity than OFDM-based ones. FBMC offers more spectral efficiency and introduces small interference to the PUs. The obtained results contribute in recommending the use of FBMC physical layer in the future cognitive radio systems. Developing a resource allocation algorithm that considers the fairness among different users as well as their quality of service (QoS) will be the guideline of our future research work towards better radio resource management.

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## Acknowledgment

This work was partially supported by the European ICT-2008-211887 project PHYDYAS.

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Shaat, M., Bader, F. Computationally Efficient Power Allocation Algorithm in Multicarrier-Based Cognitive Radio Networks: OFDM and FBMC Systems.
*EURASIP J. Adv. Signal Process.* **2010, **528378 (2010) doi:10.1155/2010/528378

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### Keywords

- Orthogonal Frequency Division Multiplex
- Cognitive Radio
- Orthogonal Frequency Division Multiplex System
- Cognitive Radio Network
- Cyclic Prefix