The IPW algorithm under Sum Transmit Power Constraint.

## 1. Introduction

Spectrum scarcity is becoming a serious problem as the rapid developments of wireless communications. However, recent spectrum measurements show that the licensed spectrum is severely underutilized by the primary users (PUs) in both the time domain and the spatial domain (FCC, 2002, Mchenry, 2005). The secondary users (SUs) are introduced to exploit the spectrum opportunities left by the PUs in order to improve the spectrum utilization. The transmission of SU is required to satisfy the PU's interference limits to protect the PU's performance. Cognitive radio (Mitola, 1999, Haykin, 2005, Zhao & Sadler, 2007) is the key technology enabling flexible, efficient and reliable spectrum utilization by adapting the radio's operating characteristics to the real-time conditions of the environment. The SUs with cognitive radio technology are able to cleverly detect and utilize the spectrum opportunities and thus realize efficient reuse of the licensed spectrum

It is extremely important to guarantee the transmission of the SU satisfy the interference limits of the PUs for the cognitive radios to be acceptable. Generally, the interference limits can be described by two metrics: the maximum allowable collision probability and the interference power limit (Zhao & Sadler, 2007). The former one is related to the spectrum sensing mechanism of the cognitive radios, which has been widely studied in both the physical (PHY) layer (Sahai et al., 2004, Oner & Jondral, 2007, Cabric et al., 2004) and the medium access control (MAC) layer (Zhao et al., 2007, Kim & Shin, 2007, Ghasemi & Sousa, 2007) in the literature. The latter one introduces power restrictions on the channel output signals in cognitive radio systems, which is different from the conventional wireless systems where the power constraints are only placed on the channel input signals. This feature motivates the studies on how to efficiently utilize the spectrum opportunities under the interference power limits of the PUs (Zhang et al., 2008, Shu et al., 2006, Sudhir & Jafar, 2007, Le & Hossain, 2007).

A cognitive radio system needs to be highly flexible in terms of the spectral shape of the transmit signal. Orthogonal frequency division multiplexing (OFDM) modulation is a promising candidate for such a flexible system because of its reconfigurable subcarrier structure (Weiss & Jondral, 2004, Berthold & Jondral, 2005). With OFDM, the SU has the ability to flexibly fill the spectral gaps left by PUs. Also, the fast Fourier transform (FFT) components at the OFDM system's receiver may also be used for the SU to execute the channel detection. In

(Weiss & Jondral, 2004), Weiss and Jondral proposed that the band of the SU covers multiple PUs' licensed spectrum, then the SU modulates zero on the subcarriers which belong to the detected PUs's licensed spectrum while utilizing other subcarriers for transmission. In OFDM-based cognitive radio systems, the adaptive subcarrier configuration should consider not only the channel state information (CSI) as in conventional OFDM systems, but also the sensing results of the SU and the interference limits of the PUs. This chapter focuses on investigating the research challenges involved in the power allocation for OFDM-based cognitive radio systems.

The optimal power allocation algorithm for conventional OFDM systems that maximizes the channel capacity is the wellknown *water-filling,* which is derived by solving a convex optimization problem subject to the sum transmit power constraint(TSE & Viswanath, 2005). In OFDM-based cognitive radio systems, the band of the SU can be divided into several subchannels, each of which is corresponding to a licensed band of one PU system. Since the interference limit of each PU introduces the subchannel transmit power constraint for the SU, the power allocation in OFDM-based cognitive radio systems should not only satisfy the sum transmit power constraint but also the subchannel transmit power constraints (Zhao & Sadler, 2007). Therefore, the conventional water-filling algorithm is not applicable in such a scenario.

The transmit power in each subchannel is comprised of the power allocated to the subcarriers inside the subchannel and the sidelodes power of the subcarriers in other subchannels. In this chapter, we first formulate the power allocation problem in the case where the effects of subcarrier sidelobes can be ignored, i.e., there is sufficient guard band between any two neighboring subchannels. Based on the convex optimization theory, an algorithm named *iterative partitioned water-filling* (IPW) is proposed to obtain the optimal power allocation that maximizes the capacity while satisfying both the sum and subchannel transmit power constraints. Then, we extend the results and address the power allocation problem in general OFDM-based cognitive radio systems, where the effects of subcarrier sidelobes are specially considered. In this case, we propose a *recursive power allocation* (RPA) algorithm to obtain the optimal power allocation by decoupling the subchannel power constraints phase-by-phase. The organization of the rest of this chapter is as follows: Sections 2 presents an overview of the cognitive radio and OFDM-based cognitive radio systems. In Section 3, the power allocation problem ignoring the effects of subcarrier sidelobes for OFDM-based cognitive radio systems is formulated and analyzed. The IPW algorithm is proposed and its optimality is proved. Then Section 4 addresses the general OFDM-based cognitive radio systems where the effects of subcarrier sidelobes are considered. The RPA algorithm is proposed and its optimality is proved. Simulation results are presented in Section 5. Section 6 discusses the future research directions and concludes the chapter.

## 2. System Model

### 2.1. Cognitive radio system and interference power limit

One of the typical cognitive radio systems is shown in Figure 1. A certain channel is licensed to the PU system. Since the PU system does not occupy the channel anywhere at any time, the channel is underutilized in both the spatial domain and the time domain. A channel is said to be a spectrum opportunity if the interference to the PU receivers caused by the SU's transmission is tolerable. The SU is permitted to access the channel if the channel is detected to be a spectrum opportunity. With the participance of the SU, the spectrum

opportunities can be identified and exploited and thus the overall spectral efficiency can be improved.

To guarantee the interference to the PU tolerable is an extremely important problem in the implementation of cognitive radio systems. Generally, the SU has to sense the channel in order to determine whether a spectrum opportunity exists before the transmission. The SU is permitted to transmit on a licensed channel unless the corresponding PU's signal is seen as absent. A detection probability has to be achieved to reduce the collisions between the SU and the PU caused by the detection errors. As in Figure 1, PU_{1}'s signal can be detected when active and thus the interference to PU_{1} receiver caused by the SU's transmission can be avoided. However, due to the signal to noise ratio limit, the SU can only detect the signal of the PU with the required detection probability within a certain region, as the reliable sensing region shown in Figure 1. Therefore, for the PU_{2} transmitter that is outside the SU's reliable sensing region, the SU is unable to detect PU_{2}'s signal with the required detection probability, as depicted in Figure 1. In this situation, as in (Zhao & Sadler, 2007), PU_{2} defines a protection area whose radius is *R* and requires the interference power at any potential receiver in this area be lower than a certain value, say 𝜂*.* Therefore, when one PU's signal is seen as absent, the SU's transmit power on the PU's licensed channel *Ptx *should be subject to a power constraint, which is given by

where *d* is the distance between the SU transmitter and the nearest undetectable PU transmitter (the PU transmitter outside the reliable sensing region is referred to undetectable PU transmitter in the following), and *β* is the path attenuation factor. Note that we only consider the distance-based path loss here for simplicity. Also, the distance between the SU and the nearest undetectable PU transmitter is assumed to be known in advance by the SU

in (1). We note that if the SU is unaware of the PU's location, some conservative scheme may be used to estimate *Ptx.* For example, *d* is set to be the radius of the reliable sensing region, i.e., the PU's transmitter is assumed to be just at the margin of the reliable sensing region.

### 2.2. OFDM-based cognitive radio systems and subchannel transmit power constraints

In order to efficiently exploit the spectrum opportunities left by different PU systems, a cognitive radio system needs to be highly flexible with respect to the spectral shape of the transmit signal. OFDM modulation is a promising candidate for such a flexible system because of its reconfigurable subcarrier structure. Also, the FFT component at the OFDM receiver can be used for spectrum sensing, which reduces the overhead for the implementation of the cognitive capability.

As in Figure 2, in an OFDM-based cognitive radio system, the spectrum that can be potentially used by the SU is divided into *M* subchannels. Each subchannel is corresponding to a PU's licensed band. The total number of the subcarriers is assumed to be *N* and *mj *denotes the index of the first subcarrier in the *j*th subchannel.

Before the transmission, the SU senses whether each subchannel is occupied firstly, by proper spectrum sensing methods. Then according to the sensing results and the CSI of each subcarrier over SU transmission link, the SU can decide the proper power allocation scheme, modulation type and other parameters for SU's transmission. Therefore, the total transceiver diagram can be shown in Figure 3. At the transmitter in Figure 3, the transmission parameters should be decided before serial-to-parallel (S/P) conversion, IFFT operation, parallel-to-serial (P/S) conversion, insertion of a Cyclic Prefix (CP) and filtering. Accordingly, at the receiver, the information on transmission parameters over the link should be obtained through signaling to receive and demodulate OFDM signals. Then the received signal of the *ith* subcarrier within one OFDM symbol can be expressed as:

where *xi *is the signal transmitted on the *i*th subcarrier by the SU transmitter, *hi *is the channel gain and *ni *is the additive white Gaussian noise with mean 0 and variance 1. If the corresponding PU transmitter is detected in a subchannel, all the subcarriers in this subchannel will be modulated by zero during the transmission, i.e., the sum power of the subcarriers in this subchannel is set to be zero. Otherwise, the SU can use this subchannel

but with a certain power constraint which has been described in Section 2.1. Assume *Gj *is the power constraint on the *j*th subchannel after the sensing, we have

where *ηj *is the interference power limit of PU* j *, *Rj *is the radius of the protection area of PU* j *, *dj *is the distance between the SU's transmitter and the nearest undetectable PU_{j}'s transmitter, and *βj *is the path attenuation factor.

## 3. Power Allocation for OFDM-based Cognitive Radio Systems without Considering Subcarrier Sidelobes

In conventional OFDM systems, given the sum transmit power constraint or the target rate, the optimal power allocation that aims at maximizing the sum rate or minimizing the required power can be achieved by the well-known water-filling algorithm. In OFDM-based cognitive radio systems, the power allocation should also satisfy the subchannel transmit power constraints which are introduced by the interference power limits of the PUs. Therefore, the water-filling algorithm needs to be modified. In this section, we first review the power allocation problem in conventional OFDM systems and then formulate that in the

cognitive radio scenario.

### 3.1. Power allocation in conventional OFDM systems

The signal model for conventional OFDM systems is as (2) shows. The power allocation problem can be classified into two categories in conventional OFDM-based systems. The first one is to optimize the power allocation across the subcarriers such that the sum rate is maximized with a given sum transmit power constraint. Assume the sum transmit power constraint of an OFDM block is P* t *, the allocated signal power on the ith subcarrier is *Pi *and the number of the subcarriers is *L*. The optimal power allocation can be given by

where (^{.})^{+} denotes max{^{.},0} and

The algorithm to obtain the optimal solution is the well-known water-filling, where the so-called water-level is calculated based on (5). The other category of power allocation problem can be formulated as optimizing the power allocation across the subcarriers such that the required transmit power is minimized while a given target rate is satisfied. Assume the target rate is R_{t}. The optimal power allocation has the same expression of (4) while the water-level

The algorithm to obtain the optimal solution is also the water-filling, where

### 3.2. Power allocation in OFDM-based cognitive radio systems

The power allocation problem in OFDM-based cognitive radio should consider the subchannel transmit power constraints in addition to the sum transmit power constraint or the target rate. Assume_{t} can be formulated as the following optimization problem:

Similarly, the power allocation problem with the target rate R_{t} can be formulated as

Note that the SU also suffers interference caused by the PU. The interference can be regarded as white noise by the SU, which may lead to different noise levels in different subchannels. However, although the noise power in each subchannel is set to be equal to each other, this fact does not affect our problem formulation since we can rescale the channel gain. Obviously, the above two optimization problems are convex optimization problems with linear constraints, which can be numerically solved by the standard optimization algorithms. However, we intend to develop analytical as well as more efficient algorithms rather than numerical methods in the following two subsections.

### 3.3. Iterative Partitioned Water-filling: Sum Transmit Power Constraint

We first consider (7), i.e., the power allocation problem with the sum transmit power constraint. The algorithm is developed by three steps. The fist step is to analyze the structure of the optimal solution of (7) based on the convex optimization theory. Then the algorithm is proposed heuristically based on the analysis in the first step. The algorithm is proved to converge to the optimal point at last. We begin with the first step in this subsection.

#### 3.3.1. Analysis of the structure of optimal solution

In (7), if

constraints and the solution is simply water-filling on the subcarriers in each subchannel individually with the corresponding subchannel transmit power constraint. Therefore, we only emphasize on the situation that

where i = 1, 2, …,N and j is the index of the subchannel which the ith subcarrier belongs to. Assume_{j} is determined by 1. for j ∈ A:

2. for j ∈ B:

Proof: The problem (7) can be reformulated into a standard convex optimization form:

The constraint conditions obviously satisfy the Slater's conditions, so the Karush-Kuhn-Tucker(KKT) conditions are sufficient and necessary for the optimal vector P (Boyd & Vandenberghe, 2004). The first two KKT conditions are the constraint conditions of (12) and the others are given by

where i =1,2,...,N, j =1,2,...,M

where i =1,2,...,N, j =1,2,...,M

where i =1,2,...,N and j is the index of the subchannel which the ith subcarrier belongs to.

I. Proof of only if part

(15) can be written as

Substituting (16) into (13a) and (14a) yields

_{i}>0 by (17). Then (18) leads to

P_{t} = 0. From (16), we also have

where i = 1, 2, …,N and j is the index of the subchannel which the ith subcarrier belongs

to.The subchannels are divided into two sets A and B. Since we assume

1) For all j ∈ A, since Fj < Gj, we have α_{j} = 0 based on (14b). Let

Therefore, we come to the equation (10b)

2) For all j ∈ B, the condition (11a) is obviously satisfied. Since α_{j}≥0, we have

Therefore, the inequality (11b) also holds.

So far, we have proved the only if part in Theorem 1.

II. Proof of if part

We need to prove that all the KKT conditions can be derived by the power allocation vector defined in Theorem 1. It is easy to see that the first two KKT conditions, i.e., the constraint conditions in (12), hold inherently.

1) Based on (10b) and A≠∅, we have

Therefore, (13c) and (14c) both hold.

2) Define

Therefore, (13b) holds. Since for j ∈ B, we have Fj = Gj, (14b) also holds.

3) Define

subcarrier belongs to.Then (15) holds inherently. If

Since we can derive_{j} and v, (26) can be written as

Then, λ=0. Therefore, given_{i} = 0 from (9). Then, we have

Hence, given

In conclusion, we have derived all of the KKT conditions and thus the if parts also holds. This completes the proof of Theorem 1. □

Comparing (9) with (4), we can find that the power allocation within each subchannel is the same to the conventional water-filling result. However, the water-levels of different subchannels may be different in (9). From Theorem 1, we know that the subchannels with the optimal power allocation can be divided into two sets: the set A, i.e., the subchannels whose allocated power is strictly smaller than the corresponding subchannel transmit power constraint and the set B, i.e., the subchannels whose allocated power is equal to the corresponding subchannel transmit power constraint. For the subchannels in A, the allocated power is the result of water-filling on all the subcarriers that belong to these subchannels with the power

A power allocation result satisfying Theorem 1 is shown in Figure 5. Within each subchannel, it is the same to the conventional water-filling result. However, the water-level of each

subchannel is not equal to each other compared to Figure 4. Subchannel 1 and 3 have the same water-level W, as each allocated power is strictly lower than the corresponding subchannel transmit power constraint. Subchannel 2 and 4 have their unique water-levels w_{2} and w_{4} respectively, as each allocated power is equal to the corresponding subchannel transmit power constraint. Also, as in Figure 5, we see that the unique water-levels w_{2},w_{4} are lower than the common water-level w.

#### 3.3.2. Iterative partitioned water-filling

From the above analysis, we can infer that if the partition of the subchannels, i.e., the elements of A and B, is known, the optimal power allocation can be obtained by first water-filling on the subcarriers in each subchannel that belongs to the set B individually with the corresponding subchannel transmit power constraint, and then water-filling on the rest of the subcarriers with the power

Divide the subchannels into two sets, say A and B and there are 2

^{M}kinds of partitions in total.For each partition, perform the conventional water-filling on the subcarriers in each subchannel that belongs to the set B individually with the corresponding subchannel transmit power constraint. Then the water-levels wj where j∈ B can be obtained.

3.Remove such partitions that

4.Verify each partition whether F_{j}< G_{j} where j ∈ A and

I. Initialization : |

II. Iterations: 1. Perform the conventional water-filling on the subcarriers in all the subchannels that belong to the set A with the target rate |

2._{j} ; 5. |

III. End |

However, the complexity of the exhaustive search algorithm is far too high. In the extreme situation, we need to consider 2^{M} kinds of partitions throughout the algorithm. Therefore, we develop a more efficient algorithm named as iterative partitioned water-filling (IPW). The basic idea is to determine the elements of A and B iteratively rather than by exhaustive search. The IPW algorithm is described in Table 1.

In the beginning of each iteration, the conventional water-filling with only sum transmit power constraint P is done on the subcarriers in all the subchannels that belong to the set A. Then those subchannels whose power exceeds its subchannel transmit power constraint are taken out from the set A and then be performed by water-filling with the corresponding subchannel transmit power constraints individually. The iteration stops when each subchannel satisfies its subchannel transmit power constraint. The IPW algorithm is more efficient than the exhaustive search algorithm. In the worst case, the algorithm converges after M iterations.

#### 3.3.3. Proof of optimality

In order to prove the optimality of the IPW, we need to explain that the IPW algorithm converges to the point that satisfies the conditions in Theorem 1. Actually, we only need to prove the inequality (11b) as other conditions are inherently satisfied.

We outline two simple lemmas about the conventional water-filling before the proof. Assume P is the power allocation vector of water-filling on N subcarriers with power 1^{T}P and the water-level is w, where 1 is the column vector of all ones.

_{n}, P

_{n}is also the power allocation vector of water-filling on the n subcarriers with the power 1

^{T}P

_{n}.

Proof: Assume in the fcth iteration, the power_{k} respectively. If we find such j that Fj ≥ Gj, then the subchannel j needs to be taken out from the set A and put into the set C_{k}. At Step II-4, the conventional water-filling is done on the subcarriers in each subchannel j that j ∈ C_{k} with the corresponding subchannel transmit power constraint Gj individually. The resulting water-levels are just the unique water-level_{k} with the power Fj individually, we also get the water-level_{j} ≥ Gj, we have

after the kth iteration.

Based on Lemma 1, for the rest of the subchannels in the set A satisfying F_{j}< G_{j}, if the conventional water-filling is done on the corresponding subcarriers with the power_{h}. In the next iteration, we need to perform the conventional water-filling on these subcarriers with the power_{h}+_{1}. Based on Lemma 2, since

When the algorithm converges after

So far, we have proved that the IPW algorithm converges to the point satisfying the conditions in Theorem 1 and thus the optimal power allocation can be obtained by the IPW algorithm.

### 3.4. Iterative Partitioned Water-filling: Target Rate Constraint

In this subsection, we consider (8), i.e., the power allocation problem with the target rate constraint. If the target rate is given, we should first verify if the target rate can be achieved under the subchannel transmit power constraints. It is easy to see that the maximum rate that can be achieved under the subchannel transmit power constraints is the result of water-filling on the subcarriers in each subchannel with the corresponding subchannel transmit power constraint individually. Therefore, the maximum achievable rate can be expressed as

where Pi is determined by

where i = 1, 2,…, N and j is the index of the subchannel which the ith subcarrier belongs to. wj can be determined by

Then, if R_{t}> R_{max}, Rt can not be achieved by any power allocation vector under the subchannel transmit power constraints. Also, the optimal power allocation can be determined by (32) and (33) when R_{t} = R_{max}. In the case R_{t}< R_{max}, the algorithm for solving (8) can be developed following the similar derivation as that in Section 3.3. A similar theorem as Theorem 1 can be described as follows.

_{t}< R

_{max}, a power allocation vector P is the solution for (8) if and only if it satisfies:

where i = 1, 2, …,N and j is the index of the subchannel which the ith subcarrier belongs to. Assume_{j} is determined by 1. for j ∈ A:

where Rj is determined by

2. for j 𝜖 B:

The proof of Theorem 3 is also similar to that of Theorem 1 and thus is omitted in this paper. Based on Theorem 3, the IPW algorithm for solving (8), which can be developed in a similar way as that in Section 3.3, is described in Table 2.

In each iteration, the conventional water-filling with only target rate constraint

I. Initialization : |

II. Iterations: 1. 1. Perform the conventional water-filling on the subcarriers in all the subchannels that belong to the set A with the target rate |

2. |

III. End |

## 4. Power Allocation for OFDM-based Cognitive Radio Systems Considering Subcarrier Sidelobes

The iterative partitioned water-filling (IPW) algorithm proposed in the above section can obtain the optimal power allocation for OFDM-based cognitive radio systems in the case where only the interference to the PU caused by the subcarriers inside the corresponding subchannel was considered. However, it is clear that the PU also suffers interference introduced by the subcarrier sidelobes of the neighboring subchannels. Therefore, the assumption in the above section actually implied that there is sufficient guard band between any two subchannels so that the effects of subcarrier sidelobes could be ignored. In this section, we extend the results in the above section and address the power allocation problem in general OFDM-based cognitive radio systems. The effects of subcarrier sidelobes are specially considered in the optimization problem. We propose the power allocation algorithm that maximizes the capacity while satisfying both the sum and subchannel transmit power constraints.

The effects of subcarrier sidelobes in OFDM-based cognitive radio systems were first addressed in (Weiss et al., 2004). The subcarrier sidelobes suppression techniques were further studied in (Cosovic et al., 2005, Pagadarai et al., 2008, Mahmoud & Arslan, 2008). In (Bansal et al., 2007), the authors proposed a power loading algorithm for OFDM-based SU systems whose bands are not overlapped with any PU's. The aim was to maximize the capacity while keeping the interference to the PUs whose licensed bands are in the proximity of the band of SU below a certain threshold.

In the following, we will firstly reformulate the system model and the optimization problem with subcarrier sidelobes in consideration. Then an algorithm will be proposed for a special case with only two non-zero weighted linear inequality constraints. Finally, this algorithm mentioned above will be extended to general cases with multiple non-zero weighted linear inequality constraints, where the extended algorithm is called recursive power allocation (RPA) algorithm.

### 4.1. Power allocation problem reformulation

The basic system model used in this section is the same as that in Figure 1 and Figure 2. The SU's

transmit power on the PU's licensed channel P_{tx} should still be subject to the power constraint as (1). However, considering the effects of subcarrier sidelobes, it is not proper to require the transmit power in the unavailable subchannel to be zero as (3), since the sidelobes of the subcarriers in other subchannels certainly lead to non-zero transmit power in the unavailable subchannel. So we set a transmit power threshold for the jth subchannel when PUj is detected, say

As mentioned in Section 3, in OFDM-based cognitive radio systems, the subchannel transmit power constraints impose further restrictions on the power allocation in addition to the sum transmit power constraint. In fact, the transmit power in one subchannel is comprised of not only the power allocated to the subcarriers inside the subchannel but also the sidelobes power of the subcarriers in other subchannels. We define Jij as the transmit power in the jth subchannel caused by the ith subcarrier with unit power. According to (Weiss et al., 2004), we have

Assume P_{i} be the power allocated to the ith subcarrier. Then the transmit power on the jth subchannel can be expressed as

Then, the original optimization problem as (7) for optimal power allocation in OFDM-based cognitive radio systems can be modified as the following one:

where P* is the optimal power allocation vector, h_{i} is the ith subcarrier's channel gain, and Pt is the maximum transmit power of the SU transmitter.

Section 3, the subchannel power constraints are approximated by ignoring the subcarrier sidelobes, i.e.,

When (41) is satisfied, it can be seen that the M subchannel power constraints in (40) are actually decoupled with each other in terms of the optimization variables. Based on this

characteristic, we manage to find an efficient algorithm named iterative partitioned water-filling. In the case where the subcarrier sidelobes are considered, however, the problem turns to be more complicated.

It is clear that (40) is a convex optimization problem with linear inequality constraints. (40) can be expressed in a compact form:

where J_{i,0} = 1 for any i = 1, 2,…,N and G_{o} = P_{t}. While the N positivity constraints are easy to handle, the remaining M + 1 constraints, which are named non-zero weighted linear inequality constraints for ease of description, introduce the main difficulties for the algorithm development.

In order to find the power allocation algorithm for (42), we start with the problem with only two non-zero weighted linear inequality constraints. Then the recursive power allocation (RPA) algorithm is proposed for the general cases, which is a recursive extension from the above simple problem.

### 4.2. Power allocation for the case of two non-zero weighted linear inequality constraints

The problem with two non-zero weighted linear inequality constraints can be expressed as

The two non-zero weighted linear inequality constraints are numbered with C_{k} and C_{l} respectively for ease of derivation.

#### 4.2.1. Degraded cases

We begin with solving the problem with C_{k} excluded. Solving this degraded problem is similar as the derivation of the conventional water-filling algorithm. The solution, which satisfies the constraint C_{l} with equality, can be expressed as

where

Substituting (44) into the excluded constraint C_{k}, if C_{k} is also satisfied, i.e.,

then (44) is also the solution of the original problem (43).

If (45) is not satisfied, we turn to consider the problem with Ci excluded. Similarly, the solution for this degraded problem can be expressed as

where

If the constraint C_{l} is also satisfied with (46), i.e.,

then (46) is also the solution of the original problem (43).

Therefore, the optimal power allocation for (43) is easy to find in the above two degraded cases, i.e., either (45) or (47) is satisfied.

#### 4.2.2.Algorithm for general cases

In order to obtain the solution in the case that neither (45) nor (47) is satisfied, we first propose the following lemma.

Lemma 3 Define a convex optimization problem (OP_{0}) with differentiate objective function and inequality constraints satisfying Slater's conditions:

and let x_{0} be any optimal point of OP_{0}. Assume x_{k} is any optimal point of a sub-problem (OP_{-k}) of OP_{0} with n — 1 inequality constraints h_{i}(x) ≤ 0,i = 1,...,k — 1,k + 1,...,n. Then if h_{k}(x_{k}) ≥ 0, we have h_{k}(x_{0}) = 0.

Proof: If h_{k}(x_{0}) ≠ 0, then we have h_{k}(x_{0}) < 0. Since x_{0} is an optimal point of the convex optimization problem OP_{0}, x_{0} satisfies all the KKT conditions of OP_{0}, which are given by

where α_{i} is the Lagrange multiplier. Since h_{k}(x_{0})<0 according to α_{k} h_{k}(x_{0}) = 0 in (49a), we have 𝛼_{k} = 0. Then (49b) can be expressed as

Based on (49a) and (50), it is clear that x_{0} also satisfies the KKT conditions of OP_{-k}. Therefore, x_{0} is also an optimal point of OP_{-k}. Consequently, we know that there is an

Step-1. Calculate the optimal power allocation P* with C k excluded based |

on (44); |

Step-2. Check if P* also satisfies C k. If C k is satisfied, exit; |

Step-3. Calculate the optimal power allocation P* with C l excluded based |

on (46); |

Step-4. Check if P* also satisfies C l . If C l is satisfied, exit; |

Step-5. Calculate the optimal power allocation P* based on (53 ) |

optimal point of OP_{-k} satisfying h_{k}(x_{k}) < 0, which contradicts the premise that h_{k}(x_{k}) > 0. Finally, we come to the conclusion that h_{k} (x_{0}) = 0.D

Based on Lemma 3, the optimal solution of (43) must satisfy the two constraints C_{k} and Ci with equalities in the case that neither (45) nor (47) holds. Therefore, solving (43) in this case is equivalent to solving the following problem:

To find the solution, we form the Lagrangian:

The optimal power allocation can be obtained by solving the first order KKT condition_{i}> 0, which is given by

where

Combining the derivations above, the power allocation algorithm for the problem with two non-zero weighted linear inequality constraints is shown in Table 3.

### 4.3. Recursive power allocation algorithm for general cases

The above algorithm can be further extended to the problem (42) including M + 1 nonzero weighted linear inequality constraints. Similarly, we first consider M + 1 degraded cases each of which excludes one non-zero weighted linear inequality constraint. Each of the M + 1 degraded optimization problems can be solved by the algorithm for the problem with M nonzero weighted linear inequality constraints. If the solution of one of these M + 1 degraded

I. Initialization : |

Number the M +1 non-zero weighted linear inequality constraints as C 0 ,C 1 ,...C M , m= M +1, A={ C 0 ,C 1 ,...C M }. |

Begin P* = PAA ( m, A ) . end function:PAA (m, A ) begin Case m =1 : Return |

problems also satisfies the corresponding excluded constraint, the optimal power allocation is obtained. Otherwise, the optimal solution of the original problem must satisfy the M + 1 constraints with equalities based on Lemma 3. Similar as that in Section 4.2.2, we form the Lagrangian:

The optimal power allocation can be obtained by solving the first order KKT condition_{i}> 0, which is given by

where

Since the algorithm for the problem including two non-zero weighted linear inequality constraints has been obtained in Section 4.2.2, the problem for the general case including multiple non-zero weighted linear inequality constraints can be solved in a recursive way, i.e., recursive power allocation (RPA) algorithm.

For ease of derivation, we define the function f (A) where A is a set comprised of all the non-zero weighted linear inequality constraints of the optimization problem with the form as (42). f (A) returns the optimal solution that satisfies all the constraints contained in A with equalities. For example, f(A) returns the value determined by (44) when A = {C_{k}} and the value determined by (53) when A = {C_{k},C_{l}}. We further assume that 𝛼_{i} denotes the ith element of A. The power allocation algorithm for general cases is shown in Table 4.

## 5. Simulation Results

In this section, two parts of simulation results are presented to verify our proposed algorithms of power allocation for OFDM-based cognitive radio systems. The first part focuses on the

IPW algorithm (as in Table 1) [1] - for OFDM-based cognitive radio systems without considering subcarrier sidelobes. The second part focuses on the RPA algorithm (as in Table 4) for OFDM-based cognitive radio systems considering subcarrier sidelobes. Part A: IPW algorithm without considering subcarrier sidelobes In this part, we set N = 16, M = 4, i.e., each subchannel has 4 subcarriers. The IPW process is shown in Figure 6. Figure 6(a) illustrates the reciprocal of the channel gain given a realization of a frequency selective fading channel, i.e., the bottom of the pool for water-filling. The power allocation result after Step in Table 1 in the first iteration is shown in Figure 6(b), where the power allocation is just the result of the conventional water-filling with only sum transmit power constraint. At Step II-2 in Table 1, we find that the power allocated to Subchannel 1 exceeds its subchannel transmit power constraint Gj.. Therefore, Subchannel 1 should be taken out from the set A and the conventional water-filling is performed on the corresponding subcarriers with the power G_{1} at the last step of the first iteration. At Step in Table 1 of the second iteration, the conventional water-filling is performed on the subcarriers in the rest of the subchannels in the set A with the power P_{t} — G_{1} and the resulting power allocation is shown in Figure 6(c). Then the power constraint on Subchannel 3 is found to be destroyed at Step II-2 in Table 1. As a result, we execute Step II-3 and Step II-4 in Table 1 again and then enter the third iteration. Figure 6(d) illustrates the power allocation after Step II-1 in Table 1 in the third iteration, where only Subchannel 2 and 4 are still remained in the set A. At Step II-2 in Table 1, each subchannel transmit power constraint is found to be satisfied and thus we may end the iterations with C = 0. Finally, the power allocation in Figure 6(d) is the optimal one that maximizing the sum rate while satisfying both the subchannel transmit power constraints and the sum transmit power constraint. Part B: RPA algorithm considering subcarrier sidelobes

In this part, the frequency selective fading channel is generated according to the Typical Urban (TU) channel model, which is comprised of six paths whose delay parameters are [0.0 0.2 0.5 1.6 2.3 5.0]ßs and power profile is [0.189 0.379 0.239 0.095 0.061 0.037]. The SU's bandwidth is 5 MHz, which is equally divided into M = 4 subchannels. The number of the subcarriers is supposed to be N = 64 and each subchannel has 16 subcarriers. We use the values 640 for P_{t}, which implies that the signal to noise ratio is about 10 dB as the noise power on each subcarrier is set to be 1. The four subchannel power constraints are specified as 80, 480, 1.6, 480 respectively. The power constraint on Subchannel 3 is about 25 dB below its neighboring subchannels, which implies that the SU may be very close to or even just inside PU_{3}'s protection area. Thus, the constraint on the transmit power in Subchannel 3 should be very rigorous in order to satisfy PU_{3}'s interference power limit. The optimal power allocation is shown in Figure 7 for a given realization of the TU channel. For comparison, the power allocation result using the IPW algorithm ignoring the effects of subcarrier sidelodes is also given in Figure 8. Though it can be seen in Figure 8 that the allocated power on each subchannel is lower than the corresponding subchannel's transmit power constraint, the actual transmit power on Subchannel 3, including the allocated power on Subchannel 3 and the sidelobe power from neighboring subchannels, is about 6 times of its transmit power constraint, which induces to serious destruction to the PU3's interference constraint. Therefore, the IPW algorithm cannot be applicable for such Subchannel 3 and its neighboring subchannels in this case. On the contrary, the RPA algorithm as shown in Figure 7 can maximize the capacity while satisfying the subchannel transmit power constraints. Comparing Figure 7 with Figure 8, we can see that the power allocated to the subcarriers near Subchannel 3 is suppressed even if the channel states are fine while more power is allocated to the subcarriers that are far away from Subchannel 3. On the other hand, it can also be found that the power allocation in Subchannel 1 is almost the same as that without considering subcarrier sidelobes, which implies that it is reasonable to ignore the effects of subcarrier sidelobes when sufficient guard band between two subchannels is present. Therefore, the IPW algorithm, with higher efficiency compared to the RPA algorithm, can be applied to the subchannels such as Subchannel 1.

## 6. Conclusions

In this chapter, we study the power allocation problem in OFDM-based cognitive radio systems.

In the cognitive radio scenario, the interference limits of the primary users introduce subchannel power constraints for the transmission of the SU. Therefore, the power allocation among the subcarriers should satisfy both the sum transmit power constraint, as that in conventional OFDM systems, as well as the subchannel power constraints, which leads to the unavailability of the well-known water-filling algorithm applied in conventional OFDM systems.

In order to find the optimal power allocation, we first propose the IPW algorithms, whose basic computation units is the conventional water-filling. The algorithm is proved to converge to the optimal power allocation, which maximizes the sum rate when the sum transmit power constraint is given or minimizes the required power when the target rate is given, after a finite number of iterations.

Furthermore, we consider the effects of subcarrier sidelobes on the power allocation in practical systems. We note that the actual transmit power in one subchannel is comprised of the allocated power as well as the subcarrier sidelobes power from the neighboring subchannels. The RPA algorithm is proposed for this scenario. The algorithm finds the optimal power allocation recursively by decoupling the subchannel power constraints phase-by-phase. Simulation results show that the IPW algorithm is unavailable when the guard bands between the subchannels are not wide enough, while the RPA algorithm can be applied in such scenario. In conventional OFDM systems, bit loading is a more practical technique for the system design. Therefore, it is interesting to study bit loading algorithms in OFDM-based cognitive radio systems, where the subchannel transmit power constraints should be considered.[1] -

## Notes

- Here, without considering subcarrier sidelobes, the allocated power is exactly the transmit power.
- Using the IPW algorithm in Table 2 has the similar results as mentioned in Section 3.4.
- This work is partially supported by International Science and Technology Cooperation Program (2008DFA12160) and National Basic Research Program of China (2007CB310608).