Gaurav Bansal

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

Adaptive Power Loading for OFDM-based Cognitive Radio Systems Gaurav Bansal, Md. Jahangir Hossain, and Vijay K. Bhargava University of British Columbia, 2356 Main Mall Vancouver, BC, V6T 1Z4, Canada Email: {gauravbs, jahangir, vijayb}@ece.ubc.ca

Abstract—Cognitive radio (CR) technology is an innovative radio design philosophy which aims to increase spectrum utilization by exploiting unused spectrum in dynamically changing environments. Orthogonal frequency division multiplexing (OFDM) is a potential modulation technique for CR networks’ air interface. In this paper, we study and explore optimal power loading algorithm for an OFDM-based CR system and the rate in each subcarrier is adjusted according to the power. As such the downlink capacity of the CR user is maximized while the interference introduced to the primary user remains within a tolerable range. We also propose two suboptimal loading algorithms that have less complexity. The performance of optimal and suboptimal schemes are compared with the performance of classical power loading algorithms that are used for conventional OFDM-based systems e.g., water-filling and uniform power but variable rate loading schemes. Presented numerical results show that for a given interference threshold the proposed optimal scheme allows the CR users to transmit more power in order to achieve higher transmission rate than the classical loading algorithms. These results also show that the proposed suboptimal schemes offer a performance close to the optimal scheme. Finally, we study the effect of subcarrier nulling mechanism on the performance of different algorithms under consideration.

Keywords: 1) Cognitive radio, 2) Opportunistic spectrum access, 3) OFDM, 4) Link adaptation, and 5) Interference reduction. I. I NTRODUCTION Radio spectrum is one of the most scarce and valuable resources for wireless communications. Given this fact, new insights into the use of spectrum have challenged the traditional approaches to the spectrum management. Actual measurements have shown that most of the allocated spectrum is largely underutilized [1]. Spectrum-Policy Task Force appointed by Federal Communications Commissions (FCC) has also reported similar views about the underutilization of the allocated spectrum [2]. Spectral efficiency can be increased significantly by giving opportunistic access of the frequency bands to a group of potential users (referred to as secondary or CR users) for whom the band has not been licensed. Cognitive radio (CR) has been proposed as a way to improve spectrum efficiency by exploiting unused spectrum in dynamically changing environments. The CR design is an innovative radio design philosophy which involves smartly sensing the swaths of spectrum and then determining the transmission characteristics (e.g., symbol rate, power, bandwidth, latency) of a group of secondary users based on the behavior of the users to whom the spectrum

has been licensed (referred to as primary users). Although opportunistic spectrum access would allow CR user to identify and access available spectrum resources, one of the main concerns is to utilize the available spectrum resources in an efficient manner. Due to great flexibility in dynamically allocating the unused spectrum among the CR users as well as the easy analysis of the spectral activity of the primary users [3], in literature orthogonal frequency division multiplexing (OFDM) has already been recognized as a potential transmission technology for CR systems. Since both CR and primary users may exist in side by side band and their access technologies may be different, the mutual interference is the limiting factor for performance of both networks. Specifically, in [4] the authors have shown that using OFDM modulation causes mutual interference between the primary and the CR users due to the non-orthogonality of the transmitted signals. The amount of interference introduced to the primary user’s band by a CR user’s subcarrier depends on the power allocated in that subcarrier as well as the spectral distance between the subcarrier and the primary user’s band. The authors have also studied the effect of subcarriers’ nulling mechanism which reduces the interference in the primary user’s band. In order to exploit the time varying nature of fading gains across the OFDM subcarriers, power loading algorithms have been proposed in literature [5]. These algorithms have maximized the transmission capacity of an OFDM system and are useful for conventional wireless networks where there is only one group of users i.e., primary users. Since there is a mutual interference between CR and primary users when both type of users co-exist in side by side band, use of the classical loading algorithms e.g., uniform power but variable rate and water-filling algorithms for CR users may result in higher mutual interference in the primary users’ band. Throughout the paper we consider CR downlink scenario where interference introduced to the primary users is the limiting factor but not the transmit power of CR user. In fact such an interference limited scenario limits the transmit power as well as the achievable transmission rate of CR users. Hence, the design problem is that given an interference threshold prescribed by the primary users, how much power should be transmitted into each CR user’s subcarrier such that the transmission rate of the CR user is maximized. According to the classical power loading schemes e.g.,

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

water-filling, more power should be loaded into the subcarrier which has higher channel gain. However the amount of interference, introduced by allowing transmission in a CR user’s subcarrier, depends on the location of the subcarrier with respect to the primary user’s spectrum. From the interference point of view, more power should be loaded into a distant subcarrier. Therefore, it requires a judicious loading policy which not only consider the fading gain of the subcarrier but also the spectral distance of the subcarrier from the primary user’s band. Motivated by the aforementioned challenging task and the interference model studied in [4], in this paper, we propose an optimal power loading scheme using Lagrange formulation. This loading scheme maximizes the downlink transmission capacity of the CR users while keeping the interference induced to the primary users below a specified threshold. As the complexity of the optimal scheme can be quite high for practical implementation, we also propose two suboptimal schemes. In a related work, the authors in [6] have proposed an unequal bit loading algorithm for a non-contiguous (NC)OFDM-based CR system. In NC-OFDM based transmission system, subcarriers, which could potentially interfere with other users’ transmission, are deactivated. Further we study the effect of subcarrier nulling mechanism on the performance of different algorithms under consideration. Selected numerical results are presented in later sections. II. S YSTEM M ODEL We consider the same side by side CR radio access model as assumed in [4]. Basically, it is assumed that the frequency band B which has been occupied by the primary user(s) is known and is located in the middle (see Fig. 1). The bandwidth sensed by the CR users for possible transmission is located on each side of the primary user band as shown in Fig. 1. OFDM modulation scheme is employed for CR users and the available bandwidth for CR transmission is divided into N subcarriers, N/2 on each side and each having a bandwidth of ∆f. We assume that each subcarrier goes under frequency flat fading and the instantaneous fading gains are perfectly known at the transmitter. The transmit power is adaptively loaded in each CR user’s subcarrier. With an ideal coding scheme, the transmission rate at ith carrier, Ri for the transmit power, Pi and channel fading gain hi is connected via the Shannon capacity formula and is given by
 2 |hi | Pi Ri (Pi , hi ) = ln 1 + 2 , (1) σ + Ji 2

where σ denotes the AWGN noise variance, and Ji denotes the interference introduced by the primary user into the CR user’s subcarrier. Due to the coexistence of primary and CR users, there are two types of interference in the system [4], one is introduced by the primary users into the CR user band and the other is introduced by the cognitive users into the primary user band.

A. Interference introduced by CR user’s signal The power density spectrum of the ith subcarrier in CR user band can be written as [4]  2 sin πf Ts , (2) φi (f ) = Pi Ts πf Ts where Pi is the total transmit power emitted by the ith subcarrier in CR user’s band and Ts is the symbol duration. The interference introduced by the ith subcarrier to the primary user band is the integration of the power density spectrum of the ith subcarrier across the primary user band and can be written as 2  di +B/2  sin πf Ts df, (3) Ii (di , Pi ) = Pi Ts πf Ts di −B/2 where di represents the distance between the ith subcarrier of CR user band and the primary user band. Ii (di , Pi ) represents the interference introduced by the ith subcarrier into the primary user band. The interference in eq. (3) should also take fading gain from the secondary user base station to the primary user receiver into account. We use a normalised fading gain of 1. B. Interference introduced by primary user’s signal The power density spectrum of the primary user signal after the M-fast Fourier transform (FFT) processing can be expressed by the following expected value of the periodogram [4]  2  π sin(w − ψ)M/2 1 φP U (ejw ) dψ, E{IN (w)} = 2πM −π sin(w − ψ)/2 (4) where w represents the frequency normalized to the sampling frequency and φP U (ejw ) is the power density spectrum of the primary user signal. Primary user signal has been taken as an elliptically filtered white noise process with an amplitude PP U [4]. The interference introduced by the primary user signal to the ith subcarrier will be the integration of the power density spectrum of the primary user signal across the ith subcarrier and can be written as  di +∆f /2 E{IN (w)}dw, (5) Ji (di , PP U ) = di −∆f /2

where Ji (di , PP U ) represents the interference introduced by the primary user signal into the ith subcarrier of CR user’s band. III. O PTIMAL SCHEME The goal is to maximize the sum capacity while keeping the instantaneous interference introduced to the primary users below a certain threshold. Therefore, we can formulate the following constrained optimization problem C = max Pi

N  i=1

Ri (Pi , hi ),

(6)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

Now using eqs. (11), (12) and (13), we can write
 N  (σ 2 + Ji )Ki 1 − = Ith . 2 λ |hi | i=1

(14)

Rearranging the above equation, λ can be expressed as N λ= (15) N Ki (σ2 +Ji ) . Ith + i=1 |h |2 i

Fig. 1.

s.t.

Distribution of primary and CR users.

N 

Ii (di , Pi ) ≤ Ith ,

(7)

i=1

and Pi ≥ 0,

(8)

where C denotes the transmission capacity of the CR users, N denotes the total number of subcarriers, Ri denotes the transmission rate in ith subcarrier and Ith denotes the interference threshold prescribed by the primary users. This optimization problem can be solved using the iterative approach. As we will see that the constraint is on the interference limits the transmit power as well as the achievable transmission rate of the CR users. Using Lagrange formulation we can write L(Pi , λ) =

N 

N  Ri (Pi , hi ) − λ( Ii (Pi , di ) − Ith ),

i=1

(9)

i=1

where λ is the Lagrange constraint. Now, substituting eq. (1) in eq. (9) and differentiating eq. (9) with respect to Pi we can write 2 ∂Ii |hi | 1 ∂L −λ = , (10) 2 2 |h | P i i ∂Pi δPi 1 + σ2 +Ji (σ + Ji ) 

Ki

2  d +B/2 where Ki = Ts dii−B/2 sinπfπfTsTs df . Equating eq. (10) to zero and after some manipulations optimal transmit power in ith subcarrier can be written as 1 σ 2 + Ji Pi∗ = − 2 . λKi |hi |

(11)

Now, the value of λ can be calculated by the following equation N  Ii (di , Pi∗ ) = Ith . (12) i=1

The interference due to ith subcarrier with power Pi can be written as [4] (13) Ii = Pi Ki .

If power comes out to be negative for some subcarriers using eqs. (11) and (15), zero power is assigned to that subcarrier for which the power comes out to be the most negative value. Then whole scheme is reiterated for the remaining subcarriers. Hence, by using the above scheme we calculate the optimal power allocation policy that maximizes the transmission capacity of the CR users while keeping the interference introduced to the primary users below the specified threshold. It is important to note that the value of λ in eq. (15) is independent of subcarrier index. If the power for some of the subcarriers comes out to be negative, several iterations may require in reaching the optimal solution. Therefore, the complexity of the optimal scheme can be quite high depending on the scenario. In what follows, we propose sub-optimal schemes namely scheme A and scheme B and also state the uniform power loading but variable rate and waterfilling schemes for CR scenario. IV. S UBOPTIMAL S CHEMES Heuristic schemes proposed in this section are based on the fact that the interference introduced to the primary user band by the CR user subcarrier’s increases as the spectral distance between them decreases. If we ignore the second 2 i term ( σ|h+J 2 ) from the power profile of eq. (11), the power i| is inversely proportional to the parameter Ki which depends on the spectral distance of the ith subcarrier from the primary user’s band. As such in order to reduce the interference, less power should be assigned to the subcarriers which are near to the primary user’s band and more power should be assigned to the subcarriers which are far from the primary user band. This suggests that the power can be distributed like a ladder profile as shown in Fig. 2. The total transmit power is determined such that the total interference introduced by all the subcarriers is equal to the interference threshold. Now, based on the step size of the ladder, we propose suboptimal schemes as follows. A. Scheme A In this scheme, the step size is fixed and is equal to the power level of the subcarrier which is nearest to the primary user band. Hence, the power in the ith subcarrier can be written as (16) PiA = P ∗ i, where P will be determined by the value of Ith . Now using eq. (16) the total interference introduced to the primary user band with this power distribution can be written as N  i=1

Ki ∗ P ∗ i = Ith .

(17)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

Using eqs. (16) and (17), the power allocated to ith subcarrier according to scheme A, PiA can be expressed in desired closedform as i ∗ Ith . (18) PiA = N i=1 Ki ∗ i B. Scheme B In this scheme, the step size of the ladder is inversely proportional to Ki . Hence, the power in the ith subcarrier can be written as PiB = P/Ki , (19)

C. Uniform Power Loading/Waterfilling Scheme The classical OFDM loading algorithms: uniform power loading and waterfilling schemes are suboptimal for such a interference limited scenario as they do not have constraint on the interference. In fact unform power loading scheme is a special case of sub-optimal scheme A when PiA is assumed as P . Therefore, for a given interference threshold Ith , power allocated to the ith subcarrier with uniform power loading, PiU can easily expressed as Ith PiU = N i=1

Ki

.

(21)

For distributing power according to waterfilling scheme we first determine the total power used by the uniform scheme in transmitting at the given interference threshold. Using this total power as the power constraint we determine power for each subcarrier using the waterfilling algorithm. V. NUMERICAL RESULTS In the numerical results presented in this section, we use the values of Ts , ∆f and B to be 4µ seconds, 0.3125 MHz and 0.3125 MHz respectively. Additive white Gaussian (AWGN) noise variance of 10−6 is assumed. The channel gain’s hi is assumed to be Rayleigh fading with an average channel power gain of 10dB. Further, we assume that there are ten subcarriers for CR users, five on each side of the primary user band. In Fig. 3, we plot the achievable transmission rate of CR user versus interference introduced to the primary user band for different schemes under considerations. From this figure, we observe that for a given interference threshold, the optimal scheme achieves the highest transmission rate for CR users. We can also see that the transmission rate that can be achieved using the sub-optimal schemes is higher than the conventional uniform-power but variable rate loading and water-filling algorithms. It is also obvious that the scheme B performs better than scheme A. It is important to mention from Fig. 3 that the uniform loading and waterfilling schemes have the lowest achievable data rate. In our results in Fig.

Fig. 2.

Ladder profile of power distribution.

55 Maximum transmitted data rate of CR user (in bits/sec.)

where P will be determined by the value of Ith as follows. Eq. (17) will hold in this case as well. By substituting eq. (17) in eq. (19) the power allocation policy for scheme B, PiB can be expressed as Ith PiB = . (20) N ∗ Ki

50

45

40

35

30 uniform−loading/water−filling scheme Scheme A

25

Scheme B Optimal Scheme

20 0.4

0.6

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −6 Interference introduced to the Primary user band (in Watts) x 10

Fig. 3. Maximum transmitted data rate of CR user vs Interference introduced to the primary user’s band.

3, the maximum transmitted data rate achieved by uniform power loading and water-filling is same as using uniform power loading does not produce any visual degradation in the performance. This is because varying both rate and power leads to a negligible higher capacity gain over varying just rate alone as reported in [7]. In Fig. 4, we present the transmit power of the CR user versus the interference introduced to the primary user’s band for various schemes under consideration. We can observe from Fig. 4 the optimal scheme allows to transmit higher power than the other schemes for a given interference threshold. The uniform power loading and waterfilling are able to load least amount of power as they do not take judiciously interference into account in their loading policy for a given interference threshold. VI. E FFECT OF S UBCARRIERS N ULLING M ECHANISM In [4], the authors have studied the effect of nulling the subcarriers and have shown that the interference introduced to the primary user band can be reduced by nulling the subcarriers which are adjacent to the primary user band. The

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

One−nulling case

−4

x 10

Uniform−loading/water−filling scheme

7

Scheme A

6

Optimal scheme

Power of CR user (in Watts)

Scheme B

5

4

3

2

1

0 0.4

Fig. 4. band.

0.6

Maximum transmitted data rate of CR user (in bits\sec.)

55

Power of CR user vs Interference introduced to the primary user’s

reason behind it is that the adjacent subcarriers produce the maximum amount of interference. It also implies that for a given interference threshold, more power can be allocated into the far apart subcarriers than the amount of power allocated into the neighboring subcarriers. As a consequence one can expect that higher transmission rate can be achieved using more power. However, nulling adjacent subcarriers loose the degrees of freedom as the adjacent subcarrier is assigned with zero power even when it has a very good channel gain. Therefore, nulling creates a trade-off. Here we study the effect of nulling on the proposed suboptimal, uniform power loading and water-filling schemes. In Fig. 5, we plot maximum transmission rate of CR user versus interference introduced to the primary user band for two proposed suboptimal schemes, uniform power loading and water-filling schemes for one nulling. Here, by one nulling we mean that we null one subcarrier from each sides of the primary user band that are nearest to it. Similar results have been plotted in Fig. 6 for two nulling case. In these we have also plotted the data rate of the optimal scheme for sake of comparison. From Figs. 5 and 6, we can observe that after nulling the performance of sub-optimal schemes and uniform power loading/water-filling scheme improves as compared to no nulling, but still the optimal scheme performs the best and transmits maximum data rate for a given interference threshold. We did not consider more number of nulling as the performance degrades and they have been checked via simulation. In Figs. 7 and 8 we present the plots for the transmit power of CR user versus interference introduced to the primary user band for all the suboptimal schemes, for one nulling and two nulling cases, respectively. The interesting observation is that as we increase the nulling for sub-optimal schemes and uniform power loading/water filling scheme they can load

50

45

40 Uniform−loading/water−filling scheme Scheme A

35

Scheme B Optimal scheme

30 0.4

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −6 Interference introduced to the primary user band (in Watts) x 10

0.6

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −6 Interference introduced to the Primary user band (in Watts) x 10

Fig. 5. Maximum transmitted data rate of CR user vs Interference introduced to the primary user band for one-nulling case.

Two−nulling case

55 Maximum transmitted data rate of CR user (in bits\sec.)

8

50

45

40

35 Uniform−loading/water−filling scheme Scheme A

30

Scheme B Optimal scheme

25 0.4

0.6

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −6 Interference introduced to the primary user band (in Watts) x 10

Fig. 6. Maximum transmitted data rate of CR user vs Interference introduced to the primary user band for two-nulling case.

more power than the optimal scheme but the transmission rate is lower than the optimal scheme. The reason of this phenomenon is that although the suboptimal and classical schemes can load more power as compared to the optimal scheme with nulling, they have less degrees of freedom as compared to the optimal scheme. In Fig. 9, we plot the maximum transmission data rate of CR user versus the interference introduced to the primary user band with two proposed suboptimal schemes and the uniform power loading/water-filling scheme for various number of nulling. From Fig. 9, we observe that both sub-optimal schemes and uniform power loading/water-filling scheme per-

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.

One−nulling case

−4

x 10

Power of the CR user (in Watts)

7

6

5

4

3 Uniform−loading/water−filling scheme

2

Scheme A Scheme B

1

Optimal scheme

0.6

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −6 Interference introduced to the primary user band (in Watts) x 10

Fig. 7. Power of CR user vs Interference introduced to the primary user band for one-nulling case.

45

40 Uniform loading−zero nulling

35

Uniform loading−one nulling Uniform loading−two nulling Scheme A−zero nulling

30

Scheme A−one nulling Scheme A−two nulling

25

Scheme B−zero nulling Scheme B−one nulling

20 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Interference introduced to the primary user band (in Watts)

2.2

2.4 −6

x 10

Fig. 9. Maximum transmitted data rate of CR user vs Interference introduced to the primary user band for various nulling.

Two−nulling case

−4

x 10

The performance of optimal and suboptimal schemes have been compared with that of conventional water-filling and uniform power loading schemes. Presented numerical results have shown that the classical loading algorithms which are used for conventional wireless networks perform the worst for CR scenario. We have also studied the effect of nulling mechanism on the performances of various schemes. Selected numerical results have shown that optimal scheme performs the best and one nulling case has achieved better data rate performance than more number of nullings and zero nulling cases.

9

8 Power of CR user (in Watts)

50

Scheme B−two nulling

0 0.4

10

Schemes under various nulling 55

Maxiumum transmitted data rate of CR user (in bits/sec.)

8

7

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4 uniform−loading/water−filling case

3

2

1 0.4

Scheme A

ACKNOWLEDGEMENT

Scheme B

This research was supported by Natural Sciences and Engineering Research Council of Canada under a Strategic Project Grant.

Optimal scheme

0.6

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 −6 Interference introduced to the primary user band (in Watts) x 10

Fig. 8. Power of CR user vs Interference introduced to the primary user band for two-nulling case.

forms the best for one nulling case. It implies that after one nulling the degradation due to decrease in degrees of freedom dominates the gain achieved by loading more power in far apart subcarriers. VII. C ONCLUSION In this paper, we have developed an optimal power loading algorithm which maximizes the downlink transmission data rate of CR user while the interference produced to the primary user remains within a given limit. We have also proposed two suboptimal power loading algorithms that have less complexity but can achieve a performance close to the optimal one.

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