Spectrum efficient power allocation schemes for OFDM cognitive radio with statistical interference constraints

Accepted Manuscript Spectrum efficient power allocation schemes for OFDM cognitive radio with statistical interference constraints Manoranjan Rai Bharti, Debashis Ghosh

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S1874-4907(17)30081-2 http://dx.doi.org/10.1016/j.phycom.2017.07.003 PHYCOM 405

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Physical Communication

Received date : 17 March 2017 Revised date : 17 June 2017 Accepted date : 17 July 2017 Please cite this article as: M.R. Bharti, D. Ghosh, Spectrum efficient power allocation schemes for OFDM cognitive radio with statistical interference constraints, Physical Communication (2017), http://dx.doi.org/10.1016/j.phycom.2017.07.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Spectrum Efficient Power Allocation Schemes for OFDM Cognitive Radio with Statistical Interference Constraints Manoranjan Rai Bharti∗, Debashis Ghosh Department of Electronics and Communication Engineering Indian Institute of Technology Roorkee Roorkee – 247 667, Uttarakhand, India

Abstract In this paper, we study the power allocation problem for an orthogonal frequency division multiplexing (OFDM)-based cognitive radio (CR) system. In a departure from the conventional power allocation schemes available in the literature for OFDM-based CR, we propose power allocation schemes that are augmented with spectral shaping. Active interference cancellation (AIC) is an effective spectral shaping technique for OFDM-based systems. Therefore, in particular, we propose AIC-based optimal and suboptimal power allocation schemes that aim to maximize the downlink transmission capacity of an OFDM-based CR system operating opportunistically within the licensed primary users (PUs) radio spectrum in an overlay approach. Since the CR transmitter may not have the perfect knowledge about the instantaneous channel quality between itself and the active PUs, the interference constraints imposed by each of the PUs are met in a statistical sense. We also study an optimal power allocation scheme that is augmented with raised cosine (RC) windowing-based spectral shaping. For a given power budget at the CR transmitter and the prescribed statistical interference constraints by the PUs, we demonstrate that although the on-the-run computational complexity of the proposed AIC-based optimal power allocation ∗ Corresponding

author Email addresses: [email protected] (Manoranjan Rai Bharti), [email protected] (Debashis Ghosh)

Preprint submitted to Physical Communication

June 17, 2017

scheme is relatively higher, it may yield better transmission rate for the CR user compared to the RC windowing-based power allocation scheme. Further, the AIC-based suboptimal scheme has the least on-the-run computational complexity, and still may deliver performance that is comparable to that of the RC windowing-based power allocation scheme. The presented simulation results also show that both the AIC-based as well as the RC windowing-based power allocation schemes lead to significantly higher transmission rates for the CR user compared to the conventional (without any spectral shaping) optimal power allocation scheme. Keywords: Cognitive radio, OFDM, Power allocation, Spectral shaping, Active interference cancellation, Raised cosine windowing

1. Introduction Orthogonal frequency division multiplexing (OFDM)-based cognitive radio (CR) has emerged as a promising technology to enhance the usage efficiency of the allocated radio spectrum, which otherwise has been reported as under5

utilized or unused at a given time in a given region [1], [2]. However, when the OFDM-based CR user opportunistically operates within the radio spectrum licensed to primary users (PUs) in an overlay approach, it may generate high out-of-band radiated interference (OBRI) to PUs coexisting in the adjacent bands [3]. This OBRI is caused by the side lobes of the OFDM signal and

10

should be kept below the interference thresholds prescribed by each of the PUs. Active interference cancellation (AIC) is an effective spectral shaping technique [4–9], employed at the OFDM-based CR transmitter, to minimize such OBRI inflicted on the PUs. In AIC technique, few of the subcarriers (called cancellation subcarriers) are reserved for OBRI reduction task and are modulated with

15

appropriate linear combinations of the data symbols transmitted on data subcarriers. However, this operation is transparent to the CR receiver which simply discards these cancellation subcarriers prior to decoding the data symbols. Another possible countermeasure to reduce the OBRI to the PUs is windowing

2

the CR transmit OFDM signal with an appropriate time domain window func20

tion [3], [10–12]. The raised cosine (RC) is a commonly used window function that has desirable characteristics, and is defined with a roll-off factor parameter which may be discretized. However, to achieve spectral shaping via RC windowing technique, few modifications are required at the CR transmitter as well as the receiver [12]. Further, since the RC windowing is applied at the cost of

25

increased OFDM symbol duration, there is a loss of bandwidth (and hence the throughput) in the CR system by a certain amount. On the other hand, power loading is an important issue for OFDM-based CR systems which helps in improving the spectral efficiency and the transmission rate of the CR user. In literature, considerable works have been done on power

30

allocation for maximizing the transmission capacity of OFDM-based CR systems [13–22]. More specifically, power loading algorithms have been proposed in [13] for an OFDM-based CR system which aim to maximize the downlink transmission capacity of a CR user subject to the constraint that the total interference induced in different PU receivers remains below a specified threshold. However,

35

the authors did not consider the constraint on total power budget available at the CR transmitter. In [14], a low complexity suboptimal power loading algorithm has been proposed for an OFDM-based CR system that uses both nonactive and active PU bands. The proposed algorithm is shown to provide significant performance improvement over the systems that use only nonactive PU bands.

40

A computationally efficient fast algorithm has been proposed in [15] to obtain an optimal power allocation in an OFDM-based CR system. The objective is to maximize the downlink data transmission rate of the CR user under the total power constraint as well as the individual interference constraints imposed by each of the PU receivers. However, in this work, the channel fading gains of

45

different links are assumed to be perfectly known at the CR transmitter. In [16], the authors have developed optimal and suboptimal power allocation schemes by taking into account the total power constraint as well as different probabilistic interference constraints prescribed by different PU receivers, while maximizing the downlink transmission rate of a single user OFDM-based CR system. A joint 3

50

rate and power allocation algorithm has been proposed in [17] that maximizes the OFDM-based CR system transmission capacity and minimizes its transmit power, while ensuring a target bit error rate (BER) per subcarrier, a total transmit power threshold for the CR user, and guaranteeing that the co-channel and adjacent channel interferences introduced to existing PUs remain below certain

55

thresholds with predefined probabilities. The problem of subcarrier and power allocation for orthogonal frequency division multiple access (OFDMA)-based CR systems has been studied in [18] where the authors have proposed subcarrier and power allocation schemes for a joint overlay-underlay spectrum access mechanism. For a given transmission power budget, the total transmission rate

60

of all the CR users is maximized while the interference induced in each of the PU receivers is kept below the corresponding prescribed threshold values with a specified probability. In [19], the problem of sum capacity maximization for multiple-input multiple-output (MIMO)-OFDMA based CR networks has been considered where the authors have proposed subcarrier and power allocation

65

algorithms that aim to maximize the downlink sum capacity of the CR network under the total interference constraint from PUs. With the objective of maximizing the transmission rate of an OFDM-based CR user under the transmission power constraints at the CR transmitter and the aggregate interference constraint of the PU system, a four-step suboptimal power loading algorithm

70

has been proposed in [20] for spectrum sharing with PUs in underlay as well as interweave approaches. In [21], the authors have studied quality of experience (QoE)-oriented subcarrier and power allocation technique for the downlink of a multiuser OFDM-based CR network. The QoE for a CR user is assessed through mean opinion score (MOS) metric which is related to the transmission

75

rate of that particular CR user. The objective is to maximize the overall QoE of the CR system under the total transmit power and interference constraints. Firstly, a heuristic user-oriented subcarrier assignment is developed. Then, for a given subcarrier assignment, a fast power allocation algorithm is proposed to maximize the overall QoE of the CR network. In [22], the authors have studied

80

the ergodic sum capacity of all CR users in an OFDMA-based CR network. Er4

godic sum capacity of fading channels is used to characterize the maximum sum data rate of all CR users averaged over all channel fading states and is an useful performance measure particularly when the available resource is of time-varying nature. By considering constraints on the long-term interference thresholds of 85

PUs, the long-term total transmit power of all CR users, and the statistical proportional fairness of all CR users, the authors have proposed a subcarrier and power allocation algorithm that can achieve higher ergodic sum capacity for CR users compared to the resource allocation algorithm in traditional OFDMA systems.

90

Despite the fact that both the AIC-based spectral shaping as well as the power loading are performed at the CR transmitter, we note that in the literature these two issues have been mostly studied independently so far. In our earlier work [23], we have shown that augmenting power allocation with AIC mechanism for spectrum overlay OFDM CR systems may lead to even better

95

utilization of the available radio spectrum. However, in [23] the instantaneous values of all the channel fading gains were assumed to be perfectly known at the CR transmitter and the optimization problem was solved without considering the total power constraint at the CR transmitter. In the present work, we propose an optimal power allocation scheme that is augmented with AIC-based

100

spectral shaping with the aim to maximize the downlink data transmission rate of the OFDM CR user while satisfying the total available transmit power budget constraint at the CR transmitter, as well as the individual statistical interference constraints prescribed by each of the active PUs. As in practice it is difficult for the CR transmitter to know precisely the instantaneous channel fading gains

105

between itself and the PU receivers, we consider that only the fading statistics of the corresponding channel gains are known at the CR transmitter; and hence, the interference constraints imposed by different PU receivers are met in a statistical sense. Since the computational complexity of the AIC-based optimal power allocation scheme may be quite high for some systems, we also

110

propose a low complexity AIC-based suboptimal power allocation scheme. As aforementioned, the spectral shaping for OFDM-based CR transmissions can, 5

alternatively, be achieved using RC windowing technique. Therefore, we also study an RC windowing-based optimal power allocation scheme and investigate its performance. We establish a reasonable value for the roll-off factor parameter 115

that may be used in RC windowing-based scheme to have its fair performance comparison with the AIC-based schemes. The performance of both the AICbased as well as the RC windowing-based schemes is also compared with the conventional (without any spectral shaping) optimal power allocation scheme in terms of the achievable transmission rates by the CR user as well as on the

120

basis of the computational complexities involved. The major contributions of this paper may therefore be summarized as • We propose an AIC-based optimal power allocation scheme that maximizes the downlink transmission capacity of the CR user in an OFDMbased CR system under the constraint on total transmission power at the

125

CR transmitter and different statistical interference constraints prescribed by different PU receivers. • We also propose an AIC-based suboptimal power allocation scheme whose computational complexity is significantly lower compared to the AICbased optimal scheme, and investigate its performance. To reduce com-

130

putational complexity, in suboptimal scheme we consider each of the constraints separately (instead of considering all of them simultaneously) and find power allocation suboptimally for each CR data subcarrier. It is worthy to mention that similar strategy was used in [16]; however, in contrast to our proposed suboptimal scheme, the work in [16] has ignored the

135

channel-to-noise-plus-interference ratio of a CR data subcarrier while allocating power to it. Moreover, although the work in [16] ensures that all the constraints are satisfied, it does not guarantee that at least one of the constraints is met strictly which is necessary to maximize the transmission capacity of the CR user.

140

• We study an RC windowing-based optimal power allocation scheme in the context of OFDM-based CR systems and investigate its performance. We 6

establish a reasonable value for the roll-off factor parameter that may be used in RC windowing-based scheme to have its fair performance comparison with the AIC-based schemes. It is pertinent to mention that 145

though the RC windowing-based power allocation scheme is a straightforward extension of the conventional optimal power loading scheme, it has not received much attention so far (in the context of OFDM-based CR systems). • We compare the performance of both the AIC-based as well as the RC

150

windowing-based schemes with the conventional (without any spectral shaping) optimal power allocation scheme in terms of the achievable transmission rates by the CR user as well as on the basis of the computational complexities involved. Through simulation results, we demonstrate that compared to RC windowing-

155

based power allocation scheme, the proposed AIC-based optimal scheme has lower on-the-run computational complexity and may yield better transmission rate for the CR user. Further, the AIC-based suboptimal scheme has the least on-the-run computational complexity, and still may deliver performance that is comparable to that of the RC windowing-based power allocation scheme. In

160

addition, the presented simulation results also show that both the AIC-based as well as the RC windowing-based power allocation schemes lead to significantly higher transmission rates for the CR user compared to the conventional (without any spectral shaping) optimal power allocation scheme. The rest of this paper is organized as follows. We discuss our system model

165

in Section 2 while the AIC-based interference model is described in Section 3. The problem formulation and the proposed AIC-based optimal power allocation scheme is presented in Section 4. In Section 5, we propose a low complexity AICbased suboptimal power allocation scheme. The RC windowing-based optimal power allocation scheme is studied in Section 6. Simulation results demonstrat-

170

ing the effectiveness of our proposed schemes are presented in Section 7. Finally, Section 8 concludes this paper. 7

2. System Model We consider the downlink of an OFDM-based CR system that operates opportunistically within the licensed PUs radio spectrum in an overlay approach. 175

We assume that the entire PUs spectrum is divided into a total of N subcarriers each of bandwidth ΔB . We consider that, at a given time in a geographical region, there are only L number of active PUs operating in different bands B1 , B2 , ..., BL , respectively, within the PUs radio spectrum. We assume that the frequency bands B1 , B2 , ..., BL , occupied by the PUs, are covered by

180

(1)

(2)

(L)

NP , NP , ..., NP

number of contiguous subcarriers, respectively. We further

assume that through its perfect spectrum sensing mechanism, the CR system can identify those subcarriers and their associated indices which have been occupied by each of the PUs. The idle frequency bands on either side of each of these active PU bands may be used by the CR system for its possible data 185

transmission. Specifically, the subcarriers in the CR system are allocated as folL (l) (l) lows: l=1 (NP + NC ) number of subcarriers, out of the total N subcarriers,

are used as cancellation subcarriers for OBRI reduction via AIC-based spectral L (l) (l) shaping and the remaining ND = N − l=1 (NP + NC ) number of subcarriers (l)

are used for data transmission by the CR transmitter. NP is the number of

190

cancellation subcarriers that are contiguous and cover the lth PU band while (l)

NC /2 is the number of additional cancellation subcarriers that lie on either side and adjacent to the lth PU band. Defining N = {1, 2, ..., N } as the index set of all the subcarriers within the radio spectrum, Nl as the index set of the

subcarriers covering the lth PU band, C as the set of indices of the cancella195

tion subcarriers and D as that of the data subcarriers, we have D = N \ C, L (l) (l) (l) |D| = ND , |C| = l=1 (NP + NC ) and |Nl | = NP , ∀l = 1, 2, ..., L. Further, let

Bl,m represents the frequency subband or the subchannel corresponding to the mth subcarrier belonging to the lth PU band, such that Bl = ∪Bl,m , ∀m ∈ Nl

and ∀l = 1, 2, ..., L. 200

For the downlink scenario under consideration, we denote the channel gain between the CR transmitter and the CR receiver in the ith data subcarrier by

8

(i)

(i)

hss , and assume that the values of hss , ∀i ∈ D are perfectly known at the CR transmitter through a feedback mechanism. The channel gain between the CR transmitter and the lth PU receiver in the mth CR subcarrier (m ∈ Nl ) 205

(l,m)

is denoted by hsp

. Since in practice it is difficult for the CR transmitter to (l,m)

perfectly know the channel gain hsp

, we assume that only the fading statistics (l,m)

(e.g., the distribution type and the mean value) of the channel gains hsp

, ∀m ∈

Nl and ∀l = 1, 2, ..., L, are known at the CR transmitter. We note that there are several mechanisms suggested in the literature through which a CR transmitter 210

(l,m)

may acquire fading statistics of these channel gains hsp

, ∀m ∈ Nl and ∀l =

1, 2, ..., L. For example, in [24], the authors argue that the CR transmitter can estimate these channel fading statistics from the pilot signals transmitted by the PU receivers. Alternatively, channel fading statistics may be obtained from (l,m)

the information about hsp 215

which may be periodically measured by a band

manager (that mediates between the PU system and the CR system) and sent to the CR transmitter via a common control channel [25–28]. Information about (l,m)

hsp

may also be estimated by listening to a beacon signal, from which the

required statistics may be obtained at the CR transmitter [27, 28]. Another example is the collaboration and exchange between the PU system and the 220

(l,m)

CR system, where information about hsp

may be directly fed back from the

active PU receiver to the CR transmitter, as proposed in [29]. In particular, an algorithm is provided in [29] that may be used by the CR system to acquire the (l,m)

estimate, and hence the statistics of hsp

. Further, we consider the non-line-of-

sight propagation conditions, so that the channel fading gains of different links 225

can be modeled as Rayleigh distributed. It is important to note that as the CR transmitter has the knowledge of the channel fading statistics instead of the (l,m)

instantaneous values for the channel gains hsp

, ∀m ∈ Nl and ∀l = 1, 2, ..., L,

the interference constraints imposed by different PU receivers can be guaranteed in a statistical sense, i.e., we will propose power allocation schemes which ensure 230

that the expectation of the interference induced in each of the PU receivers remains below the corresponding prescribed threshold value.

9

3. AIC-based Interference Model In this section, we formulate the amount of interference introduced in each of the PU receivers after spectral shaping the OFDM-based CR transmissions. As discussed before, we make use of AIC technique for spectral shaping purpose; which helps in reducing OBRI, and thus provides more protection to PUs against interference. To implement AIC technique, we consider that the OFDM-based CR transmitter transmits data symbols using ND subcarriers (henceforth called as data subcarriers) belonging to the set D and some linear combination of data L (l) (l) symbols using l=1 (NP + NC ) subcarriers (henceforth called as cancellation

subcarriers) belonging to the set C. Defining d as the ND × 1 vector of data L (l) (l) symbols modulating the data subcarriers, c as the l=1 (NP + NC ) × 1 vector of cancellation coefficients modulating the cancellation subcarriers, and x as the

N × 1 vector representing the frame of uncorrelated data symbols along with cancellation coefficients to be transmitted in one OFDM symbol and modulating all the N subcarriers, we can write √ T x = [x1 x2 · · · xN ] = αU Pd + Vc,

(1)

where α is a scaling factor used to regulate the available transmit power between the data and the cancellation subcarriers, and is an user selectable parameter 235

with 0 < α ≤ 1. The power values Pi , ∀i ∈ D, contained in the ND × ND

diagonal matrix P, are scaled with parameter α2 at the CR transmitter for transmitting data symbols on data subcarriers. IN is an N × N identity matrix,

240

U is an N × ND matrix that contains ND columns of IN corresponding to the L (l) (l) set of data subcarriers D, and V is an N × l=1 (NP + NC ) matrix containing the remaining columns of IN corresponding to the set of cancellation subcarriers

C. It should be noted that there will be an SNR loss of 10 log10 α2 dB at the √ CR receiver with respect to the case wherein x = U Pd (in which α = 1 and c = 0, i.e., where null subcarriers are used instead of active cancellation). Thus, the choice of scaling factor α involves a trade-off between the OBRI reduction 245

at the CR transmitter and the bit-error-rate at the CR receiver.

10

Following [5], the spectrum corresponding to the nth (n ∈ N ) modulated subcarrier can be expressed as Xn (f ) = xn Ts sinc ((f − nΔB ) Ts ) , where sinc (z) 

sin(πz) πz ,

(2)

f is the frequency variable and Ts represents the OFDM

symbol duration. Using (2) and defining ψn (f ) = sinc ((f − nΔB ) Ts ), the total CR transmitted spectrum can be written as X (f ) =

N 

Xn (f ) = Ts

n=1

N 

xn ψn (f ) = Ts xT ψ (f ) ,

(3)

n=1 T

where ψ (f )  [ψ1 (f ) ψ2 (f ) ... ψN (f )] . From (1) and (3), we observe that in AIC technique the problem amounts to finding the appropriate cancellation coefficients c so that the resulting spectrum X (f ) in the frequency range B1 ∪ B2 ∪ · · · ∪ BL occupied by the PUs is minimum. We address this problem by minimizing the interference power radiated over the frequency bands occupied by the PUs with a constraint on the total transmit power at the CR transmitter. Following [7], the cancellation coefficients c can be generated as a linear combination of data symbols, i.e., √ c = W Pd, where W is the

L

(l) l=1 (NP

(4)

(l)

+ NC ) × ND weight matrix which needs to be

optimized. Substituting (4) in (1), we get x = (αU + VW)

√

√ Pd = A Pd,

(5)

where A  αU + VW. The power spectral density (PSD) Sx (f ) of the transmitted signal can be expressed as [30]    1  2 E |X (f )| = Ts ψ H (f ) E xxH ψ (f ) . Sx (f ) = Ts

(6)

Using (5) and assuming that the elements of the data vector d are independent   and identically distributed with zero-mean and variance given by E ddH = IND , the PSD of the CR transmitted signal can finally be written as   Sx (f ) = Ts × tr PAH Ψ (f ) A , 11

(7)

where Ψ (f )  ψ (f ) ψ H (f ) is the Hermitian matrix. ∞ By defining ΨT = −∞ Ψ (f ) df , the total power transmitted with AIC at

the CR transmitter may be expressed as ∞   Sx (f ) df = Ts × tr PAH ΨT A .

(8)

−∞

On the other hand, the interference power radiated in the frequency subband Bl,m (corresponding to the mth subcarrier belonging to the lth PU band or more concisely m ∈ Nl ) may be obtained as   Sx (f ) df = Ts × tr PAH ΨBl,m A ,

where ΨBl,m 



(9)

Bl,m

Bl,m

Ψ (f ) df . Considering all the subcarriers covering the lth

PU band, the interference power radiated in the band Bl occupied by the lth PU may be written as

Sx (f ) df



=

Bl

m∈Nl



=

m∈Nl 250



Sx (f ) df Bl,m



  Ts × tr PAH ΨBl,m A .

(10)

Thus, the total interference power radiated in the frequency range B1 ∪ B2 ∪ · · · ∪ BL occupied by the PUs can be expressed as L  l=1

where ΨB 

Sx (f ) df Bl

=

m∈Nl

L  

l=1 m∈Nl

=

L  l=1

  Ts × tr PAH ΨBl,m A

  Ts × tr PAH ΨB A ,

(11)

ΨBl,m for notational convenience.

We note that the interference power, given in (11), is a function of the weight matrix W. We aim to minimize this OBRI subject to a constraint on the total power transmitted in (8), i.e.,   min Ts × tr PAH (W) ΨB A (W) , W

subject to

  Ts × tr PAH (W) ΨT A (W) ≤ PT , 12

(12)

(13)

where PT is the available transmit power budget at the CR transmitter. Notice that the equations (12) and (13) define the least squares problem with a quadratic constraint, which can be efficiently solved by using the method of Lagrange multipliers and resorting to the generalized singular value decomposition (GSVD) tools [31]. After some mathematical manipulations, the problem in hand can be written as        min α2 Ts tr PUH ΨB U +Ts tr PWH BH BW +2αTs tr  PWH BH Q , W

(14)

subject to        α2 Ts tr PUH ΨT U +Ts tr PWH TH TW +2αTs tr  PWH TH S ≤ PT , (15)

where we have introduced the following matrices 1/2

Q = B−1 VH ΨB U,

(16)

1/2

S = T−1 VH ΨT U,

(17)

B = EH B ΛB EB , T = EH T ΛT ET ,

with EB , ET as the unitary matrices and ΛB , ΛT as the diagonal matrices given by the eigen-decompositions of the Hermitian matrices VH ΨB V and VH ΨT V as follows V H ΨB V = EH B ΛB EB ,

V H ΨT V = E H T Λ T ET .

(18)

With power equally distributed among data subcarriers so that the matrix P is invertible, and by equating to zero the gradient of the Lagrangian associated with the problem defined in (14) and (15) with respect to W, we get   BH BW + αBH Q + λ TH TW + αTH S = 0,

(19)

where λ is the Lagrange multiplier.

Now consider the GSVD of the matrices B and T, given by [31, Ch. 12] B = YDB X−1 ,

T = ZDT X−1 ,

(20)

where Y, Z are the unitary matrices, X is invertible, and DB , DT are the diagonal and positive semidefinite matrices with D2B + D2T = I. Using (19) and 13

(20), the solution to the optimization problem formulated in (12) and (13) may be written as −1    DB YH Q + λDT ZH S . W = −αX D2B + λD2T

255

(21)

The Lagrange multiplier λ can be determined by substituting (21) in (15) with   PT IND , and solving the resulting equation at the boundary. Even P = N D

though such an equation is nonlinear, its structure can be exploited to efficiently find a numerical solution, as shown in [31].

Now, the interference power radiated by the downlink CR transmissions

260

over the frequency subband Bl,m , m ∈ Nl , may be obtained using (9) as Ts ×   tr PAH (W ) ΨBl,m A (W ) . Considering the channel gain between the CR

transmitter and the lth PU receiver, the interference induced in the mth subcar     (l,m) 2 rier belonging to the lth PU receiver will be hsp  Ts ×tr PAH (W ) ΨBl,m A (W ) , and the total interference induced in the lth PU receiver after AIC-based spectral       (l,m) 2 shaping would be m∈Nl hsp  Ts × tr PAH (W ) ΨBl,m A (W ) .

265

4. Problem Formulation and AIC-based Optimal Power Allocation (i)

For the given values of instantaneous channel fading gains hss , ∀i ∈ D, (l,m)

given channel fading statistics for hsp

, ∀m ∈ Nl and ∀l = 1, 2, ..., L, the (l)

given interference thresholds prescribed by each of the PU receivers Ith , ∀l = 1, 2, ..., L, and the available transmit power budget PT at the CR transmitter, our design objective is to determine the optimal power values Pi , ∀i ∈ D, such that the downlink data transmission rate R of the CR user is maximized subject to the statistical interference constraints imposed by each of the PUs as well as the total power constraint at the CR transmitter. Mathematically, the problem may be formulated as follows ⎛

  ⎞  (i) 2 α Pi hss  ⎟  ⎜ ΔB log2 ⎝1 + R = max ⎠, Pi σ 2 + Ji i∈D

14

2

(22)

subject to   2     (l) (l,m)  H   E ≤ Ith , ∀l = 1, 2, ..., L, hsp  Ts × tr PA (W ) ΨBl,m A (W ) m∈Nl





(23)

Ts × tr PAH (W ) ΨT A (W ) ≤ PT ,

(24)

Pi ≥ 0, ∀i ∈ D,

(25)

where R denotes the maximum achievable data transmission rate of the CR user, σ 2 represents the additive white Gaussian noise (AWGN) variance, and Ji denotes the sum total of the interference introduced in the ith CR data subcarrier by all the PU transmitters. In a practical system with large number of active 270

PUs, the interference Ji may be approximated as AWGN from the central limit theorem. As aforementioned, the factor α is used to scale down the power values Pi , ∀i ∈ D, so that α2 Pi represents the amount of power allocated to the ith CR data subcarrier.

   (l,m)  Considering that hsp  is Rayleigh distributed with a known parameter    (l,m) 2 ηl,m , hsp  will follow exponential distribution with its mean value given by

2 . Thus, the interference constraints in (23) may be expressed as 2ηl,m



m∈Nl

  (l) 2 2ηl,m × Ts × tr PAH (W ) ΨBl,m A (W ) ≤ Ith , ∀l = 1, 2, ..., L. (26)

By introducing Kl 



m∈Nl

  2 2ηl,m × Ts × AH (W ) ΨBl,m A (W ) as an

ND × ND matrix, the interference constraints expressed in (26) can further be manipulated as tr {PKl } =



i∈D

(l)

Pi Ki,l ≤ Ith , ∀l = 1, 2, ..., L,

(27)

where Ki,l is an element on the main diagonal of the matrix Kl corresponding to   its ith row and ith column. Similarly, by defining F = Ts × AH (W ) ΨT A (W ) as an ND × ND matrix, the total power constraint in (24) may be written as tr {PF} =



i∈D

15

Pi Fi ≤ PT ,

(28)

275

where Fi is an element on the main diagonal of the matrix F corresponding to its ith row and ith column. By using (27) and (28), the problem defined in (22), (23), (24) and (25) may be expressed in a standard convex optimization form, which can be solved by using the Karush-Kuhn-Tucker (KKT) conditions as in [32]. By introducing the Lagrange multipliers (ξi , γl and μ for the inequality constraints given in (25), (27) and (28), respectively), the KKT conditions for the convex problem may be written as follows [32] 

i∈D

(l)

Pi Ki,l ≤ Ith , ∀l = 1, 2, ..., L, 

i∈D

γl





i∈D

−

ΔB ln 2

(30)

− Pi ≤ 0, ∀i ∈ D,

(31)

γl ≥ 0, ∀l = 1, 2, ..., L,

(32)

μ ≥ 0,

(33)

ξi ≥ 0, ∀i ∈ D, 

(34)

(l)

Pi Ki,l − Ith μ



Pi Fi ≤ PT ,





i∈D

= 0, ∀l = 1, 2, ..., L,

Pi Fi − PT



= 0,

ξi Pi = 0, ∀i ∈ D,

1 σ2 +Ji  (i) 2 α2 hss 

(29)

+ Pi

+

L  l=1

γl Ki,l + μFi − ξi = 0, ∀i ∈ D.

(35)

(36) (37) (38)

From (38), we have ξi = −



ΔB ln 2



1 σ2 +Ji  (i) 2 α2 hss 

+ Pi

+

16

L  l=1

γl Ki,l + μFi , ∀i ∈ D.

(39)

By substituting (39) into (34) and (37), we obtain L 

μFi +

l=1

Pi μFi + Pi

L  l=1

When μFi +

L

l=1

γl Ki,l ≥



ΔB ln 2



ΔB ln 2

γl Ki,l −

γl Ki,l <

 ΔB  ln 2

⎛ ⎝



1

1 σ2 +Ji  (i) 2 α2 hss 

+ Pi

 , ∀i ∈ D,

(40)

 = 0, ∀i ∈ D.

(41)

Pi

σ2 +Ji  (i) 2 α2 hss 

σ 2 +Ji (i) 2 α2 hss

⎞,

+ Pi

then (40) will hold only if Pi > 0

⎠

| |  B 2 i 1 (since Ki,l > 0). In this case, (41) leads to Pi = Δ − σ2  +J . (i) 2 ln 2 μFi + L γ K l i,l l=1 α hss    L 1 B ⎛ ⎞ , then (41) will On the other hand, when μFi + l=1 γl Ki,l ≥ Δ ln 2 ⎝

σ 2 +Ji (i) 2 α2 hss

⎠

| | not hold for Pi > 0. Since Pi < 0 would violate (31), in this case we will have

280

Pi = 0. Thus, the AIC-based power profile (Pi , ∀i ∈ D) that maximizes the downlink data transmission rate of the CR user, for the given set of constraints in (23), (24) and (25), and for the given weight matrix W, may be written as ⎡ ⎤+

2 1 σ + Ji ⎥ ⎢ ΔB − Pi = ⎣   ⎦ , ∀i ∈ D,  ln 2 μFi + L  (i) 2 γ K l=1 l i,l α2 hss 

(42)

where the Lagrange multipliers μ and γl , ∀l = 1, 2, ..., L, can be determined

using the Newton’s method [32]. The interior point method can also be used to

285

solve the problem formulated in (22), (23), (24) and (25) with a computational  3 [32]. complexity of the order O ND

It should be noted that, in the AIC-based power allocation scheme described

above, the weight matrix W is obtained first, assuming a uniform power profile   PT IND (as discussed in Section 3); and then, with this computation of P= N D W in place, the AIC-based power profile in (42) is sought. However, since the

optimum W can be easily obtained for a given P (not necessarily uniform), we 290

can now iterate in the optimization of W and P (by taking the other parameter as fixed) so as to obtain the AIC-based optimal power profile Pi∗ , ∀i ∈ D. 17

The AIC-based optimal power allocation procedure is outlined in detail in Algorithm 1. Algorithm 1 AIC-based optimal power allocation algorithm   P 1: Initialization: P = NT IND , k = 1. D 2:

3:

Substitute (21) in (15) and solve at the boundary to find λ corresponding   PT IND . Let λk = λ. to the uniform power profile P = N D do {

(a) Compute weight matrix W using (21) with λ = λk ; (b) Obtain P (i.e., Pi∗ , ∀i ∈ D) using (42) corresponding to the weight matrix W found in Step 3(a); (c) k ⇐ k + 1; (d) Substitute (21) in (15) and solve at the boundary to find λ corresponding to the power profile P found in Step 3(b). Let λk = λ; } while (|λk − λk−1 | > ); where > 0 is a given small constant. 4:

Compute the vector of cancellation coefficients c using (4) corresponding to the optimum weight matrix W found in Step 3(a) and the AIC-based optimal power profile P (i.e., Pi∗ , ∀i ∈ D) obtained in Step 3(b). In Algorithm 1 (AIC-based optimal power allocation algorithm), we note

295

that the Steps 1 and 2 may be computed offline, and hence their computations do not impact the online complexity of the algorithm. Thus, the on-the-run computational complexity of the algorithm is mainly due to the execution of Steps 3 and 4. In Step 3, we iterate in the optimization of W and P matrices to obtain AIC-based optimal power profile. Specifically, Step 3(a) involves

300

computation of weight matrix W for a given λ, and has complexity of the order 2   (l) (l) L , where κ1 is the number of iterations in N O κ1 N D × + N P C l=1 the do-while loop of the algorithm. The power profile P in Step 3(b), for a

given weight matrix W, may be obtained using the interior point method involv  3 [32]. The computational ing computational complexity of the order O κ1 ND 18

305

complexity of Step 3(c) is O (κ1 ). In Step 3(d), the value of λ is computed. As discussed before (in Section 3), for a given P, the value of λ can be determined by substituting (21) in (15), and solving the resulting equation at the boundary. Since the function over which to search for λ is a monotonically decreasing function for λ > 0 [31], the value of λ can be found by applying any of the

310

standard root-finding techniques, such as the bisection method or the Newton’s method [31]. The computational complexity of Step 3(d) in finding λ numer 2   (l) (l) L + N + N ically using the bisection method is O κ1 κ2 ND × P C l=1     (l) (l) L 2 O κ1 κ2 ND , where κ2 is the number of iterations in× l=1 NP + NC

volved in the bisection method. Step 4 of the algorithm is related to AIC-based 315

spectral shaping. It should be noted that, once the matrices W and P have been optimized, the online complexity of the AIC-based spectral shaping mainly comes from the computation of cancellation coefficients for all the PU bands  L  (l) (l) using (4) [7], which requires only 3 × ND × l=1 NP + NC arithmetic op-

320

erations. Thus, the on-the-run computational complexity involved in perform  L  (l) (l) ing AIC-based spectral shaping is of the order O ND × l=1 NP + NC

[7]. Considering all the steps together, the overall on-the-run computational

complexity of the AIC-based optimal power allocation scheme, detailed in Algo2     L  (l) (l) (l) (l) L 2 N N + + N κ N × + N rithm 1, is O κ1 κ2 ND × +O κ 1 2 D P C P C l=1 l=1   3 O κ1 N D . 325

5. AIC-based Suboptimal Power Allocation The AIC-based optimal power profile, derived in Section 4, is iterative in nature, and depends on the Lagrange multipliers λ, μ and γl , ∀l = 1, 2, ..., L. Solving for these Lagrange multipliers is computationally intensive, and therefore, the computational complexity of the AIC-based optimal power allocation

330

scheme may be quite high for some systems. We note that such a complexity of the optimal scheme is a result of the fact that all the L + 1 constraints (L interference constraints specified in (23) and the total power constraint given in (24)) are to be satisfied simultaneously while maximizing the transmission data 19

rate of the CR user. Moreover, the process of iterating in the optimization of 335

W and P matrices (for obtaining the AIC-based optimal power profile) further adds to the computational complexity. In this section, we propose a low complexity AIC-based suboptimal power allocation scheme which is based on the power profile derived in (42) for a fixed weight matrix W (i.e., we do not iterate in the optimization of W and P matrices). In the proposed suboptimal scheme,

340

instead of considering all the L + 1 constraints simultaneously, we consider each of them separately and find the power allocation suboptimally corresponding to each of the CR data subcarriers. For example, by considering the lth interference constraint, we show that the power allocation corresponding to the ith data subcarrier is a function of the factor Ki,l (which depends on the spectral

345

distance of the ith data subcarrier from the mth subcarrier belonging to the lth (l,m)

PU band as well as on the statistics of the channel gains hsp , ∀m ∈ Nl ) and 2 |h(i) | the channel-to-noise-plus-interference ratio σ2ss +Ji associated with that particular data subcarrier. This is in contrast to the suboptimal algorithm proposed in [16] which does not consider the channel-to-noise-plus-interference ratio on 350

a CR data subcarrier while allocating power to it. Furthermore, although the suboptimal power loading algorithm given in [16] ensures that all the constraints are satisfied, it does not guarantee that at least one of the constraints is met strictly to maximize the transmission rate of the CR user. Our proposed AIC-based suboptimal power loading scheme is as follows.

355

(l)

Let Pi

denotes the power allocation corresponding to the ith data subcarrier (L+1)

due to the lth interference constraint, and Pi

denotes the power allocation

corresponding to the ith data subcarrier due to the total power constraint. Using this notation, the power allocated in the ith data subcarrier due to the lth (l)

interference constraint would be α2 Pi 360

(L+1)

would be α2 Pi

and due to the total power constraint

. Thus, corresponding to a given data subcarrier i ∈ D, we (l)

(L+1)

will have L + 1 power allocations (Pi , l = 1, 2, ..., L, and Pi

) and we will

select only one of these L + 1 power allocations such that all the constraints are satisfied while the transmission rate of the CR user is also maximized. (l)

Firstly, we find the power allocation Pi , l = 1, 2, ..., L, due to each of 20

the L interference constraints. Considering the lth interference constraint, the problem may be expressed as

max (l)

Pi



i∈D

subject to

⎛

 2 ⎞ 2 (l)  (i)  P α h ss  i ⎜ ⎟ ΔB log2 ⎝1 + ⎠, σ 2 + Ji 

i∈D

(l)

(44)

≥ 0, ∀i ∈ D.

(45)

Pi Ki,l ≤ Ith ,

(l)

Pi

(l)

(43)

The solution to this problem can be easily obtained using the KKT conditions and may be written as ∗,(l)

Pi

⎫ ⎧ ⎪ ⎪ ⎨ Δ 1 2 σ + Ji ⎬ B , ∀i ∈ D. = max 0, −   ⎪ ln 2 γ˜l Ki,l  (i) 2 ⎪ ⎩ α2 hss  ⎭

(46)

where γ˜l is the nonnegative Lagrange multiplier associated with the lth interference constraint in (44). The value of γ˜l can be computed by substituting (46) in (44) and solving at the boundary. It should be noted that the computational ∗,(l)

complexity involved in finding the power allocation Pi

, ∀i ∈ D due to the lth

interference constraint using (46) is not very high. However, since we need to find such power allocation for each of the L interference constraints, the complexity may become of concern for large values of L. To reduce this complexity, we propose power allocation corresponding to the ith data subcarrier due to lth interference constraint as follows. From (46) we observe that the power allocation is inversely proportional to the factor Ki,l . In addition, from (46) we also 2 α2 |h(i) ss | for a data subcarrier i ∈ D, more will observe that higher the value of σ2 +J i be the power allocation. This suggests that the power allocation corresponding to the ith data subcarrier due to the lth interference constraint may be written as (l)

Pi

(l)

=

P Ki,l

⎛

 2 ⎞ 2  (i)  α h ss  ⎜ ⎟ ⎝ 2 ⎠ ∀i ∈ D, ∀l = 1, 2, ..., L, σ + Ji 21

(47)

where P (l) can be calculated by substituting (47) in (44) and solving at the boundary. This gives (l)

P (l) =



i∈D

I

th 

 ,  (i) 2 α2 hss 

∀l = 1, 2, ..., L.

(48)

σ 2 +Ji

(l)

Now, using (47) and (48), we can determine Pi as ⎤ ⎡ 2 (i) 2 α |hss | ⎥ (l) ⎢ σ 2 +Ji ⎥ Ith ⎢ (l) ⎢ ⎥ , ∀i ∈ D, ∀l = 1, 2, ..., L.

 Pi =   ⎥  (i) 2 2 Ki,l ⎢ α hss  ⎣ ⎦

(49)

σ 2 +Ji

i∈D

(L+1)

Next, we require to find power allocation Pi

due to the total power

constraint. In this case, the problem may be expressed as ⎛  2 ⎞ 2 (L+1)  (i)  α Pi hss  ⎟  ⎜ max ΔB log2 ⎝1 + ⎠, 2 (L+1) σ + Ji Pi

(50)



Fi ≤ PT ,

(51)

≥ 0, ∀i ∈ D.

(52)

i∈D

subject to

(L+1)

Pi

i∈D

(L+1)

Pi

This problem can be easily solved using the KKT conditions and the solution may be written as (L+1)

Pi

⎧ ⎫ ⎪ ⎪ ⎨ Δ 1 2 σ + Ji ⎬ B , ∀i ∈ D. = max 0, −   ⎪ ln 2 μ ˜Fi  (i) 2 ⎪ ⎩ α2 hss  ⎭

(53)

where μ ˜ is the nonnegative Lagrange multiplier associated with the total power 365

constraint in (51). The value of μ ˜ can be determined by substituting (53) in (51) and solving at the boundary. Having calculated (using (49) and (53)) all the L + 1 power allocations corresponding to a data subcarrier, the next step is to choose the minimum of

22

these L + 1 power allocations corresponding to that particular data subcarrier, i.e.,

  (1) (2) (L+1) , ∀i ∈ D. Pimin = min Pi , Pi , ..., Pi

(54)

By selecting the minimum of the L + 1 power allocations corresponding to each data subcarrier, we ensure that all the constraints are satisfied. However, there is a possibility that none of the constraints is met strictly. Therefore, to maximize the data transmission rate of the CR user, we scale the power values obtained from (54) as follows sub,(l) Pi

=

Pimin



sub,(L+1)

Pi

 (l) Ith  , ∀i ∈ D, ∀l = 1, 2, ..., L, min K i,l i∈D Pi

PT , ∀i ∈ D. = Pimin  min F i i∈D Pi

(55)

(56)

Finally, we set the final power allocation corresponding to the ith data subcarrier to the minimum of L + 1 power values obtained in (55) and (56), i.e.,   sub,(1) sub,(2) sub,(L+1) , ∀i ∈ D. (57) , Pi , ..., Pi Pisuboptimal = min Pi

Thus, the proposed AIC-based suboptimal power allocation scheme ensures that

all the constraints are satisfied, as well as at least one of the constraints is met strictly, while the transmission capacity of the CR user is also maximized. 370

The AIC-based suboptimal power allocation procedure is summarized in Algorithm 2. We note that the Steps 1, 2 and 3 of Algorithm 2 may be executed offline, and therefore, the on-the-run computational complexity of this algorithm is mainly due to the computations involved in Steps 4 through 9. Specifically, Steps 4 to 8

375

of the algorithm execute the suboptimal power allocation procedure for a given W, and have complexity of the order O (ND log ND )+O ((L + 1) ND ). Step 9 of the algorithm is related to AIC-based spectral shaping, and has computational   L  (l) (l) . Thus, the overall oncomplexity of the order O ND × l=1 NP + NC

the-run computational complexity of the AIC-based suboptimal power alloca380

tion scheme, summarized in Algorithm 2, is O (ND log ND ) + O ((L + 1) ND ) +   L  (l) (l) . O ND × l=1 NP + NC 23

Algorithm 2 AIC-based suboptimal power allocation algorithm   P 1: Initialization: P = NT IND . D 2:

3: 4:

Substitute (21) in (15) and solve at the boundary to find λ corresponding   PT IN D . to the uniform power profile P = N D Compute weight matrix W using (21). (l)

Find power allocation (Pi , ∀i ∈ D, ∀l = 1, 2, ..., L) due to each of the L interference constraints using (49).

5:

(L+1)

Find power allocation (Pi

, ∀i ∈ D) due to the total power constraint

using (53). 6:

Select the minimum of the L + 1 power allocations for each data subcarrier, i.e., find Pimin , ∀i ∈ D, using (54).

7:

1, 2, ..., L, and 8:

sub,(l)

Scale the power values obtained in Step 6, i.e., find Pi sub,(L+1) , Pi

, ∀i ∈ D, ∀l =

∀i ∈ D, using (55) and (56), respectively.

Obtain the AIC-based suboptimal power profile Pisuboptimal , ∀i ∈ D, using (57).

9:

Compute the vector of cancellation coefficients c using (4) corresponding to the weight matrix W found in Step 3 and the AIC-based suboptimal power profile Pisuboptimal , ∀i ∈ D, obtained in Step 8.

24

6. RC Windowing-based Power Allocation In this section, we present an RC windowing-based power allocation scheme for OFDM-based CR systems. We start with a brief discussion on time domain 385

windowing technique from existing literature, and then move on to developing the RC windowing-based power allocation scheme for OFDM-based CR systems. We note that in literature, power allocation augmented with RC windowingbased spectral shaping for OFDM-based CR systems has not received much attention so far. The CR transmitted OFDM signal may cause high OBRI to PUs coexisting in the adjacent bands. As aforementioned, this OBRI is due to the side lobes of the sinc shaped spectra on each OFDM subcarrier. In fact, the sinc shape of the subcarrier spectra is a result of the sharp transitions among successive OFDM symbols. Such a sinc shaped spectrum may be avoided if we allow smooth transitions between consecutive OFDM symbols (by making the amplitude of each OFDM symbol decay smoothly to zero at the symbol boundaries). This may be achieved by windowing the CR transmit OFDM signal with an appropriate time domain windowing function. One such windowing function that has desirable temporal characteristics and has been widely used in practice is the RC window

390

w(t) defined as [3] ⎧ 1 ⎪ ⎪ 2 + ⎪ ⎪ ⎪ ⎪ ⎨ 1, w (t) = 1 ⎪ ⎪ ⎪ 2 + ⎪ ⎪ ⎪ ⎩ 0,

1 2

1 2

 cos π + cos



πt βTs

π(t−Ts ) βTs





,

,

for 0 ≤ t < βTs for βTs ≤ t < Ts for Ts ≤ t < (1 + β) Ts

(58)

otherwise,

where β is called as the roll-off factor that can take values in the range 0 ≤ β ≤ 1. It is important to mention that, due to RC windowing, the effective duration of each OFDM symbol increases from Ts to (1 + β)Ts [3], [12], which amounts to a bandwidth loss of

β 1+β

in frequency domain [11].

As a result of windowing in time domain, the sharp transitions among suc395

cessive OFDM symbols are smoothed and each OFDM subcarrier has better

25

spectral characteristics with reduced side lobes compared to a sinc shaped spectrum as β increases. Thus, the time domain RC windowing can be considered as an alternative technique for spectral shaping, which can help in reducing OBRI due to OFDM-based CR transmissions. 400

6.1. RC Windowing-based Interference Model In this subsection, we formulate the amount of interference that is received by each of the PU receivers after RC windowing-based spectral shaping of the OFDM-based CR transmissions. Since the windowing technique does not use any cancellation subcarrier, from system model (discussed in Section 2) we will L (l) (l) have α = 1, C = ∅, NC = 0, ∀l = 1, 2, ..., L, and ND = N − l=1 NP , where (l)

is the number of contiguous subcarriers which are covering the lth PU *L band. Further, D = N \ ( l=1 Nl ), |D| = ND , and |C| = 0. Thus, in the case NP

of RC windowing-based spectral shaping, the N × 1 vector g representing the

frame of uncorrelated data symbols, to be transmitted in one OFDM symbol and modulating all the N subcarriers, may be written as √ T g = [g1 g2 · · · gN ] = U Pd,

(59)

and the magnitude spectrum of the nth (n ∈ N ) modulated subcarrier, after windowing with the RC window defined in (58), may be written as

 cos (βπ (f − nΔB ) Ts ) Gn (f ) = gn Ts sinc ((f − nΔB ) Ts ) × . 2 1 − 4β 2 (f − nΔB ) Ts2

(60)

Thus, the total spectrum of the CR transmit windowed signal becomes G (f ) =

N 

Gn (f ) = Ts

n=1

where ϕn (f )  sinc ((f − nΔB ) Ts )×

N 

gn ϕn (f ) = Ts gT ϕ (f ) ,

(61)

n=1



cos(βπ(f −nΔB )Ts ) 1−4β 2 (f −nΔB )2 Ts2



T

and ϕ (f )  [ϕ1 (f ) ϕ2 (f ) ... ϕN (f )] .

Now, similar to (7), the PSD of the CR transmitted windowed signal may be obtained as

  Sg (f ) = Ts × tr PUH Φ (f ) U , 26

(62)

where Φ (f )  ϕ (f ) ϕH (f ). Finally, taking into account the channel gains between the CR transmitter and the lth PU receiver, the amount of interference

405

introduced to the lth PU receiver (by the downlink CR transmissions) after RC       (l,m) 2 windowing-based spectral shaping will be m∈Nl hsp  Ts ×tr PUH ΦBl,m U ,  where ΦBl,m  Bl,m Φ (f ) df . 6.2. RC Windowing-based Power Allocation In using RC windowing technique to achieve spectral shaping, while the duration of each OFDM symbol increases by a factor of (1+β), the transmission rate of the CR user decreases by the same factor [12]. Further, since α = 1, no scaling of power values Pi , ∀i ∈ D is required, and therefore in this case,

Pi will represent the amount of power transmitted on the ith data subcarrier. Mathematically, the RC windowing-based power allocation problem may be formulated as max Pi

ΔB (1 + β)



i∈D

⎛

  ⎞  (i) 2 P h ss  i ⎜ ⎟ log2 ⎝1 + 2 ⎠, σ + Ji

subject to   2      (l) (l,m) E ≤ Ith , ∀l = 1, 2, ..., L, hsp  Ts × tr PUH ΦBl,m U

(63)

(64)

m∈Nl



i∈D

Pi ≤ PT ,

(65)

Pi ≥ 0, ∀i ∈ D.

(66)    (l,m)  As before, by considering the magnitude of each of the channel gains, hsp  , ∀m ∈ Nl and ∀l = 1, 2, ..., L, as Rayleigh distributed with ηl,m as the corresponding pa-

rameter, we can manipulate the interference constraints given in (64) and write them as follows tr {POl } =



i∈D

(l)

Pi Oi,l ≤ Ith , ∀l = 1, 2, ..., L,

(67)

where Oi,l , ∀i ∈ D are the elements on the main diagonal of ND × ND matrix    2 × Ts × UH ΦBl,m U . Ol  m∈Nl 2ηl,m 27

Now, the power allocation problem in hand can easily be converted into the standard convex optimization form and may be solved by invoking the KKT conditions [32]. By following the procedure developed in Section 4, for a given β, the RC windowing-based optimal power profile Pi∗ , ∀i ∈ D, may be obtained as

⎡

⎢ Pi∗ = ⎣



ΔB (1 + β) ln 2

μ ˆ+

1 L

2

ˆl Oi,l l=1 γ

⎤+

σ + Ji ⎥ −   ⎦ ,  (i) 2 hss 

(68)

where μ ˆ and γˆl , ∀l = 1, 2, ..., L, are the deterministic Lagrange multipliers as410

sociated with the inequality constraints in (65) and (67), respectively, which may be found by using the Newton’s method [32]. The problem may also be solved by using the interior point method with a computational complexity of  3 [32]. the order O ND It is shown in [3] and [11] that the sidelobes of the CR transmitted OFDM

415

signal decrease in magnitude as β increases. Thus, on one hand, the OBRI reduction performance improves with increase in β, the maximum transmission rate of the CR user tends to decrease (with increase in β) on the other. Therefore, it becomes important to use the smallest β value possible, in order to avoid any unnecessary loss in the throughput of the CR user. We study this effect of

420

β on the CR user throughput in Section 7.

7. Simulation Results For the simulation results presented in this section, we consider an OFDMbased CR overlay system operating within the PUs radio spectrum that can be divided into N = 128 subcarriers (with indices from #1 to #128) each 425

of bandwidth ΔB = 0.3125 MHz. We consider that there are L = 3 active PUs operating in different bands B1 , B2 , B3 , respectively, within the (1)

PUs radio spectrum, with B1 covering NP B3 covering

(3) NP

(2)

= 5, B2 covering NP

= 9, and

= 14 subcarriers bandwidth. We assume that the indices

of the subcarriers covering the first PU band B1 are from #26 to #30, sec430

ond PU band B2 are from #56 to #64, and third PU band B3 are from 28

(i)

#90 to #103. Following [33], we model the channel fading gains hss ’s, be-

435

tween the CR transmitter and the CR receiver, as complex-valued circularly+  ,    (i) 2 (i) = 0.1 symmetric Gaussian random variables with E hss = 0, E hss  , +     ∗   (i) 2 (i ) (i ) 2 = E hss  / 1 + [|i1 − i2 | ΔB /Δfc ] , where (·)∗ deand E hss1 hss2

notes the complex conjugate operation and Δfc is the coherence bandwidth

of the wireless channel which we take as Δfc = 2.5 MHz [33]. According (i)

and the correlation is given by to this channel model, + hss ’s,are correlated  ∗   2   (i ) (i ) (i) 2 = 1/ 1 + [|i1 − i2 | ΔB /Δfc ] . The magnitude /E hss  E hss1 hss2 (i)

of hss is assumed to follow Rayleigh distribution with an average power gain

440

of −10 dB. Similarly, the magnitudes of the channel fading gains between the CR transmitter and the PU receivers are assumed to be Rayleigh distributed with average power gains in the range −10 dB to −5 dB. The values for Ji are randomly generated using uniform distribution with an average of 10−6 W. The

values of Ts and σ 2 have been taken as 4 μs and 10−8 W, respectively. The 445

(1)

(2)

prescribed values of the interference thresholds Ith and Ith have been considered to be 1 × 10−7 W and 1.8 × 10−7 W, respectively. For each of the schemes under consideration, the performance results have been averaged over 10,000 independent simulation runs. To demonstrate the effectiveness of our proposed AIC-based optimal and

450

suboptimal power allocation schemes, we compare their performance with that of the RC windowing-based power allocation scheme as well as the conventional power loading scheme (which is without any spectral shaping) on the basis of the achievable data transmission rate for the CR user. As discussed in Section 6.2, the maximum achievable data transmission rate of the CR user using

455

RC windowing-based power allocation scheme is a function of the roll-off factor β. Therefore, before proceeding for the performance comparison among different schemes under consideration, we need to establish a reasonable choice of the factor β for the RC windowing-based power allocation scheme. To investigate the dependence of CR user throughput on the roll-off factor

460

β, in case of the RC windowing-based power allocation scheme, we plot the

29

Maximum transmitted data rate (bits/sec)

4

×10 7

3.5 3 2.5 2 P T = 20 mW P T = 15 mW P T = 10 mW P T = 5 mW P T = 1 mW

1.5 1 0.5 0

0

0.1

0.2

β

0.3

0.4

0.5

Figure 1: RC windowing-based power allocation scheme: Maximum transmitted data (3)

rate vs. β for different values of PT ; α2 = 0.97, Ith = 2.8 × 10−7 W.

Maximum transmitted data rate (bits/sec)

3

×10 7

2.5

2

1.5

1 I (3) = 1.5 μW th I (3) = 1 μW th

0.5

I (3) = 0.5 μW th I (3) = 0.1 μW th

0

I (3) = 0.01 μW th

0

0.1

0.2

β

0.3

0.4

0.5

Figure 2: RC windowing-based power allocation scheme: Maximum transmitted data (3)

rate vs. β for different values of Ith ; α2 = 0.97, PT = 10 mW.

maximum transmitted data rate as a function of β in Fig. 1 for different values of available transmit power budget PT . We take α2 = 0.97 and the value for the

30

(3)

interference threshold Ith = 2.8 × 10−7 W. From this figure, we note that for a given PT , there is an optimal value of β which yields the highest transmission 465

rate. When β is increased beyond this optimal value, the loss in bandwidth incurred due to increased OFDM symbol duration outweigh the OBRI reduction benefits, and hence the transmission rate of the CR user decreases. For example, when PT = 10 mW, the highest transmission rate is achieved at β = 0.1. We also note that as PT is increased, the optimal value of β tends to increase.

470

Similarly, the plots in Fig. 2 illustrate the variation of maximum transmitted (3)

data rate against β for different values of the interference threshold Ith . We keep α2 = 0.97 and PT is fixed at 10 mW. From this figure, we observe that the optimal value of β, at which the highest transmission rate is achieved, tends (3)

to decrease with an increase in Ith value. This is expected since by increasing 475

(3)

Ith , we are in fact relaxing the corresponding interference constraint, and as such, the highest transmission rate may be achieved at a relatively lower value of β. From Fig. 2, we also note that although the optimal value of β depends (3)

on Ith , the transmission capacity has a maximum point at or in the vicinity of (3)

β = 0.1 over the range of Ith values considered in the simulations. Thus, from 480

the results presented in Fig. 1 and Fig. 2, it is reasonable to take β = 0.1 for the RC windowing-based power allocation scheme, while we make the performance comparison among different power allocation schemes under consideration. In Fig. 3, we have plotted the maximum achievable data transmission rate of the CR user versus the available transmit power budget (PT ) at the CR

485

transmitter for different schemes. We take α2 = 0.97 and the prescribed value (3)

for the interference threshold Ith = 2.8 × 10−7 W. Without loss of generality, (1)

we consider NC

(2)

= NC

(3)

= NC

= NC . From this figure, we observe that

at low values of the power budget (where the power constraint dominates the interference constraints and the CR system operates in a power limited region), 490

all the schemes under consideration yield approximately the same transmission rates for the CR user. Further, when the CR system is operating in such a power limited scenario, the transmission rate of the CR user increases as the power

31

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

5

×10 7 AIC-based Optimal, Nc=8 AIC-based Suboptimal, Nc=8 AIC-based Optimal, Nc=6 AIC-based Suboptimal, Nc=6 With RC Windowing, β=0.1 AIC-based Optimal, Nc=4 AIC-based Optimal, Nc=2 Without Spectral Shaping

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

0.005

0.01

0.015

0.02

Available transmit power budget at CR transmitter (Watts)

Figure 3: Maximum transmitted data rate of CR user vs. available transmit power (3)

budget (PT ) at CR transmitter for different schemes; α2 = 0.97, Ith = 2.8 × 10−7 W, (1)

(2)

(3)

and NC = NC = NC = NC .

Total power allocated to CR user (Watts)

0.025

AIC-based Optimal, Nc=8 AIC-based Suboptimal, Nc=8 AIC-based Optimal, Nc=6 AIC-based Suboptimal, Nc=6 With RC Windowing, β=0.1 AIC-based Optimal, Nc=4 AIC-based Optimal, Nc=2 Without Spectral Shaping

0.02

0.015

0.01

0.005

0

0

0.005

0.01

0.015

0.02

Available transmit power budget at CR transmitter (Watts)

Figure 4: Total power allocated to CR user vs. available transmit power budget (PT ) (3)

(1)

at CR transmitter for different schemes; α2 = 0.97, Ith = 2.8 × 10−7 W, and NC = (2)

(3)

NC = N C = N C .

32

budget is increased for each scheme. This is due to the fact that in the power limited region, the total power allocated to the CR user also increases with an 495

increase in the power budget (cf. Fig. 4). From Fig. 3, we also observe that as the power budget is increased beyond a certain value for a given scheme, the interference constraints become more dominant and the achievable data transmission rate of the CR user tends to saturate for that particular scheme, i.e., does not increase with the power budget. This is due to the reason that,

500

in this region, the CR system operates in an interference limited scenario (for that particular scheme). For a given power budget in the interference limited region, Fig. 3 shows that the proposed AIC-based (optimal and suboptimal) power allocation schemes, as well as the RC windowing based power allocation scheme with β = 0.1, can deliver significantly higher transmission rates for the

505

CR user as compared to the conventional optimal power allocation scheme which is without any spectral shaping. This is because of the fact that, for a given power budget in the interference limited region, the proposed AIC-based as well as the RC windowing-based power allocation schemes are able to allocate much higher power to the CR user compared to the conventional power allocation

510

scheme which is without spectral shaping, as shown in Fig. 4. From Fig. 3, we further note that the RC windowing-based power allocation scheme performs much better than the AIC-based optimal scheme with NC = 2 or NC = 4 in terms of the achievable CR user throughput. However, for higher values of NC , i.e., for NC = 6 and NC = 8, the performance of the AIC-

515

based optimal scheme improves and becomes better than the RC windowingbased power allocation scheme with β = 0.1. For a given power budget in the interference limited region, we see that the proposed AIC-based optimal scheme with NC = 8 is able to achieve the highest data transmission rate for the CR user. We also observe that the performance of the AIC-based suboptimal power

520

allocation scheme with NC = 8 is comparable to that of the RC windowingbased power allocation scheme. In Fig. 5, we have plotted the maximum transmission data rate that can be (3)

achieved by the CR user versus the interference threshold Ith prescribed by the 33

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

4

×10 7

3.5 3 2.5 2 1.5 AIC-based Optimal, Nc=8 AIC-based Suboptimal, Nc=8 AIC-based Optimal, Nc=6 AIC-based Suboptimal, Nc=6 With RC Windowing, β=0.1 AIC-based Optimal, Nc=4 AIC-based Optimal, Nc=2 Without Spectral Shaping

1 0.5 0 10 -8

10 -7

10 -6

Interference threshold prescribed by 3rd PU receiver (Watts)

(3)

Figure 5: Maximum transmitted data rate of CR user vs. interference threshold Ith

prescribed by 3rd PU receiver for different schemes; α2 = 0.97, PT = 10 mW, and (1)

(2)

(3)

NC = N C = N C = N C .

Total power allocated to CR user (Watts)

0.012

0.01

0.008

AIC-based Optimal, Nc=8 AIC-based Suboptimal, Nc=8 AIC-based Optimal, Nc=6 AIC-based Suboptimal, Nc=6 With RC Windowing, β=0.1 AIC-based Optimal, Nc=4 AIC-based Optimal, Nc=2 Without Spectral Shaping

0.006

0.004

0.002

0 10 -8

10 -7

10 -6

Interference threshold prescribed by 3rd PU receiver (Watts)

(3)

Figure 6: Total power allocated to CR user vs. interference threshold Ith prescribed (1)

(2)

by 3rd PU receiver for different schemes; α2 = 0.97, PT = 10 mW, and NC = NC = (3)

NC = N C .

34

3rd PU receiver for different schemes under consideration. We take α2 = 0.97 525

and the value for the available transmit power budget PT = 10 mW. From (3)

this figure we observe that, for a particular scheme, when the value of Ith is increased, the maximum transmitted data rate of the CR user also increases until the total power constraint or any of the interference constraints is (are) exceeded. We also observe that the transmission rate of the CR user tends to 530

(3)

saturate as the interference threshold Ith is increased beyond a certain value. (3)

This is due to the reason that when the interference threshold Ith is relaxed by increasing its value, the other constraints (total power constraint due to PT , and (1)

(2)

the interference constraints due to Ith and Ith ) become more dominant. For (3)

example, in case of AIC-based optimal scheme with NC = 8, we see that as Ith 535

is increased, the power allocated to the CR user, and hence the transmission rate also increases. However, when the allocated power becomes equal to 10 mW, the total power constraint due to PT is strictly met, and the achievable transmission (3)

rate of the CR user saturates with any further increase in Ith (cf. Fig. 5 and Fig. 6). Further, since RC windowing-based power allocation scheme is able 540

to utilize all the available transmit power budget PT even at the lowest of the (3)

considered Ith values (see Fig. 6), the transmission rate versus interference threshold curve corresponding to this scheme is almost flat, as can be seen in Fig. 5. (3)

From Fig. 5 we again observe that, for a given interference threshold Ith , the 545

proposed AIC-based (optimal and suboptimal) power allocation schemes, as well as the RC windowing based power allocation scheme with β = 0.1, lead to much higher transmission rates for the CR user compared to the conventional optimal power loading scheme which is without spectral shaping. Further, at low values (3)

of interference threshold Ith , the RC windowing-based power allocation scheme 550

with β = 0.1 performs much better than the AIC-based schemes. However, at (3)

relatively higher values of Ith , the AIC-based optimal scheme with NC = 8 and NC = 6 performs even better than the RC windowing-based scheme. We also observe that the performance of the proposed AIC-based suboptimal scheme with NC = 8 and NC = 6 is comparable to that of the RC windowing-based 35

555

(3)

power allocation scheme at higher values of Ith .

Maximum transmitted data rate (bits/sec)

4.5

×10 7

4

3.5

3

2.5 P T = 20 mW P T = 15 mW P T = 10 mW

2 0.93

0.94

0.95

0.96

α

0.97

0.98

0.99

(1)

Figure 7: AIC-based optimal power allocation scheme with NC

1

(2)

(3)

= NC = NC = 8: (3)

Maximum transmitted data rate vs. α for different values of PT ; Ith = 2.8 × 10−7 W.

To study the effect of scaling parameter α on the CR user throughput in case (1)

of the proposed AIC-based optimal power allocation scheme, we take NC (2)

=

(3)

NC = NC = 8 and plot transmission data rate versus α in Fig. 7 for different (3)

values of PT . The value of Ith is fixed at 2.8×10−7 W. From this figure, we note 560

that, the transmission data rate (for a particular value of PT ) initially increases with α, and then starts decreasing when α is increased beyond a certain value. This is due to the reason that by increasing α, although the total power allocated to CR data subcarriers increases, the total power allocated to CR cancellation subcarriers decreases at the same time. Thus, when α is increased beyond a

565

certain value, the total power allocated to the cancellation subcarriers becomes insufficient to suppress the OBRI to within prescribed limits. Therefore, the total power allocated to CR data subcarriers is reduced to satisfy the interference constraints, resulting in a decrease of the maximum transmitted data rate. From Fig. 7, we further observe that the value of α which yields the highest

570

transmission rate, decreases with the increase in PT . The reason is that when 36

PT is increased, the total power allocated to the CR user tends to increase and may generate higher OBRI. In order to keep the OBRI within prescribed threshold limits, more power need to be allocated to cancellation subcarriers. This may be achieved by lowering the value of α. 575

The computational complexities of different power allocation schemes under consideration are presented in Table 1. We observe that for a given scheme, the computational complexity is largely dependent on the effective number of data subcarriers available in that particular scheme. It is important to note

580

that the effective number of data subcarriers in the case of AIC-based schemes L L (l) (l) (l) is N − l=1 (NP +NC ), which is less than the effective number of N − l=1 NP data subcarriers available in the RC windowing-based or the conventional (without spectral shaping) scheme. From Table 1, we also observe that the proposed AIC-based optimal power allocation scheme has a relatively higher on-the-run computational complexity compared to the RC windowing-based power alloca-

585

tion or the conventional (without spectral shaping) optimal power allocation scheme. Further, the computational complexity of the proposed AIC-based suboptimal power allocation scheme is significantly low compared to all other schemes under consideration.

8. Conclusion 590

Optimal and suboptimal power allocation schemes augmented with AICbased spectral shaping have been proposed for the OFDM-based CR system that operates in a spectrum overlay approach within the radio spectrum licensed to PUs. The proposed schemes aim at maximizing the data transmission rate of the CR user subject to a set of statistical interference constraints imposed by

595

each of the active PUs, as well as the total power constraint at the CR transmitter. Instead of the instantaneous values for the channel fading gains between the CR transmitter and the active PU receivers, the proposed schemes require the knowledge of the channel fading statistics at the CR transmitter. We have also studied an optimal power allocation scheme that is augmented with RC

37

Table 1: Computational complexity of different power allocation schemes

Scheme

Proposed AIC-based optimal scheme

Proposed AIC-based suboptimal scheme RC windowing-based optimal scheme Optimal scheme (conventional) without spectral shaping

600

Computational complexity 2   (l) (l) L N O κ 1 κ2 N D × + N P C l=1     (l) (l) L 2 × l=1 NP + NC +O κ1 κ2 ND   3 , +O κ1 ND L (l) (l) where ND = N − l=1 (NP + NC )    L (l) (l) O ND × l=1 NP + NC

+O (ND log ND ) + O ((L + 1) ND ), L (l) (l) where ND = N − l=1 (NP + NC )  3 , O ND L (l) where ND = N − l=1 NP  3 , O ND

where ND = N −

L

l=1

(l)

NP

windowing-based spectral shaping. Through simulation results, we have shown that both the AIC-based as well as the RC windowing-based power allocation schemes deliver much higher transmission rates for the CR user compared to the conventional (without spectral shaping) optimal power allocation scheme. We have demonstrated that although the AIC-based optimal scheme has a rela-

605

tively higher computational complexity, it may yield better CR user throughput compared to the RC windowing-based power allocation scheme. We have also shown that the AIC-based suboptimal scheme is computationally more efficient and may exhibit performance comparable to that of the RC windowing-based power allocation scheme.

610

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43

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