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Journal of Electrical and Computer Engineering
Volume 2010, Article ID 419430, 5 pages
http://dx.doi.org/10.1155/2010/419430
Research Article

Resource Allocation in a Cognitive Radio System with Imperfect Channel State Estimation

1School of Computer Engineering, Nanyang Technological University, Singapore 639798
2Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, B.C., Canada V6T 1Z4
3School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798

Received 17 October 2009; Accepted 23 February 2010

Academic Editor: Adam Panagos

Copyright © 2010 Tao Qin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Cognitive radio (CR) is a promising concept for improving the utilization of scarce radio spectrum resources. Orthogonal frequency division multiplexing (OFDM) is regarded as a technology which is well matched for CR systems. It is shown that channel estimation errors can result in a severe performance degradation in a multiuser OFDM CR system. A simple back-off scheme is proposed, and simulation results are provided which show that the proposed scheme is very effective in mitigating the negative impact of channel estimation errors.

1. Introduction

It is believed that the cognitive radio (CR) concept can be used to greatly improve spectral utilization by allowing secondary (unlicensed) users (SUs) to use frequency bands not currently being used by the primary (licensed) users (PUs) in a certain location [1, 2]. Orthogonal frequency division multiplexing (OFDM) is regarded as a good modulation scheme for CR systems due to its flexibility in allocating resources among SUs [3]. In CR systems, it is important to manage mutual interference problems effectively since primary and secondary users will often simultaneously use adjacent bands.

The problem of power, bit, and subchannel loading for multiuser OFDM CR systems has been studied in [4], in which it is assumed that perfect channel state information is available. In practice, this assumption is often unrealistic, and it is therefore important to study the performance degradation due to imperfect channel state estimation. In [5], the effect of partial channel information in a non-CR multiuser MIMO-OFDM system is discussed. However, the effect of mutual interference which would arise between primary and secondary users in a CR system is not addressed.

A reduced complexity (RC) resource allocation (RA) scheme is proposed for a multiuser OFDM CR system in [4], and it is shown that the scheme provides good performance when perfect channel estimates are available. In this paper, the performance degradation due to imperfect channel state estimation for the RC RA scheme is shown to be quite severe. A simple back-off scheme is proposed and found to be very effective in reducing this degradation.

2. System Model

The system model used in this paper is the same as in [4] and is summarized here for the convenience of the reader. We consider the problem of allocating resources on the downlink of an OFDM CR system in which a CR base station (CRBS) serves one primary and 𝑀 secondary users. The PU band is 𝑊𝑝 Hz wide. On each side of the PU band, there are 𝐾/2 OFDM subchannels, each of width 𝑊𝑠 Hz. As the CRBS can transmit simultaneously to the PU and SUs, the PU signal can cause interference to the SUs and vice versa.

The baseband power spectral density (PSD) of the 𝑘th subchannel SU signal is modeled as [6] Φ𝑘(𝑓)=𝑃𝑘𝑇𝑠sin𝜋𝑓𝑇𝑠𝜋𝑓𝑇𝑠2,(1) where 𝑃𝑘 is the transmit power of the 𝑘th subchannel signal and 𝑇𝑠 is the symbol duration. The interference power introduced by this signal into the PU band is 𝐼𝑘𝑑𝑘,𝑃𝑘=𝑃𝑘𝐼𝐹𝑘,(2) where 𝐼𝐹𝑘=𝑑𝑘+𝑊𝑝𝑑/2𝑘𝑊𝑝/2||𝑔𝑘||2𝑇𝑠sin𝜋𝑓𝑇𝑠𝜋𝑓𝑇𝑠2𝑑𝑓(3) is the interference factor for the 𝑘th subchannel. In (2) and (3), 𝑔𝑘 is the channel gain from the CRBS to the PU for the 𝑘th subchannel, and 𝑑𝑘 is the spectral distance between the 𝑘th subchannel and the center frequency of the PU band.

The interference power introduced by the PU signal into the 𝑘th subchannel band at SU 𝑚 is𝑆𝑚𝑘𝑑𝑘=𝑑𝑘+𝑊𝑠𝑑/2𝑘𝑊𝑠/2||𝑚𝑘||2ΦPU(𝑓)𝑑𝑓,(4) where 𝑚𝑘 is the subchannel 𝑘 gain from the CRBS to SU 𝑚, and ΦPU(𝑓) is the PSD of the signal destined for the PU.

It is assumed that each subchannel can be used for transmission to at most one SU at any given time. Let 𝑃𝑚𝑘 denote the transmit power allocated to subchannel 𝑘 of SU 𝑚. From [7], the maximum number of bits in a symbol transmitted on this subchannel is set to𝑏𝑚𝑘=log2||1+𝑚𝑘||2𝑃𝑚𝑘Γ𝑁0𝑊𝑠+𝑆𝑚𝑘,(5) where denotes the floor function, 𝑁0 is the one-sided noise PSD, and 𝑆𝑚𝑘 is given by (4). For convenience, the parameter Γ is set to unity in the remainder of this paper.

Let 𝑎𝑚𝑘{0,1} be a subchannel allocation indicator, that is, 𝑎𝑚𝑘=1 if and only if subchannel 𝑘 is allocated to SU 𝑚. Our objective is to maximize the total bit rate for all SUs subject to total transmit power, fairness, and PU interference constraints. Specifically, the optimization problem is expressed as follows: max𝑊𝑠𝑀𝑚=10𝑥0200𝑑𝐾𝑘=10𝑥0200𝑑𝑎𝑚𝑘𝑏𝑚𝑘(6) subject to 𝑎𝑚𝑘{0,1},𝑚,𝑘(7)𝑀𝑚=1𝑎𝑚𝑘𝑃1,𝑘,(8)𝑚𝑘0,𝑚,𝑘,(9)𝑀𝑚=10𝑥0200𝑑𝐾𝑘=10𝑥0200𝑑𝑎𝑚𝑘𝑃𝑚𝑘𝑃total,(10)𝑀𝐾𝑚=1𝑘=10𝑥0200𝑑𝑎𝑚𝑘𝑃𝑚𝑘𝐼𝐹𝑘𝐼th,(11) where 𝑃total is the total SU power budget, and 𝐼th is the PU's maximum tolerable interference power. Inequality (8) follows from the assumption that a subchannel can be allocated to at most one SU. Inequalities (10) and (11) correspond to the power and interference constraints, respectively. The nominal bit rate weight (NBRW) for SU 𝑚 is denoted by 𝜆𝑚 so that 𝜆𝑚/𝑀𝑖=1𝜆𝑖 is the fraction of the total number of SU bits loaded that is to be fairly allocated to SU 𝑚. It is also convenient to denote the total number of bits per symbol period allocated to SU 𝑚 by 𝐵𝑚𝐾𝑘=1𝑏𝑚𝑘 and define the total bit rate, 𝑅𝑚, of SU 𝑚 as 𝑅𝑚𝑊𝑠𝐵𝑚. The total bit rate for all SUs is 𝑅𝑠𝑀𝑚=1𝑅𝑚.

Channel state estimation errors are modeled as follows: let denote the actual (complex) gain of a channel. This channel could correspond to the 𝑘th OFDM subchannel from the CRBS to SU 𝑚 or to the PU. For simplicity, all channels are assumed to be independently Rayleigh faded, that is, their complex gains are drawn from circularly symmetric, complex Gaussian distributions. The estimated (complex) channel gain is given by =+𝑒,(12) where 𝑒 is the channel estimation error. For the simulation results presented below, 𝑒 is assumed to be the outcome of an independent, circularly symmetric, complex Gaussian random variable.

The impact of channel estimation errors on the total SU bit rate, 𝑅𝑠, can be described as follows. The maximum bit rate, 𝑅𝑚𝑘𝑊𝑠𝑏𝑚𝑘, that can be achieved for SU 𝑚 on subchannel 𝑘 depends on the channel gain, 𝑚𝑘, the transmit power, 𝑃𝑚𝑘, and the total interference-plus-noise power, as shown in (5). However, the CRBS knows only 𝑚𝑘 and not 𝑚𝑘. It thus calculates an estimated maximum transmit bit rate, 𝑅𝑚𝑘. If 𝑅𝑚𝑘<𝑅𝑚𝑘, then the opportunity for a higher transmit bit rate is lost. On the other hand, if 𝑅𝑚𝑘>𝑅𝑚𝑘, then 𝑅𝑚𝑘 exceeds the channel capacity and the actual achieved transmit bit rate is zero. Therefore, the channel estimation errors, if not carefully taken into account in the design of the RA scheme, can result in a severe throughput degradation.

3. A Scheme for Mitigating Throughput Degradation

In order to reduce the overall throughput degradation caused by the use of inaccurate channel gain values, we introduce a back-off factor, 𝐵𝐺,0𝐵𝐺1, such that the RA algorithm uses 𝐵𝐺×|𝑚𝑘|2 instead of |𝑚𝑘|2 as the channel power gain in calculating the estimated maximum transmit bit rate, 𝑅𝑚𝑘. Therefore, (4) and (5) are modified accordingly as 𝑆𝑚𝑘𝑑𝑘=𝐵𝐺𝑑𝑘+𝑊𝑠𝑑/2𝑘𝑊𝑠/2|||𝑚𝑘|||2ΦPÛ𝑏(𝑓)𝑑𝑓,(13)𝑚𝑘=log2𝐵1+𝐺|||𝑚𝑘|||2𝑃𝑚𝑘Γ𝑁0𝑊𝑠+𝑆𝑚𝑘.(14)

Let 𝐼PU𝐾𝑘=1𝑃𝑘𝐼𝐹𝑘 be the total interference power introduced into the PU band by SU signals. To control the probability, 𝑃𝑜, that 𝐼PU exceeds 𝐼th, the proposed scheme uses a second back-off factor, 𝐵𝐼, such that the RA algorithm uses 𝐵𝐼×𝐼th instead of 𝐼th as the target PU interference power threshold value. A lower value of 𝑃𝑜 generally requires a lower 𝐵𝐼 value.

From (14), the incremental power required for transmitting one bit to SU 𝑚 on subchannel 𝑘 is given by Δ𝑃𝑚𝑘=𝑁0𝑊𝑠+𝑆𝑚𝑘𝐵𝐺|||𝑚𝑘|||22̂𝑏𝑚𝑘.(15) From (2) and (15), the incremental interference power generated by such a transmission to the primary user is Δ𝐼𝑚𝑘=Δ𝑃𝑚𝑘𝐼𝐹𝑘.(16)

The MP, MI, and RC RA algorithms were proposed in [4] to improve the throughput in a multiuser OFDM-based CR system. In this paper, modified versions, referred to as m-MP, m-MI, and m-RC, are proposed to mitigate the negative impact of channel estimation errors. Pseudocode listings of these three algorithms are provided below. In the algorithms, 𝐵𝑚 is the estimated total number of bits allocated to SU 𝑚 and 𝑃SU is the total transmit power of SUs.

The m-MP algorithm is used to determine the interference power, 𝐼MP, introduced into the PU band if, at each bit loading, we choose the subchannel which minimizes the incremental power needed for the selected SU.

Algorithm m-MP
(1) Step 1—Initialization (a)Set 𝑃SU=0,𝐼MP=0. (b)Set 𝐵𝑚=0 for 𝑚{1,2,,𝑀}. (c)Set ̂𝑏𝑚𝑘=0, and calculate Δ𝑃𝑚𝑘 as in (15), for 𝑚{1,2,,𝑀} and 𝑘{1,2,,𝐾}.
(2) Step 2(a)Determine 𝑚=argmin𝑚𝐵𝑚/𝜆𝑚; ties are first broken in decreasing order of 𝜆 then randomly.(b)Determine 𝑘𝑃=argmin𝑘Δ𝑃𝑚𝑘.(c)If 𝑃SU+Δ𝑃𝑚𝑘𝑃𝑃total, perform the following updates: 𝐵𝑚=𝐵𝑚+1,𝑃SU=𝑃SU+Δ𝑃𝑚𝑘𝑃, ̂𝑏𝑚𝑘𝑃=̂𝑏𝑚𝑘𝑃+1, calculate Δ𝑃𝑚𝑘𝑃 as in (15), Δ𝑃𝑚𝑘𝑃=,𝑚𝑚, and go to step 2(a).(d)If 𝑃SU+Δ𝑃𝑚𝑘𝑃>𝑃total, then set 𝑚 to be the user with the next higher value of 𝐵𝑚/𝜆𝑚 and go to step 2(b). Stop if all users have been considered.

Similarly, the m-MI algorithm is used to determine the total power, 𝑃MI, required for transmitting to the SUs if, at each bit loading, we choose the subchannel which minimizes the incremental interference power introduced into the PU band.

Algorithm m-MI
(1) Step 1—Initialization (a)Set 𝑃MI=0,𝐼PU=0. (b)Set 𝐵𝑚=0 for 𝑚{1,2,,𝑀}. (c)Set ̂𝑏𝑚𝑘=0 and calculate Δ𝐼𝑚𝑘 as in (16), for 𝑚{1,2,,𝑀} and 𝑘{1,2,,𝐾}.
(2) Step 2(a)Determine 𝑚=argmin𝑚𝐵𝑚/𝜆𝑚; ties are first broken in decreasing order of 𝜆 then randomly.(b)Determine 𝑘𝐼=argmin𝑘Δ𝐼𝑚𝑘.(c)If 𝐼PU+Δ𝐼𝑚𝑘𝐼𝐵𝐼𝐼th, perform the following updates: 𝐵𝑚=𝐵𝑚+1, 𝐼PU=𝐼PU+Δ𝐼𝑚𝑘𝐼,𝑃MI=𝑃MI+Δ𝐼𝑚𝑘𝐼/𝐼𝐹𝑘𝐼, ̂𝑏𝑚𝑘𝐼=̂𝑏𝑚𝑘𝐼+1, calculate Δ𝐼𝑚𝑘𝐼 as in (16), set Δ𝐼𝑚𝑘𝐼=,𝑚𝑚, and go to step 2(a).(d)If𝐼PU+Δ𝐼𝑚𝑘𝐼>𝐵𝐼𝐼th, then set𝑚 to be the user with the next higher value of𝐵𝑚/𝜆𝑚 and go to step 2(b). Stop if all users have been considered.

The relative importance of the SU power and PU interference is measured using 𝑃𝑉𝑃=MI𝑃total𝑃total𝐼,𝑉𝐼=MP𝐵𝐼𝐼th𝐵𝐼𝐼th,(17) respectively. Note that 𝑉𝑃 is negative when 𝑃total>𝑃MI, and 𝑉𝐼 is negative when 𝐵𝐼𝐼th>𝐼MP.

The m-RC algorithm uses 𝑉𝐼 and 𝑉𝑃 as follows.

Algorithm m-RC
(1) Step 1—Initialization(a)Set 𝑃SU=0,𝐼PU=0. (b)Set 𝐵𝑚=0 for 𝑚{1,2,,𝑀}. (c)Set ̂𝑏𝑚𝑘=0, and calculate Δ𝑃𝑚𝑘 as in (15) and Δ𝐼𝑚𝑘 as in (16), for 𝑚{1,2,,𝑀} and 𝑘{1,2,,𝐾}.
(2) Step 2(a)Determine 𝑚=argmin𝑚𝐵𝑚/𝜆𝑚; ties are first broken in decreasing order of 𝜆 then randomly.(b)Determine 𝑘𝑃=argmin𝑘Δ𝑃𝑚𝑘.(c)Determine 𝑘𝐼=argmin𝑘Δ𝐼𝑚𝑘.(d)Compute 𝑋=𝑉𝐼(Δ𝑃𝑚𝑘𝑃𝐼𝐹𝑘𝑃Δ𝐼𝑚𝑘𝐼)/Δ𝐼𝑚𝑘𝐼 and 𝑌=𝑉𝑃(Δ𝐼𝑚𝑘𝐼/𝐼𝐹𝑘𝐼Δ𝑃𝑚𝑘𝑃)/Δ𝑃𝑚𝑘𝑃(e)If 𝑋𝑌, set 𝑘=𝑘𝐼; otherwise set 𝑘=𝑘𝑃.(f)If 𝑃SU+Δ𝑃𝑚𝑘𝑃total and 𝐼PU+Δ𝐼𝑚𝑘𝐵𝐼𝐼th, perform the following updates: 𝐵𝑚=𝐵𝑚̂𝑏+1,𝑚𝑘=̂𝑏𝑚𝑘+1,𝐼PU=𝐼PU+Δ𝐼𝑚𝑘,𝑃SU=𝑃SU+Δ𝑃𝑚𝑘, calculate Δ𝑃𝑚𝑘 as in (15) and Δ𝐼𝑚𝑘 as in (16), set Δ𝑃𝑚𝑘=,Δ𝐼𝑚𝑘=,and𝑚𝑚, and go to step 2(a).(g)If 𝑃SU+Δ𝑃𝑚𝑘>𝑃total or 𝐼PU+Δ𝐼𝑚𝑘>𝐵𝐼𝐼th, then set 𝑚 to be the user with the next higher value of 𝐵𝑚/𝜆𝑚, and go to step 2(b). Stop if all users have been considered.(h)Calculate 𝑏𝑚𝑘 as in (5),If ̂𝑏𝑚𝑘>𝑏𝑚𝑘, then set 𝑏𝑚𝑘=0. Otherwise, set 𝑏𝑚𝑘=̂𝑏𝑚𝑘. Update 𝐵𝑚.

Good values for the back-off factors, 𝐵𝐺 and 𝐵𝐼, are chosen as follows (1)For given values of 𝑃total and 𝐼th, both 𝑃𝑜 and the throughput, 𝑅𝑠, increase with 𝐵𝐼, whereas 𝐵𝐺 has little effect on 𝑃𝑜. We therefore choose to use the largest value of 𝐵𝐼 which can satisfy the 𝑃𝑜 requirement.(2)Once the value of 𝐵𝐼 is chosen, we determine the throughput for different values of 𝐵𝐺 and select the 𝐵𝐺 value which yields the highest throughput.

4. Results

Computer simulations were run with the proposed m-RC RA algorithm assuming that the PU signal is an elliptically filtered white noise process and using the same parameter values as in [4]. In addition, let 𝑉𝑒 denote the variance of the real (or imaginary) part of the channel estimation error in (12).

Figure 1 shows the total SU bit rate, 𝑅𝑠, as a function of 𝑃total with 𝐼th=6×106 W for two different values of 𝑉𝑒 and three cases: (1) perfect channel estimation, (2) channel estimation errors with the RC RA algorithm in [4], and (3) channel estimation errors with the proposed m-RC RA algorithm. It can be seen that in the presence of channel estimation errors, the RC RA algorithm in [4] shows a big drop in 𝑅𝑠 compared to the perfect channel estimation case. Furthermore, 𝑃𝑜 for the RC RA algorithm in [4] can be as high as 6%. Using the m-RC RA algorithm with 𝐵𝐺=0.84 and 𝐵𝐼=0.815, the loss relative to the perfect channel estimation case is small, and 𝑃𝑜 is negligibly small. The results show that the proposed m-RC RA is quite robust against channel estimation errors.

419430.fig.001
Figure 1: Total bit rate, 𝑅𝑠, as a function of 𝑃total with 𝐼th=6×106W.

Figure 2 shows 𝑅𝑠 for the m-RC RA algorithm as a function of 𝑃total for different values of 𝐼th. It was found from the simulation results that a 𝐵𝐺 value of 0.84 is near-optimal for maximizing 𝑅𝑠. In this figure, 𝐵𝐼 was set to 0.905 to achieve a 𝑃𝑜 of 103. As expected, 𝑅𝑠 increases with 𝐼th. For comparison, the perfect channel estimation curves are also included in Figure 2. It can be observed that the 𝑅𝑠 value for m-RC RA is less than 5% lower.

419430.fig.002
Figure 2: Total bit rate, 𝑅𝑠, as a function of 𝑃total for the m-RC RA algorithm with 𝑃𝑜=103.

5. Conclusion

A simple back-off scheme was proposed to counter the deleterious effect of channel estimation errors in a multiuser OFDM CR system. Simulation results show that the proposed scheme can greatly reduce the loss in the total bit rate to SUs.

Acknowledgment

This research was supported in part by the Singapore Ministry of Education Grant no. RGM24/06.

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