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

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 𝑊_𝑝 Hz wide. On each side of the PU band, there are 𝐾/2 OFDM subchannels, each of width 𝑊_𝑠 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] Φ_𝑘(𝑓)=𝑃_𝑘𝑇_𝑠sin𝜋𝑓𝜋𝑇𝑓_𝑠𝑇^𝑠²,(1) where 𝑃_𝑘 is the transmit power of the 𝑘th subchannel signal and 𝑇_𝑠 is the symbol duration. The interference power introduced by this signal into the PU band is 𝐼_𝑘𝑑_𝑘,𝑃_𝑘=𝑃_𝑘𝐼𝐹_𝑘,(2) where 𝐼𝐹_𝑘=^{𝑑𝑘+𝑊𝑝𝑑/2}_{𝑘−𝑊𝑝/2}||𝑔_𝑘||²𝑇_𝑠sin𝜋𝑓𝜋𝑇𝑓_𝑠𝑇^𝑠²𝑑𝑓(3) is the interference factor for the 𝑘th subchannel. In (2) and (3), 𝑔_𝑘 is the channel gain from the CRBS to the PU for the 𝑘th subchannel, and 𝑑_𝑘 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 is𝑆_𝑚𝑘𝑑_𝑘=^{𝑑𝑘+𝑊𝑠𝑑/2}_{𝑘−𝑊𝑠/2}||ℎ_𝑚𝑘||²Φ_PU(𝑓)𝑑𝑓,(4) where ℎ_𝑚𝑘 is the subchannel 𝑘 gain from the CRBS to SU 𝑚, and Φ_PU(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 𝑃_𝑚𝑘 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 to𝑏_𝑚𝑘=log₂1+||ℎ^𝑚𝑘||²𝑃_𝑚𝑘Γ𝑁₀𝑊_𝑠+𝑆_𝑚𝑘,(5) where ⌊⋅⌋ denotes the floor function, 𝑁₀ is the one-sided noise PSD, and 𝑆_𝑚𝑘 is given by (4). For convenience, the parameter Γ is set to unity in the remainder of this paper.

Let 𝑎_𝑚𝑘∈{0,1} be a subchannel allocation indicator, that is, 𝑎_𝑚𝑘=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: max𝑊_𝑠𝑀𝑚₌₁0𝑥0200𝑑^𝐾_𝑘=10𝑥0200𝑑𝑎_𝑚𝑘𝑏_𝑚𝑘(6) subject to 𝑎_𝑚𝑘∈{0,1},∀𝑚,𝑘(7)_𝑀𝑚₌₁𝑃𝑎_𝑚𝑘≤1,∀𝑘,(8)_𝑚𝑘≥0,∀𝑚,𝑘,(9)_𝑀𝑚₌₁0𝑥0200𝑑^𝐾_𝑘=10𝑥0200𝑑𝑎_𝑚𝑘𝑃_𝑚𝑘≤𝑃_total,(10)_𝑀𝐾_𝑚=1_𝑘=10𝑥0200𝑑𝑎_𝑚𝑘𝑃_𝑚𝑘𝐼𝐹_𝑘≤𝐼_th,(11) where 𝑃_total is the total SU power budget, and 𝐼_th 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 𝜆_𝑚 so that 𝜆_𝑚/∑^𝑀𝑖=1𝜆_𝑖 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 𝐵_𝑚≜∑^𝐾𝑘=1𝑏_𝑚𝑘 and define the total bit rate, 𝑅_𝑚, of SU 𝑚 as 𝑅_𝑚≜𝑊_𝑠𝐵_𝑚. The total bit rate for all SUs is 𝑅_𝑠≜∑^𝑀𝑚=1𝑅_𝑚.

Channel state estimation errors are modeled as follows: let ℎ 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 ℎ=ℎ+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, 𝑅_𝑠, can be described as follows. The maximum bit rate, 𝑅_𝑚𝑘≜𝑊_𝑠𝑏_𝑚𝑘, that can be achieved for SU 𝑚 on subchannel 𝑘 depends on the channel gain, ℎ_𝑚𝑘, the transmit power, 𝑃_𝑚𝑘, and the total interference-plus-noise power, as shown in (5). However, the CRBS knows only ℎ_𝑚𝑘 and not ℎ_𝑚𝑘. It thus calculates an estimated maximum transmit bit rate, 𝑅_𝑚𝑘. If 𝑅_𝑚𝑘<𝑅_𝑚𝑘, then the opportunity for a higher transmit bit rate is lost. On the other hand, if 𝑅_𝑚𝑘>𝑅_𝑚𝑘, then 𝑅_𝑚𝑘 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, 𝐵_𝐺,0≤𝐵_𝐺≤1, such that the RA algorithm uses 𝐵_𝐺×|ℎ_𝑚𝑘|² instead of |ℎ_𝑚𝑘|² as the channel power gain in calculating the estimated maximum transmit bit rate, 𝑅_𝑚𝑘. Therefore, (4) and (5) are modified accordingly as 𝑆_𝑚𝑘𝑑_𝑘=𝐵_𝐺^{𝑑𝑘+𝑊𝑠𝑑/2}_{𝑘−𝑊𝑠/2}|||ℎ_𝑚𝑘|||²𝑏̂Φ_PU(𝑓)𝑑𝑓,(13)_𝑚𝑘=⎢⎢⎢⎢⎣log₂⎛⎜⎜⎜⎝1+𝐵_𝐺|||ℎ_𝑚𝑘|||²𝑃_𝑚𝑘Γ𝑁₀𝑊_𝑠+𝑆_𝑚𝑘⎞⎟⎟⎟⎠⎥⎥⎥⎥⎦.(14)

Let 𝐼_PU≜∑^𝐾𝑘=1𝑃_𝑘𝐼𝐹_𝑘 be the total interference power introduced into the PU band by SU signals. To control the probability, 𝑃_𝑜, that 𝐼_PU exceeds 𝐼_th, the proposed scheme uses a second back-off factor, 𝐵_𝐼, such that the RA algorithm uses 𝐵_𝐼×𝐼_th instead of 𝐼_th as the target PU interference power threshold value. A lower value of 𝑃_𝑜 generally requires a lower 𝐵_𝐼 value.

From (14), the incremental power required for transmitting one bit to SU 𝑚 on subchannel 𝑘 is given by Δ𝑃_𝑚𝑘=𝑁⁰𝑊_𝑠+𝑆_𝑚𝑘𝐵_𝐺|||ℎ_𝑚𝑘|||²2𝑏̂^𝑚𝑘.(15) From (2) and (15), the incremental interference power generated by such a transmission to the primary user is Δ𝐼_𝑚𝑘=Δ𝑃_𝑚𝑘𝐼𝐹_𝑘.(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, 𝐵_𝑚 is the estimated total number of bits allocated to SU 𝑚 and 𝑃_SU is the total transmit power of SUs.

The m-MP algorithm is used to determine the interference power, 𝐼_MP, 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 𝑃_SU=0,𝐼_MP=0. (b)Set 𝐵_𝑚=0 for 𝑚∈{1,2,…,𝑀}. (c)Set 𝑏̂_𝑚𝑘=0, and calculate Δ𝑃_𝑚𝑘 as in (15), for 𝑚∈{1,2,…,𝑀} and 𝑘∈{1,2,…,𝐾}.
(2) Step 2(a)Determine 𝑚^∗=argmin_𝑚𝐵_𝑚/𝜆_𝑚; ties are first broken in decreasing order of 𝜆 then randomly.(b)Determine 𝑘_𝑃=argmin_𝑘Δ𝑃_𝑚∗𝑘.(c)If 𝑃_SU+Δ𝑃_{𝑚∗𝑘𝑃}≤𝑃_total, perform the following updates: 𝐵_𝑚∗=𝐵_𝑚∗+1,𝑃_SU=𝑃_SU+Δ𝑃_{𝑚∗𝑘𝑃}, 𝑏̂_{𝑚∗𝑘𝑃}=𝑏̂_{𝑚∗𝑘𝑃}+1, calculate Δ𝑃_{𝑚∗𝑘𝑃} as in (15), Δ𝑃_𝑚𝑘𝑃=∞,∀𝑚≠𝑚^∗, and go to step 2(a).(d)If 𝑃_SU+Δ𝑃_{𝑚∗𝑘𝑃}>𝑃_total, then set 𝑚^∗ to be the user with the next higher value of 𝐵_𝑚/𝜆_𝑚 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, 𝑃_MI, 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 𝑃_MI=0,𝐼_PU=0. (b)Set 𝐵_𝑚=0 for 𝑚∈{1,2,…,𝑀}. (c)Set 𝑏̂_𝑚𝑘=0 and calculate Δ𝐼_𝑚𝑘 as in (16), for 𝑚∈{1,2,…,𝑀} and 𝑘∈{1,2,…,𝐾}.
(2) Step 2(a)Determine 𝑚^∗=argmin_𝑚𝐵_𝑚/𝜆_𝑚; ties are first broken in decreasing order of 𝜆 then randomly.(b)Determine 𝑘_𝐼=argmin_𝑘Δ𝐼_𝑚∗𝑘.(c)If 𝐼_PU+Δ𝐼_{𝑚∗𝑘𝐼}≤𝐵_𝐼𝐼_th, perform the following updates: 𝐵_𝑚∗=𝐵_𝑚∗+1, 𝐼_PU=𝐼_PU+Δ𝐼_{𝑚∗𝑘𝐼},𝑃_MI=𝑃_MI+Δ𝐼_{𝑚∗𝑘𝐼}/𝐼𝐹_𝑘𝐼, 𝑏̂_{𝑚∗𝑘𝐼}=𝑏̂_{𝑚∗𝑘𝐼}+1, calculate Δ𝐼_{𝑚∗𝑘𝐼} as in (16), set Δ𝐼_𝑚𝑘𝐼=∞,∀𝑚≠𝑚^∗, and go to step 2(a).(d)If𝐼_PU+Δ𝐼_{𝑚∗𝑘𝐼}>𝐵_𝐼𝐼_th, then set𝑚^∗ to be the user with the next higher value of𝐵_𝑚/𝜆_𝑚 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 𝑉𝑃=𝑃^MI−𝑃_total𝑃_total,𝑉𝐼=𝐼^MP−𝐵_𝐼𝐼_th𝐵_𝐼𝐼_th,(17) respectively. Note that 𝑉𝑃 is negative when 𝑃_total>𝑃_MI, and 𝑉𝐼 is negative when 𝐵_𝐼𝐼_th>𝐼_MP.

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

Algorithm m-RC
(1) Step 1—Initialization(a)Set 𝑃_SU=0,𝐼_PU=0. (b)Set 𝐵_𝑚=0 for 𝑚∈{1,2,…,𝑀}. (c)Set 𝑏̂_𝑚𝑘=0, and calculate Δ𝑃_𝑚𝑘 as in (15) and Δ𝐼_𝑚𝑘 as in (16), for 𝑚∈{1,2,…,𝑀} and 𝑘∈{1,2,…,𝐾}.
(2) Step 2(a)Determine 𝑚^∗=argmin_𝑚𝐵_𝑚/𝜆_𝑚; ties are first broken in decreasing order of 𝜆 then randomly.(b)Determine 𝑘_𝑃=argmin_𝑘Δ𝑃_𝑚∗𝑘.(c)Determine 𝑘_𝐼=argmin_𝑘Δ𝐼_𝑚∗𝑘.(d)Compute 𝑋=𝑉𝐼(Δ𝑃_{𝑚∗𝑘𝑃}𝐼𝐹_𝑘𝑃−Δ𝐼_{𝑚∗𝑘𝐼})/Δ𝐼_{𝑚∗𝑘𝐼} and 𝑌=𝑉𝑃(Δ𝐼_{𝑚∗𝑘𝐼}/𝐼𝐹_𝑘𝐼−Δ𝑃_{𝑚∗𝑘𝑃})/Δ𝑃_{𝑚∗𝑘𝑃}(e)If 𝑋≥𝑌, set 𝑘^∗=𝑘_𝐼; otherwise set 𝑘^∗=𝑘_𝑃.(f)If 𝑃_SU+Δ𝑃_𝑚∗𝑘^∗≤𝑃_total and 𝐼_PU+Δ𝐼_𝑚∗𝑘^∗≤𝐵_𝐼𝐼_th, perform the following updates: 𝐵_𝑚∗=𝐵_𝑚∗+1,𝑏̂_𝑚∗𝑘^∗=𝑏̂_𝑚∗𝑘^∗+1,𝐼_PU=𝐼_PU+Δ𝐼_𝑚∗𝑘^∗,𝑃_SU=𝑃_SU+Δ𝑃_𝑚∗𝑘^∗, calculate Δ𝑃_𝑚∗𝑘^∗ as in (15) and Δ𝐼_𝑚∗𝑘^∗ as in (16), set Δ𝑃_𝑚𝑘^∗=∞,Δ𝐼_𝑚𝑘^∗=∞,and∀𝑚≠𝑚^∗, and go to step 2(a).(g)If 𝑃_SU+Δ𝑃_𝑚∗𝑘^∗>𝑃_total or 𝐼_PU+Δ𝐼_𝑚∗𝑘^∗>𝐵_𝐼𝐼_th, then set 𝑚^∗ to be the user with the next higher value of 𝐵_𝑚/𝜆_𝑚, and go to step 2(b). Stop if all users have been considered.(h)Calculate 𝑏_𝑚𝑘 as in (5),If 𝑏̂_𝑚𝑘>𝑏_𝑚𝑘, then set 𝑏_𝑚𝑘=0. Otherwise, set 𝑏_𝑚𝑘=𝑏̂_𝑚𝑘. Update 𝐵_𝑚.

Good values for the back-off factors, 𝐵_𝐺 and 𝐵_𝐼, are chosen as follows (1)For given values of 𝑃_total and 𝐼_th, both 𝑃_𝑜 and the throughput, 𝑅_𝑠, increase with 𝐵_𝐼, whereas 𝐵_𝐺 has little effect on 𝑃_𝑜. We therefore choose to use the largest value of 𝐵_𝐼 which can satisfy the 𝑃_𝑜 requirement.(2)Once the value of 𝐵_𝐼 is chosen, we determine the throughput for different values of 𝐵_𝐺 and select the 𝐵_𝐺 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 𝑉_𝑒 denote the variance of the real (or imaginary) part of the channel estimation error in (12).

Figure 1 shows the total SU bit rate, 𝑅_𝑠, as a function of 𝑃_total with 𝐼_th=6×10⁻⁶ W for two different values of 𝑉_𝑒 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 𝑅_𝑠 compared to the perfect channel estimation case. Furthermore, 𝑃_𝑜 for the RC RA algorithm in [4] can be as high as 6%. Using the m-RC RA algorithm with 𝐵_𝐺=0.84 and 𝐵_𝐼=0.815, the loss relative to the perfect channel estimation case is small, and 𝑃_𝑜 is negligibly small. The results show that the proposed m-RC RA is quite robust against channel estimation errors.

Figure 1 

Total bit rate, 𝑅𝑠, as a function of 𝑃total with 𝐼th=6×10−6W.

Figure 2 shows 𝑅_𝑠 for the m-RC RA algorithm as a function of 𝑃_total for different values of 𝐼_th. It was found from the simulation results that a 𝐵_𝐺 value of 0.84 is near-optimal for maximizing 𝑅_𝑠. In this figure, 𝐵_𝐼 was set to 0.905 to achieve a 𝑃_𝑜 of 10⁻³. As expected, 𝑅_𝑠 increases with 𝐼_th. For comparison, the perfect channel estimation curves are also included in Figure 2. It can be observed that the 𝑅_𝑠 value for m-RC RA is less than 5% lower.

Figure 2 

Total bit rate, 𝑅𝑠, as a function of 𝑃total for the m-RC RA algorithm with 𝑃𝑜=10−3.

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.