Journal of Electrical and Computer Engineering

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 secondary users. The PU band is  Hz wide. On each side of the PU band, there are OFDM subchannels, each of width  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 th subchannel SU signal is modeled as [6] where is the transmit power of the th subchannel signal and is the symbol duration. The interference power introduced by this signal into the PU band is where is the interference factor for the th subchannel. In (2) and (3), is the channel gain from the CRBS to the PU for the th subchannel, and is the spectral distance between the th subchannel and the center frequency of the PU band.

The interference power introduced by the PU signal into the th subchannel band at SU is where is the subchannel gain from the CRBS to SU , and 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 denote the transmit power allocated to subchannel of SU . From [7], the maximum number of bits in a symbol transmitted on this subchannel is set to where denotes the floor function, is the one-sided noise PSD, and is given by (4). For convenience, the parameter is set to unity in the remainder of this paper.

Let be a subchannel allocation indicator, that is, if and only if subchannel is allocated to SU . 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: subject to where is the total SU power budget, and 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 is denoted by so that is the fraction of the total number of SU bits loaded that is to be fairly allocated to SU . It is also convenient to denote the total number of bits per symbol period allocated to SU by and define the total bit rate, , of SU as . The total bit rate for all SUs is .

Channel state estimation errors are modeled as follows: let denote the actual (complex) gain of a channel. This channel could correspond to the th OFDM subchannel from the CRBS to SU 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 where is the channel estimation error. For the simulation results presented below, 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, , can be described as follows. The maximum bit rate, , that can be achieved for SU on subchannel depends on the channel gain, , the transmit power, , and the total interference-plus-noise power, as shown in (5). However, the CRBS knows only and not . It thus calculates an estimated maximum transmit bit rate, . If , then the opportunity for a higher transmit bit rate is lost. On the other hand, if , then 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, , such that the RA algorithm uses instead of as the channel power gain in calculating the estimated maximum transmit bit rate, . Therefore, (4) and (5) are modified accordingly as

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

From (14), the incremental power required for transmitting one bit to SU on subchannel is given by From (2) and (15), the incremental interference power generated by such a transmission to the primary user is

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, is the estimated total number of bits allocated to SU and is the total transmit power of SUs.

The m-MP algorithm is used to determine the interference power, , 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
() Step —Initialization (a)Set . (b)Set for . (c)Set , and calculate as in (15), for and .
() Step (a)Determine ; ties are first broken in decreasing order of then randomly.(b)Determine .(c)If , perform the following updates: , , calculate as in (15), , and go to step 2(a).(d)If , then set to be the user with the next higher value of 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, , 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
() Step —Initialization (a)Set . (b)Set for . (c)Set and calculate as in (16), for and .
() Step (a)Determine ; ties are first broken in decreasing order of then randomly.(b)Determine .(c)If , perform the following updates: , , , calculate as in (16), set , and go to step 2(a).(d)If, then set to be the user with the next higher value of 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 respectively. Note that is negative when , and is negative when .

The m-RC algorithm uses and as follows.

Algorithm m-RC
() Step —Initialization(a)Set . (b)Set for . (c)Set , and calculate as in (15) and as in (16), for and .
() Step (a)Determine ; ties are first broken in decreasing order of then randomly.(b)Determine .(c)Determine .(d)Compute and (e)If , set ; otherwise set .(f)If and , perform the following updates: , calculate as in (15) and as in (16), set , and go to step 2(a).(g)If or , then set to be the user with the next higher value of , and go to step 2(b). Stop if all users have been considered.(h)Calculate as in (5),If , then set . Otherwise, set . Update .

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

Figure 1 shows the total SU bit rate, , as a function of with  W for two different values of and three cases: () perfect channel estimation, () channel estimation errors with the RC RA algorithm in [4], and () 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 compared to the perfect channel estimation case. Furthermore, for the RC RA algorithm in [4] can be as high as . Using the m-RC RA algorithm with and , the loss relative to the perfect channel estimation case is small, and is negligibly small. The results show that the proposed m-RC RA is quite robust against channel estimation errors.

Figure 1: Total bit rate, , as a function of with .

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

Figure 2: Total bit rate, , as a function of for the m-RC RA algorithm with .

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.