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Research Letters in Communications
Volume 2008 (2008), Article ID 420478, 4 pages
http://dx.doi.org/10.1155/2008/420478
Research Letter

Performance of Equal Power Allocation in Multiuser OFDM-Based Cognitive Radio Systems

Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC, Canada V6T 1Z4

Received 22 July 2008; Accepted 15 September 2008

Academic Editor: Adam Panagos

Copyright © 2008 Yonghong Zhang and Cyril Leung. 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

The performance difference between a simple equal subchannel power allocation and the optimal water-filling subchannel power allocation schemes is studied for a multiuser OFDM-based cognitive radio (CR) system. It is shown that this difference depends on the average subchannel gain variations among the CR users as well as the activity levels of the primary users.

1. Introduction

Cognitive radio (CR) [13] is a concept which can potentially alleviate the pending spectrum shortage crisis. As discussed in [2], orthogonal frequency division multiplexing (OFDM) is an attractive modulation candidate for CR systems. It is well known that the optimal solution to the problem of determining the capacity of a set of 𝑀 parallel additive white Gaussian noise (AWGN) subchannels, each of which may have a different noise power level, subject to a total input signal power constraint, has a nice water-filling interpretation [4]. We will refer to this as optimal water-filling (OWF). In OWF, the signal powers allocated to different subchannels are in general different and no power is allocated to “silent” subchannels on which the noise power exceeds a certain threshold (water level). The scheme in which the total available signal power is shared equally among all (silent and nonsilent) subchannels is referred to as plain equal power allocation (PEPA).

Simulation results in [5] indicate that the difference between OWF and PEPA is quite small in a multiple user system with Rayleigh fading when each subchannel is assigned to the user with the best channel quality for that subchannel. Analytical results in [6] show that the performance difference between OWF and PEPA decreases with the number of users and average signal-to-noise ratio (SNR). In this paper, we study the performance difference between PEPA and OWF in a multiuser OFDM-based CR system. It is found that PEPA performs almost as well as OWF when there is little variation in CR user (CRU) average subchannel gains or PU activity level is high.

2. System Model

We consider a CR system with a total bandwidth of 𝑊 Hz and 𝐿 PUs; PU 𝑙,𝑙=1,2,,𝐿 has a bandwidth allocation of 𝑊𝑙 Hz. Frequency bands carrying PU signals are referred to as active; nonactive bands are also termed spectrum holes. In order to reduce the mutual interference between secondary CRUs and PUs to acceptable levels, some subchannels adjacent to active PU bands are not used by the CRUs.

We are interested in downlink transmissions from one CR base station (CRBS) to 𝐾 CRUs. It is assumed that the CRBS and the CRUs are able to accurately locate the spectrum holes. The system bandwidth of 𝑊 Hz can accommodate 𝑀OFDM subbands (or subchannels), each with noise power 𝜎20. Interference among the subchannels is assumed to be negligible.

The system is time-slotted with a slot duration equal to an OFDM symbol duration (𝑇𝑠). The subchannels are modelled in discrete time, with the gain for subchannel 𝑚 and time slot 𝑡 from the CRBS to CRU 𝑘 denoted by 𝑔𝑡𝑘,𝑚. For simplicity, it is assumed that for any given value of 𝑘, {𝑔𝑘,𝑚,𝑚=1,2,,𝑀} are identically distributed random variables (RVs) with a common probability function (pdf) and cumulated distribution function (cdf) denoted as 𝑓𝐺𝑘(𝑔𝑘) and 𝐹𝐺𝑘(𝑔𝑘), respectively.

At each time slot 𝑡, each subchannel within 𝑡, the set of available subchannels at time slot 𝑡, can be used by the CRBS to transmit to at most one CRU. We use 𝑓𝐺(𝑔) and 𝐹𝐺(𝑔) to denote the pdf and the cdf of the selected CRUs, respectively. The number (𝑟𝑡𝑚) of bits per OFDM symbol which can be supported by subchannel 𝑚 in time slot 𝑡 is given by [7] 𝑟𝑡𝑚=log2𝑔(1+𝑡𝑚𝑠𝑡𝑚Γ𝜎20),(1) where 𝑔𝑡𝑚 is the subchannel gain of the selected CRU, 𝑠𝑡𝑚 is the power allocated to subchannel 𝑚 at time slot 𝑡, and Γ is an SNR gap parameter which indicates how far the system is operating from capacity. The available power constraint implies that 𝑚𝑡𝑠𝑡𝑚𝑆𝑡,(2) where 𝑆 is the total power per time slot.

The availability of a PU band is modelled by a two-state Markov chain. During a time slot 𝑡, a PU band can be in one of two modes: active or inactive [8]. A PU band can change mode once every 𝑇state slots. At a transition time, the probability of a PU band changing from active to inactive mode is 1𝑝𝑎, and the probability of changing from inactive to active mode is 1𝑝𝑛. The number (𝑙CR,𝑡) of available PU bands at time slots {𝑡,𝑡=1,2,} then forms a Markov chain, with a transition probability matrix 𝐐={𝑞𝑖𝑗},𝑖,𝑗=0,1,2,,𝐿, where state 𝑖 corresponds to the event that the number of available PU bands is equal to 𝑖 and the probability (𝑞𝑖𝑗) of moving from state 𝑖 to state 𝑗 is given by 𝑞𝑖𝑗=𝐿𝑛=0𝑖𝑛𝑖𝑛1𝑝𝑛𝑛𝑝𝑛𝑖𝑛𝐿𝑖𝑛𝑖+𝑗𝐿𝑖𝑛𝑖+𝑗1𝑝𝑎𝑛𝑖+𝑗𝑝𝑎𝐿𝑗𝑛.(3)

The steady-state probability column vector 𝚷=(𝜋0,𝜋1,,𝜋𝐿)𝑇 is given by [9] 𝚷=𝐔1𝐕,(4) where 𝐔=1,1,1,10,1𝑞00+𝑞10,1𝑞00+𝑞20,1𝑞00+𝑞𝐿00,𝑞01+1𝑞11,𝑞01𝑞21,𝑞01𝑞𝐿10,𝑞0𝐿𝑞1𝐿,𝑞0𝐿𝑞2𝐿,𝑞0𝐿+1𝑞𝐿𝐿,(5) and 𝐕=(1,1𝑞00,𝑞01,,𝑞0𝐿)𝑇. If each PU band can accommodate an equal number (𝑀/𝐿) of subchannels, the probability of having 𝑚 available subchannels is 𝑏𝑚=𝜋𝑙,if𝑚=𝑙𝑀𝐿,𝑙=0,1,2,,𝐿0,otherwise.(6)

3. Achievable Bit Rates for OWF and PEPA

Although OWF provides the optimal solution for subchannel power allocation, it is more complex-to-implement than PEPA. We now derive expressions to compare the bit rates achievable by the two schemes.

3.1. Upper Bounds on Achievable Bit Rate for OWF

Assuming that OWF is applied to the 𝑚CR,𝑡 subchannel gains of the selected CRUs at time slot 𝑡, the average bit rate (ABR) (𝐵OWFΔ=lim𝑇𝐵(𝑇)OWF) is given by [4] 𝐵(𝑇)OWF=1𝑇𝑠𝑇𝑇𝑡=1𝑚𝑡Γ𝜎20/𝑔𝑡𝑚𝐿𝑡OWFlog2𝐿𝑡OWF𝑔𝑡𝑚Γ𝜎20.(7) In (7), Γ𝜎20/𝑔𝑡𝑚 can be viewed as the equivalent noise power on subchannel 𝑚 and 𝐿𝑡OWF is the water level at time 𝑡.

Let 𝒯𝑖 be the set of time slots with 𝑚CR,𝑡=𝑖, and 𝑇𝑖 be the number of elements in set 𝒯𝑖. Grouping the time slots with 𝑚CR,𝑡=𝑖, we can rewrite (7) as 𝐵(𝑇)OWF=1𝑇𝑠𝑇𝑀𝑖=1𝑡𝑖𝒯𝑖𝑚𝑡𝑖Γ𝜎20/𝑔𝑡𝑖𝑚𝐿𝑡𝑖OWF,𝑖log2(𝐿𝑡𝑖OWF,𝑖𝑔𝑡𝑖𝑚Γ𝜎20),(8) where 𝐿𝑡𝑖OWF,𝑖 is the water level at time slot 𝑡𝑖.

When 𝑚CR,𝑡=𝑖, the ABR if OWF is applied at each time slot 𝑡𝑖 is smaller than that if OWF is applied in one shot to all the 𝑖𝑇𝑖 subchannel gains of the selected CRUs over the 𝑇𝑖 time slots, that is, 𝐵(𝑇)OWF1𝑇𝑠𝑇𝑀𝑖=1𝑡𝑖𝒯𝑖𝑚𝑡𝑖Γ𝜎20/𝑔𝑡𝑖𝑚𝐿𝑖)(𝑇OWF,𝑖log2(𝐿(𝑇𝑖)OWF,𝑖𝑔𝑡𝑖𝑚Γ𝜎20),(9) where 𝐿(𝑇𝑖)OWF,𝑖 is the global water level for set 𝒯𝑖.

Letting 𝑇𝑖=𝑏𝑖𝑇, and taking the limit as 𝑇, we have 𝐵OWF1𝑇𝑠𝑀𝑖=1𝑖𝑏𝑖Γ𝜎20/𝐿()OWF,𝑖log2(𝐿()OWF,𝑖𝑔Γ𝜎20)𝑓𝐺(𝑔)𝑑𝑔.(10)

In (9), 𝐿(𝑇𝑖)OWF,𝑖, is lower than the level calculated by including all subchannels at time slots 𝑡𝒯𝑖 because there may exist some subchannels with above water level noises. Therefore, 𝐿(𝑇𝑖)OWF,𝑖𝑡𝒯𝑖𝑚𝑡Γ𝜎20/𝑔𝑡𝑚𝑖𝑇𝑖+𝑆𝑖.(11)Taking the limit as 𝑇𝑖 in (11) yields 𝐿()OWF,𝑖1𝐸𝐺Γ𝜎20+𝑆𝑖.(12)

Substituting for the two occurrences of 𝐿()OWF,𝑖 in (10) by the RHS of (12), we obtain 𝐵OWF1𝑇𝑠𝑀𝑖=1𝑖𝑏𝑖1𝐹𝐺1𝐿𝑖log2𝐿𝑖+1/𝐿𝑖log2(𝑔)𝑓𝐺,(𝑔)𝑑𝑔(13) where 𝐿𝑖=𝐋𝑖 and 𝐋𝑖𝐸{1/𝐺}+𝑆/(𝑖Γ𝜎20).

The upper bound for 𝐿()OWF,𝑖 in (12) can be quite loose since it includes every subchannel, regardless of its equivalent noise power Γ𝜎20/𝑔𝑡𝑚. As 𝑔𝑡𝑚0, Γ𝜎20/𝑔𝑡𝑚 increases without bound. Consequently, the bound in (13) is also loose. We obtain a tighter bound by turning off any subchannel 𝑚𝑡 for which Γ𝜎20/𝑔𝑡𝑚 is greater than the RHS of (12), that is, we consider only the subchannels for which 𝑔𝑡𝑚>1/𝐋𝑖 in calculating the water levels. The resulting water level is still higher than 𝐿()OWF,𝑖 so that 𝐿()OWF,𝑖Γ𝜎201/𝐋𝑖(1/𝑔)𝑓𝐺(𝑔)𝑑𝑔+𝑆/𝑖1𝐹𝐺1/𝐋𝑖.(14)

Substituting 𝐿()OWF,𝑖 in (10) by the RHS of (14), we obtain a tighter bound, namely, (13) with 𝐿𝑖=1/𝐋𝑖(1/𝑔)𝑓𝐺(𝑔)𝑑𝑔+𝑆/𝑖Γ𝜎201𝐹𝐺1/𝐋𝑖.(15)

3.2. Achievable Bit Rate for PEPA

The ABR for PEPA is 𝐵PEPAΔ=lim𝑇𝐵(𝑇)PEPA, where 𝐵(𝑇)PEPA=1𝑇𝑠𝑇𝑇𝑡=1𝑚𝑡log2(1+𝑆𝑔𝑡𝑚𝑚CR,𝑡Γ𝜎20).(16) Grouping the time slots with 𝑚CR,𝑡=𝑖, we have 𝐵(𝑇)PEPA=1𝑇𝑠𝑇𝑀𝑖=1𝑡𝒯𝑖𝑚𝑡log2(1+𝑆𝑔𝑡𝑚𝑖Γ𝜎20).(17) Letting 𝑇𝑖, we obtain 𝐵PEPA=1𝑇𝑠𝑀𝑖=1𝑖𝑏𝑖𝐸{log2𝑆(1+𝑖Γ𝜎20𝐺)}.(18)

4. Rayleigh Fading Channel

In Section 3, we studied the ABR for OWF and PEPA for arbitrary pdf's and cdf's of the subchannel gains of the selected CRUs. In this section, we obtain the pdf and cdf of the subchannel gains of the selected CRUs for two different subchannel allocation strategies. The subchannel gains of the CRUs are assumed to be Rayleigh-distributed, that is, the power gains are exponentially distributed.

4.1. Opportunistic Subchannel Assignment

Suppose that at each time 𝑡, each of the 𝑀 subchannels is assigned to the CRU with the highest gain for that subchannel. If the average subchannel power gains for all CRUs are equal, the pdf of the power gain for the CRU assigned to any subchannel is readily obtained using a standard result in order statistics [10], that is, 𝑓𝐺(𝑔)=𝐾1𝑒𝑔/𝐸𝐺𝐾1𝑒𝑔/𝐸𝐺𝐸𝐺(19) with corresponding cdf 𝐹𝐺(𝑔)=1𝑒𝑔/𝐸𝐺𝐾.(20)

4.2. A Fairer Subchannel Assignment Scheme

If the average subchannel gains for CRUs are quite different, assigning a subchannel to the CRU with the highest gain may be too unfair to CRUs with poor average subchannel gains. A fairer scheme [11] is to select, for each subchannel, the CRU with the best channel gain relative to its own mean gain,that is, 𝑘(𝑡)=argmax𝑘𝑔𝑘,𝑚(𝑡)𝐸𝐺𝑘.(21) The distribution of a CRU's subchannel gain relative to its own mean is exponential with a mean of 1. Thus, the probability of selecting CRU 𝑖 is 1/𝐾, that is, 𝑃(𝑘=𝑖)=1/𝐾,𝑖=1,2,,𝐾. The cdf of the power gain of the selected CRU for a subchannel is 𝐹𝐺=(𝑔)=𝑃(𝐺𝑔)𝐾𝑖=1𝑃𝐺𝑔𝑘𝑃𝑘=𝑖=1=𝑖𝐾𝐾𝐾𝑖=1𝑗=1𝑃(𝐺𝑗𝐺𝑔𝐸𝑗𝐸𝐺𝑖)=1𝐾𝐾𝑖=11𝑒𝑔/𝐸𝐺𝑖𝐾.(22) The corresponding pdf is 𝑓𝐺(𝑔)=𝐾𝑖=11𝑒𝑔/𝐸𝐺𝑖(𝐾1)𝑒𝑔/𝐸𝐺𝑖𝐸𝐺𝑖.(23)

5. Numerical Results

To compare the ABR for OWF and PEPA in a multiuser OFDM-based CR system, the expressions in (7), (18), and (13) with 𝐿𝑖 equal to the RHS of (15) are evaluated. The two subchannel allocation strategies in Sections 4.1 and 4.2, hereafter referred to as Case A and Case B, respectively, are considered. In Case A, the average subchannel power gain for each CRU is chosen as 2×1013. In Case B, we increase the number of CRUs by six at a time. The average subchannel power gains of the six CRUs are chosen as follows: one with value 1012, two with value 1013 and three with value 1014. The resulting overall average subchannel gain for the six CRUs is 2×1013. In our calculations, we also use the following parameter values: Γ=1, 𝜎20=1016, 𝑆=0.1 W, 𝑊=2 MHz, 𝑊𝑙=250 kHz, 𝑙{1,2,,𝐿}, 𝐿=8, 𝑀=64, 𝑝𝑛=0.9, and 𝑇𝑠=40𝜇s.

Figure 1 shows the ABR for OWF and PEPA as a function of 𝑝𝑎 for 𝐾=6 CRUs. For both cases, the ABR for OWF and PEPA decreases with 𝑝𝑎 due to the reduced number of available subchannels. In Case A, the improvement of OWF over PEPA is 0.1% at 𝑝𝑎=0.1 and 0.05% at 𝑝𝑎=0.9. In Case B, the improvement of OWF over PEPA is 14% at 𝑝𝑎=0.1 and 2% at 𝑝𝑎=0.99. The difference between OWF and PEPA decreases with 𝑝𝑎 because with a fixed total power, the average SNR for the available subchannels increases. The difference between OWF and PEPA is known to decrease with average SNR [6]. For both cases, the proposed upper bound for OWF is very close to the actual OWF curve and the difference decreases with 𝑝𝑎. The results show that the relative performance of PEPA depends on the activity level of the PUs and the variations in average subchannel gains among the CRUs.

420478.fig.001
Figure 1: ABR as a function of 𝑝𝑎 for OWF and PEPA with 𝐾=6.

The ABRs for OWF and PEPA were also determined as a function of the number (𝐾) of CRUs. For both cases, the ABRs of OWF and PEPA increase with 𝐾 as a result of multiuser diversity. The ABR difference between OWF and PEPA in Case A is negligible; in Case B, the improvement of OWF over PEPA is 9% for 𝐾=6 and 2% for 𝐾=48.

6. Conclusions

The performance difference between the PEPA and OWF subcarrier power allocation schemes in a multiuser OFDM-based CR system was studied. A proposed upper bound for OWF was shown to be tight. When the PU activity is high or the CRU average gains are similar, the simpler PEPA scheme suffers little loss relative to OWF.

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