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

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) [1–3] 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 𝑊_𝑙 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 (𝑇_𝑠). 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 𝑔^{𝑡𝑘,𝑚}. For simplicity, it is assumed that for any given value of 𝑘, {𝑔_𝑘,𝑚,𝑚=1,2,…,𝑀} are identically distributed random variables (RVs) with a common probability function (pdf) and cumulated distribution function (cdf) denoted as 𝑓_𝐺𝑘(𝑔_𝑘) and 𝐹_𝐺𝑘(𝑔_𝑘), respectively.

At each time slot 𝑡, each subchannel within ℳ_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 𝑓_𝐺(𝑔) and 𝐹_𝐺(𝑔) to denote the pdf and the cdf of the selected CRUs, respectively. The number (𝑟^𝑡𝑚) of bits per OFDM symbol which can be supported by subchannel 𝑚 in time slot 𝑡 is given by [7] 𝑟^𝑡𝑚=log₂(1+𝑔^𝑡𝑚𝑠^𝑡𝑚Γ𝜎²⁰),(1) where 𝑔^𝑡𝑚 is the subchannel gain of the selected CRU, 𝑠^𝑡𝑚 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 _𝑚_∈ℳ𝑡𝑠^𝑡𝑚≤𝑆∀𝑡,(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 𝑇_state slots. At a transition time, the probability of a PU band changing from active to inactive mode is 1−𝑝_𝑎, and the probability of changing from inactive to active mode is 1−𝑝_𝑛. The number (𝑙_CR,𝑡) of available PU bands at time slots {𝑡,𝑡=1,2,…} then forms a Markov chain, with a transition probability matrix 𝐐={𝑞_𝑖𝑗},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 (𝑞_𝑖𝑗) of moving from state 𝑖 to state 𝑗 is given by 𝑞_𝑖𝑗=^𝐿_𝑛=0𝑛𝑖𝑛𝑖1−𝑝^𝑛^𝑛𝑝^{𝑛𝑖−𝑛}𝑛𝐿−−𝑖+𝑖𝑗𝑛𝐿−−𝑖+𝑖𝑗1−𝑝^𝑎^{𝑛−𝑖+𝑗}𝑝^{𝑎𝐿−𝑗−𝑛}.(3)

The steady-state probability column vector 𝚷=(𝜋₀,𝜋₁,…,𝜋_𝐿)^𝑇 is given by [9] 𝚷=𝐔⁻¹𝐕,(4) where 𝐔=⎡⎢⎢⎢⎢⎢⎢⎢⎣10,,1−𝑞1⁰⁰,1,⋯1+𝑞₁₀,1−𝑞₀₀+𝑞₂₀,⋯1−𝑞₀₀+𝑞_𝐿00,𝑞₀₁+1−𝑞₁₁,𝑞₀₁−𝑞₂₁,⋯𝑞₀₁−𝑞_𝐿1⋯0,𝑞_0𝐿⋯⋯⋯⋯−𝑞_1𝐿,𝑞_0𝐿−𝑞_2𝐿,⋯𝑞_0𝐿+1−𝑞_𝐿𝐿⎤⎥⎥⎥⎥⎥⎥⎥⎦,(5) and 𝐕=(1,1−𝑞₀₀,𝑞₀₁,…,𝑞_0𝐿)^𝑇. If each PU band can accommodate an equal number (𝑀/𝐿) of subchannels, the probability of having 𝑚 available subchannels is 𝑏_𝑚=⎧⎪⎨⎪⎩𝜋^𝑙,if𝑚=𝑙𝑀0,otherwis𝐿e.,𝑙=0,1,2,…,𝐿(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 𝑚_CR,𝑡 subchannel gains of the selected CRUs at time slot 𝑡, the average bit rate (ABR) (𝐵=_OWF^Δlim_𝑇→∞𝐵^(𝑇)_OWF) is given by [4] 𝐵^(𝑇)_OWF=1𝑇_𝑠𝑇^𝑇_𝑡=1_{𝑚∈ℳ𝑡∶Γ𝜎20/𝑔𝑡𝑚≤𝐿𝑡OWF}log₂𝐿^𝑡OWF𝑔^𝑡𝑚Γ𝜎²⁰.(7) In (7), Γ𝜎²⁰/𝑔^𝑡𝑚 can be viewed as the equivalent noise power on subchannel 𝑚 and 𝐿^𝑡_OWF is the water level at time 𝑡.

Let 𝒯_𝑖 be the set of time slots with 𝑚_CR,𝑡=𝑖, and 𝑇_𝑖 be the number of elements in set 𝒯_𝑖. Grouping the time slots with 𝑚_CR,𝑡=𝑖, we can rewrite (7) as 𝐵^(𝑇)_OWF=1𝑇_𝑠𝑇^𝑀_𝑖=1_{𝑡𝑖∈𝒯𝑖}_{𝑚∈ℳ𝑡𝑖∶Γ𝜎20/𝑔𝑡𝑖𝑚≤𝐿𝑡𝑖OWF,𝑖}log₂(𝐿^{𝑡𝑖OWF,𝑖}𝑔^𝑡𝑖_𝑚Γ𝜎²⁰),(8) where 𝐿^𝑡𝑖_OWF,𝑖 is the water level at time slot 𝑡_𝑖.

When 𝑚_CR,𝑡=𝑖, the ABR if OWF is applied at each time slot 𝑡_𝑖 is smaller than that if OWF is applied in one shot to all the 𝑖𝑇_𝑖 subchannel gains of the selected CRUs over the 𝑇_𝑖 time slots, that is, 𝐵^(𝑇)_OWF≤1𝑇_𝑠𝑇^𝑀_𝑖=1_{𝑡𝑖∈𝒯𝑖}_{𝑚∈ℳ𝑡𝑖∶Γ𝜎20/𝑔𝑡𝑖𝑚≤𝐿𝑖)(𝑇OWF,𝑖}log₂(𝐿^{(𝑇𝑖)OWF,𝑖}𝑔^𝑡𝑖_𝑚Γ𝜎²⁰),(9) where 𝐿^(𝑇𝑖)_OWF,𝑖 is the global water level for set 𝒯_𝑖.

Letting 𝑇_𝑖=𝑏_𝑖𝑇, and taking the limit as 𝑇→∞, we have 𝐵_OWF≤1𝑇_{𝑠𝑀𝑖=1}𝑖𝑏_𝑖^∞Γ𝜎_20/𝐿^(∞)_OWF,𝑖log₂(𝐿^{(∞)OWF,𝑖}𝑔Γ𝜎²⁰)𝑓_𝐺(𝑔)𝑑𝑔.(10)

In (9), 𝐿^(𝑇𝑖)_OWF,𝑖, is lower than the level calculated by including all subchannels at time slots 𝑡∈𝒯_𝑖 because there may exist some subchannels with above water level noises. Therefore, 𝐿^(𝑇𝑖)_OWF,𝑖≤∑^{𝑡∈𝒯𝑖}∑_{𝑚∈ℳ𝑡}Γ𝜎²⁰/𝑔^𝑡𝑚𝑖𝑇_𝑖+𝑆𝑖.(11)Taking the limit as 𝑇_𝑖→∞ in (11) yields 𝐿^(∞)_OWF,𝑖≤𝐸1𝐺Γ𝜎²⁰+𝑆𝑖.(12)

Substituting for the two occurrences of 𝐿^(∞)_OWF,𝑖 in (10) by the RHS of (12), we obtain 𝐵_OWF≤1𝑇_{𝑠𝑀𝑖=1}𝑖𝑏_𝑖1−𝐹^𝐺1𝐿_𝑖log²𝐿^𝑖+^∞1/𝐿_𝑖log₂(𝑔)𝑓_𝐺(𝑔)𝑑𝑔,(13) where 𝐿_𝑖=𝐋_𝑖 and 𝐋_𝑖≜𝐸{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/𝑔^𝑡𝑚. As 𝑔^𝑡𝑚→0, Γ𝜎²⁰/𝑔^𝑡𝑚 increases without bound. Consequently, the bound in (13) is also loose. We obtain a tighter bound by turning off any subchannel 𝑚∈ℳ_𝑡 for which Γ𝜎²⁰/𝑔^𝑡𝑚 is greater than the RHS of (12), that is, we consider only the subchannels for which 𝑔^𝑡𝑚>1/𝐋_𝑖 in calculating the water levels. The resulting water level is still higher than 𝐿^(∞)_OWF,𝑖 so that 𝐿^(∞)_OWF,𝑖≤Γ𝜎²⁰∫^∞1/𝐋_𝑖(1/𝑔)𝑓_𝐺(𝑔)𝑑𝑔+𝑆/𝑖1−𝐹_𝐺1/𝐋^𝑖.(14)

Substituting 𝐿^(∞)_OWF,𝑖 in (10) by the RHS of (14), we obtain a tighter bound, namely, (13) with 𝐿_𝑖=∫^{∞1/𝐋𝑖}(1/𝑔)𝑓_𝐺(𝑔)𝑑𝑔+𝑆/𝑖Γ𝜎²⁰1−𝐹_𝐺1/𝐋^𝑖.(15)

3.2. Achievable Bit Rate for PEPA

The ABR for PEPA is 𝐵=_PEPA^Δlim_𝑇→∞𝐵^(𝑇)_PEPA, where 𝐵^(𝑇)_PEPA=1𝑇_𝑠𝑇^𝑇_𝑡=1_{𝑚∈ℳ𝑡}log₂(1+𝑆𝑔^𝑡𝑚𝑚_CR,𝑡Γ𝜎²⁰).(16) Grouping the time slots with 𝑚_CR,𝑡=𝑖, we have 𝐵^(𝑇)_PEPA=1𝑇_𝑠𝑇^𝑀_𝑖=1_{𝑡∈𝒯𝑖}_{𝑚∈ℳ𝑡}log₂(1+𝑆𝑔^𝑡𝑚𝑖Γ𝜎²⁰).(17) Letting 𝑇_𝑖→∞, we obtain 𝐵_PEPA=1𝑇_{𝑠𝑀𝑖=1}𝑖𝑏_𝑖𝐸{log₂(1+𝑆𝑖Γ𝜎²⁰𝐺)}.(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, 𝑓_𝐺(𝑔)=𝐾1−𝑒^−𝑔/𝐸^𝐺^𝐾−1𝑒^−𝑔/𝐸^𝐺𝐸𝐺(19) with corresponding cdf 𝐹_𝐺(𝑔)=1−𝑒^−𝑔/𝐸^𝐺^𝐾.(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)=argmax_𝑘𝑔_𝑘,𝑚(𝑡)𝐸𝐺_𝑘.(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 𝐹_𝐺(𝑔)==𝑃(𝐺≤𝑔)_𝐾_𝑖=1=1𝑃𝐺≤𝑔∣𝑘^∗=𝑖𝑃𝑘^∗=𝑖𝐾^𝐾_𝐾𝑖=1_𝑗=1𝑃(𝐺_𝑗≤𝑔𝐸𝐺^𝑗=1𝐸𝐺_𝑖)𝐾^𝐾_𝑖=11−𝑒^−𝑔/𝐸^𝐺𝑖^𝐾.(22) The corresponding pdf is 𝑓_𝐺(𝑔)=^𝐾_𝑖=11−𝑒^−𝑔/𝐸^𝐺𝑖^(𝐾−1)𝑒^−𝑔/𝐸^𝐺𝑖𝐸𝐺_𝑖.(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 𝐿_𝑖 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 2×10⁻¹³. 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 10⁻¹², two with value 10⁻¹³ and three with value 10⁻¹⁴. The resulting overall average subchannel gain for the six CRUs is 2×10⁻¹³. In our calculations, we also use the following parameter values: Γ=1, 𝜎²⁰=10⁻¹⁶, 𝑆=0.1 W, 𝑊=2 MHz, 𝑊_𝑙=250 kHz, 𝑙∈{1,2,…,𝐿}, 𝐿=8, 𝑀=64, 𝑝_𝑛=0.9, and 𝑇_𝑠=40𝜇s.

Figure 1 shows the ABR for OWF and PEPA as a function of 𝑝_𝑎 for 𝐾=6 CRUs. For both cases, the ABR for OWF and PEPA decreases with 𝑝_𝑎 due to the reduced number of available subchannels. In Case A, the improvement of OWF over PEPA is 0.1% at 𝑝_𝑎=0.1 and 0.05% at 𝑝_𝑎=0.9. In Case B, the improvement of OWF over PEPA is 14% at 𝑝_𝑎=0.1 and 2% at 𝑝_𝑎=0.99. The difference between OWF and PEPA decreases with 𝑝_𝑎 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 𝑝_𝑎. 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.

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.