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

Authors: Yonghong Zhang; Cyril Leung
Journal: Research Letters in Communications
 (2008)

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## 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.


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## Body

## 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 ofW 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) toK 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 σ02.
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 gk,mt.
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 slott,
each subchannel within ℳt,
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 (rmt) of bits per OFDM symbol which can be
supported by subchannel m in time slot t is given by [7] (1)rmt=log2(1+gmtsmtΓσ02), where gmt is the subchannel gain of the selected CRU, smt 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 (2)∑m∈ℳtsmt≤S∀t, 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 slott,
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 1−pa,
and the probability of changing from inactive to active mode is 1−pn.
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 (3)qij=∑n=0L(in)in(1−pn)npni−n(L−in−i+j)L−in−i+j(1−pa)n−i+jpaL−j−n.The steady-state probability column vectorΠ=(π0,π1,…,πL)T is given by [9] (4)Π=U−1V, where (5)U=[1,1,1,⋯10,1−q00+q10,1−q00+q20,⋯1−q00+qL00,q01+1−q11,q01−q21,⋯q01−qL1⋯⋯⋯⋯⋯0,q0L−q1L,q0L−q2L,⋯q0L+1−qLL], and V=(1,1−q00,q01,…,q0L)T.
If each PU band can accommodate an equal number (M/L) of subchannels, the probability of having m available subchannels is (6)bm={πl,ifm=lML,l=0,1,2,…,L0,otherwise.


## 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 themCR,t subchannel gains of the selected CRUs at time
slot t,
the average bit rate (ABR) (BOWF=ΔlimT→∞BOWF(T)) is given by [4] (7)BOWF(T)=1TsT∑t=1T∑m∈ℳt:Γσ02/gmt≤LOWFtlog2(LOWFtgmtΓσ02). In (7), Γσ02/gmt can be viewed as the equivalent noise power on subchannel m and LOWFt is the water level at time t.Let𝒯i 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 (8)BOWF(T)=1TsT∑i=1M∑ti∈𝒯i∑m∈ℳti:Γσ02/gmti≤LOWF,itilog2(LOWF,itigmtiΓσ02), where LOWF,iti is the water level at time slot ti.WhenmCR,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, (9)BOWF(T)≤1TsT∑i=1M∑ti∈𝒯i∑m∈ℳti:Γσ02/gmti≤LOWF,i(Ti)log2(LOWF,i(Ti)gmtiΓσ02), where LOWF,i(Ti) is the global water level for set 𝒯i.LettingTi=biT,
and taking the limit as T→∞,
we have (10)BOWF≤1Ts∑i=1Mibi∫Γσ02/LOWF,i(∞)∞log2(LOWF,i(∞)gΓσ02)fG(g)dg.In (9), LOWF,i(Ti),
is lower than the level calculated by including all subchannels at time slots t∈𝒯i because there may exist some subchannels with
above water level noises. Therefore, (11)LOWF,i(Ti)≤∑t∈𝒯i∑m∈ℳt(Γσ02/gmt)iTi+Si.Taking the limit as Ti→∞ in (11) yields  (12)LOWF,i(∞)≤E{1G}Γσ02+Si.Substituting for the two occurrences ofLOWF,i(∞) in (10) by the RHS of (12), we
obtain (13)BOWF≤1Ts∑i=1Mibi[(1−FG(1Li))log2(Li)+∫1/Li∞log2(g)fG(g)dg], where Li=Li and Li≜E{1/G}+S/(iΓσ02).The upper bound forLOWF,i(∞) in (12) can be quite loose since it includes
every subchannel, regardless of its equivalent noise power Γσ02/gmt.
As gmt→0, Γσ02/gmt increases without bound. Consequently, the
bound in (13) is also loose. We obtain a tighter bound by turning off any
subchannel m∈ℳt for which Γσ02/gmt is greater than the RHS of (12), that is, we
consider only the subchannels for which gmt>1/Li in calculating the water levels. The resulting
water level is still higher than LOWF,i(∞) so that (14)LOWF,i(∞)≤Γσ02∫1/Li∞(1/g)fG(g)dg+S/i1−FG(1/Li).SubstitutingLOWF,i(∞) in (10) by the RHS of (14), we obtain a
tighter bound, namely, (13) with (15)Li=∫1/Li∞(1/g)fG(g)dg+S/(iΓσ02)1−FG(1/Li).


### 3.2. Achievable Bit Rate for PEPA

The ABR for
PEPA isBPEPA=ΔlimT→∞BPEPA(T),
where (16)BPEPA(T)=1TsT∑t=1T∑m∈ℳtlog2(1+SgmtmCR,tΓσ02). Grouping the time slots with mCR,t=i,
we have (17)BPEPA(T)=1TsT∑i=1M∑t∈𝒯i∑m∈ℳtlog2(1+SgmtiΓσ02). Letting Ti→∞,
we obtain (18)BPEPA=1Ts∑i=1MibiE{log2(1+SiΓσ02G)}.


## 3.1. Upper Bounds on Achievable Bit  Rate for OWF

Assuming that
OWF is applied to themCR,t subchannel gains of the selected CRUs at time
slot t,
the average bit rate (ABR) (BOWF=ΔlimT→∞BOWF(T)) is given by [4] (7)BOWF(T)=1TsT∑t=1T∑m∈ℳt:Γσ02/gmt≤LOWFtlog2(LOWFtgmtΓσ02). In (7), Γσ02/gmt can be viewed as the equivalent noise power on subchannel m and LOWFt is the water level at time t.Let𝒯i 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 (8)BOWF(T)=1TsT∑i=1M∑ti∈𝒯i∑m∈ℳti:Γσ02/gmti≤LOWF,itilog2(LOWF,itigmtiΓσ02), where LOWF,iti is the water level at time slot ti.WhenmCR,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, (9)BOWF(T)≤1TsT∑i=1M∑ti∈𝒯i∑m∈ℳti:Γσ02/gmti≤LOWF,i(Ti)log2(LOWF,i(Ti)gmtiΓσ02), where LOWF,i(Ti) is the global water level for set 𝒯i.LettingTi=biT,
and taking the limit as T→∞,
we have (10)BOWF≤1Ts∑i=1Mibi∫Γσ02/LOWF,i(∞)∞log2(LOWF,i(∞)gΓσ02)fG(g)dg.In (9), LOWF,i(Ti),
is lower than the level calculated by including all subchannels at time slots t∈𝒯i because there may exist some subchannels with
above water level noises. Therefore, (11)LOWF,i(Ti)≤∑t∈𝒯i∑m∈ℳt(Γσ02/gmt)iTi+Si.Taking the limit as Ti→∞ in (11) yields  (12)LOWF,i(∞)≤E{1G}Γσ02+Si.Substituting for the two occurrences ofLOWF,i(∞) in (10) by the RHS of (12), we
obtain (13)BOWF≤1Ts∑i=1Mibi[(1−FG(1Li))log2(Li)+∫1/Li∞log2(g)fG(g)dg], where Li=Li and Li≜E{1/G}+S/(iΓσ02).The upper bound forLOWF,i(∞) in (12) can be quite loose since it includes
every subchannel, regardless of its equivalent noise power Γσ02/gmt.
As gmt→0, Γσ02/gmt increases without bound. Consequently, the
bound in (13) is also loose. We obtain a tighter bound by turning off any
subchannel m∈ℳt for which Γσ02/gmt is greater than the RHS of (12), that is, we
consider only the subchannels for which gmt>1/Li in calculating the water levels. The resulting
water level is still higher than LOWF,i(∞) so that (14)LOWF,i(∞)≤Γσ02∫1/Li∞(1/g)fG(g)dg+S/i1−FG(1/Li).SubstitutingLOWF,i(∞) in (10) by the RHS of (14), we obtain a
tighter bound, namely, (13) with (15)Li=∫1/Li∞(1/g)fG(g)dg+S/(iΓσ02)1−FG(1/Li).


## 3.2. Achievable Bit Rate for PEPA

The ABR for
PEPA isBPEPA=ΔlimT→∞BPEPA(T),
where (16)BPEPA(T)=1TsT∑t=1T∑m∈ℳtlog2(1+SgmtmCR,tΓσ02). Grouping the time slots with mCR,t=i,
we have (17)BPEPA(T)=1TsT∑i=1M∑t∈𝒯i∑m∈ℳtlog2(1+SgmtiΓσ02). Letting Ti→∞,
we obtain (18)BPEPA=1Ts∑i=1MibiE{log2(1+SiΓσ02G)}.


## 4. Rayleigh Fading Channel

In Section3,
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 timet,
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, (19)fG(g)=K[1−e−g/E{G}]K−1e−g/E{G}E{G} with corresponding
cdf (20)FG(g)=[1−e−g/E{G}]K.


### 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, (21)k*(t)=argmaxkgk,m(t)E{Gk}. 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  (22)FG(g)=P(G≤g)=∑i=1KP(G≤g∣k*=i)P(k*=i)=1K∑i=1K∏j=1KP(Gj≤gE{Gj}E{Gi})=1K∑i=1K(1−e−g/E{Gi})K. The corresponding pdf
is (23)fG(g)=∑i=1K(1−e−g/E{Gi})(K−1)e−g/E{Gi}E{Gi}.


## 4.1. Opportunistic Subchannel Assignment

Suppose that at
each timet,
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, (19)fG(g)=K[1−e−g/E{G}]K−1e−g/E{G}E{G} with corresponding
cdf (20)FG(g)=[1−e−g/E{G}]K.


## 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, (21)k*(t)=argmaxkgk,m(t)E{Gk}. 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  (22)FG(g)=P(G≤g)=∑i=1KP(G≤g∣k*=i)P(k*=i)=1K∑i=1K∏j=1KP(Gj≤gE{Gj}E{Gi})=1K∑i=1K(1−e−g/E{Gi})K. The corresponding pdf
is (23)fG(g)=∑i=1K(1−e−g/E{Gi})(K−1)e−g/E{Gi}E{Gi}.


## 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 2×10−13.
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, σ02=10−16, S=0.1 W, W=2 MHz, Wl=250 kHz, l∈{1,2,…,L}, L=8, M=64, pn=0.9,
and Ts=40μs.Figure1 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.Figure 1
ABR as a
function ofpa 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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*Source: 420478-2008-10-19.xml*