Adaptive rate transmission for spectrum sharing system with quantized channel state information
ABSTRACT The capacity of a secondary link in spectrum sharing systems has been recently investigated in fading environments. In particular, the secondary transmitter is allowed to adapt its power and rate to maximize its capacity subject to the constraint of maximum interference level allowed at the primary receiver. In most of the literature, it was assumed that estimates of the channel state information (CSI) of the secondary link and the interference level are made available at the secondary transmitter via an infiniteresolution feedback links between the secondary/primary receivers and the secondary transmitter. However, the assumption of having infinite resolution feedback links is not always practical as it requires an excessive amount of bandwidth. In this paper we develop a framework for optimizing the performance of the secondary link in terms of the average spectral efficiency assuming quantized CSI available at the secondary transmitter. We develop a computationally efficient algorithm for optimally quantizing the CSI and finding the optimal power and rate employed at the cognitive transmitter for each quantized CSI level so as to maximize the average spectral efficiency. Our results give the number of bits required to represent the CSI sufficient to achieve almost the maximum average spectral efficiency attained using full knowledge of the CSI for Rayleigh fading channels.

Article: Throughput Maximization in Cognitive Radio Under Peak Interference Constraints With Limited Feedback
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ABSTRACT: A spectrumsharing scenario in a cognitive radio (CR) network where a secondary user (SU) shares a narrowband channel with $N$ primary users (PUs) is considered. We investigate the SU ergodic capacity maximization problem under an average transmit power constraint on the SU and $N$ individual peak interference power constraints at each primaryuser receiver (PURx) with various forms of imperfect channelstate information (CSI) available at the secondaryuser transmitter (SUTx). For easy exposition, we first look at the case when the SUTx can obtain perfect knowledge of the CSI from the SUTx to the secondaryuser receiver link, which is denoted as $g_{1}$, but can only access quantized CSI of the SUTx to PURx links, which is denoted as $g_{0i}$, $i = \hbox{1}, \ldots, N$, through a limitedfeedback link of $B = \log_{2}L$ b. For this scenario, a locally optimum quantized power allocation (codebook) is obtained with quantized $g_{0i}$, $i = \hbox{1}, \ldots, N$ information by using the Karush–Kuhn–Tucker (KKT) necessary optimality conditions to numerically solve the nonconvex SU capacity maximization problem. We derive asymptotic approximations for the SU ergodic capacity performance for the case when the number of feedback bits grows large $(B \rightarrow \infty)$ and/or there is a large number of PUs $(N \rightarrow \infty)$ that operate. For the interferencelimited regime, where the average transmit power constraint is inactive, an alternative locally optimum scheme, called the quantizedrate allocation strategy, based on the quantizedratio $g_{1}/\max_{i}g_{0i}$ information, is proposed. Subsequently, we relax the strong assumption of fullCSI knowledge of $g_{1}$ at the SUTx to imperfect $g_{1}$ knowledge that is also available at the SUTx. Depending on the way the SUTx obtains the $g_{1}$ information, the following two different suboptimal quantized power codebooks are derived for the SU ergodic capacity maximization problem: 1) the power codebook with noisy $g_{1}$ estimates and quantized $g_{0i}$, $i = \hbox{1}, \ldots, N$ knowledge and 2) another power codebook with both quantized $g_{1}$ and $g_{0i}$, $i = \hbox{1}, \ldots, N$ information. We emphasize the fact that, although the proposed algorithms result in locally optimum or strictly suboptimal solutions, numerical results demonstrate that they are extremely efficient. The efficacy of the proposed asymptotic approximations is also illustrated through numerical simulation results.IEEE Transactions on Vehicular Technology 01/2012; 61(3):12871305. · 2.06 Impact Factor
Page 1
Adaptive Rate Transmission for Spectrum Sharing
System with Quantized Channel State Information
Mohamed Abdallah∗, Ahmed Salem∗∗, MohamedSlim Alouini§, Khalid A. Qaraqe∗
∗Electrical and Computer Engineering, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar
Email: {mohamed.abdallah, khalid.qaraqe}@qatar.tamu.edu
∗∗Department of Electronics and Electrical Communication, Cairo University, Giza, Cairo
§Electrical Engineering Program, KAUST, Thuwal, Saudi Arabia
Email: mohamed.alouini@kaust.edu.sa
Abstract—The capacity of a secondary link in spectrum shar
ing systems has been recently investigated in fading environments.
In particular, the secondary transmitter is allowed to adapt
its power and rate to maximize its capacity subject to the
constraint of maximum interference level allowed at the primary
receiver. In most of the literature, it was assumed that estimates
of the channel state information (CSI) of the secondary link
and the interference level are made available at the secondary
transmitter via an infiniteresolution feedback links between
the secondary/primary receivers and the secondary transmitter.
However, the assumption of having infinite resolution feedback
links is not always practical as it requires an excessive amount of
bandwidth. In this paper we develop a framework for optimizing
the performance of the secondary link in terms of the average
spectral efficiency assuming quantized CSI available at the
secondary transmitter. We develop a computationally efficient
algorithm for optimally quantizing the CSI and finding the
optimal power and rate employed at the cognitive transmitter
for each quantized CSI level so as to maximize the average
spectral efficiency. Our results give the number of bits required
to represent the CSI sufficient to achieve almost the maximum
average spectral efficiency attained using full knowledge of the
CSI for Rayleigh fading channels.
I. INTRODUCTION
Spectrumsharing system has been recently introduced as an
efficient means for utilizing the scarce spectrum by allowing
the secondary users to share the spectrum with licensed
primary users. In particular, the secondary user is allowed to
use the spectrum of the primary link under the constraint of
not increasing the average interference level at the primary re
ceiver above a predetermined value. Adaptive modulation can
be utilized as a powerefficient technique for maximizing the
capacity of the secondary link while satisfying the interference
constraint by allowing the secondary user to adapt its power
and rate according to the channel state information (CSI) of
the secondary channel between the secondary transmitter and
the secondary receiver, and the interference channel between
the secondary transmitter and the primary receiver.
The capacity of the secondary link in spectrum sharing sys
tems has been first studied in [1] under the constraint of aver
age interference level. Then expressions for the capacity results
have been developed under various sets of constraints such
This work is supported by Qatar National Research Fund (QNRF) grant
through National Priority Research Program (NPRP) No. 29674. QNRF is
an initiative of Qatar Foundation.
as peak/average transmit power and peak/average interference
level in [2]. In both papers, it was assumed that the secondary
transmitter has full knowledge of the CSI of the secondary
and interference links. Assuming imperfect CSI available at
the secondary transmitter, the capacity of the secondary link
under the constraint of peak and average interference outage
level has been developed [3], [4]. It was assumed that the
imperfect CSI is represented by an infinite resolution estimate
of the CSI in addition to an estimation error. Such assumption
requires the need for an infinite resolution feedback link which
is not always practical as it requires an excessive amount of
bandwidth. In [4], the effect of quantizing the CSI of the
interference channel on the capacity of the secondary link was
studied, however, a midrise uniform quantizer was assumed
which is not necessarily the optimal quantizer that maximizes
the capacity.
In this paper, we consider the problem of maximizing
the average spectral efficiency of a secondary link in fading
environment assuming quantized CSI of the secondary and
interference links. Under the assumption of quantized CSI
made available at the secondary transmitter, our objective is
to find the optimal CSI quantizers employed at the primary
and the secondary receivers and the associated discrete power
and rate level associated with each quantized CSI level so
as to maximize the performance of the secondary link in
terms of the average spectral efficiency. This problem has been
well investigated for the case of single slowly fading wireless
channel under the assumption of average transmit power. It
was shown that the optimal design of the CSI quantizer and
the associated power and rates results in achieving average
spectral efficiency values almost close to the Shanon capacity
[5], [6], [7].
We note that the problem of interest has been first addressed
in [8] where a framework was developed for finding optimal
power and rates assuming a fixed set of CSI quantizers of
the secondary and interference links. Then numerical search
techniques were applied for finding the optimal quantizer
thresholds that are not computationally efficient for high
number of quantizer bits. In addition, the CSI quantizer for
the secondary link were assumed to be fixed and the search
focused on optimal quantizer for the interference link and the
associated discrete power and rates levels.
9781424498482/11$26.00©2011 IEEE
Page 2
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Figure 1.
quantized channel state information .
System block diagram for the spectrum sharing system based on
However, in this paper, we develop a framework for finding
the optimal CSI quantizers of the both the secondary and
interference links. We also develop a computationally efficient
algorithm for jointly finding the optimal CSI quantizers of the
secondary and interference links and the associated discrete
power and rate values. We also find the number of bits
sufficient to achieve the average spectral efficiency almost
attained using full knowledge of CSI of the secondary and
interference links.
The remainder of this paper is organized as follows. In Sec.
II and Sec. III, we present the system model as well as the
problem formulation. In Sec. IV, we present iterativebased
techniques for finding the optimal rates and the optimal CSI
quantizers of both the secondary and interference links. In Sec.
V, we present numerical results to assess the performance of
the developed techniques and determine the number of bits
required to achieve average spectral efficiency values attained
using full knowledge of the CSI for Rayleigh fading channels.
Finally, we conclude the paper in Sec. VI.
II. SYSTEM MODEL
In this paper, we consider a network setting depicted in
Fig.II whereby a secondary transmitter is communicating
with its secondary destination over a secondary channel. The
secondary transmitter is allowed to share the spectrum used
by a primary network. Such simultaneous transmission results
in an interference power level observed at the primary receiver
due to the existence of a channel between the secondary
transmitter and the primary receiver; namely, the interference
channel. We assume discretetime Rayleigh fading channels
where the signaltonoise ratio (SNR) of the secondary and
interference link are given by γs and γp, respectively, with
mean values given by ¯ γs and ¯ γp, respectively. We assume
that the CSI of the secondary and interference channels are
estimated at the secondary and primary receivers, respectively.
The CSI of the secondary and interference channels are
then quantized using nbit quantizer and mbit quantizer,
respectively, then fed back to the secondary transmitter over
errorfree finiteresolution feedback channels. We note that
our assumption of relying only on the quantized CSI of
the interference link stems from the fact that the primary
base station will be required to estimate the channels of all
secondary nodes involved in communication in addition to its
primary transmitter. Therefore, in order to limit the amount
of feedback bandwidth needed for conveying the CSI of the
interference channels, it is desirable to rely on quantized CSI.
We note that the assumption of the primary receiver aids
in estimating the interference channel has been considered in
almost all of the literature related to spectrumsharing systems
[1], [2], [3], [9] as it is a crucial assumption in order to satisfy
the interference power level constraint at the primary receiver.
An example of a method for obtaining these estimates by
the primary receiver is shown in [9] whereby the secondary
transmitter is allowed to transmit a pilot signal every frame
to the primary receiver. This pilot signal can be used by
the primary receiver to estimate the CSI of the interference
channel.
A. Quantizer Model
We consider quantizing γs of the secondary channel and
γp of the interference channel using nbit quantizer with
N = 2nquantizer thresholds denoted by {γs,i} and mbit
quantizer with M = 2mquantizer thresholds denoted by
{γp,j}, respectively. As a result, the quantizer regions for the
secondary channel are given by
Ii= [γs,i,γs,i+1)
(1)
with quantizer thresholds satisfying γs,0= 0 < γs,1< ··· <
γs,N+1= ∞. While the quantizer regions for the interference
channel are given by
Ij= [γp,j,γp,j+1)
(2)
with quantizer thresholds satisfying γp,0= 0 < γp,1< ··· <
γp,M+1 = ∞. We adopt the scenario of zero information
outage where the secondary user is not allowed to transmit
below γs,1given any value of γp[7]. In addition we assume,
without loss of generality, no transmission is allowed above
γp,M .
By combining the quantizer regions defined in equations (1)
and (2), we define the following two dimensional quantizer
regions
Ii,j= ([γs,i,γs,i+1),[γp,j,γp,j+1)),
(3)
that represents the quantized CSI of the secondary and interfer
ence channels. Our objective in this paper is to find the optimal
discrete rate and power employed at each region Ii,jand the
associated optimal quantizer thresholds so as to maximize
the average spectral efficiency subject to interference power
constraint at the primary receiver.
III. PROBLEM FORMULATION
In this paper, we adopt the average spectral efficiency as
a performance metric for our problem. We assume that the
secondary employs discrete power pi,jand discrete rates Ri,j
2
Page 3
at each interval Ii,j. Assuming capacityachieving code the
discrete rate Ri,jis given by
Ri,j= log2(1 + pi,jγs,i).
(4)
By averaging the discrete rates over all possible quantizer
regions, the average spectral efficiency can be given as follows
¯ η=
N
�
i=1
M−1
�
j=0
Ri,j[Fs(γs,i+1)
−
Fs(γs,i)]
(5)
where Fs(·) and Fp(·) are the cumulative distribution functions
(CDF) of the secondary and interference SNRs, respectively.
We consider the problem of maximizing the average spectral
efficiency subject to average interference power allowed at the
primary receiver, i.e.
max
{γs,i,γp,j,pi,j}¯ η
subject to the constraints
N
�
i=1
M−1
�
j=0
[pi,j[Fs(γs,i+1) − Fs(γs,i)]
[Hp(γp,j+1) − Hp(γp,j)]] ≤ Q,
pi,j≥ 0,
γpfp(γp)dγp, and Q is the maximum
average interference power allowed at the primary receiver.
The above problem is a multidimensional optimization
problem that involves finding the optimal thresholds as well as
the discrete rates that maximize the average spectral efficiency.
Unfortunately, we can show that the above problem is not
convex and hence can not be solved using standard convex
optimization technique. In the next section, we develop an
iterativebased algorithm that exhibits fast convergence which
jointly find the discrete power levels as well as the optimal
quantizer thresholds that maximize the average spectral effi
ciency and satisfy the interference constraint.
(6)
where Hp(γp,j) =´γp,j
0
IV. ITERATIVEBASED SYSTEM DESIGN ALGORITHM
In this section, we present an efficient and fast algorithm
to find the optimal discrete power values pi,j and the quan
tizer thresholds. By introducing the Lagrangian multiplier λ
associated with the interference power constraint, we focus
on solving the dual problem
λminmax
{γs,i,γp,j,pi,j}
N
�
i=1
M−1
�
j=0
log2(1 + pi,jγs,i)[Fs(γs,i+1)
−Fs(γs,i)][Fp(γp,j+1) − Fp(γp,j)] − λ
N
�
i=1
M−1
�
j=0
[pi,j
[Fs(γs,i+1) − Fs(γs,i)][Hp(γp,j+1) − Hp(γp,j)]]. (7)
In solving the problem in (7), we first find the discrete values
of pi,jassuming a fixed set of quantizer thresholds {γs,i,γp,j}.
By differentiating (7) with respect to pi,j and setting the
derivative to zero the value of optimal pi,j is given by the
following waterfilling algorithm
�log2(e)[Fp(γp,j+1)−Fp(γp,j)]
where the value of λ can be obtained by applying the interfer
ence constraint and [·]+= max(·,0). In the next step, we fix
the values of pi,j and λ and the quantizer thresholds for the
secondary channel {γs,i}, then find the values of the quantizer
thresholds for the interference channel {γp,j} that maximize
(7). In particular, by setting the first derivative of (7) with
respect to γp,j, we obtain the following equation
�
pi,j=
λ[Hp(γp,j+1) − Hp(γp,j)]
−
1
γs,i
�+
(8)
γp,j=
i∈Ilog2
λ�
�1+pi,j−1γs,i
i∈I(pi,j−1− pi,j)(Fs(γs,i+1) − Fs(γs,i))
where I is the set of the index i for which pi,j are strictly
positive. By careful investigation of Eq. (9), we note that, at
each iteration, the value of each quantizer threshold for the
interference channel does not depend on the values of the
other interference thresholds. Instead it depends only on the
values of pi,j and the quantizer thresholds for the secondary
channel{γs,i}. This results in reducing the complexity of
finding the values of the interference thresholds.
In the final step, similarly by fixing the values of pi,j and
λ and the quantizer thresholds for the interference channel
{γp,j}, we can find the values of the quantizer thresholds for
the secondary channels that maximize (7) as follows
1+pi,jγs,i
�
(Fs(γs,i+1) − Fs(γs,i))
,
(9)
Fs(γs,i+1) = Fs(γs,i) +
fs(γs,i)
pi,j[Fp(γp,j+1)−Fp(γp,j)]
1+pi,jγs,i
log2(e)�M−1
j=0
+ λ
M−1
�
j=0
log2(
1 + pi,jγs,i
1 + pi−1,jγs,i−1)[Fp(γp,j+1) − Fp(γp,j)]
M−1
�
j=0
[pi,j− pi−1,j][Hp(γp,j+1) − Hp(γp,j)]
(10)
where fs(.) denotes the probability density function of the
secondary SNR. The above equation can be used to find the
quantizer thresholds for the secondary thresholds by applying
a simple numerical iterative technique by exploiting the fact
that γs,0= 0 and γs,N+1= ∞. Specifically, we can first an
initial value for γs,1 then using (10) to find the rest of the
quantizer thresholds γs,i iteratively. Since γs,N+1 = ∞, we
check whether Fs(γs,N+1) is equal to one. If not, we can either
increase or decrease the initial value of γs,1if Fs(γs,N+1) is
less or greater than one, respectively. We repeat these steps
until we find the quantizer thresholds.
After finding the values of pi,j, γs,i and γp,j using the
above three steps, these steps are repeated until these values
converge. These steps for solving (7) are described by the
following iterative algorithm
Our numerical results show that the algorithm converges to
the same optimal values of pi,j, γp,jand γs,iindependent of
the initial values. We finally note that the above algorithm does
3
Page 4
Algorithm 1 Iterative algorithm for finding the optimal pi,j,
γs,iand γp,j
Initialize k = 0, fix arbitrary γ0
solve for p0
repeat
Fix pk
solve for γk
Fix pk
solve for γk
increment k
until pi,j, γp,jand γs,iconverge.
p,jand γ0
s,i,
i,jand λ0using (8) by the waterfilling algorithm.
i,j, λkand γk−1
p,jusing (9);
i,j, λkand γk−1
s,iusing (10)
s,i,
p,j,
not guarantee that the obtained rates and thresholds are glob
ally optimal. However, based on comparing our results with
those achieved using an exhaustive search shown in [8], we
can argue that the iterative algorithm achieves the maximum
average spectral efficiency attained using an exhaustive search
method.
V. NUMERICAL RESULTS
In this section, we present numerical results that assess
the performance of the proposed iterative algorithm. We also
determine the optimal number of quantizer bits required to
represent the secondary and interference channels so as to
achieve the maximum average spectral efficiency attained
using full knowledge of the CSI.
Fig. 2 and Fig. 3 illustrate the convergence behavior of the
secondary and interference thresholds, respectively, attained
using the iterative algorithm for the values of N = 4 (n = 2)
bits, M = 4 (m = 2) bits, Q = 0 dB and ¯ γp = 1 dB
and ¯ γs= 10 dB. It is evident that the algorithm shows fast
convergence as it requires about 30 iterations compared to
the exhaustive search method shown in [8]. We note that the
algorithm shows even faster convergence for lower values of
Q.
Fig. 4 depicts the maximum average spectral efficiency
as a function of M (the number of quantizer levels for the
interference channel) for different values of N (the number of
quantizer levels for the secondary channel) at ¯ γp = 1 dB
and ¯ γs = 10 dB. The figure shows two sets of curves;
the upper and lower set of curves represent the case for
Q = 10 dB (weak interference constraint) and Q = 0 dB
(strong interference constraint), respectively. For the sake of
comparison, the average spectral efficiency is compared to
the capacity of the system assuming full knowledge of CSI
available at the secondary transmitter developed in [1]. Fig. V
shows that the N = 16 (4bit quantizers) and M = 8 (3bit
quantizers) suffice to achieve an average spectral efficiency
almost close to the capacity values developed in [1] assuming
continuous rate adaptation for Rayleigh fading channels. The
figure also shows that such selection for the number of bits is
not affected by the value of Q. We also note that for strong
interference constraint (Q = 0 dB) which is considered a
practical scenario, selecting M = 4 (2bit quantizer) for the
interference channel almost attain the same average spectral
efficiency compared to M = 8.
5 1015 202530
0
5
10
15
20
25
30
No. of Iterations
Secondary Thresholds (γs,i)
γs,1
γs,2
γs,3
γs,4
Figure 2.
secondary channel obtained using algorithm (1) for N = 4(n = 2) bits,
M = 4(m = 2) bits, Q = 0dB and ¯ γp= 1dB and ¯ γs= 10dB..
Convergence behavior of the quantizer thresholds {γs,i} for the
5101520 2530
0
1
2
3
4
5
6
7
8
No. of Iterations
Interference Thresholds (γp,j)
γp,1
γp,2
γp,3
γp,4
Figure 3.
interference channel obtained using algorithm (1) for N = 4 (n = 2) bits,
M = 4 (m = 2) bits, Q = 0 dB and ¯ γp= 1 dB and ¯ γs= 10 dB.
Convergence behavior of the quantizer thresholds {γp,j} for the
Fig. 5 depicts the average spectral efficiency as a function of
the average SNR of the interference channel ¯ γpfor different
values of the average SNR of the secondary channel ¯ γs =
5,10,15 dB, and N = 16, M = 8 and Q = 0 dB. The
figures compare the spectral efficiency against the capacity of
continuous rate spectrum sharing system. As our results reveal,
the selection of N = 16 and M = 8 is still sufficient to
represent the CSI of the secondary and interference channels,
respectively for different values of ¯ γsand ¯ γp. The figure also
shows that the average spectral efficiency decreases almost
linearly with ¯ γpin dB.
Fig. 6 depicts the probability of no transmission (outage
probability) defined as
Pout= 1 − (1 − Pr(γs< γs,1))(1 − Pr(γp> γp,M)) (11)
4
Page 5
as a function of ¯ γpfor different values of ¯ γs= 5,10,15 dB,
and N = 16, M = 8 and Q = 0 dB. As the figure reveals, the
outage probability increases as ¯ γpincreases for a fixed value
of ¯ γs. In addition as the values of ¯ γs decreases, the outage
probability suffers from higher increase as ¯ γpincreases.
2468101214
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
M
Spectral Efficiency(bps/hz)
N = 16
N = 8
N = 4
Continuous
Q =10dB
Q = 0dB
Figure 4. Average spectral efficiency as a function of M for different values
of N = 4,8,16 at ¯ γp = 1 dB and ¯ γs = 10 dB. The upper and lower set of
curves represent the case for Q = 10 dB (weak interference constraint) and
Q = 0 dB (strong interference constraint), respectively.
123456789
1
1.5
2
2.5
3
3.5
4
4.5
5
γp(dB)
Spectral Efficiency (bps/hz)
N = 16, M = 8 ,γs = 15dB
N = 16, M = 8 , γs = 10dB
N = 16, M = 8 , γs = 5dB
Continuous , γs = 15 dB
Continuous , γs = 10 dB
Continuous , γs = 5 dB
Figure 5.
the interference channel ¯ γp for different values of the average SNR of the
secondary channel ¯ γs= 5,10,15 dB and N = 16, M = 8 and Q = 0 dB.
VI. CONCLUSIONS
In this paper, we consider the problem of maximizing the
average spectral efficiency of a secondary link employing
discrete power and rate in spectrumsharing systems assuming
quantized CSI of the secondary and interference channels. We
presented an iterative algorithm for finding the optimal quan
tizers of the secondary and interference CSI and the discrete
power and rate employed within each quantizer levels. We
have shown that a four bit level representation of the secondary
CSI and a three bit level representation of the interference CSI
attains values of average spectral efficiency almost close to the
capacity values of spectrumsharing systems.
Average spectral efficiency as a function of the average SNR of
24681012
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
γp (dB)
Outage Probability
γs = 5 dB
γs = 10 dB
γs = 15 dB
Figure 6. Outage probability versus ¯ γpfor different values of ¯ γs= 5,10,15
dB, N = 16, M = 8 and Q = 0 dB.
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5
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