In wireless communications, the assumption that the transmitter has perfect channel state information (CSI) is often unreasonable, due to feedback delays, estimation errors, and quantization errors. In order to accurately assess system performance, a more careful analysis with imperfect CSI is needed. In this paper, the impact of partial CSI due to feedback delays in a multiuser Orthogonal Frequency Division Multiplexing (MU-OFDM) cognitive radio (CR) system is investigated. The effect of partial CSI on the bit error rate (BER) is analyzed. A relationship between the transmit power and the number of bits loaded on a subcarrier is derived which takes into account the target BER requirement. With this relationship, existing resource allocation schemes which are based on perfect CSI being available can be applied when only partial CSI is available. Simulation results are provided to illustrate how the system performance degrades with increasingly poor CSI.
1. Introduction
In performance analyses of wireless communication systems, it is often assumed that perfect channel state information (CSI) is available at the transmitter. This assumption is often not valid due to channel estimation errors and/or feedback delays. To ensure that the system can satisfy target quality of service (QoS) requirements, a careful analysis which takes into account imperfect CSI is required [1].
Cognitive radio (CR) is a relatively new concept for improving the overall utilization of spectrum bands by allowing unlicensed secondary users (also referred to as CR users or CRUs) to access those frequency bands which are not currently being used by licensed primary users (PUs) in a given geographical area. In order to avoid causing unacceptable levels of interference to PUs, CRUs need to sense the radio environment and rapidly adapt their transmission parameter values [2‐6].
Anzeige
Orthogonal frequency division multiplexing (OFDM) is a modulation scheme which is attractive for use in a CR system due to its flexibility in allocating resources among CRUs. The problem of optimal allocation of subcarriers, bits, and transmit powers among users in a multiuser-(MU-) OFDM system is a complex combinatorial optimization problem. In order to reduce the computational complexity, the problem is solved in two steps by many suboptimal algorithms [7‐10]: determine the allocation of subcarriers to users and determine the allocation of bits and transmit powers to subcarriers. Resource allocation algorithms for MU-OFDM systems have been studied in [11‐14]. These algorithms are designed for non-CR MU-OFDM systems in which there are no PUs.
In an MU-OFDM CR system, mutual interference between PUs and CRUs needs to be considered. The problem of optimal allocation of subcarriers, bits, and transmit powers among users in an MU-OFDM CR system is more complex. It is commonly assumed that perfect CSI is available at the transmitter [15, 16]. As noted earlier, this assumption is often not reasonable. In this paper, we investigate the problem of resource allocation in an MU-OFDM CR system when only partial CSI is available at the CR base station (CRBS). We assume that CSI is acquired perfectly at the CRUs and fed back to the CRBS with a delay of seconds. The channel experiences frequency-selective fading. The objective is to maximize the total bit rate while satisfying BER, transmit power, and mutual interference constraints.
The rest of the paper is organized as follows. The system model is described in Section 2. Based on the system model, a constrained multiuser resource allocation problem is formulated in Section 3. A suboptimal algorithm for solving the problem is discussed in Section 4. Simulation results are presented in Section 5 and the main findings are summarized in Section 6.
2. System Model
We consider the problem of allocating resources on the downlink of an MU-OFDM CR system with one base station (BS) serving one PU and CRUs. The basic system model is the same as that described in [15] and is summarized here for the convenience of the reader.
Anzeige
The PU channel is Hz wide and the bandwidth of each OFDM subchannel is Hz. On either side of the PU channel, there are OFDM subchannels. The BS has only partial CSI and allocates subcarriers, transmit powers, and bits to the CRUs once every OFDM symbol period. The channel gain of each subcarrier is assumed to be constant during an OFDM symbol duration.
Suppose that is the transmit power allocated on subcarrier and is the channel gain of subcarrier from the BS to the PU. The resulting interference power spilling into the PU channel is given by
(1)
where
(2)
represents the interference factor for subcarrier , is the spectral distance between the center frequency of subcarrier and that of the PU channel, and denotes the normalized baseband power spectral density (PSD) of each subcarrier.
Let be the channel gain of subcarrier from the BS to CRU , and let be the baseband PSD of the PU signal. The interference power to CRU on subcarrier is given by
(3)
Let denote the transmit power allocated to CRU on subcarrier . For QAM modulation, an approximation for the BER on subcarrier of CRU is [13]
(4)
where is the one-sided noise PSD and is given by (3). Rearranging (4), the maximum number of bits per OFDM symbol period that can be transmitted on this subcarrier is given by
(5)
where and denotes the floor function.
Equation (4) shows the relationship between the transmit power and the number of bits loaded on the subcarrier for a given BER requirement when perfect CSI is available at the transmitter. We now establish an analogous relationship when only partial CSI is available.
The imperfect CSI that is available to the BS is modeled as follows. We assume that perfect CSI is available at the receiver. The channel gain, hnk, for subcarrier n and CRU k is the outcome of an independent complex Gaussian random variable, that is, [17], corresponding to Rayleigh fading. For clarity, we will denote random variables and their outcomes by uppercase and lowercase letters, respectively.
For notational simplicity, we will use to denote an arbitrary channel gain. The BS receives the CSI after a feedback delay , where is the OFDM symbol duration. We assume that the noise on the feedback link is negligible. Suppose that is the channel gain information that is received at the BS, then . From [18], the correlation between and is given by
(6)
where the correlation coefficient, , is given by
(7)
In (6) and (7), denotes the zeroth-order Bessel function of the first kind, is the Doppler frequency, is the expectation operator, and denotes the complex conjugate of .
The minimum mean square error (MMSE) estimator of based on is given by [19]
(8)
From (6), the actual channel gain can be written as [20] follows:
(9)
where with .
3. Formulation of the Multiuser Resource Allocation Problem
Based on the partial CSI available at the BS, we wish to maximize the total CRU transmission rate while maintaining a target BER performance on each subcarrier and satisfying PU interference and total BS CRU transmit power constraints. Let denote the average BER on subcarrier , and let represent the prescribed target BER. The optimization problem can be expressed as follows:
(10)
subject to
(11)
(12)
(13)
(14)
(15)
(16)
(17)
where is the total power budget for all CRUs, is the maximum interference power that can be tolerated by the PU, and is a subcarrier assignment indicator, that is, if and only if subcarrier is allocated to CRU . The term represents the nominal bit rate weight (NBRW) for CRU , and
(18)
denotes the total bit rate achieved by CRU . Constraint (11) ensures that the average BER for each subcarrier is below the given BER target. Constraint (12) states that the total power allocated to all CRUs cannot exceed , while constraint (14) ensures that the interference power to the PU is maintained below an acceptable level . Constraint (15) results from the assumption that each subcarrier can be assigned to at most one CRU. Constraint (17) ensures that the bit rate achieved by a CRU satisfies a proportional fairness condition.
Based on (9), we calculate the average of the right-hand side (RHS) of (4), treating as an outcome of an independent complex Gaussian variable. For an arbitrary vector , we have [21] the following:
(19)
where denotes the identity matrix. Applying (19) to (4), we obtain
(20)
where , , and denotes the channel gain that is fedback to the BS.
From (20), an explicit relationship between minimum transmit power and number of transmitted bits cannot be easily derived. However, since in (20) is a monotonically decreasing function of , we obtain the minimum power requirement while satisfying the constraint in (11) by setting .
We now derive a simpler, albeit approximate, relationship between the required transmit power, , and the number of loaded bits.
When setting , , , and , the RHS of (20) has the form
(21)
with . The function is Rician distributed with Rician factor [20]. A Rician distribution with can be approximated by a Nakagami- distribution [22] as follows:
(22)
with , where . Therefore, we approximate the RHS of (20) by
(23)
Then, from (23), we obtain
(24)
where . From (24), we obtain
(25)
where .
4. Resource Allocation with Partial Csi
Note that the joint subcarrier, bit, and power allocation problem in (10)–(17) belongs to the mixed integer nonlinear programming (MINP) class [23]. For brevity, we use the term "bit allocation" to denote both bit and power allocation. Since the optimization problem in (10)–(17) is generally computationally complex, we first use a suboptimal algorithm, which is based on a greedy approach, to solve the subcarrier allocation problem in Section 4.1. After subcarriers are allocated to CRUs, we apply a memetic algorithm (MA) to solve the bit allocation problem in Section 4.2.
4.1. Subcarrier Allocation
From (17), it can be seen that the subcarrier allocation depends not only on the channel gains, but also on the number of bits allocated to each subcarrier. Moreover, allocation of subcarriers close to the PU band should be avoided in order to reduce the interference power to the PU to a tolerable level. Therefore, we use a threshold scheme to select subcarriers for CRUs.
Suppose that subcarriers are available for allocating to CRUs. We assume equal transmit power for each subcarrier. Let
(26)
(27)
If a subcarrier is assigned to CRU , the maximum number of bits which can be loaded on the subcarrier is given by
(28)
Using (26)–(28), we can determine the number of subcarriers assigned to each CRU as follows. Let be the number of subcarriers allocated to CRU . Assuming that the same number of bits is loaded on every subcarrier assigned to a given CRU, the objective in (10) is equivalent to finding a set of subcarriers to maximize
(29)
subject to
(30)
(31)
(32)
where is the total transmit power allocated to all subcarriers and is the total interference power experienced by the PU due to CRU signals. The subcarrier allocation problem in (29)–(32) can be solved using the SA algorithm proposed in [24]. Note that we need to make use of (24) in the SA algorithm if only partial CSI is available. A pseudocode listing for the SA algorithm is shown in Pseudocode 1. The algorithm has a relatively low computational complexity . After subcarriers are allocated to CRUs, we then determine the number, , of bits allocated to subcarrier .
4.2. Bit Allocation
Memetic algorithm (MAs) are evolutionary algorithms which have been shown to be more efficient than standard genetic algorithms (GAs) for many combinatorial optimization problems [25‐27]. Using (24), the bit allocation problem can be solved using the MA algorithm proposed in [24]. It should be noted that the chosen genetic operators and local search methods greatly influence the performance of MAs. The selection of these parameters for the given optimization problem is based on the results in [24]. A pseudocode listing of the proposed memetic algorithm is shown in Pseudocode 2.
Let be the chromosome of member in a population, expressed as
(33)
where denotes the population size. A brief description of the MA algorithm in [24] is now provided.
(1)
The function selects a set, , of chromosomes from in a roulette wheel fashion, that is, selection with replacement.
(2)
Crossover: suppose that . Let denote the crossover probability, and let , denote the outcome of an independent random variable which is uniformly distributed in , then is selected as a candidate for crossover if and only if . Suppose that we have such candidates, we then form disjoint pairs of candidates (parents).
For each pair of parents and ,
(34)
we first generate a random integer , then we obtain the (possibly identical) chromosomes of two children as follows:
(35)
(3)
Mutation: let denote the mutation probability. For each chromosome in , we generate , where denotes the outcome of an independent random variable which is uniformly distributed in . Then for each component for which , we substitute the value with a randomly chosen admissible value.
(4)
Selection of surviving chromosomes: we select the chromosomes of parents and offsprings with the best fitness values as input for the next generation.
5. Results
In this section, performance results for the proposed algorithm described in Section 4 are presented. In the simulation, the parameters of the MA algorithm were chosen as follows: population size, ; number of generations = 20; crossover probability, ; mutation probability, .
We consider a system with one PU and CRUs. The total available bandwidth for CRUs is 5 MHz and supports 16 subcarriers with MHz. We assume that and an OFDM symbol duration, of s. In order to understand the impact of the fair bit rate constraint in (17) on the total bit rate, three cases of user bit rate requirements with were considered. In addition, three cases of partial CSI with were studied. It is assumed that the subcarrier gains and , for are outcomes of independent identically distributed (i.i.d.) Rayleigh-distributed random variables (rvs) with mean square value . The additive white Gaussian noise (AWGN) PSD, , was set to W/Hz. The PSD, , of the PU signal was assumed to be that of an elliptically filtered white noise process. The total CRU bit rate, , results were obtained by averaging over 10,000 channel realizations. The confidence intervals for the simulated results are within of the average values shown.
Figure 1 shows the average total bit rate, , as a function of the total CRU transmit power, , for , and 1 with , W, and a PU transmit power, , of 5 W. As expected, the average total bit rate increases with the maximum transmit power budget . It can be seen that the average total bit rate, , varies greatly with . For example, at W, increases by a factor of 2 as increases from 0.7 to 0.9. This illustrates the big impact that inaccurate CSI may have on system performance. The curves level off as increases due to the fixed value of the maximum interference power that can be tolerated by the PU.
Figure 1
Average total CRU bit rate,, versus total CRU transmit power,, with W, W, and.
×
Corresponding results for and are plotted in Figures 2 and 3, respectively. The average total bit rate, , decreases as the NBRW distribution becomes less uniform; the reduction tends to increase with .
Figure 2
Average total CRU bit rate,, versus total CRU transmit power,, with W, W, and.
Figure 3
Average total CRU bit rate,, versus total CRU transmit power,, with W, W, and.
×
×
Figure 4 shows as a function of for three different cases of with , W, and W. As to be expected, increases with . It can be seen that for is larger than for , and for is larger than for . When the bit rate requirements for CRUs become less uniform, decreases due to a decrease in the benefits of user diversity. With W, increases by about when changes from to . Results for are shown in Figure 5 and are qualitatively similar to those in Figure 4.
Figure 4
Average total CRU bit rate,, versus total CRU transmit power,, with W, W, and.
Figure 5
Average total CRU bit rate,, versus total CRU transmit power,, with W, W, and.
×
×
The average total bit rate, , is plotted as a function of the maximum PU tolerable interference power, , with W and W, for and 0.7 in Figures 6 and 7, respectively. As expected, increases with and decreases as the CRU bit rate requirements become less uniform. The curves level off as increases due to the fixed value of the total CRU transmit power, .
Figure 6
Average total CRU bit rate,, versus maximum PU tolerable interference power,, with W, W, and.
Figure 7
Average total CRU bit rate,, versus maximum PU tolerable interference power,, with W, W, and.
×
×
6. Conclusion
The assumption of perfect CSI being available at the transmitter is often unreasonable in a wireless communication system. In this paper, we studied an MU-OFDM CR system in which the available partial CSI is due to a delay in the feedback channel. The effect of partial CSI on the BER was investigated; a relationship between transmit power, number of bits loaded, and BER was derived. This relationship was used to study the performance of a resource allocation scheme when only partial CSI is available. It is found that the performance varies greatly with the quality of the partial CSI.
Pseudocode 1: Pseudocode for subcarrier allocation algorithm.
Algorithm: SA
for to number of subcarriers do
find which maximizes
;
Using (25), calculate the number of bits loaded on
Subcarrier
as with ;
initialize to 0;
ifthen
subcarrier is available; increment by 1;
else
subcarrier is not available;
end if
end for
For each , initialize the number, , of
subcarriers allocated to CRU to 0
calculate using (28);
for to do
find the value, , of which minimizes
;
allocate subcarrier to CRU ;
increment by one.
end for
Pseudocode 2: Pseudocode for the memetic algorithm.
Algorithm: MA
initialize Population ;
;
for to Number_of_Generatio do
;
;
;
add to ;
;
;
add to ;
;
end for
return.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
determine the allocation of subcarriers to users and
determine the allocation of bits and transmit powers to subcarriers. Resource allocation algorithms for MU-OFDM systems have been studied in [11‐14]. These algorithms are designed for non-CR MU-OFDM systems in which there are no PUs.
seconds. The channel experiences frequency-selective fading. The objective is to maximize the total bit rate while satisfying BER, transmit power, and mutual interference constraints.
CRUs. The basic system model is the same as that described in [15] and is summarized here for the convenience of the reader.
Hz wide and the bandwidth of each OFDM subchannel is
Hz. On either side of the PU channel, there are
OFDM subchannels. The BS has only partial CSI and allocates subcarriers, transmit powers, and bits to the CRUs once every OFDM symbol period. The channel gain of each subcarrier is assumed to be constant during an OFDM symbol duration.
is the transmit power allocated on subcarrier
and
is the channel gain of subcarrier
from the BS to the PU. The resulting interference power spilling into the PU channel is given by
,
is the spectral distance between the center frequency of subcarrier
and that of the PU channel, and
denotes the normalized baseband power spectral density (PSD) of each subcarrier.
be the channel gain of subcarrier
from the BS to CRU
, and let
be the baseband PSD of the PU signal. The interference power to CRU
on subcarrier
is given by
denote the transmit power allocated to CRU
on subcarrier
. For QAM modulation, an approximation for the BER on subcarrier
of CRU
is [13]
is the one-sided noise PSD and
is given by (3). Rearranging (4), the maximum number of bits per OFDM symbol period that can be transmitted on this subcarrier is given by
and
denotes the floor function.
[17], corresponding to Rayleigh fading. For clarity, we will denote random variables and their outcomes by uppercase and lowercase letters, respectively.
to denote an arbitrary channel gain. The BS receives the CSI after a feedback delay
, where
is the OFDM symbol duration. We assume that the noise on the feedback link is negligible. Suppose that
is the channel gain information that is received at the BS, then
. From [18], the correlation between
and
is given by
, is given by
denotes the zeroth-order Bessel function of the first kind,
is the Doppler frequency,
is the expectation operator, and
denotes the complex conjugate of
.
with
.
denote the average BER on subcarrier
, and let
represent the prescribed target BER. The optimization problem can be expressed as follows:
is the total power budget for all CRUs,
is the maximum interference power that can be tolerated by the PU, and
is a subcarrier assignment indicator, that is,
if and only if subcarrier
is allocated to CRU
. The term
represents the nominal bit rate weight (NBRW) for CRU
, and
. Constraint (11) ensures that the average BER for each subcarrier is below the given BER target. Constraint (12) states that the total power allocated to all CRUs cannot exceed
, while constraint (14) ensures that the interference power to the PU is maintained below an acceptable level
. Constraint (15) results from the assumption that each subcarrier can be assigned to at most one CRU. Constraint (17) ensures that the bit rate achieved by a CRU satisfies a proportional fairness condition.
as an outcome of an independent complex Gaussian variable. For an arbitrary vector
, we have [21] the following:
denotes the identity matrix. Applying (19) to (4), we obtain
,
, and
denotes the channel gain that is fedback to the BS.
in (20) is a monotonically decreasing function of
, we obtain the minimum power requirement while satisfying the constraint in (11) by setting
.
, and the number of loaded bits.
,
,
, and
, the RHS of (20) has the form
. The function
is Rician distributed with Rician factor
[20]. A Rician distribution with
can be approximated by a Nakagami-
distribution [22] as follows:
, where
. Therefore, we approximate the RHS of (20) by
. From (24), we obtain
.
subcarriers are available for allocating to CRUs. We assume equal transmit power for each subcarrier. Let
, the maximum number of bits which can be loaded on the subcarrier is given by
be the number of subcarriers allocated to CRU
. Assuming that the same number of bits is loaded on every subcarrier assigned to a given CRU, the objective in (10) is equivalent to finding a set of
subcarriers to maximize
is the total transmit power allocated to all subcarriers and
is the total interference power experienced by the PU due to CRU signals. The subcarrier allocation problem in (29)–(32) can be solved using the SA algorithm proposed in [24]. Note that we need to make use of (24) in the SA algorithm if only partial CSI is available. A pseudocode listing for the SA algorithm is shown in Pseudocode 1. The algorithm has a relatively low computational complexity
. After subcarriers are allocated to CRUs, we then determine the number,
, of bits allocated to subcarrier
.
be the chromosome of member
in a population, expressed as
denotes the population size. A brief description of the MA algorithm in [24] is now provided.
function selects a set,
, of chromosomes from
in a roulette wheel fashion, that is, selection with replacement.
. Let
denote the crossover probability, and let
,
denote the outcome of an independent random variable which is uniformly distributed in
, then
is selected as a candidate for crossover if and only if
. Suppose that we have
such candidates, we then form
disjoint pairs of candidates (parents).
and
,
, then we obtain the (possibly identical) chromosomes of two children as follows:
denote the mutation probability. For each chromosome in
, we generate
, where
denotes the outcome of an independent random variable which is uniformly distributed in
. Then for each component
for which
, we substitute the value with a randomly chosen admissible value.
chromosomes of parents and offsprings with the best fitness values as input for the next generation.
; number of generations = 20; crossover probability,
; mutation probability,
.
CRUs. The total available bandwidth for CRUs is 5 MHz and supports 16 subcarriers with
MHz. We assume that
and an OFDM symbol duration,
of
s. In order to understand the impact of the fair bit rate constraint in (17) on the total bit rate, three cases of user bit rate requirements with
were considered. In addition, three cases of partial CSI with
were studied. It is assumed that the subcarrier gains
and
, for
are outcomes of independent identically distributed (i.i.d.) Rayleigh-distributed random variables (rvs) with mean square value
. The additive white Gaussian noise (AWGN) PSD,
, was set to
W/Hz. The PSD,
, of the PU signal was assumed to be that of an elliptically filtered white noise process. The total CRU bit rate,
, results were obtained by averaging over 10,000 channel realizations. The
confidence intervals for the simulated
results are within
of the average values shown.
, as a function of the total CRU transmit power,
, for
, and 1 with
,
W, and a PU transmit power,
, of 5 W. As expected, the average total bit rate increases with the maximum transmit power budget
. It can be seen that the average total bit rate,
, varies greatly with
. For example, at
W,
increases by a factor of 2 as
increases from 0.7 to 0.9. This illustrates the big impact that inaccurate CSI may have on system performance. The
curves level off as
increases due to the fixed value of the maximum interference power that can be tolerated by the PU.
, versus total CRU transmit power,
, with
W,
W, and
.
and
are plotted in Figures 2 and 3, respectively. The average total bit rate,
, decreases as the NBRW distribution becomes less uniform; the reduction tends to increase with
.
, versus total CRU transmit power,
, with
W,
W, and
.
, versus total CRU transmit power,
, with
W,
W, and
.
as a function of
for three different cases of
with
,
W, and
W. As to be expected,
increases with
. It can be seen that
for
is larger than for
, and
for
is larger than for
. When the bit rate requirements for CRUs become less uniform,
decreases due to a decrease in the benefits of user diversity. With
W,
increases by about
when
changes from
to
. Results for
are shown in Figure 5 and are qualitatively similar to those in Figure 4. 
, versus total CRU transmit power,
, with
W,
W, and
.
, versus total CRU transmit power,
, with
W,
W, and
.
, is plotted as a function of the maximum PU tolerable interference power,
, with
W and
W, for
and 0.7 in Figures 6 and 7, respectively. As expected,
increases with
and decreases as the CRU bit rate requirements become less uniform. The
curves level off as
increases due to the fixed value of the total CRU transmit power,
.
, versus maximum PU tolerable interference power,
, with
W,
W, and
.
, versus maximum PU tolerable interference power,
, with
W,
W, and
.
to number of subcarriers do
which maximizes
;
as
with
;
to 0;
then
is available; increment
by 1;
is not available;
, initialize the number,
, of
to 0
using (28);
to
do
, of
which minimizes
;
to CRU
;
by one.
;
;
to Number_of_Generatio do
;
;
;
to
;
;
;
to
;
;
.
