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Resource Allocation in MU-OFDM Cognitive Radio Systems with Partial Channel State Information
EURASIP Journal on Wireless Communications and Networking volume 2010, Article number: 189157 (2010)
Abstract
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].
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.
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
where
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
Let 
denote the transmit power allocated to CRU 
on subcarrier 
. For QAM modulation, an approximation for the BER on subcarrier 
of CRU 
is [13]
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
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
where the correlation coefficient, 
, is given by
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]
From (6), the actual channel gain can be written as [20] follows:
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:
subject to
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
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:
where 
denotes the identity matrix. Applying (19) to (4), we obtain
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
with 
. The function 
is Rician distributed with Rician factor 
[20]. A Rician distribution with 
can be approximated by a Nakagami-
distribution [22] as follows:
with 
, where 
. Therefore, we approximate the RHS of (20) by
Then, from (23), we obtain
where 
. From (24), we obtain
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
If a subcarrier is assigned to CRU 
, the maximum number of bits which can be loaded on the subcarrier is given by
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
subject to
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
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.
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.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.
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 
.
Average total CRU bit rate,
, versus total CRU transmit power,
, with
 W,
 W, and
.
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.
Average total CRU bit rate,
, versus total CRU transmit power,
, with
 W,
 W, and
.
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, 
.
Average total CRU bit rate,
, versus maximum PU tolerable interference power,
, with
 W,
 W, and
.
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;
  if
then
   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
. 
 
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Huang, D., Shen, Z., Miao, C. et al. Resource Allocation in MU-OFDM Cognitive Radio Systems with Partial Channel State Information. J Wireless Com Network 2010, 189157 (2010). https://doi.org/10.1155/2010/189157
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DOI: https://doi.org/10.1155/2010/189157