In this section, we will first define a cooperative cognitive radio system model, and then formulate the WSR maximization problem by introducing the objective function, problem constraints, and the optimization variables based on the system model.
We consider an OFDM-based multicarrier cooperative cognitive radio system, in which both primary and secondary users share the same bandwidth for their transmissions. The secondary system model contains a base station (BSS), a single user (SU) and a DF relay node (R), which transmits simultaneously with a primary system also containing a base station (BSP) and one user (PU). The secondary transmission consists of two hops. In first hop (time slot) BSS transmits over subcarrier k, while both the relay and SU receive the information. In second hop the half-duplex relay retransmits the information which has been received in the first time slot. The relay uses a Decode-and-Forward (DF) method, and transmits over a different subcarrier m, in second hop. Therefore the secondary system transmission lasts two time slots using a Subcarrier Pair denoted as SP (k, m). So if we divide the total frequency band to M subcarriers we will have M subcarrier pairs for secondary system transmission. The secondary system’s downlink destination (SU) receives the information from BSS and the relay node using a Maximal Ratio Combiner (MRC) receiver to exploit spatial diversity.
As shown in 
denote the secondary BS (Source) to SU (Destination) downlink channel gain on subcarrier k, 
and 
denote channel gains of BSS to Relay on subcarrier k, and relay to SU on subcarrier m respectively. Also let 
denote the primary system downlink channel gain on subcarrier f. The channel gains from BSS and the relay to PU are defined as 
and 
on subcarriers k and m respectively. As shown in 
, 
, 
, and 
are the desired channel gains (solid lines) while 
and 
causes unwanted interference to PU (dashed line). Also let 
denote interfering channel gain from primary BS (BSP) to SU. It’s assumed that all channel gains are independent from each other, and remain constant in a two-slot period. Moreover, all the desired channel gains of the secondary system and the interfering channel gains from secondary to primary system which are
, 
, 
, 
, and 
respectively, are assumed to be perfectly known in BSS. The BSS performs subcarrier pairing and power allocation, and informs the relay node and the SU of the corresponding parameters through a control signaling prior to data transmission. In next section we will formulate our optimization problem based on the system model shown in
The aim of this paper is to maximize the downlink throughput of secondary system. To formulate our optimization problem we have to define our objective function, constraints, and optimization variables. Similar to [
for a given subcarrier pair SP (k, m). Also let 
and 
denote the transmission power of secondary BS in the first time slot and the relay in the second time slot, respectively. Without loss of generality and to make the mathematical formulas simpler in notation, we assume that all the channel gains shown in 
as
where 
is the channel gain from secondary BS to SU on subcarrier k and 
is the variance of Additive White Gaussian Noise (AWGN) on the corresponding subcarrier.
Since a relay node is used in secondary transmissions, the subcarrier pair SP (k, m), may works in two different modes depending on relay’s activity: direct-link mode and relay mode. In the relay mode the DF relay will forward the message in the second time slot on subcarrier m, while in the direct-link mode the message will be sent only via subcarrier k of SD link in the first time slot, and the relay is not used since it will not improve the throughput.
Now that the parameters are defined, we can formulate the achievable weighted rate of secondary downlink for SP (k, m) as [14,15]
where 
is a weighting factor to represent Quality-of-Service (QoS) requirements on subcarrier k. Also the rate is scaled by 
since the transmission lasts two time slots.
Introducing total power constraint in secondary system as 
for SP (k, m), it is shown [
Similar to [
and 
in (2)
and defining the equivalent secondary channel gain 
as
the expression in (2) can be re-written in a unified format as [
Also we define 
which indicates whether SP (k, m) is selected or not. Now that we have defined our objective function and the required variables, the WSR optimization problem can be formulated over variables p and t as follows
where p and t are 
matrices with entries 
and 
for each SP (k, m), and 
is the total power constraint of secondary system, and 
is the maximum tolerable interference vector for primary user over each subcarrier. Constraints (8)-(12) are corresponded to secondary system limitations, while constraints (13) and (14) are the primary user interference temperature thresholds.
to be an integer (either 0 or 1), it will be relaxed to
as the total power constraint. To satisfy constraints (13) and (14) and make it in correspondence with
, let 
denote the maximum total allowed power over subcarrier pair SP (k, m). Using expressions in (4) and combining constraints (13) and (14), 
can be immediately derived as
in [P1], the relaxed problem becomes
. The new relaxed problem is a convex optimization problem since its objective function is concave, and the constraints are convex sets [
may not be an integer. Using the same approach in [
which satisfies (11). This solution will be an upper bound for problem (17).
is the Lagrange dual variable, and 
is the dual Lagrange vector with entries
. The dual problem will be
.
, by maximizing 
over 
will be
. Also called as “multi-level water-filling” because of weighting factor wk.
, we will rewrite the Lagrangian function (20) as (24)
(24)
and 
are independent of t. 
represents the achievable rate for selected subcarrier pair SP(k,m) including cost of selecting subcarrier m in second time slot (
) and price of power consumption (the last term in
). Hence, the optimum pair for subcarrier k in first time slot is subcarrier m in second time slot which maximizes
. It can be readily derived that
and 
are given we derived 
and 
in (23) and (25) respectively, and the interior part of our dual optimization problem is solved. The next step is to solve the exterior part which is finding the optimum values of 
and 
which satisfy constraints (8) and (9). The following iterative equations will obtain the optimum values of 
and 
using the sub-gradient method
is the step size in the ith iteration. Note that the step size of (26) and (27) can be different. In each iteration the values of 
and 
can be updated using the new values obtained for 
and 
in (23) and (25). As the iteration advances, (26) and (27) will converge and the optimum dual variables will be obtained.
) [
and
, assuming an initial value for dual Lagrange variables, and second part is finding the optimum values for the
and 
to minimize (21) with the values of 
and 
obtained in the first part. In section 3.1 a closed form solution for the first part is derived in equations (23) and (25). While for the second part a sub-gradient method is discussed to find the optimum values of 
and 
using iterative equations (26) and (27). In Algorithm 1, we will first initialize the required variables. In each iteration, we will first calculate the optimum value of 
and 
using equations (23) and (25), and then find new values for 
and 
with the corresponding equations. The loop will continue until the dual Lagrange parameters converge to their optimum values which minimize the constraints cost. Since constraint (9) is guaranteed by equation (25), while constraint (10) may be violated, an amendment is used to avoid selecting the same relay subcarrier for more than one source subcarriers, which results in more than one “1”s in a column of t and violate constraint (10) [
to maximize WSR for secondary system while the interference threshold of the primary network is not violated. The optimum subcarrier pairing scheme and power allocation which maximize WSR of secondary system while preserving QoS of primary network in an OFDM based cooperative cognitive network is obtained by using this algorithm.
, 
, 
, 
, 
, 
, and 
are independent of each other, and remain constant in a two-slot period. Also it is assumed that the channel gains are independent and identically distributed (i.i.d.) over subcarriers and follow a Rayleigh distribution with parameter b differs from one link to another corresponding to the mean square of subcarrier channel gains in different scenarios. The variance of Additive White Gaussian Noise (AWGN) is assumed to be one.
, and
. The convergence criterion is
, and the step size for the subgradient method is set to be
, where i is the iteration index. In Un-Weighted Sum Rate (UWSR) scenario, the weighting factor 
is set to be all “1” for different subcarriers, and in Weighted Sum Rate (WSR) scenario, it obeys the equation
. The results are obtained based on 1000 randomly generated channel realizations.
, 
, and
.
[Watts], and the maximum tolerable interference for primary user is equal to 
over all subcarriers. In this figure the WSR of the Fixed Subcarrier Pairing (FSP) method—where 
for every subcarrier pair
—is compared to the Joint Subcarrier Pairing and Power Allocation (JSPPA) approach. In JSPPA1 we only apply the maximum power constraint 
for Constrained Water-Filling in the last part of Algorithm 1, while in JSPPA2 the 
is considered in both subcarrier pairing and power allocation to find optimum values 
and 
respectively. As it is shown in 
subcarriers. 
, 
, and
.
subcarriers is 9%.
for 
subcarriers. And 
subcarriers and fixed total power 
[Watts]. Both Figures 6 and 7 show that the JSPPA method outperforms the FSP.