Statistical analysis of linear spatial holes estimators in cognitive radio
© Kazemi et al; licensee Springer. 2012
Received: 17 May 2011
Accepted: 5 February 2012
Published: 5 February 2012
One of key features of cognitive radio (CR) networks is environment awareness which is having knowledge of such parameters as spatial holes. This information is employed to exploit the spatial resources more efficiently and limit the interference to the primary users to an admissible level. In order to evaluate the performance of a spatial holes estimation algorithm, statistical characteristics of its estimation error must be compared to a benchmark such as Cramer-Rao lower bound (CRLB). In this article, the performance of cognitive RSS-WLS algorithm which is an important linear spatial hole estimation algorithm in CR systems has been analyzed by obtaining the closed form expression for mean and covariance of its estimation. Then its performance is compared with CRLB and it is shown that cognitive RSS-WLS estimator is asymptotically efficient.
Increasing demand of radio bandwidth in recent years plus inefficient use of licensed bands has encouraged the researchers to devise a mechanism to allocate the radio bandwidth in a more efficient way. One of the most promising ideas to overcome this problem was cognitive radio (CR) which was first introduced by Mitola . A CR secondary network tries to coexist with a primary network. The primary network is a network that owns a licensed radio band and the secondary network is a network which wants to use the same band without causing any harmful interference to the primary network.
One of key features of CR networks is environment awareness which is having knowledge of such parameters as spatial holes. By employing spatial holes like location and power information of primary network users, secondary users can reduce their interference to the primary network by efficient utilization of this information in beamforming and power control algorithms. To choose the best estimator when implementing the cognitive system, or to utilize the hybrid estimators, the error of each spatial holes estimation as a performance measure, should be analyzed by comparing them with a common benchmark like Cramer-Rao lower bound (CRLB).
In this article, it is assumed that there is no cooperation or known signaling between secondary and primary users. Hence, algorithms which need cooperation between primary and secondary networks cannot be used. For example, because of the need of synchronization between primary and secondary networks, time-based algorithms like time of arrival (TOA) algorithm cannot be used in the CR case. The problem which appears in angle of arrival (AOA) algorithms is that although they have no problem estimating the locations of the primary users, they are unable to estimate their transmission powers. Hence among all of the common location estimation algorithms only the received signal strength (RSS) algorithms can be used for simultaneous estimation of locations and powers of the primary users in a CR network. To estimate the location of the primary user, RSS algorithms measure and process the received power of the primary user's signal at the secondary users' receivers.
There have not been a lot of works in the field of simultaneous estimation of location and power, especially in CR networks. In , a linear RSS algorithm based on weighted least square (WLS) method is introduced and analyzed which is called RSS-WLS. In WLS as a generalization of LS method, the matrix of coefficients is multiplied by a weighting matrix to let the system of linear equations be solved using LS method. In , a spatial holes estimation algorithm, namely RoTPE, is proposed based on RSS-WLS algorithm which can estimate primary users' powers and locations simultaneously. This algorithm rearranges RSS-WLS algorithm to add the capability of estimating the power of the primary user in a CR network setup. In , Kazemi et al. introduce cognitive RSS-WLS which improves  by taking input noise and error of path loss exponent estimation into account in addition to shadowing. Although [3, 4] propose an RSS-WLS based simultaneous location and power estimation algorithm for CR setup, they do not do any exact or approximate analytical performance analysis.
In this article, the performance of cognitive RSS-WLS approach of  in an AWGN channel has been analyzed by obtaining the closed form expression for mean and covariance of its estimation error and comparing it with CRLB. in this article, first CRLB of RSS-based algorithms for CR setup in an AWGN channel, is obtained. Then, the closed form expression for mean and covariance of estimation error of cognitive RSS-WLS algorithm is calculated. At the end, based on the results of the previous sections, an analytical performance evaluation of cognitive RSS-WLS algorithm is done and it is shown that cognitive RSS-WLS estimator is asymptotically efficient.
2 Cramer-Rao lower bound (CRLB)
where d i is the distance between the primary user and the i th secondary user, and p is the transmission power of primary user which due to lack of cooperation between primary and secondary networks, is assumed unknown. In Equation (1), α is the path loss exponent, and K i is all other factors that affect the received signal power including the antennas gain and height and the signal carrier frequency.
3 Statistical characteristics of cognitive RSS-WLS algorithm
where F = [2X 2Y ∧ -1 M ], s = X ⊙ X + Y ⊙ Y, φ = [x y P ρ ].
Equation (16) is a system of linear equations in matrix form where F is the coefficients matrix, φ is the column vector of variables, and s is the column vector of solutions. Although path loss exponent is naturally an unknown variable, depending on the chosen system model, it is considered to be known based on a predefined table. For example, in log-distance system model for urban area cellular radio, its value is set to 3. As a result, path loss exponent is not taken as an unknown variable in this article. In presence of the RSS measurement error, variable vector, φ, in (16) can be obtained using WLS method, which the details are discussed in . In brief, in WLS method the weighted squared error due to inequality of both sides of (16) is minimized.
where J(v) is a continuous function of v.
The approximations in (22) and (23) are for sufficiently small noise power variances, means the variance of error of RSS measurements is small enough which is considered in this work. To derive the final relation for (22) and (23) the following relation needs to be computed for each of the expected value term used in (22) and (23).
where the first two rows and columns of C v consist of covariance matrix of positioning error, while the last diagonal element is the covariance of power estimation error. Since cognitive RSS-WLS estimator is unbiased, covariance matrix of estimation error of (36) is also covariance matrix of the estimator. It should be noted that since WLS is a generalization of LS method, the error statistics of RSS-LS method for the same scenario can be obtained by substituting the weighting matrix W with the identity matrix in (23) and (24).
Comparing (14) and (38) shows that covariance of location estimation using cognitive RSS-WLS method is equal to CRLB. since we have obtained this result by assuming the input noise to be sufficiently small, we can say that cognitive RSS-WLS estimator is asymptotically efficient.
To conclude, cognitive RSS-WLS algorithm can be used to simultaneously estimate location and power of the primary user in CR system in an AWGN channel. Through this article, first CRLB of RSS-based algorithms for CR setup in an AWGN channel is obtained, then the closed form expression for mean and covariance of estimation error of cognitive RSS-WLS algorithm is calculated. At the end, based on the results of the previous sections, an analytical performance evaluation of cognitive RSS-WLS algorithm is done and it is analytically shown that cognitive RSS-WLS estimator is asymptotically efficient.
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