Elaborate analysis and design of filter-bank-based sensing for wideband cognitive radios
© Maliatsos et al.; licensee Springer. 2014
Received: 12 December 2013
Accepted: 1 May 2014
Published: 3 June 2014
The successful operation of a cognitive radio system strongly depends on its ability to sense the radio environment. With the use of spectrum sensing algorithms, the cognitive radio is required to detect co-existing licensed primary transmissions and to protect them from interference. This paper focuses on filter-bank-based sensing and provides a solid theoretical background for the design of these detectors. Optimum detectors based on the Neyman-Pearson theorem are developed for uniform discrete Fourier transform (DFT) and modified DFT filter banks with root-Nyquist filters. The proposed sensing framework does not require frequency alignment between the filter bank of the sensor and the primary signal. Each wideband primary channel is spanned and monitored by several sensor subchannels that analyse it in narrowband signals. Filter-bank-based sensing is proved to be robust and efficient under coloured noise. Moreover, the performance of the weighted energy detector as a sensing technique is evaluated. Finally, based on the Locally Most Powerful and the Generalized Likelihood Ratio test, real-world sensing algorithms that do not require a priori knowledge are proposed and tested.
Spectrum sensing has been brought into the center of research activities due to its application in the context of cognitive radio (CR) . Cognitive radio and dynamic spectrum access have been identified as the means to maximize spectrum exploitation and efficiency. The cognitive radios share the available spectrum with a licensed primary system (PS) and have the responsibility not to adversely affect the PS user operation by causing interference. Spectrum sensing is used to identify and consequently avoid co-existing primary signals. Several spectrum sensing techniques have been derived and studied [2, 3]. These algorithms present pros and cons concerning the need for a priori knowledge of PS signals features, the computational complexity, the robustness in channel variations and coloured noise etc.
In , the introduction of filter-bank-based sensing was made and further analysis was provided in . The main advantage of this technique lies in the fact that since the CR networks will use multicarrier modulations for transmission, the analysis filters that are intended for receiver operation, e.g. in OFDM-OQAM [6, 7], in discrete wavelet multiTone  and in filtered multiTone , could also be used for sensing without extra computational workload. In , filter-bank-based physical layer design for CR systems was introduced, where simultaneous spectrum sensing and transmission may be possible using the filter bank. In  and , Fahrang-Boroujeny examined the filter bank operation as an estimator of the power Spectral density (PSD) incorporating a spectrum analyser in the receiver structure. A comparison in the performance of the estimation with the non-parametric multitaper method  was also made. It was confirmed that the use of a filter bank can similarly achieve remarkable spectral analysis with the use of a larger set of samples but much less computational complexity.
However, in [4, 5], no detectors were presented that could be practically used to identify primary emissions in a given band of interest. Since then, several studies were made regarding filter-bank-based sensing. Most studies consider simplified models for the operation of the filter bank and do not attain optimality [12, 13]. In , filter bank sensing is performed using data-aided feature detection achieving results in extremely low signal-to-noise ratio (SNR); however, the algorithm required knowledge for specific features of the PS signal. Some studies focus on the reduction of the computational workload [15–17], while others propose new filters suitable for sensing [18, 19]. In addition, special issues have been addressed that concern the application of filter banks in spectrum sensing [20, 21]. Moreover, in [22, 23], efficient implementations of filter bank sensors are presented.
This study is motivated by the absence of strong theoretical description for filter-bank-based sensors. A variety of detectors are presented for uniform and modified discrete Fourier transform (uniform DFT-MDFT) filter banks and especially root-Nyquist filters. As in , filter banks are implemented using the polyphase structure in order to perform simultaneous parallel sensing on all subchannels; however, the proposed algorithms are not based on specific signal feature extraction. Optimality based on the Neyman-Pearson theorem  is achieved. Moreover, the designed sensors are based on the approach that there is no ‘1-1’ matching of the receiver filters and the PS signal bandwidth. This is a common flaw among the majority of filter-bank-based sensing studies. Primary signals are in their vast majority wideband, while the CR filter bank has the objective to divide the monitored spectrum into narrowband channels. Therefore, in a common configuration, a primary channel is spanned and analysed by a number of CR filters. The use of multiple CR filters on a single PS transmission also eliminates the need for frequency alignment between the primary signal and the CR detector. An additional advantage is that the filter bank can partially exploit the radio channel frequency selectivity without the need for complicated equalization procedures on the primary signal. In  and , multiple CR subchannels are also used to span the primary signal. In the first study, a technique for SNR estimation of the PS signal is proposed using simple approximations, while in  the authors introduce a weighted energy detector scheme that is able to efficiently scan Bluetooth channels.
In Section 2, the used system model is presented. In Section 3, the Neyman-Pearson optimal detectors for uniform DFT banks are extracted, while in Section 4, the optimal detectors in the output of an OFDM-OQAM receiver (using an MDFT bank) are studied. In Section 5, the weighted energy detector for uniform DFT banks is analysed, and in Section 6, the extension of the filter bank sensing application in coloured noise is presented. It is noted that coloured noise has been identified  as a significant challenge for spectrum sensing applications. Finally in Section 7, practical implementations of the detectors - the Locally Most Powerful (LMP) test and the Generalized Likelihood Ratio Test (GLRT) - are proposed. Simulation results are presented in Section 8.
2 System model
Moreover, the assumed prototype filter is real and symmetric ensuring linear phase response. Deeper analysis is performed for filter banks using a root-Nyquist prototype filter . The root-Nyquist filters are typically used for pulse shaping in radio communication systems due to the InterSymbol Interference (ISI)-free transmission properties. It is noted that the necessary and sufficient condition for MDFT-PR filters suitable for OFDM-OQAM also leads to root-Nyquist filters .
where is the impulse response for the l th subchannel of the filter bank. In (6), it is assumed that the signal is maximally decimated at the output. Without loss of generality, L is assumed even for presentation purposes. No additional information on primary signal features and specifications is assumed. The next step is the definition of the detectors.
3 Neyman-Pearson optimal detectors for maximally decimated signals
In this paper, the energy detector is used as a reference for the evaluation of the extracted algorithms.
When referring to maximally decimated signals, it is considered that the rate of the signals at the filter output is reduced to the minimum as it is defined by the Nyquist sampling theorem. This means that the specific detector operates at the output of a uniform DFT filter bank (or the output of one of the parallel banks in an OFDM-OQAM system before the real-imaginary separation - Figure 2b). The sensing mechanism can be considered as a vector detection problem with vector observations from L different sensors. Assuming that the k0th primary channel is observed, the output from the filter paths with indexes k0L+l, l=0,…,L-1 is collected.
The first step is to define a single observation vector with the use of a reordering technique for the selected vectors. For reasons that will be cleared out at the next steps, the following reordering is selected:
The observation vectors are divided in two groups:
Group 0 contains the even subchannels.
Group 1 contains the odd subchannels.
Spatial reordering (or row rollout ) is performed (choosing first the group 0 vector). Therefore,(9)
where y l is given in (5). For simplicity reasons, the common term k0L of the vector subscripts was omitted. Subvectors y i represent the observation vector per filter path, i.e. per subchannel.
where f is the probability density function (PDF) for the random variable (operand ‘ /’ indicates a conditional probability). Threshold θ is selected in order to achieve the desired PFA. In order to define the threshold, the distributions of the vector random variables, as well as the distributions of the final detection metric must be defined. Based on the initial assumptions, y is a multidimensional Gaussian variable. The first task is to determine its covariance matrix. Similarly to the energy detector, the SNR, or equivalently the signal and noise powers, should be known. More specifically, the investigated detector should be aware of the SNR per subchannel that is given by . Initially, this information is assumed known. In a real-world design of the detector, an estimation procedure is performed before or during the detection (Section 7).
3.1 Distribution under
It remains to calculate the cross-correlation between the output samples of adjacent filters (l=-1,1).
3.2 Distribution under
If the known (or estimated) quantity is the SNR γ i , the channel coefficients can be calculated from .
3.3 The detector metric
3.4 Metric distribution under
The determinant of Λ can be omitted. In order to move forward, the approach described below is followed:
The inverse of C-1 is also an hermitian matrix.
Thus, the matrix can be seen as the sum of two covariance matrices of multidimensional Gaussian variables. It is known from probability theory  that if two random variables are independent, the covariance matrix of their sum is equal to the sum of the covariance matrices. Let us assume a random variable that follows the distribution and a second random variable that follows the distribution . In addition, the two variables are considered independent. It is assumed that the variables are added and applied to the selected filter bank. Then, the covariance matrix of their sum at the output of the bank will be provided by matrix V.
However, for independent Gaussian variables a different approach can be used that produces equivalent results. The sum of two independent zero-mean Gaussian random variables is also a zero-mean Gaussian random variable. The variance of the sum is equal with the sum of variances . Therefore, instead of considering the sum of two independent random variables applied to the filter bank, it is equivalent to assume a single random variable that follows the distribution .
Based on the second approach, the covariance matrix at the output of the filter will be given by(48)
The extraction of the covariance matrix in this case is performed using the same procedure that was followed in (39), (40) and (41).
Since the two procedures are equivalent for Gaussian random variables, it is expected that the equivalence will also stand between the covariance matrices V and W and therefore they can be used interchangeably. The followed approach is valid, since the eigenvalues extracted by V are applied on the Gaussian random variable . In addition, the validity of the procedure was verified with extended tests.
Thus, instead of calculating the determinant of V, we proceed using matrix W. From (48) and given the fact that the determinant of C is non-zero (since C is invertible), it is extracted that(49)
With the use of V- W equivalence, the metric matrix is considered to have L discrete eigenvalues with N degrees of multiplicity each. The eigenvector matrix does not play any role in the procedure and it does not need to be calculated. Since U is unitary, the random variable will follow the same distribution as z. Therefore, the metric under is given by(50)
In order to verify the validity of the predescribed equivalence, extended tests were performed for various filters and radio channels. The eigenvalues of matrix V were numerically extracted and used directly for metric calculation. It was observed that in all cases, the eigenvalues of V and W were very similar. However, the most important observation was that they produced, as expected, identical distributions when applied to z for metric calculation. Therefore, the equivalence was also verified via simulations.
Threshold θ is calculated for a given PFA with numerical inversion of the function.
Therefore, if the number of coefficients is estimated using (60), then there will be 99.99% probability that the approximated distribution has successfully converged. It is noted that with the use of the described procedure, a very accurate estimate of the PFA from (58) is achieved with absolute error less than 10-5 for PFA>0.25.
Given the fact that based on the aforementioned analysis, the selected threshold θ′ is also discrete, an approximation of the PFA for any threshold can be calculated using linear interpolation.
3.5 Metric distribution under
which leads to .
The calculation of the metric distribution under can be made with the same techniques as before:
Using the analytical solution (55) and (56), where the used eigenvalues are
With FFT-based approximation of the PDF from the characteristic function given by(65)
When the truncation of the infinite sum is done using the ratio of the maximum over the minimum eigenvalue and the empirical formula of (60), then the approximation of the PD value is extremely accurate (absolute error less than 10-4).
4 Detector at the OFDM-OQAM receiver output
The metric distribution can be extracted by
With the use of the analytical method of (55), (56) and (58) using as eigenvalues the detector weights, i.e.(82)
Or using the FFT-based numerical integration with for the outcomes m=0,1.
5 Energy detection in the uniform DFT filter bank
The optimum detector for an MDFT filter bank in Section 4 is proven to be a weighted (per subchannel) energy detector. Although the performance is identical, the uniform DFT optimum detector is much more complicated since it involves multiplications with the matrix C-1. In this section, the use of the weighted energy detector in a uniform DFT is investigated. It can be proved that the weighted energy detector is optimal if R=0; however, this is not possible in a DFT filter bank with a root-Nyquist prototype filter.
The weighted energy detector as a filter bank-based algorithm was presented in , where it was assumed that the subchannel filter outputs are uncorrelated and that the metric performance is determined using Gaussian distributions. It is also claimed that the weighted energy detector operation with optimal weights is similar to the maximum ratio combining (MRC) for independent variables; however, the optimal weights are not determined. The MRC consideration is proved to be the correct and optimum approach for the OFDM-OQAM detector in Section 4, where filter outputs are uncorrelated. In this section, an extension and generalization of the weighted energy detector is performed. The subchannel filter outputs are considered correlated, a fact that is inevitable for uniform DFT filter banks. Moreover, accurate and analytic distributions for the metric under and are extracted based on probability theory.
Since independence between samples from the adjacent filters yi+1 and yi-1 is not possible, the performance of the detector will be inferior to the detector in Section 3. The hysteresis of the detector depends on the transition bandwidth and the roll-off of the prototype filter.
where the q n s are the normal correlations for variables (zk 1,zk 2). Jensen’s result is valid for real random variables with standard normal distribution. However, it can easily be proved through the correlation coefficient that the same formula is valid for 2N-length real normal variables with μ=0,σ2=1/2. Since a N-complex normal vector is equivalent with a 2N-real vector, the Jensen’s corollary can be used for the current analysis.
The correlation coefficient is equal to the mean of the R2 eigenvalues, i.e.
6 Coloured noise
where BW is the overall bandwidth.
The result in (94) shows that with simple weighting of the observation samples by , the problem is transformed to the equivalent Neyman-Pearson detector with AWGN. More specifically, the equivalent AWGN problem is defined with and .
7 Practical implementation of the detectors
7.1 The locally most powerful test
The main disadvantage of the sensing techniques described in the previous sections is that the detector should know the signal variance for each subchannel (the received signal power and the channel coefficient). In real-world implementations, the optimal detectors can be approximated with the use of estimates; however, these detectors are only optimal asymptotically. In case the information for the signal variance is missing or primarily when the subchannel variances are extremely small (linearly expressed SNR →0), then the locally most powerful test can be used for detection. A composite detection problem is considered to have a uniformly LMP detector, when both the metric and the threshold for a given PFA do not depend on unknown parameters (in this case the variances). Unfortunately, these cases are very rare.
Since θ remains unknown, the accurate calculation of PD is not possible for the LMP test.
which is the simple energy detector. The PFA can be calculated from (103) with σ w =1. Therefore, the MDFT filter bank has a uniformly LMP test.