Model Order Selection and Meta Analysis-Based Cooperative Wideband Spectrum Sensing

This letter introduces two innovative solutions for cooperative wideband spectrum sensing (WSS) that obviate the requirement for prior knowledge of noise power at the sensors and primary users (PUs) signals. The first method employs an information theoretic criteria (ITC)-based approach, presenting a threshold-free solution. The second method harnesses sensor cooperation through a novel mixture detector based on meta-analysis, a statistical method that combines results from multiple independent tests. To evaluate the efficacy of the proposed detectors, we conduct a comprehensive case study that considers shadowing effects and frequency-selective multipath channels between PUs and sensors. Our results demonstrate that the two WSS methods exhibit remarkable detection performance, particularly in low signal-to-noise ratio (SNR) regimes, outperforming a set of machine learning-based state-of-the-art solutions.


I. INTRODUCTION
I N RECENT years, the escalating demand for higher data rates has driven wireless technologies to their limits, resulting in congestion within the limited spectrum resources.In this context, spectrum sharing emerged as a promising solution to enhance spectrum efficiency.This approach hinges on the user's ability to detect unused spectrum bands, a process commonly referred to as spectrum sensing.In particular, the problem of evaluating the occupancy state of a large number of portions of the spectrum, commonly known as wideband spectrum sensing (WSS), has arisen [1].A large number of WSS techniques have been proposed in the last decade.Various high computational complexity sub-Nyquist techniques for cooperative WSS that achieve good performance in a high signal-to-noise ratio (SNR) regime and under certain sparsity constraints can be found in the literatures [1] and [2].In [3], a cooperative narrowband energy detection method exploiting the observations of multiple secondary users (SUs) having different noise power is proposed.Each SU sends its locally derived basic mass assignment (BMA) to the fusion center (FC), which performs combining via Dempster fusion rule.Recently, artificial intelligence (AI)-based techniques for spectrum sensing have gained momentum.In [4], several classical machine learning techniques for cooperative narrowband The authors are with the Department of Electrical, Electronic, and Information Engineering "Guglielmo Marconi" (DEI), CNIT, University of Bologna, 40126 Bologna, Italy (e-mail: luca.arcangeloni2@unibo.it; enrico.testi@unibo.it;andrea.giorgetti@unibo.it).
Digital Object Identifier 10.1109/LCOMM.2024.3375124spectrum sensing are proposed, including K-means clustering and Gaussian mixture model (GMM).In [5], an unsupervised deep clustering approach for cooperative narrowband spectrum sensing is proposed.It consists of a sparse autoencoder to learn hidden features from energy levels locally computed at each SU and aggregated at the FC.The autoencoder is then followed by GMM-based clustering and binary hypothesis test.However, this interesting approach does not consider SUs with different noise power and requires a priori identification of the clusters corresponding to the noise and the primary user (PU) signals.In [6], two parallel convolutional neural networks (CNNs) for WSS are proposed.The I/Q received samples are split into two parts and processed concurrently by the two CNNs.Their outputs are fed into two fully connected layers, and WSS is approached as a supervised classification problem.Therefore, we present two cooperative WSS methods that detect the spectrum holes with high accuracy.The aim is to maximise spectrum utilization whilst minimising the interference between PU and SU.The contributions of this work are the following: • We propose two novel solutions for cooperative WSS that do not require sparsity assumptions nor prior sensors' noise power and PUs signals knowledge.The first leverages an information theoretic criteria (ITC)-based approach that leads to a threshold-free solution, while the second exploits sensor cooperation through a mixture detector based on meta-analysis, i.e., a statistical method that combines the outcomes of multiple independent tests.• To assess the performance of the proposed solutions, we investigate a case study that accounts for shadowing and frequency-selective multipath channels between the PUs and the sensors, proving that the proposed WSS methods achieve high detection performance in the low SNR regime, compared to the state-of-the-art solutions.
Throughout the letter, capital and lowercase boldface letters denote matrices and vectors, respectively.With v i,j , v i,: , and v :,j , we represent, respectively, the element, the ith row, and the jth column of the matrix V. X ∼ χ 2 m is a central chi squared distributed random variable (r.v.) with m degrees of freedom (d.o.f.), X ∼ N (µ, σ 2 ) is a Gaussian r.v. with mean µ and variance σ 2 , and Z ∼ CN (0, σ 2 ) is a zero-mean circularly symmetric complex Gaussian r.v. with variance σ 2 .We denote the cumulative distribution function (CDF) of the r.v.X with F X (x) and the expectation operator with E[X].We use Q(x) for the Q-function value in x.The operations (•) T , and (•) H indicate transpose and conjugate transpose, respectively.We use the big O notation, O(•), to denote the computational complexity of algorithms.

II. SYSTEM MODEL
Let us consider a scenario where N R sensors observe the same frequency band and share the frequency domain representations of the received signals.Their goal is to understand which frequency band segments are currently utilized by PUs and to identify those that can be classified as spectrum holes.We assume that the frequency domain representations are shared with a FC, which makes the final decision.The proposed cooperative WSS techniques are based on the observation of N S independent frequency domain vectors collected by repeated measurements from each sensor,1 each with N B frequency components, x j,NB ) T with i = 1, . . ., N S and j = 1, . . ., N R .Without loss of generality, we consider equivalent low-pass signals.The received signal, in time-domain, at the ith observation in the jth sensor is denoted as the vector of length N B r (i) where ξ (i) j,: and ν (i) j,: are the aggregation of PUs signals and additive white Gaussian noise (AWGN) with power σ2 j , respectively.The corresponding output of the discrete Fourier transform (DFT) at the ith observation in the jth sensor is denoted as where s , where σ2 j = N B σ2 j is the noise power per frequency bin.An overview of the data gathering process is presented in Fig. 1.If PU signals are present in the observed frequency band we consider that they occupy k * bins, while the remaining N B − k * contain only noise.The objective of cooperative WSS is to identify the k * occupied bins.

III. MODEL ORDER SELECTION
For many communication signals, such as the widely adopted orthogonal frequency-division multiplexing (OFDM), where Q is the set of occupied bins with cardinality |Q| = k, σ 2 j,q is the qth diagonal element in Σ j , X j ∈ C NS×NB is the matrix whose ith row is x (i) j,: , and θ (k) j is the vector of unknown parameters of length ϕ j (k), in which the dependence on k is emphasized. 3In our problem, we have θ

A. Hard Decision Combining
In this section, we perform per bin spectrum sensing independently at each sensor and then combine the hard decisions (HDs) at the FC.At the sensor, to detect the frequency bins containing the PU signal we formulate the problem as model order selection (MOS) in which k * is the order of the model.For solving the MOS problem, we adopt ITC, a statistical approach for choosing the most suitable model for the observed data among a set [8].Let us assume that at least one of the observed frequency bins contains only noise, such that the kth model corresponds to the case in which the first k ordered bins are occupied, with k ∈ {0, . . ., N B − 1}. 4  The proposed scheme estimates k * at each sensor and also identifies the occupied bins for that sensor.Let us consider the operations performed at the jth sensor S j .
Let us sort all vectors x (i) j,: according to the estimated power of each bin, σ 2 j,q = (1/N S ) 2 , so that are arranged in descending order.We denote with X j ∈ C NS×NB the ordered matrix whose ith row is x(i) j,: .According to ITC, the model that better fits the observed data is the one that minimizes the penalized likelihood [8], [9] 3 When q / ∈ Q then σ 2 j,q is equal to the noise power σ 2 j in the jth sensor. 4We refer to the kth model also as the kth hypothesis.
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is the vector of the estimated parameters in the kth hypothesis, and P(k) is the ITC penalty.The vector of unknown parameters can be estimated as θ j,q | 2 , and Therefore, the log-likelihood in (4) can be expressed as [9] ln f X j ; θ where we omitted all the terms that are not a function of k.
By incorporating ( 5) and ( 6) in ( 4) and performing the minimization, MOS provides an estimate of k * , k, which allows to construct the vector ω j composed by the concatenation of k ones and N B − k zeros.Lastly, by means of reverse ordering, we obtain the occupancy vector ω j estimated at the jth sensor.The qth frequency bin is declared occupied if the corresponding element in ω j , namely ω j,q , equals 1.
With hard decision combining by K-out-of-N rule, the qth element of the occupancy vector ω (see Fig. 1) is calculated as where ω th ∈ [0, N R − 1] is a threshold to control the trade-off between missed detections and false alarms.The computational complexity for HD combining is O(N R N S N 2 B ).

B. Soft Decision Combining
A possible alternative to HD is soft decision (SD), where the FC performs a unique decision based on the soft information received from the sensors.In this scenario, each sensor computes the vector σ 2 j,1 , σ 2 j,2 , . . ., σ 2 j,NB and sends it to the FC.Then, the FC combines the estimated powers, e.g., by calculating the sum σ 2 q = NR j=1 σ 2 j,q for q = 1, . . ., N B .Similarly to the HD case, the vector of unknown parameters can be estimated as θ Therefore, the log-likelihood can be expressed as where X is the ordered tensor with X j the corresponding frontal slice.By minimizing −2 ln f X; θ + P(k) with respect to k, as in (4), followed by reverse ordering, we can find an estimate of k * , k, which allows to construct ω in Fig. 1.The computational complexity for SD combining is O(N S N 2 B ).In both SD and HD, the choice of the penalty P(k) defines the criterion adopted, determining its performance and complexity [8].In this work, we consider the Akaike information criterion (AIC) where P AIC (k) = 2 (k+1) and the Bayesian information criterion (BIC) with penalty P BIC (k) = (k + 1) ln N S .Furthermore, we consider a penalty in the form P GIC (k) = (k + 1) ν, called generalized information criterion (GIC), where ν can be a constant or a function of the system parameters [9].
IV. META-ANALYSIS Meta-analysis refers to the synthesis of data from multiple independent tests.In this section, we combine the detection performed by each sensor in a mixture detector by using metaanalysis.In this scenario, each sensor performs a statistical hypothesis test based on the frequency domain observations of a single frequency bin; then, the outcomes of N R binary hypotheses tests are combined to determine the presence or absence of a signal in that bin.The procedure is repeated for each frequency bin.The meta-analysis relies on the evaluation of the p-values, which represent the probability of observing data that is at least as extreme as the data currently observed, assuming that the null hypothesis H 0 is correct.They are indicators of the level of statistical significance achieved by a hypothesis test.The p-value for test j at bin k is p j,k = P(V ≥ v|H 0 ), where V is the r.v.representing the test statistic and v is its observation.
In the following, we derive closed-form expressions for the p-values and then we present two different meta-analysis strategies.From (2), a binary hypotheses model for the jth sensor and the kth bin can be formulated As already mentioned, for many communication signals (e.g., OFDM), the received samples can be modeled as zero-mean complex Gaussian r.v.s, such that x Let us assume that we are able to estimate the noise power, e.g., from the N B th frequency bin obtained after ordering as described in Section III.Then, a test statistic is constructed where we chose to approximate V to a shifted Gamma distributed r.v.[10].Thus, G ∼ G(v, ψ) is a Gamma r.v. with shape parameter v and scale parameter ψ, and α is a constant.The parameters are obtained via moments matching as in [9,Appendix].Since the observation dataset is finite, we compute the p-value via the cumulative distribution function (c.d.f.) as

A. Fisher's Method
According to Fisher's method, the p-values are combined as [11] and [12] where denotes the mixture detector test statistic for the kth bin.It is possible to compute the mixture p-value Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
via the c.d.f. as p Finally, for each frequency bin the following test is carried out where p FA is the predefined false alarm probability set during system design.

B. Weighted Z-Transform
According to the z-transform method, each p-value, p j,k ∈ . 5 Then, the z-values are combined to obtain the mixture detector test statistic where w j is the weight assigned to z j,k . 6Finally, the mixture p-value is computed as and binary hypothesis test is carried out as for Fisher's method Both Z-transform and Fisher methods have a computational complexity of O(3N S N B ), which is mostly determined by the computation of V in (10).

V. CASE STUDY AND NUMERICAL RESULTS
In this section, we evaluate the performance of the proposed cooperative WSS techniques in a realistic scenario, accounting for multiple PUs and channel impairments such as path-loss, multipath propagation (frequency-selectivity), shadowing, and unequal noise power at the sensors.
Without loss of generality, we consider all frequencies normalized with respect to the sampling frequency at the receivers.The scenario consists of two PUs which emit independent band-limited Gaussian processes mimicking the ubiquitous OFDM signals, with normalized center frequencies f 1 = 0.3 and f 2 = 0.8, respectively, and normalized bandwidths B 1 = B 2 = 0.3.The N R sensors experience AWGN with different noise powers.The parameter ∆σ 2 N,dB represents the noise power spread among the N R receivers, i.e., the difference among the noise power of sensor S 1 and sensor S NR is ∆σ 2 N,dB , and such spread is equally distributed among the remaining sensors.For example, if ∆σ 2 N,dB = 2 and N R = 5 then the noise power at the receivers in ascending order from sensor S 1 till sensor S 5 is σ 2 N,dB + [1, 0.5, 0, −0.5, .1]where σ 2 N,dB is the nominal noise power.
Because of the wideband nature of the signals and receivers involved, we consider frequency-selective multipath channels between the PUs and the sensors.In particular, we consider the International Telecommunication Union (ITU) Extended pedestrian A channel model, which consists of 5 Could negative opinions from male soldiers affect women's pride in wearing the U.S. Army uniform?To answer this question, Stouffer proposed the z-transform method in footnote 15 of [14, p. 45]. 6The classical z-transform method can be obtained assigning equal weights to all the z-values, such that seven independent Rayleigh distributed paths, with average power gains [0, −1, −2, −3, −8, −17.2, −20.8] dB and normalized (with respect to the sampling frequency) path delays [0, 30,70,80,110,190, 410] • 10 −3 , respectively [15].Each PU-sensor link is also subject to log-normal shadowing with parameter σ S,dB . 7Moreover, to account for different path-loss experienced by PU-sensor links we define a nominal SNR, SNR, and the SNR spread, ∆SNR dB , i.e., the maximum difference among the SNRs at receivers.For MOS HD and SD combining, the GIC penalty was set to ν = 12.25 based on the proposed method in [16], considering P MAX over = 0.1.1) MOS With HD: Fig. 2 shows the accuracy of MOS with HD for different values of ω th .In particular, ω th = 0 and ω th = 4 mean that a bin is considered occupied if at least one or all the 5 sensors detect a PU signal, respectively.AIC penalty provides good performance when ω th = 1.On the contrary, both BIC and GIC reach the best accuracy for ω th = 0.In particular, for low SNR values BIC performs better than GIC.Hence, in the following tests we compare the performance of the meta analysis with that of BIC with HD and ω th = 0.
2) MOS With SD: The accuracy for AIC, BIC and GIC, using SD combining is shown in Fig. 2. It is evident that AIC penalties provide the best performance, although it does not outperform BIC with HD.
3) Meta-Analysis: Fig. 3a shows the P d of the meta analysis-based methodology varying SNR for P fa = 0.05.For the Z-transform, three variants are considered: equal weights for all the sensors, w 1 = [0.05,0.1, 0.2, 0.25, 0.4], and w 2 = [0.1,0.1, 0.2, 0.3, 0.3].The weights are assigned sequentially, such that the sensor with the highest noise power has the lowest weight.The Fisher's method yields higher detection probability compared to the Z-transform, which exhibits a slight performance increase when using weights.
4) State-of-the-Art: We compared our WSS algorithms against three state-of-the-art counterparts, namely the GMM and K-means-based solutions proposed in [4], and the cascade of an autoencoder and a GMM proposed in [5].The algorithms from [4] require prior knowledge of the number of PUs and  are fed as input with the matrix of received power samples P ∈ R NB×NR , where p k,j = 1 NS NS i=1 |x (i) j,k | 2 .Fig. 3a and Fig. 3b show how our methods outperform the considered solutions for a large range of SNR values, confirming that Fisher meta-analysis approach provides the best performance.

B. Number of SUs
The performance of cooperative WSS strongly depends on the number of SUs performing joint detection.In Fig. 3c, P all d and P d varying the number of sensors from 2 to 10 with SNR = −7 dB are shown.We kept ∆SNR dB and ∆σ 2 N,dB constant, such that while the number of SUs increases, the maximum difference among the SNRs at receivers remains constant.As expected, performance improves as the number of SUs increases; again, BIC with HD and ω th = 0, and Fisher meta-analysis approach achieve the best performance.
VI. CONCLUSION In this letter, we propose two novel methods for cooperative WSS based on MOS and meta-analysis.Extensive results on the detection performance varying the most sensitive parameters, including nominal SNR values and the number of sensors, are provided.The results demonstrate that Fisher meta-analysis approach attains the best performance, achieving a detection probability P d over 90% in the low-SNR regime (i.e., SNR = −9 dB).Moreover, remarkable performance is also obtained with the MOS-based method when HD is combined with BIC, achieving a detection probability P d over 90% for SNR = −8 dB.Both solutions with appropriate parameter settings outperform a set of state-of-the-art algorithms.

Manuscript received 26
February 2024; accepted 6 March 2024.Date of publication 7 March 2024; date of current version 10 May 2024.The associate editor coordinating the review of this letter and approving it for publication was M. Morales-Céspedes.(Corresponding author: Enrico Testi.)

Fig. 1 .
Fig.1.The wideband spectrum sensing scheme: an illustration of the data gathering process and decision making.
For example, if ∆SNR dB = 10 and N R = 5 then the SNRs at the sensors are SNR dB + [5, 2.5, 0, −2.5, −5].For each parameter setup, 10 3 Monte Carlo trials are carried out over the channel realizations (noise, multipath, shadowing) to obtain averaged performance.The performance of the proposed cooperative WSS techniques is assessed considering the following metrics: P d and P fa are the per bin detection and false alarm probabilities, respectively, P all d is the probability that all the bins containing the PUs signals are detected, and β is the accuracy computed by dividing the number of correct detections by the total number of detections made.If not otherwise specified, in the following results we consider N S = 100, N B = 512 bins, N R = 5 sensors, ∆SNR dB = 10, ∆σ 2 N,dB = 4, and σ S,dB = 3. A. Performance Varying the SNR In Fig. 2, Fig. 3a, and Fig. 3b, P d , P all d and β are shown for different nominal SNR values.

Fig. 2 .
Fig. 2. Accuracy of the MOS methods with HD and SD rules for different SNR values.

Fig. 3 .
Fig. 3. (a) P d with meta-analysis is shown for different SNR values.We also plot the performance for BIC-HD with ω th = 0 and AIC-SD.In orange, violet and pink there are the performance for state-of-the-art solutions.(b) P all d as a function of SNR.(c) P d and P all d for different N R values.