In this section, the novel Max–Min EDbased CSS method is proposed. This sensing technique, considered earlier only basic singlestation sensing, is robust to the NU and yet reduces the complexity in comparison with existing methods with such robustness. The proposed sensing techniques outperform the other advance spectrum sensing methods under NU condition.
A matter of primary interest is to design algorithms that can deliver acceptable spectrum sensing performance with reduced complexity and reliability in terms of detection and false alarm performance. Existing spectrum sensing techniques are not satisfying in this respect. Particularly, sensing in low SNR range, i.e., (− 25 dB, − 10 dB), is challenging due to the noise susceptibility issues and the hidden node problem also exists. To counteract these issues, the spectrum sensing technique has to be more robust to NU, while exhibiting realistic computational complexity for practical implementation. In this section, a novel cooperative Max–Min ED scheme is proposed which reduces complexity and NU. Literature in [7, 16] covers the idea of the frequency diversity gain exploitation with the help of the statistics of the energy spectral density (ESD). Differentiation stage is suggested in the literature as a solution to the NU. However, studies in our group have proposed a solution without the differentiation stage, while maintaining the robustness to NU and exhibiting reduced computational complexity. Results show that the proposed solution outperforms the traditional ED and other detection algorithms. Figure 4 shows the steps implemented in different variants of the Max–Min ED algorithm.
Proposed Max–Min EDbased CSS method
In this subsection, an enhanced Max–Min ED method is considered, which is less complex than existing methods which are robust to NU. Illustration of the methods is as seen in Fig. 4. The maximum and minimum energies of the subbands are utilized for constructing the decision statistics. These statistics are used to estimate the presence and absence of PU. Actually, we consider here three alternative schemes, which are utilizing the subband energies in a different manner [6, 7]. These methods consist of following steps:

SED to calculate subband energies,

Ordering of the determined subband energies,

Differentiation of the ordered subband energy sequence,

Quantification of the maximum and minimum energies level,

Calculation of a threshold and the implementation of decision device.
The proposed method removes the ordering and differentiation blocks from Fig. 4. Hence, the proposed method is less complex yet outperforms the other sensing methods.
The FFT operation on blocks of \(N_{FFT}\) input samples is applied. Alternatively, AFB with \(N_{FFT}\) subbands can be used and this choice is preferred in high dynamic range scenarios. The subband signals are formulated as in Eq. (5). Frequency variability of ESD is featured in Max–Min ED algorithms as depicted in Fig. 4 and the process is summarized as, \(U_k=\frac{1}{L_t} \sum _{m=1}^{L_t}Y_k[m]^2\), where \(L_t=N/N_{FFT}\) represents length of the window. From the central limit theorem, \(U_k\) for both \(\mathcal {H}_0\) and \(\mathcal {H}_1\) hypotheses is expressed as,
$$\begin{aligned} U_k={\left\{ \begin{array}{ll} \mathcal {N}\big (\sigma _{w,k}^2,\frac{2}{L_t}\sigma _{w,k}^4 \big ), &{} \mathcal {H}_0\\ \mathcal {N}\bigg (H_k^2\sigma _{x,k}^2+\sigma _{w,k}^2, \frac{2}{L_t}\big ( H_k^2,\sigma _{x,k}^2+\sigma _{w,k}^2\big )^2 \bigg ). &{} \mathcal {H}_1 \end{array}\right. } \end{aligned}$$
(18)
Maximum and minimum energies are estimated as depicted in Fig. 4 and the test statistics is calculated from the energy values. Test statistic is then compared with a predetermined threshold that is obtained from the target \(P_{FA}\) with the aid of Gumbel distribution. The presence and absence status of the PU signal is determined by comparing the threshold and test statistics. Analytical approach to calculate the thresholds will be given later in Sect. 4.2.
It is noted that differential subband energybased scheme in [16] utilizes the following additional steps upon knowledge of the subband energies:
 \(*\):

Ordering: This step requires placing of the subband energies in the order of magnitude. This has no effect on the statistical properties of the ordered sequence, \(\widehat{U}_{k}\), which follows the distribution in (18).
 \(*\):

Differentiation: The ordered subband energy sequence is differentiated such that \({ D_{k}=\widehat{U}_{k+1}\widehat{U}_{k}}\). This operation can be interpreted as a subtraction of two normally distributed random variables, as shown in (18), yielding
$$\begin{aligned} {D_{k}} \simeq \left\{ \begin{array}{l} \mathcal {N}\left( {0,\frac{4}{L_{t}}\sigma _{w,k}^4} \right) , \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \mathcal {H}_0\\ \mathcal {N}\left( {E[\widehat{U}_{k}]E[\widehat{U}_{k1}],\frac{4}{L_{t}}{{\left( {{{\left H_k \right }^2}\sigma _{x,k}^2 + \sigma _{w,k}^2} \right) }^2}} \right) . \,\,\,\,\mathcal {H}_1 \end{array} \right. \end{aligned}$$
(19)
Under \(\mathcal {H}_0\) hypothesis, the above expression yields a normal distribution with zero mean and twice the variance in (18). It is also noted that when the observed PU is white, the mean value reduces to zero and all subband energies follow a zeromean Gaussian distribution also under \(\mathcal {H}_1\) hypothesis. As a consequence, the algorithm fails to sense the PU.
The energy threshold \(\gamma\) is calculated as in the simplified method. Finally, if \({D_{max}  D_{min}} > \gamma\), the PU signal is assumed present, otherwise only noise is assumed present.
In terms of the computational complexity, the ordering and differentiation steps in the differentiationbased approach bring additional \(\mathcal {O}(N_{FFT})\) and \(\mathcal {O}(N_{FFT})\) complexities, respectively, compared to our proposed simplified method.
Analytical models for Max–Minbased energy detector
In this section, novel analytic expressions for the Max–Min EDbased CSS are formulated. Later, the derived analytical results are compared to simulation results, and a very good match is found between them [7].
Probability of false alarm and energy threshold
Recalling earlier studies, the test statistics depend on the maximum and minimum values of \(U_k\). The statistics of maximum and minimum distribution is characterized by the von Mises theorem [7]. Following these statistics, the Gumbel distribution [23] is used for efficient representation of the extreme values of an arbitrary distribution namely,
$$\begin{aligned} f_{min}(x)=\frac{1}{\beta }e^\frac{x\alpha }{\beta }e^{e^\frac{x\alpha }{\beta }} \end{aligned}$$
(20)
and
$$\begin{aligned} f_{max}(x)=\frac{1}{\beta }e^{\frac{x\alpha }{\beta }}e^{e^{\frac{x\alpha }{\beta }}} \end{aligned}$$
(21)
here \(\alpha\) and \(\beta\) represent the location and scale parameters of the distribution. The expected value and standard deviation of the difference of maximum and minimum values are derived from Eqs. (20) and (21), respectively. Based on the above equation and earlier studies of our group [7], both detection and false alarm probabilities for each sensing station are formulated as Eqs. (23) and (25), respectively.^{Footnote 1}
Using Gumbel distribution with mean and variance values of \(U_k\) in (18) for the \(\mathcal {H}_0\) hypothesis one obtains
$$\begin{aligned} {U_{\max  \min }} _{\mathcal {H}_0}&\sim \mathcal{Q_G} \left( \alpha _{\mathcal {H}_0}, \beta _{\mathcal {H}_0} \right) \\&\sim \mathcal{Q_G}\left( {\frac{{\sigma _{w,k}^2}}{2} + C\sqrt{\frac{6}{{{L_t}}}} \frac{{\sigma _{w,k}^2}}{\pi },\sqrt{\frac{6}{{{L_t}}}} \frac{{\sigma _{w,k}^2}}{\pi }} \right) \end{aligned}$$
(22)
where \(\mathcal Q_G(\alpha ,\beta )\) denotes the Gumbel distribution and C = 0.577215665 is the Euler’s constant. The standard Gumbel complementary distribution function is given by \(\mathcal{G }\left( \frac{x\alpha }{\beta }\right) = 1  {e^{  {e^{  \frac{{x  \alpha }}{\beta }}}}}\) [23,24,25,26,27,28,29].
In the practical environment, both expressions \(P_{FA}\) and \(P_D\) have to consider the NU. It is recalled that the noise distribution is summarized in the range by \(\sigma _{w,k}^2\in [\frac{1}{\rho }\sigma _{n,k}^2,\rho \sigma _{n,k}^2]\) where \(\rho\) is the corresponding NU parameter. Hence, the worstcase false alarm probability is expressed as follows:
$$\begin{aligned} P_{FA}&= \underset{\sigma _{w,k}^2\in [\frac{1}{\rho }\sigma _{n,k}^2,\rho \sigma _{n,k}^2 ]}{\mathrm {max}} \mathcal {G} \Bigg ( \frac{\gamma \bigg (\frac{\sigma ^2_{w,k}}{2}+C\sqrt{\frac{6}{L_t}} \frac{\sigma ^2_{w,k}}{\pi }\bigg )}{\big (\frac{6}{L_t}\big )^{1/4}\frac{\sigma _{w,k}}{\sqrt{\pi }}} \Bigg ) \\&= \mathcal {G} \Bigg ( \frac{\gamma \bigg (\frac{\rho \sigma ^2_{n,k}}{2}+C\sqrt{\frac{6}{L_t}} \frac{\rho \sigma ^2_{n,k}}{\pi }\bigg )}{\big (\frac{6}{L_t}\big )^{1/4}\sqrt{\frac{\rho }{\pi }}\sigma _{n,k}}\Bigg ), \end{aligned}$$
(23)
where \(\rho\) is corresponding uncertainty parameter. Based on Eq. (23), threshold is formulated as,
$$\begin{aligned} \gamma =\mathcal {G}^{1}\big (P_{FA}\big )\bigg (\frac{6}{L_t}\bigg )^{1/4}\sqrt{\frac{\rho }{\pi }}\sigma _{n,k}+\frac{\rho \sigma ^2_{n,k}}{2}+C \sqrt{\frac{6}{L_t}} \frac{\rho \sigma ^2_{n,k}}{\pi }. \end{aligned}$$
(24)
Probability of detection
Similarly, detection probability is derived for \(\mathcal {H}_1\) hypothesis from Eq. (18) as follows,
$$\begin{aligned} P_D&= \underset{\sigma _{w,k}^2\in [\frac{1}{\rho }\sigma _{n,k}^2,\rho \sigma _{n,k}^2 ]}{\mathrm {min}} \mathcal {G} \Bigg ( \frac{\gamma \bigg (\frac{\kappa }{2}+C\sqrt{\frac{6}{L_t}} \frac{\kappa }{\pi }\bigg )}{\big (\frac{6}{L_t}\big )^{1/4}\frac{\kappa }{\sqrt{\pi }}} \Bigg ) \\&= \mathcal {G} \Bigg ( \frac{\gamma \bigg (\frac{\widehat{\kappa }}{2}+C\sqrt{\frac{6}{L_t}} \frac{\widehat{\kappa }}{\pi }\bigg )}{\big (\frac{6}{L_t}\big )^{1/4}\sqrt{\frac{\rho }{\pi }}\widehat{\kappa }}\Bigg ) \end{aligned}$$
(25)
where \(\kappa =E_{max}E_{min}+\sigma ^2_{w,k}\) and \(\widehat{\kappa }=E_{Max}E_{Min}+\sigma ^2_{n,k}/\rho\). \(E_{max}\) and \(E_{min}\) are evaluated as \(E_{max}=\underset{k}{m}ax \big (H_k^2E_k\big )\) and \(E_{min}=\underset{k}{m}in \big (H_k^2E_k\big )\). \(H_k\) and \(E_k\) are PU channel gain and PU signal energy in subband k.
NU introduces severe effects in basic EDbased spectrum sensing methods. Since the observed primary signal PSD is frequency dependent and the noise is additive white Gaussian noise, the proposed maximum–minimum approach eliminates the noise floor. Removal of the noise floor minimizes the uncertainty effects, and hence, the proposed Max–Minbased CSS method is robust is to NU. Later, numerical results for the variation of the detection threshold \(\gamma\) based on Eq. (24) on the proposed Max–Min ED are shown in Sect. 5.2.
Cooperative maximum–minimum energy detection
Analytical detection and false alarm probabilities with Max–Min ED for an individual sensing station are obtained from Eqs. (25) and (23), respectively. Linear fusion rules are applied to combine the sensing results from the sensing stations, each of which applies the Max–Min ED. Here, linear fusion rules for hard decision combining are applied at FC using AND rule, OR rule and Majority rule. Details of these linear fusion rules have been covered in Sect. 2. Cooperative probabilities after implementation of linear fusion rules are obtained from Eqs. (1), (2), and (3).