Elsevier

Physical Communication

Volume 32, February 2019, Pages 172-184
Physical Communication

Full length article
SNR wall for generalized energy detector in the presence of noise uncertainty and fading

https://doi.org/10.1016/j.phycom.2018.11.013Get rights and content

Abstract

The performance of energy detection (ED) degrades under low SNR, noise uncertainty (NU) and fading. The generalized energy detector (GED) is obtained by changing the squaring operation in ED by an arbitrary positive number p. In this paper, we investigate the signal to noise ratio (SNR) wall for GED under diversity considering NU as well as fading by considering p-Law combining (pLC) and plaw selection (pLS) diversity. First, the SNR walls considering AWGN channel are derived. It is shown that for pLC diversity, increasing p results in lower SNR wall. It is also shown that under no diversity and pLS diversity, the SNR wall is independent of p. The analysis is then extended to the channel with Nakagami fading where it is shown that the SNR wall increases significantly. As a byproduct of this work, we also study the effect of NU and fading on the detection performance and show that above certain value, the effect of NU is more severe when compared to the fading. The effect of p on the performance is analyzed and it is shown that the performance is the best for values of p close to 2. The performance of pLC and pLS is also compared.

Introduction

The radio frequency (RF) resource is scarce since most RF bands are already licensed to the primary users (PU). It has been observed that most of the licensed bands are underutilized [1] resulting in the wastage of resources. Cognitive radio (CR) can provide the solution to spectrum scarcity by allowing the unlicensed secondary users (SU) to access the licensed bands when the PUs are not using them [2]. Recent research shows the applications of CR in Internet of Things (IoT) and fifth generation (5G) wireless communication [3]. To access the unoccupied licensed band, SU needs to obtain the occupancy status of the PU, which is called spectrum sensing (SS) in the literature. Spectrum sensing is one of the crucial tasks of the CR.

In literature different techniques for spectrum sensing have been investigated [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. The method of energy detection (ED) [13], [14], [15], [16] does not require prior knowledge of the PU and is easy to implement. Due to its simplicity, ED represents popular SS technique even though other techniques may exhibit better performance. The performance of ED under fading channels and diversity receptions are widely studied [14], [15], [19], [20], [21], [22], [23] but they do not consider noise uncertainty. In [24], the conventional energy detection (CED) which is commonly referred to as ED is generalized by replacing the squaring operation of received signal amplitude by an arbitrary positive power p, which is referred to as the generalized energy detector (GED) [25], [26] or the improved energy detector [24], [27], [28] or p-norm detector [29], [30]. In [24], [28], [31], it is shown that performance of the CED can be improved by choosing a suitable value of p which depends on the probability of false alarm, the signal to noise ratio (SNR) as well as on the number of samples used for computing the decision statistic.

In GED, the decision on the occupancy of a channel is made based on the predefined threshold, which can be determined by the noise variance and hence it plays an important role in determining the performance of the detector. One has to know true noise variance to determine the value of this threshold. If known exactly, it is possible to sense the occupancy of PU even at very low SNR if the sensing time is made sufficiently large [4], i.e., a large number of samples (N) are used in sensing. In practice the noise variance varies with time as well as location and hence it is not possible to find its exact value. Due to this, there exists unpredictability about the true variance of noise which is known as noise uncertainty (NU). The effect of worst case noise uncertainty is discussed in [32], [33]. In [32], a phenomenon called SNR wall is studied for CED, which says that if the noise variance is not known exactly and is confined to an interval, it is not possible to achieve targeted detection performance when the SNR falls below certain value regardless of sensing time. This makes ED an inefficient sensing method.

The effect of uniformly distributed NU on the performance of CED is studied in [34], [35]. The authors in [34] derive the expression for SNR wall for CED assuming a uniform distribution for NU. In [36], an asymptotic analysis of noise power estimation is performed for CED. Here, the condition for existence of SNR wall is obtained and the effect of noise power estimation on the performance of CED is studied. In [25], [26], the performance of GED is studied under NU where in the authors in [25] show that under worst case of NU the SNR wall is not dependent on value of p. It is also shown that under the assumption of uniform distribution of NU, the CED represents the optimum ED. The expression for SNR wall is obtained in [26] where the NU is once again chosen as uniformly distributed. It is also shown that the SNR wall does not depend on the value of p. The study of the detection performance is then extended to NU having log normal distribution and the SNR wall for the same is calculated numerically. Note that, authors in [25], [26] have derived the SNR wall for GED under NU without considering the diversity. The SNR wall for cooperative spectrum sensing (CSS) with CED assuming the same SNRs and NUs for all the cooperating secondary users (CSUs) is discussed in [37], [38]. Recently in [39], the SNR wall for GED under CSS without considering fading is studied assuming different NUs and SNRs for CSUs. Thus, we notice that the existing analysis is only limited to no diversity case or CSS. Also, the available analysis do not consider NU with fading. We know that, in CSS multiple SUs have to cooperate in order to decide on the occupancy status of the PU and also require additional control channels to communicate, thus making it difficult to implement in practice. Hence, the diversity is better suited to combat the adverse effects of multi-path fading as well as shadowing [14], [21], [22], [40].

Two new diversity schemes namely plaw combining (pLC) and plaw selection (pLS) are proposed in [29], in which GED is used at each of the diversity branches. Here, they consider diversity and fading without considering NU. They show that GED with p=2.5 achieves higher probability of detection than energy detector, i.e., p=2. Also, GED with p=3.3 performs the second best. This motivates us to study the performance of energy detector considering different values of p under noise uncertainty and fading. Note that the experiments in [29] are carried out for very small sample size (N) and they do not consider noise uncertainty as done in our case. The problem with use of small N for detection is the low probability of detection since it is directly proportional to N. Hence, one has to consider sufficiently large N even when we consider noise uncertainty. To this end, we first derive the SNR wall considering AWGN channel and then extend it to the channel with Nakagami fading. Nakagami fading represents a generalized fading model that includes Rayleigh, Rice and Hoyt fading models as its special cases and can be used to model propagation in urban as well as sub-urban areas by setting the Nakagami parameter m. To the best of our knowledge, researchers have not worked on GED under diversity, noise uncertainty and fading. As the first study on GED with diversity in the presence of NU and fading, analysis in this paper is limited to pLC and pLS diversity only. We also study the effect of p on the detection performance. Finally, the effects on noise uncertainty and fading on the detection performance are studied.

The key contributions of this paper are

  • We first derive the expressions for average probability of false alarm (P̄F) and detection (P̄D) considering AWGN channel under NU with very large value of collected samples (N) for detection, i.e., N. These expressions are derived for no diversity, pLC and pLS diversity. Making use of these expressions, we derive the expressions for SNR wall in each case. The conclusions are drawn based on these results and the advantages of using diversity on the sensing performance (in terms of SNR wall) are also discussed.

  • We next derive P̄F and P̄D considering both noise uncertainty and fading for N. The derived expressions suggest that, deriving the SNR wall in this case is mathematically not tractable and hence we obtain the SNR wall using the numerical methods in this case. Here, we show that, when we consider fading, the value of SNR wall significantly increases resulting in the poor detection performance. All our analytical results are verified by using the Monte Carlo simulations.

  • The effect of p on the detection performance of GED is studied considering the noise uncertainty and the fading. We show that for sufficiently high sample size (N), the GED with the value of p close to 2 performs the best. We show that pLC performs better than the pLS diversity scheme and as N, the detection performance becomes independent of the value of p and it remains the same for both pLC and pLS schemes.

  • As a byproduct of this work, we study the effects of noise uncertainty and fading on the detection performance. Receiver operating characteristic (ROC) plots are shown to study the same. It is found that when the noise uncertainty is above certain value, its effect on detection performance is more severe when compared to the fading. The performance becomes worst when both the fading and the noise uncertainty exist.

The rest of the paper is organized as follows. In Section 2 and Section 3, we provide system and noise uncertainty models, respectively. The derivation of SNR wall for AWGN channel is then discussed in Section 4 for no diversity as well as by considering diversity. This is then extended to the case where both the noise uncertainty and the fading are considered and the same issue is dealt with in detail in Section 5. The analytical results and Monte-Carlo simulations are provided in Section 6 and finally, Section 7 concludes the paper.

Section snippets

System model

In cognitive radio, the signal received at the SU can be written as y(n)=w(n);H0,h(n)s(n)+w(n);H1,where h(n),s(n) and w(n) correspond to nth sample of the complex fading channel gain, PU signal and the noise, respectively with n=1,2,,N. The signal and noise samples are independent and identically distributed (iid) with s(n)CN(0,σs2)1

Noise uncertainty model

The characterization of additive white Gaussian noise (AWGN), i.e., w(n) in Eq. (1) depends on its variance. In general for many detection methods it is assumed that the true noise variance is known a priori. However, the noise variance depends on calibration error, variations in thermal noise and changes in low nose amplifier (LNA) gain. In practice, the true noise variance may vary over time and location, thus giving rise to a phenomenon called noise uncertainty [32], [34], which makes it

SNR wall for AWGN channel

For a given SNR >0, if there exists a threshold for which limNP̄F=0andlimNP̄D=1,then the sensing scheme is considered as unlimitedly reliable [34]. In other words, if the channel is sensed for sufficiently long time, i.e., N, one can achieve desired target probability of false alarm and the probability of detection at any SNR level. However, this is possible only when there is no noise uncertainty. When there exists noise uncertainty, it is not possible to achieve this performance even

SNR wall for fading channel

In this section we first derive P̄D under Nakagami fading and noise uncertainty for both no diversity and diversity cases. Since we know that P̄F is independent of SNR, here we give the derivation for P̄D only. We then discuss the SNR wall for each case.

Results and discussions

In this section, we validate the theoretical analysis that is carried out using Monte Carlo simulations. For carrying out the simulation, we generate both the PU signal and the noise as complex Gaussian and the results are averaged over 105 iterations. In all our simulations, we use the expected value of noise variance (σˆw2) as 1.5

Conclusion

In this paper, we study the SNR wall for generalized energy detector under diversity reception in the presence of both the noise uncertainty and the fading. We derive closed form expressions for P̄F and P̄D for no diversity, pLC and pLS diversity considering very large sample size. Using those expressions, we first derive the SNR wall for AWGN channel case under noise uncertainty. This analysis is then extended to the case when there exist Nakagami fading in addition to noise uncertainty in

Kamal Captain received B.E degree from Mahatma Gandhi Institute of Technical Education and research center (MGITER), Navsari in 2009. In 2011, he received M.Tech degree from Sardar Vallabhbhai National Institute of Technology (SVNIT), Surat. Currently, he is pursuing Ph.D from Dhirubhai Ambani Institute of Information and Communication Technology (DA-IICT), Gandhinagar.

His research interest includes statistical signal processing, machine learning and cognitive radio. His main research is

References (47)

  • FCC, Spectrum policy task force report (ET Docket no. 02-135), Nov...
  • MitolaI.

    Software radios: Survey, critical evaluation and future directions

    IEEE Aerosp. Electron. Syst. Mag.

    (1993)
  • LiuX. et al.

    A novel multi-channel internet of things based on dynamic spectrum sharing in 5g communication

    IEEE Internet Things J.

    (2018)
  • KayS.
  • ChenH.S. et al.

    Spectrum sensing using cyclostationary properties and application to IEEE 802.22 wran

  • KimK. et al.

    Cyclostationary approaches to signal detection and classification in cognitive radio

  • LundenJ. et al.

    Spectrum sensing in cognitive radios based on multiple cyclic frequencies

  • ZengY. et al.

    Covariance based signal detections for cognitive radio

  • ZengY. et al.

    Spectrum-sensing algorithms for cognitive radio based on statistical covariances

    IEEE Trans. Veh. Technol.

    (2009)
  • ZengY. et al.

    Maximum-minimum eigenvalue detection for cognitive radio

  • ZengY. et al.

    Eigenvalue-based spectrum sensing algorithms for cognitive radio

    IEEE Trans. Commun.

    (2009)
  • KobeissiH. et al.

    On the performance evaluation of eigenvalue-based spectrum sensing detector for mimo systems

  • UrkowitzH.

    Energy detection of unknown deterministic signals

    Proc. IEEE

    (1967)
  • DighamF. et al.

    On the energy detection of unknown signals over fading channels

    IEEE Trans. Commun.

    (2007)
  • KostylevV.

    Energy detection of a signal with random amplitude

  • AtapattuS. et al.

    Unified analysis of low-snr energy detection and threshold selection

    IEEE Trans. Veh. Technol.

    (2015)
  • KobeissiH. et al.

    Simple and accurate closed-form approximation of the standard condition number distribution with application in spectrum sensing

  • LiuX. et al.

    Optimal resource allocation in simultaneous cooperative spectrum sensing and energy harvesting for multichannel cognitive radio

    IEEE Access

    (2017)
  • AtapattuS. et al.

    Energy detection of primary signals over η-μ fading channels

  • SofotasiosP. et al.

    Energy detection based spectrum sensing over κ -μ and κ -μ extreme fading channels

    IEEE Trans. Veh. Technol.

    (2013)
  • CaptainK.M. et al.

    Energy detection based spectrum sensing over η-λ-μ fading channel

  • AtapattuS. et al.

    Performance of an energy detector over channels with both multipath fading and shadowing

    IEEE Trans. Wireless Commun.

    (2010)
  • HerathS.P. et al.

    Energy detection of unknown signals in fading and diversity reception

    IEEE Trans. Commun.

    (2011)
  • Cited by (12)

    • Signal detection with spectrum windows

      2022, Heliyon
      Citation Excerpt :

      The detection threshold setting is of great interest. One reason is that the energy detector is known to be sensitive to uncertainties in noise level in Gaussian and generalized Gaussian noise [13] and in fading channels [14]. Sliding window energy detection in the time domain using other than rectangular windows and related constant false alarm rate (CFAR) threshold setting are considered in [15] that implemented a few methods using software defined radios but did not provide any theoretical performance results.

    • CR-IoTNet: Machine learning based joint spectrum sensing and allocation for cognitive radio enabled IoT cellular networks

      2021, Ad Hoc Networks
      Citation Excerpt :

      This method is widely adopted due to its simplicity in structure and efficiency in determining the energy levels in the received signals. However, it is pertinent to mention that the classical ED is highly sensitive to noise uncertainty and SNR wall at low SNRs regimes [33–35]. Moreover, the training energy vectors are not always linearly separable [27] and may require a non-linear mapping function to map them into a higher-dimensional feature space in order to obtain the linearly separable training samples [27].

    • A review of the noise uncertainty impact on energy detection with different OFDM system designs

      2019, Computer Communications
      Citation Excerpt :

      The results of the simulation demonstrate that the proposed GED under the NU can reduce the SNR wall problem and achieve a lower probability of error compared to the conventional ED. In [31], the authors study the SNR wall under diversity reception in the presence of the NU and the fading for GED obtained through changing the squaring operation in ED by an arbitrary positive number p. It is shown that above a certain value, the effect of NU is more severe when compared to the fading.

    • A Total Blind Multi-antenna Spectrum Sensing Algorithm Based on Difference of Extreme Eigenvalues

      2023, Hunan Daxue Xuebao/Journal of Hunan University Natural Sciences
    View all citing articles on Scopus

    Kamal Captain received B.E degree from Mahatma Gandhi Institute of Technical Education and research center (MGITER), Navsari in 2009. In 2011, he received M.Tech degree from Sardar Vallabhbhai National Institute of Technology (SVNIT), Surat. Currently, he is pursuing Ph.D from Dhirubhai Ambani Institute of Information and Communication Technology (DA-IICT), Gandhinagar.

    His research interest includes statistical signal processing, machine learning and cognitive radio. His main research is focused on developing spectrum sensing techniques for cognitive radio.

    Manjunath V. Joshi (M’05) is serving as a Professor with the Dhirubhai Ambani Institute of Information and Communication Technology, Gandhinagar, India. He has been involved in research in the areas cognitive radio and machine learning and has several publications in quality journals and conferences. He has co-authored two books entitled Motion-Free Super Resolution (Springer, New York) and Digital Heritage Reconstruction using Super-resolution and Inpainting (Morgan and Claypool). Dr. Joshi was the recipient of the Dr. Vikram Sarabhai Award for the year 2006–2007 in the field of information technology constituted by the Government of Gujarat, India. He has visited Italy, France, Hong Kong, USA, Canada, South Korea, Indonesia and contributed to research in his area of expertise.

    View full text