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Spectrum Sensing, Measurement, and Modeling

  • Ghaith HattabEmail author
  • Danijela Cabric
Living reference work entry
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Abstract

Modeling spectrum sensing is a critical step that paves the way to (i) identify the key impairments that affect the detection performance and (ii) help develop algorithms and receiver architectures that mitigate these impairments. In this chapter, realistic and practical sensing models are presented beyond those developed for classical detection theory. These models capture the impact of different sensing receiver impairments on several detectors such as the energy, the pilot, and the cyclostationarity detectors. Several receiver nonidealities are investigated, including noise uncertainty, imperfect synchronization, and cyclic frequency offsets. In addition, challenges and impairments pertaining to wideband sensing are analyzed, including the presence of strong adjacent interferers as well as the nonlinearities of the receiver RF front-end. From these models, several mitigation techniques are developed to compensate for the presence of the different sensing receiver impairments. Measurements and simulation results are presented throughout the chapter to show the negative impact of such impairments and validate that the developed mitigation techniques provide tangible performance gains.

Keywords

Spectrum Sensing Techniques Cyclostationarity Detection Wideband Sensor Offset Frequency Imperfect Synchronization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction

A standard cognitive radio (CR) system seeks to identify channels that are not occupied by primary systems so it can access them. Such cognitive and dynamic approach promises to enhance spectrum utilization. For this reason, the CR receiver must be equipped with a spectrum sensor that helps scan a single (in case of narrowband) or multiple (in case of wideband) spectrum bands. The objective of the spectrum sensing receiver is to employ detection algorithms to quickly and reliably detect primary systems and identify available spectral resources.

Among the most popular spectrum sensing techniques proposed in the literature are the energy, pilot, and cyclostationarity detectors. The theoretical detection performance of these detectors has been thoroughly investigated in the literature, yet the derived expressions are assumed to hold under ideal assumptions irrespective of the signal-to-noise ratio (SNR) at the sensing receiver front-end, as will be discussed in section “Spectrum Sensing Techniques”. Indeed, measurements have verified that in negative SNR regimes, many of these assumptions do not hold. In this chapter, more accurate spectrum sensing models are presented, where several receiver impairments are included to better capture the performance attained via experimental studies. Specifically, the energy detector requires noise power estimation, which is commonly assumed to be perfect. Such assumption is dropped, and the detection performance is analyzed in the presence of noise uncertainty. Similarly, pilot and cyclostationarity detectors require tight synchronization to reap the coherent gains achieved via signal feature exploitation. This synchronization is difficult to attain in practice, where frequency, cyclic frequency, and sampling clock offsets are inevitable. Modeling these impairments and studying their impact on energy detection, pilot detection, and cyclostationarity detection will be discussed in details in sections “Energy Detection Under Noise Uncertainty,” “Pilot Detection Under Frequency Offsets,” and “Cyclostationarity Detection Under Imperfect Synchronization”, respectively. Several mitigation algorithms are also presented in their corresponding sections.

While narrowband sensing is fundamental, wideband sensing is a highly desirable feature since it enables the CR receiver to explore more spectral resources and switch between different channels in case some of them become occupied by primary systems. To this end, modeling the wideband sensing problem has been reduced to modeling several narrowband sensing problems by dividing the wideband into many narrowbands. Such approach typically assumes an ideal channelization process, which is infeasible in practice. Indeed, two major bottlenecks arise in wideband sensing. First, a band that is adjacent to other bands with strong signals can suffer from high interference due to the nonideal filter mask in practice, which is commonly assumed to be a brick wall in theory. In addition, strong signals can saturate the RF front-end components such as the low-power amplifier (LNA). This pushes the LNA to operate in a nonlinear region, introducing spurious terms that can affect the detection performance. These two challenges and the mitigation techniques to overcome them will be presented in section “Wideband Sensing: Challenges and Solutions”.

The different sensing impairments require revisiting the sensing models for two reasons. First, it is important to understand how the presence of these impairments affects the performance. Second, by identifying the key parameters that affect the performance, it becomes feasible to develop compensation algorithms and architectures to mitigate these impairments. The design procedure, which will be followed throughout this chapter, is summarized below:
  1. 1.

    Include the impairment in the sensing model. Such impairment may be identified through measurements or more practical modeling.

     
  2. 2.

    For a given detection algorithm, derive the detection performance in the presence of the impairment. The theoretical derivations help identify the key parameters that affect the detection performance.

     
  3. 3.

    Develop a compensation algorithm that mitigates the issues introduced by the impairment.

     

Spectrum Sensing Techniques

The CR receiver must have sensing capabilities to decide whether a channel is occupied by other users or not. Generally, physical layer sensing relies on estimating parameters that convey information about the channel such as the signal energy in that channel or the presence of signal features or pilots. In essence, the spectrum sensing problem can be viewed as a classical binary hypothesis test, where H 0 stands for the absence of primary user signals, i.e., noise-only samples, and H 1 stands for the presence of users, i.e., both noise and signal samples are present. This is a digital implementation, where a test statistic, Λ, is used to process N samples and estimate a desired parameter. Then, the statistic is compared to a predetermined decision threshold. Mathematically, this is expressed as
$$\displaystyle{ \varLambda \mathop{\mathop{\gtrless }\limits_{ H_{0}}}\limits^{H_{1}}\lambda, }$$
(1)
where λ is a threshold that can be optimized to attain a certain objective, e.g., meet a false alarm constraint.

There are a plethora of spectrum sensing techniques [2, 11, 23, 28], but the most prominent candidates for practical implementation are the energy detector, the pilot detector, and the cyclostationarity detector, which will be reviewed next.

Energy Detection

This is one of the simplest forms of detection because the CR receiver does not require any knowledge about the received samples beforehand. Specifically, the objective is to process the received samples to compute the energy level in the channel. Let r(n) denote the n-th sample of the received signal; then the energy detector is expressed as
$$\displaystyle{ \varLambda _{E} = \frac{1} {N}\sum _{n=0}^{N-1}\vert r(n)\vert ^{2}. }$$
(2)
The detection performance of this test statistic is well-investigated in the literature for different signal and noise models. For instance, it can be shown that under the additive white Gaussian noise (AWGN) channel, the receiver operating characteristic (ROC) performance is expressed as [24]
$$\displaystyle{ P_{d} = Q\left ( \frac{1} {1 +\mathop{ SNR}\nolimits }\left [Q^{-1}(P_{ f}) -\sqrt{N}\mathop{SNR}\nolimits \right ]\right ), }$$
(3)
where P d is the probability of detection, P f is the probability of false alarm, Q(⋅ ) is the Q-function, and Q −1(⋅ ) is the inverse Q-function. It can be observed that the performance improves for higher SNR or longer sensing times, i.e., larger N. Note that this expression is assumed to be valid for any SNR value .

Pilot Detection

While the energy detector is a universal detector, since it does not need any specific signal structure, some practical communication systems deliberately embed signal features to either perform synchronization and acquisition or help improve signal decoding. For instance, in some broadcast communication systems, e.g., digital television, sinewave pilot tones are transmitted for data frame synchronization. Mathematically, the transmitted primary signal can be expressed as [10, 24]
$$\displaystyle{ x(n) = \sqrt{\varepsilon }x_{p}(n) + \sqrt{1-\varepsilon }x_{d}(n), }$$
(4)
where x p (n) is a known pilot tone, x d (n) is the data-carrying signal, and ɛ is the pilot power factor, i.e., the fraction of the total power allocated to the pilot tone. For example, pilots in digital TV signals are 11 dB weaker than the average signal power, i.e., ɛ ≈ 0. 1 [1].
Pilot detection infers the occupancy of a channel by utilizing prior knowledge about pilots embedded in transmitted signals. One common approach for pilot detection is the following test statistic [5, 24]
$$\displaystyle{ \varLambda _{P} = \frac{1} {N}\sum _{n=0}^{N-1}\hat{x}_{ p}^{{\ast}}(n)r(n), }$$
(5)
where \(\hat{\mathbf{x}}_{p} = [\hat{x}_{p}(1),\hat{x}_{p}(2),\cdots \,,\hat{x}_{p}(N)]^{T}\) is a unit vector in the direction of the pilot tone. The ROC performance of this statistic in an AWGN channel can be shown to be [10]
$$\displaystyle{ P_{d} = Q\left (Q^{-1}(P_{ f}) -\sqrt{N\varepsilon \mathop{ SNR}\nolimits }\right ). }$$
(6)
It can be observed that the performance depends on the power allocated to the pilot tone.

Cyclostationarity Detection

In the absence of deterministic pilot tones, the CR receiver can instead utilize the inherit features of modulated signals, which exhibit periodic statistical properties [7, 8]. Specifically, many of the modulated signals are second-order cyclostationary, i.e., their means and autocorrelation functions are periodic, where the period depends on the symbol period and the carrier frequency of the signal [21].

Cyclostationarity properties can be observed via the spectral correlation function, a two-dimensional complex transform [8]. This transform is actually a generalization of the power spectral density function, and it maintains several key advantages. First, it preserves phase and frequency information related to certain parameters in modulated signals. Second, features that overlap in the power spectrum are nonoverlapping features in the spectral correlation domain, making it easier to detect them. Third, different modulation schemes, e.g., BPSK and QPSK, have identical power spectral density functions, yet they can have highly distinct spectral correlation functions [4]. Last, noise samples are typically uncorrelated, and hence noise does not exhibit any cyclic features, making the detection in low SNR regime robust. Indeed, the measured spectral correlation of a receiver sensing noise-only samples is illustrated in Fig. 1, which confirms that noise does not have any peaks in the spectral correlation function except at zero cyclic frequencies.
Fig. 1

Measured spectral correlation function of the noise at the 2.4 GHz receiver

The detection of cyclic features in the received signal is typically done by computing the cyclic autocorrelation function (CAF) , where the received signal is correlated with a frequency-shifted version of itself [7]. More formally, the CAF can be estimated using N samples as follows [14, 21]
$$\displaystyle{ R_{r}^{\alpha }(\tau ) = \frac{1} {N}\sum _{n=0}^{N-1}r(n)r^{{\ast}}(n-\tau )e^{-j2\pi \alpha nT_{s} }, }$$
(7)
where α is the cyclic frequency and T s is the sampling period. Once the CAF is computed at the CR receiver, the following test statistic can be performed:
$$\displaystyle{ \varLambda _{\alpha } = \vert R_{r}^{\alpha }(\tau )\vert. }$$
(8)
As stated, different modulation schemes have peaks at different cyclic frequencies, and hence by varying α, it can be possible to not only detect signals but also classify them [9, 22].

Beyond Classical Detection Theory

Modeling the binary hypothesis testing problem in classical detection theory generally includes many ideal assumptions. For instance, it is commonly assumed that noise samples are generated from a white Gaussian wide-stationary process with a noise variance that is precisely known. This means that the threshold used for the energy detector can be accurately optimized to achieve any desired detection performance. Similarly, for pilot detection, tight synchronization is assumed between the transmitter and the sensing receiver to properly correlate the received signal with a replica of the pilot tone, whereas frequency and clock offsets are neglected in the analysis of cyclostationarity detection.

Such ideal assumptions can be warranted if detection is done in good SNR conditions, where noise estimation and receiver synchronization are more reliable. However, primary user systems require protection even in the worst-case scenarios when the received signal at a CR receiver could be far below noise floor. For example, for a cognitive radio operation in licensed TV bands, IEEE 802.22 working group defined required SNR sensitivities for primary user signals to be − 22 dB for DTV signals and − 10 dB for wireless microphones [6]. Hence, spectrum sensing must be reliable in negative SNR regimes.

In addition to the reliable operation under stringent SNR requirements, the CR receiver must seek spectral opportunities over a wide swath of the spectrum, elevating the need for wideband spectrum sensing. The problem of wideband sensing has been typically approached by breaking the spectrum into many narrowband channels, and hence the problem is converted into several binary hypotheses tests, one per channel [11, 12, 16, 17, 18]. Such simplification, however, neglects many design challenges inherited with wideband sensing, including the impact of strong interferers in some channels [26, 27], spectral leakage due to nonideal filters [26], or the presence of spurious harmonics generated from nonlinearities in the receiver front-end [19].

The aforementioned design challenges require revisiting the sensing models for two reasons. First, it is important to accurately understand the impact of operating in negative SNR regions with nonideal wideband receiver front-ends on the detection performance. Second, by identifying the key parameters that affect the detection performance, the sensing algorithms can be enhanced to compensate for the different impairments that affect the detection reliability.

Energy Detection Under Noise Uncertainty

In an unoccupied channel, the CR receiver collects noise-only samples. From these samples, the noise variance can be estimated using the energy detector to optimize the detection threshold λ in order to achieve any desired probability of detection. Indeed, it can be observed from the ROC performance of the energy detector in (3) that controlling the sensing time N can help meet any specified (P d , P f ) pair. Specifically, the following relation holds
$$\displaystyle{ N = \frac{\left [Q^{-1}(P_{f}) - Q^{-1}(P_{d})(1 +\mathop{ SNR}\nolimits )\right ]^{2}} {\mathop{SNR}\nolimits ^{2}}. }$$
(9)
In other words, the theoretical analysis shows that the number of samples asymptotically scales as \(1/\mathop{SNR}\nolimits ^{2}\), which follows using the approximation \(1 +\mathop{ SNR}\nolimits \approx 1\) in negative SNR regimes. To verify this scaling law, an experimental study is performed using a real CR test bed [3]. In this study, the objective is to detect a QPSK signal under different SNR values. For each value, two different sets of energy values are collected: one in the absence of the signal and one in its presence. When the signal is absent, the noise-only samples are used to estimate the detection threshold λ that achieves a false alarm of P f = 0. 05. The threshold is then applied to compute P d . From these measurements, the sensing time for a given SNR is derived to achieve a detection performance of P d = 0. 9 as shown in Fig. 2. It can be observed that the theoretical scaling law holds for SNR values above − 20 dB. However, as the signal becomes weaker, detection becomes progressively harder, and when the signal is below − 23 dB, the detector cannot sense the signal irrespective of the sensing time, leading to a phenomenon known as the SNR wall [24]. Such deviation illustrates that the energy detector lacks robustness in negative SNR regimes. In addition, the measurements show that the existing sensing model must be revisited to accurately capture the SNR wall phenomenon.
Fig. 2

Measured sensing time versus SNR for energy detector

To understand the deviation of the measured sensing time curve from the predicted one, it is imperative to address the assumptions used in the existing sensing model. Specifically, there are two strong assumptions used here. First, noise is assumed to be an additive white Gaussian wide-sense stationary process with zero-mean and known variance. However, noise is an aggregation of various sources including not only thermal noise at the receiver and underlined circuits but also interference due to nearby unintended emissions, weak signals from transmitters very far away, etc. Second, by assuming that the noise variance is perfectly known, the detection threshold can be optimized with infinite precision. However, in the actual implementation, this is practically impossible as noise could vary over time due to temperature change, ambient interference, filtering, etc. Indeed, Fig. 3 shows that the measured noise power level in the receiver used for testing of energy detection varies over time. The impact of the time-varying nature of the noise process on detection becomes tangible when the signal strength is below the estimation error of the noise variance. Hence, these temporal changes must be captured in the sensing model, particularly when the receiver operates in negative SNR regimes .
Fig. 3

Measured noise power at the CR receiver

Modeling Noise Uncertainty

In a standard sensing model, it is common to model the noise-only samples as \(w(n) \sim \mathcal{ N}(0,\sigma _{w}^{2})\), i.e., a Gaussian random variable with zero-mean and perfectly known variance σ w 2. However, it is more accurate to assume that such noise samples are instead generated from \(\tilde{w}(n) \sim \mathcal{ N}(0,\tilde{\sigma }_{w}^{2})\) such that
$$\displaystyle{ \tilde{\sigma }_{w}^{2} \in \left [\frac{1} {\rho } \sigma _{w}^{2},\rho \sigma _{ w}^{2}\right ], }$$
(10)
where ρ ≥ 1 is a parameter that quantifies the noise uncertainty. Note that ρ = 1 implies perfect knowledge of the noise variance. In other words, the CR receiver estimates that the noise variance is σ w 2, whereas the actual variance is \(\tilde{\sigma }_{w}^{2}\).
To understand how such noise uncertainty leads to an SNR wall, consider the worst-case scenario. Specifically, the highest false alarm probability occurs when the actual noise variance is \(\tilde{\sigma }_{w}^{2} =\rho \sigma _{ w}^{2}\) since in this case σ w 2 becomes an underestimate of the true variance, forcing the CR receiver to more frequently decide that the channel is occupied. Similarly, the lowest probability of detection occurs when the actual noise variance is \(\tilde{\sigma }_{w}^{2} = (1/\rho )\sigma _{w}^{2}\). In this case, the CR receiver overestimates the true variance, leading to increasing the frequency of declaring a channel to be empty. Under such worst-case scenario, it can be shown that the ROC of the energy detector becomes [24]
$$\displaystyle{ P_{d} = Q\left (\frac{\rho Q^{-1}(P_{f}) -\sqrt{N}(\mathop{SNR}\nolimits +1/\rho -\rho )} {\mathop{SNR}\nolimits +1/\rho } \right ). }$$
(11)
Clearly, for ρ = 1, (11) simplifies to (3). From this expression, the sensing time can be derived to be
$$\displaystyle{ N = \frac{\left [\rho Q^{-1}(P_{f}) - Q^{-1}(P_{d})(1/\rho +\mathop{ SNR}\nolimits )\right ]^{2}} {(\mathop{SNR}\nolimits +1/\rho -\rho )^{2}}. }$$
(12)
It is evident that N when \(\mathop{SNR}\nolimits +1/\rho -\rho = 0\), i.e., there exists an SNR wall where detection below that SNR value becomes impossible. More formally, for ρ > 1, the SNR wall occurs at
$$\displaystyle{ \mathop{SNR}\nolimits _{\mathop{wall}\nolimits } = \frac{\rho ^{2} - 1} {\rho }. }$$
(13)
Figure 4 illustrates the theoretical sensing time in the presence and absence of noise uncertainty where ρ dB = 10log10(ρ). It is evident that even small uncertainty makes the energy detector poor in highly negative SNR regions. It is also observed that the modified sensing model captures the behavior observed in the experimental study illustrated in Fig. 2.
Fig. 4

Sensing time in the presence of noise uncertainty

Compensating Noise Uncertainty

Frequent noise power estimation becomes imperative, particularly in low SNR regimes. To this end, the CR receiver must collect noise-only samples. These samples, denoted as w(n), can be used to estimate the noise variance using the following maximum likelihood (ML) estimator [13]
$$\displaystyle{ \hat{\sigma }_{w}^{2} = \frac{1} {M}\sum _{m=0}^{M-1}\vert w(m)\vert ^{2}. }$$
(14)
The key issue here is the availability of such samples or inferring that the collected samples are actually noise-only samples instead of samples of a weak signal. One solution to this issue is to infrequently trigger a fine-sensing stage, where a feature detector is used [15]. Specifically, during the fine-sensing stage, if the decision is H 0, then the collected samples can, with high accuracy, be declared as noise-only samples, and hence they can be used to update \(\hat{\sigma }_{w}^{2}\) for subsequent energy detection. Such approach can be implemented using the following detector:
$$\displaystyle{ \begin{array}{rl} \varLambda _{E}^{{\prime}}& = \frac{\varLambda _{E}} {\hat{\sigma }_{w}^{2}} \\ & = \frac{M} {N} \frac{\sum _{n=0}^{N-1}\vert r(n)\vert ^{2}} {\sum _{m=0}^{M-1}\vert w(m)\vert ^{2}}. \end{array} }$$
(15)
Figure 5 illustrates the enhanced energy detector. It consists of two stages: fine sensing of duration M, that is triggered infrequently, and energy sensing for duration N, which is triggered frequently. The former is motivated to reliably update the noise variance estimate, which will be used for subsequent fast energy detection.
Fig. 5

The enhanced energy detector utilizes the noise variance estimate, which is periodically updated

The detection performance of this detector can be approximated as [15]
$$\displaystyle{ P_{d} = Q\left ( \frac{1} {1 +\mathop{ SNR}\nolimits }\left [Q^{-1}(P_{ f}) -\sqrt{ \frac{N \cdot M} {N + M}}\mathop{SNR}\nolimits \right ]\right ). }$$
(16)
From this expression, the sensing time of the energy detector is derived as follows:
$$\displaystyle{ \sqrt{ \frac{N \cdot M} {N + M}} = \frac{Q^{-1}(P_{f}) - Q^{-1}(P_{d})(1 +\mathop{ SNR}\nolimits )} {\mathop{SNR}\nolimits }. }$$
(17)
For instance, if M = N, the sensing time needed to achieve a specific (P d , P f ) pair is
$$\displaystyle{ N = \frac{2\left [Q^{-1}(P_{f}) - Q^{-1}(P_{d})(1 +\mathop{ SNR}\nolimits )\right ]^{2}} {\mathop{SNR}\nolimits ^{2}}. }$$
(18)
That is, there is an SNR penalty of \(10\log _{10}(\sqrt{2}) \approx 1.5\) dB in comparison with the ideal energy detector. Note, however, that if M = N 2, then \(\sqrt{ \frac{N\cdot M} {N+M}} \approx \sqrt{N}\), i.e., the performance of Λ E reaches the ideal performance of Λ E in the negative SNR regime.
Alternatively, the minimum SNR for which the detection is possible can be derived as [15]
$$\displaystyle{ \mathop{SNR}\nolimits _{\mathop{min}\nolimits } = \frac{1 + \sqrt{\frac{M+N} {N\cdot M}} Q^{-1}(P_{ f})} {1 + \sqrt{\frac{M+N} {N\cdot M}} Q^{-1}(P_{d})} - 1. }$$
(19)
It is clear that if M = N or M = N 2, then \(\mathop{SNR}\nolimits _{\mathop{min}\nolimits } \rightarrow 0\) as N, i.e., there is no SNR wall. However, if M is a constant, e.g., M = 100, then \(\mathop{SNR}\nolimits _{\mathop{min}\nolimits }> 0\) as N.
Figure 6 illustrates the sensing time needed to achieve P f = 0. 05 and P d = 0. 9 when Λ E is used. As expected, having longer sensing periods to estimate the noise variance, i.e., larger M, improves the performance of the energy detector. However, to mitigate the SNR wall, the noise power estimation period should scale with the duration of the energy detector, e.g., M = N and M = N 2.
Fig. 6

Performance of the enhanced energy detector

Pilot Detection Under Frequency Offsets

The simplicity of the energy detector comes at the expense of a poor performance in negative SNR regimes. To circumvent this, pilot detection exploits certain signal features to robustify the detection performance, and particularly it relies on pilot tones that are sent alongside data-carrying signals. Indeed, the processing of the received samples via correlation provides coherent gains that make detection of very weak signals possible. However, this coherent processing requires the CR receiver to be in perfect synchronization with the pilot in the received signal.

Consider the ROC performance of the pilot detector in (6); then the sensing time needed to achieve any desired (P d , P f ) pair can be shown to be
$$\displaystyle{ N = \frac{\left [Q^{-1}(P_{f}) - Q^{-1}(P_{d})\right ]^{2}} {\varepsilon \mathop{SNR}\nolimits }. }$$
(20)
Thus, the theoretical scaling law of the sensing time is \(N \sim 1/\mathop{SNR}\nolimits\). Comparing this scaling law with the one achieved using the energy detector, then it can be observed that the sensing time under pilot detection is a lower bound on that achieved using energy detection as long as \(\varepsilon>\mathop{ SNR}\nolimits\).
To verify the scaling law of the pilot detector, an experimental study is performed on a sinewave pilot, with signal levels varying from − 110 to − 136 dB. The measured sensing time is shown in Fig. 7 in the presence of different frequency offsets. It is observed that for strong pilot tones, the measured sensing time follows the theoretical scaling law. However, as the pilot power decreases, the sensing time deviates from the theoretical curve, leading to the SNR wall phenomenon.
Fig. 7

Measured sensing time with variations of the sine wave signal power under different frequency offsets

The deviation of the experimental result from the theoretical curve is explained as follows. Practical receivers have imperfect thus inaccurate oscillators and circuitry, deeming perfect synchronization near impossible, particularly in negative SNR regimes. Typically, synchronization loops can estimate and reliably correct frequency offsets when the SNR at the receiver is positive. However, in negative SNRs these loops are driven by noise and cannot perform robust synchronization. The imperfect synchronization can severely affect the coherent processing gains achieved by correlating the received signal with the pilot tone .

Modeling Frequency Offsets

Consider the sinewave pilot tone x p (n) = exp(j(ω o n + θ)), where ω 0 is the carrier frequency. Suppose there exists a frequency offset, ψ, between the primary transmitter and the CR receiver. This can be modeled by assuming that the pilot tone replica used at the receiver is equal to \(\hat{x}_{p}(n) = x_{p}(n)\exp (j\psi n)\). Using the pilot detector, it can be shown that under H 1
$$\displaystyle{ \begin{array}{rl} \varLambda _{P}& = \frac{1} {N}\sum _{n=0}^{N-1}\hat{x}_{ p}(n)^{{\ast}}r(n) \\ & \approx \frac{\sqrt{\varepsilon }} {N}\sum _{n=0}^{N-1}\exp (-j\psi n).\end{array} }$$
(21)
If the sensing time N becomes comparable or larger than the period of the frequency offset, then the pilot detector loses its coherent processing gain. In other words, in the presence of frequency offsets, the pilot detector can suffer from the SNR wall, explaining the measured curve in Fig. 7.

Compensating Frequency Offsets

As discussed in the previous section, the presence of frequency offsets can be detrimental if the sensing time is in the order of frequency offset time period. Thus, it is intuitive to break down the sensing time into shorter time periods to help achieve partial coherent processing gains. This motivates the following enhanced pilot detector [5]:
$$\displaystyle{ \varLambda _{P}^{{\prime}} = \frac{1} {K}\sum _{k=0}^{K-1} \frac{1} {M}\left [\sum _{m=0}^{M-1}\hat{x}_{ p}^{{\ast}}(kM + m)r(kM + m)\right ]^{2}. }$$
(22)
This detector can be interpreted as a two-stage pilot detector. Specifically, in the first stage, the CR receiver correlates the received signal with a replica of the pilot tone, but this time it is done over a short period, i.e., MN. This process is repeated K times, and hence in the second stage, the CR receiver noncoherently averages over these collected K blocks, making the total sensing time N = K ⋅ M. The receiver architecture of the two-stage pilot detector is shown in Fig. 8.
Fig. 8

The two-stage pilot detector complements coherent processing with noncoherent processing

It can be shown that the performance of the enhanced pilot detector is
$$\displaystyle{ P_{d} = Q\left ( \frac{1} {\sqrt{1 + 2M\varepsilon \mathop{ SNR}\nolimits }}\left [Q^{-1}(P_{ f}) -\sqrt{\frac{K} {2}} M\varepsilon \mathop{SNR}\nolimits \right ]\right ). }$$
(23)
There are several key observations here. First, if \(\varepsilon \mathop{SNR}\nolimits \ll 1\), then the performance can be approximated as
$$\displaystyle{ P_{d} \approx Q\left (Q^{-1}(P_{ f}) -\sqrt{\frac{K} {2}} M\varepsilon \mathop{SNR}\nolimits \right ). }$$
(24)
In this case, increasing K improves the performance with a similar scaling to the energy detector (cf. (3)). Yet, the coherent processing effectively improves the SNR by 10log10(M)dB. In other words, comparing (24) with (3), it can be observed that under noise uncertainty, which affects nocoherent processing, the enhanced pilot detector moves the SNR wall by 10log10(M)dB.
Second, if \(\varepsilon \mathop{SNR}\nolimits \gg 1\), then the performance can be approximated as
$$\displaystyle{ P_{d} \approx Q\left ( \frac{Q^{-1}(P_{f})} {\sqrt{2M\varepsilon \mathop{ SNR}\nolimits }} -\sqrt{\frac{KM\varepsilon \mathop{ SNR }\nolimits } {4}} \right ). }$$
(25)
In this case, the performance of the enhanced pilot detector is similar to the performance of the ideal pilot detector (cf. (6)).
Figure 9 shows the detection performance of the enhanced pilot detector in negative SNR regimes. The enhanced detector is compared with the ideal energy and pilot detectors in (2) and (5), respectively. It is assumed that P f = 0. 05, ɛ = 0. 1, and N = 10, 000, where K = 10 and M = 1000. It is clear that the enhanced pilot detector benefits from the coherent processing gains in very negative SNRs, and since MN, the robustness against frequency offsets improves.
Fig. 9

The detection performance of the enhanced pilot detector

Cyclostationarity Detection Under Imperfect Synchronization

Similar to pilot detection, where prior knowledge about pilot tones is needed, cyclostationarity detection requires knowledge about the cyclic frequency of the modulated signal. Such feature exploitation helps robustify detection in negative SNR regimes. However, it is critical to analyze the detection performance when such knowledge is not perfectly known.

Consider the zero-lag CAF, i.e., τ = 0, which is expressed as
$$\displaystyle{ R_{r}^{\alpha }(0) = \frac{1} {N}\sum _{n=0}^{N-1}r(n)r^{{\ast}}(n)e^{-j2\pi \alpha nT_{s} }. }$$
(26)
Clearly, for α = 0, the cyclostationarity detector simplifies to the energy detector in (2). From this, one can interpret this detector for α ≠ 0 as computing the energy of the received signal at a cyclic frequency α. In the presence of noise-only samples, it can be shown that as N then R r α (0) → 0 for α ≠ 0 at all SNRs. That is, the detector can theoretically suppress noise at the negative SNR regime by increasing the sensing time N due to averaging a stationary noise process.
The robustness of the cyclostationarity detector in negative SNR regimes is attained when α is perfectly known, yet this is difficult to achieve in practice. Specifically, the presence of Doppler shifts, imperfect estimation of carrier frequencies, and the frequency mismatch due to local oscillators all introduce cyclic frequency offsets (CFOs) that can degrade the performance. Such performance degradation is verified experimentally [20]. In particular, a BPSK signal with a symbol period T = 10μs is generated. At the receiver end, the detector in (26) is implemented at α = 1∕T = 100 KHz with a sampling frequency 1∕T s = 2 MHz. The cyclic feature of the BPSK signal is analyzed in the presence of different frequency offsets Δ α given in units per million (ppm). Figure 10 shows the impact of increasing the sensing time N on the cyclic feature. Interestingly, as the sensing time increases, it becomes harder to detect the cyclic feature. Thus, although increasing N averages the noise process in negative SNR regimes, the cyclic feature becomes harder to detect, leading to an SNR wall phenomenon. The decay in the cyclic feature is theoretically investigated in the next section.
Fig. 10

Normalized cylic feature in the presence of CFOs. Markers denote the experimental results and curves denote the theoretical expression in (29)

Modeling Cyclic Frequency Offsets

To model the CFO, it can be assumed that the test statistic is computed at a cyclic frequency that is deviated from the correct one by Δ α , i.e., the cyclic frequency used is
$$\displaystyle{ \hat{\alpha }=\alpha (1 +\varDelta _{\alpha }). }$$
(27)
Note that the CFO will not affect the noise process since it is stationary. Hence, noiseless signals are considered in the subsequent analysis.
For illustration purposes, consider a single-carrier signal r(n) = m a(mT b )p(nT s mT)exp(−2πf 0 nT s ), where a(mT b ) are data symbols with period T b , e.g., for BPSK a(mT b ) ∈ {+1, −1}, p(⋅ ) is a pulse shaping filter, and f 0 is the carrier frequency. Then,
$$\displaystyle{ \begin{array}{rl} R_{r}^{\hat{\alpha }}(0)& = \frac{1} {N}\sum _{n=0}^{N-1}r(n)r^{{\ast}}(n)e^{-j2\pi \hat{\alpha }nT_{s}} \\ & = \frac{1} {N}\sum _{n=0}^{N-1}\left \vert \sum _{ m}a(mT_{b})p(nT_{s} - mT)\right \vert ^{2}e^{-j2\pi \alpha nT_{s}}e^{-j2\pi \alpha \varDelta _{\alpha }nT_{s}} \\ & \approx \frac{R_{r}^{\alpha }(0)} {N} \frac{e^{-j2\pi \alpha \varDelta _{\alpha }NT_{s}}-1} {e^{-j2\pi \alpha \varDelta _{\alpha }T_{s}}-1} \\ & = R_{r}^{\alpha }(0)e^{-j2\pi \alpha \varDelta _{\alpha }(N-1)T_{s}}\frac{\sin (\pi \alpha \varDelta _{\alpha }NT_{s})} {N\sin (\pi \alpha \varDelta _{\alpha }T_{s})}. \end{array} }$$
(28)
Note that \(R_{r}^{\hat{\alpha }}(0) \rightarrow R_{r}^{\alpha }(0)\) as Δ α → 0. The test statistic to detect the cyclic feature is shown to be [20, 29]
$$\displaystyle{ \vert R_{r}^{\hat{\alpha }}(0)\vert = \vert R_{ r}^{\alpha }(0)\vert \left \vert \frac{\sin (\pi \alpha \varDelta _{\alpha }NT_{s})} {N\sin (\pi \alpha \varDelta _{\alpha }T_{s})}\right \vert. }$$
(29)
It can be observed that for Δ α ≠ 0, the cyclic feature decays as N increases. This makes \(\vert R_{r}^{\hat{\alpha }}(0)\vert\) under H 1 to be similar to \(\vert R_{r}^{\hat{\alpha }}(0)\vert\) under H 0, making detection very difficult. Figure 10 shows the theoretical curves of the normalized cyclic feature in the presence of CFOs, which match the results obtained via the experimental study.

Compensating Frequency Offsets

It is shown that long sensing windows severely impact the cyclostationarity detector in the presence of CFOs. However, long sensing windows are necessary to average noise, particularly in negative SNR regimes. This suggests a two-stage sensing detector similar to the approach used in improving the pilot detector. Specifically, the N samples are broken into K blocks, where each one is of length M. In each block of samples, cyclostationarity detection is done, yet M < N, and hence the impact of the CFO is reduced. At the same time, K > 1 in order to average noise. Mathematically, the two-stage detector is given as [21]
$$\displaystyle{ R_{r}^{\alpha }(0)^{{\prime}} = \frac{1} {K \cdot M}\sum _{k=0}^{K-1}\sum _{ m=0}^{M-1}r(kM + m)r^{{\ast}}(kM + m)e^{-j2\pi \alpha mT_{s} }. }$$
(30)
The enhanced cyclostationarity detector is shown in Fig. 11. Similar to the analysis given for \(R_{r}^{\hat{\alpha }}(0)\), it can be shown that the impact of the CFO on R r α (0) is given by
$$\displaystyle{ \vert R_{r}^{\hat{\alpha }}(0)^{{\prime}}\vert = \vert R_{ r}^{\alpha }(0)^{{\prime}}\vert \left \vert \frac{\sin (\pi \alpha \varDelta _{\alpha }MT_{s})} {M\sin (\pi \alpha \varDelta _{\alpha }T_{s})} \cdot \frac{\sin (\pi \alpha KMT_{s})} {K\sin (\pi \alpha MT_{s})}\right \vert. }$$
(31)
Fig. 11

The enhanced cyclostationarity detector

Figure 12a shows the normalized cyclic frequency under the enhanced detector for different number of frames. It is evident that breaking the N samples into several frames can significantly reduce the rate at which the cyclic feature decays, making it more reliable to detect in negative SNR regimes. Figure 12b illustrates the cyclic feature with variations of the CFO, where N = 20, 000. It can be observed that increasing K does not always provide the highest gains. For instance, the cyclic feature is weak for K = 5 when the CFO is small. This emphasizes that the number of samples and how these samples are divided into blocks both affect the performance. This suggests an optimization framework that maximizes the cyclic feature by optimizing K and M [20].
Fig. 12

Normalized cyclic frequency under the enhanced cyclostationarity detector

So far, it is assumed that there are no phase offsets between the different blocks, which occur when MαT s is an integer, i.e., the estimation of the CAF is done over an integer number of periods of the cyclic frequency. However, sampling clock offsets (SCOs) resulted in the analog-to-digital conversion stage may prevent coherent integration of the different blocks.

One way to model the SCO is to rewrite the sampling period as
$$\displaystyle{ \hat{T}_{s} = T_{s}(1+\delta ). }$$
(32)
Here, it is assumed that Δ α = 0 to explicitly understand how the SCO affects the performance of the enhanced cyclostationarity detector. Hence, and similar to the analysis done for the CFO, it can be shown that
$$\displaystyle{ \vert \hat{R}_{r}^{\alpha }(0)^{{\prime}}\vert = \vert R_{ r}^{\alpha }(0)^{{\prime}}\vert \left \vert \frac{\sin (\pi \alpha KMT_{s}(1+\delta ))} {K\sin (\pi \alpha MT_{s}(1+\delta ))}\right \vert, }$$
(33)
where \(\hat{R}_{r}^{\alpha }(0)^{{\prime}}\) denotes the enhanced detector in the presence of the SCO.
The impact of the SCO on the enhanced detector is studied for K = 5. Figure 13a shows the normalized cyclic frequency with variations of the number of samples, whereas Fig. 13b shows the impact of δ, where N = 10, 000. It is assumed that α = 1∕T = 100 KHz and 1∕T s = 2 MHz. It is clear that sampling offsets have detrimental effects on the performance.
Fig. 13

Normalized cyclic frequency in the presence of SCOs

The enhanced detector, in general, requires optimizing K and M to limit the performance loss due to CFOs and SCOs [21]. An optimization framework can be formulated when these two impairments are modeled as random variables. Figure 14 illustrates the detection performance of the enhanced detector, where both impairments are modeled as zero-mean Gaussian random variables with variances σ cfo 2 = 2 × 10−4 and σ sco 2 = 1 × 10−5. The signal to be detected is assumed to be BPSK with α = 5 MHz, and 1∕T s = 10 MHz. Figure 14a shows the simulated ROC performance for different number of frames, where \(\mathop{SNR}\nolimits = -5\) dB. The total sensing window is fixed at N = K ⋅ M = 5000 samples. It can be observed that the way by which the samples are split is critical to the detector’s performance. Figure 14b shows the detection performance for K = 12 under different SNR regimes. Overall, the detector provides robust performance in negative SNR regimes if K and M are optimized .
Fig. 14

Detection performance of the enhanced detector in the presence of CFOs and SCOs

Wideband Sensing: Challenges and Solutions

To realize a full-scale uptake of cognitive radio systems, it is imperative to explore a wide swath of the spectrum in order to identify as many spectral opportunities as possible. Hence, it is critical to equip CR receivers with wideband sensing capabilities, i.e., scanners that can scan many channels in parallel. Not only this provides more bandwidth, and hence more throughput, but also enables the receiver to move from one channel to another when a primary system reappears.

Typically in wideband sensing, the received wideband signal is fed into a filter bank to channelize it into nonoverlapping subbands. In this case, the wideband sensing model becomes a collection of narrowband sensing models. Hence, the impairments discussed in the previous sections can still occur for each subband, e.g., noise uncertainty, imperfect synchronization, frequency offsets, etc.

Besides the aforementioned impairments, there are other impairments and design challenges inherited in wideband sensing. For instance, consider the general wideband sensing architecture shown in Fig. 15. Observe that the digital signal processor (DSP) has the additional block channelization (or filtering) to convert the wideband signal into several narrowbands. Such filtering procedure is nonideal, causing leakage and interference in the channel of interest. In addition, due to the high bandwidth of the signal, the low-power amplifier (LNA) in the RF front-end may be pushed to operate in a nonlinear region, causing the wideband signal to be distorted. These two impairments will be the focus of this section.
Fig. 15

A typical wideband sensing architecture

Adjacent Band Interfering Power

Consider a wideband signal that is composed of several nonoverlapping narrowband primary users, where for simplicity all bands are assumed to be of equal bandwidth and modulation scheme. Figure 16 illustrates an example of the power spectral density (PSD) of three nonoverlapping signals. Consider detecting the weak primary signal, which is adjacent to two strong primary signals. In an ideal architecture, each signal can be processed and detected independently of the other signals. However, in a practical receiver, there are two issues that arise. First, the filters in frequency domain are not perfectly rectangular, with sharp edges. Hence, the tail of a strong adjacent signal may introduce interference to the band of interest. In other words, even if the band of interest is unoccupied, the interference present in that channel may increase false alarms. Second, the channelization of the time-domain samples can introduce spectral leakage to other channels. These two issues will be collectively referred as adjacent band interfering power [26].
Fig. 16

Nonideal channelization and filtering result in adjacent interfering power

Modeling Adjacent Interference Power

The discrete-time domain received wideband signal r(n) is a superposition of primary signals corrupted with noise. In other words, let \(w(n) \sim \mathcal{ N}(0,\sigma _{w}^{2})\) be the noise samples; then r(n) = l = 1 L x l (n) + w(n), where x l (n) is the transmitted primary signal over the l-th channel. The received signal can be decoupled into narrowband signals using the frequency domain representation of r(n), which is computed by the normalized fast Fourier transform (FFT) as follows
$$\displaystyle{ R_{k}[m] = \frac{1} {N_{F}}\sum _{n=0}^{N_{F}-1}r(n + kN_{ F})e^{-2\pi nm/N_{F} }, }$$
(34)
where k is the FFT block index, m is the frequency bin, and N F is the FFT size. Each channel is represented by M bins. Using Parseval’s theorem, the signal power in the l-th channel can be computed in frequency domain as
$$\displaystyle{ \varLambda _{E,l} = \frac{1} {KM}\sum _{k=0}^{K-1}\sum _{ m}\vert R_{k}[m]\vert ^{2}, }$$
(35)
where the second sum term is over the l-th channel bins. The issue here is that R k, l = m | R k [m] |2 is not the power of noise-only samples in case of H 0 (or noise-plus-signal samples in case of H 1) since now these samples are corrupted by the adjacent interference. In this case, the variance of the samples under H 0 is higher than σ w 2, increasing false alarms if the threshold is not corrected. As a result, it is critical to not only reduce the interference but also to estimate it in order to correct the decision threshold.

Mitigating Adjacent Interference Power

The high-level overview of the procedure that mitigates the adjacent interference power is shown in Fig. 17. In the first step, the RF antennas are switched off to calibrate the noise power. In the second step, a coarse estimation of the PSD is performed, where each channel is sensed using an energy detector with the same number of samples. By measuring the power in each adjacent channel to the band of interest, the interfering power is estimated in the third step.
Fig. 17

A high-level procedure to mitigate adjacent interference power

Once the interference power, σ I, l 2, is estimated for each channel, the sensing time needed for each channel to perform a fine PSD estimation is optimized. The PSD estimation can be robustified against power leakages using a windowed FFT instead of using (34). In the windowed FFT, the received samples r(n) are weighted first by a normalized window coefficient ω(n) before computing the FFT. This weighting, however, comes at the expense of reducing the spectral resolution. To maintain a high spectral resolution, a multitap-windowed FFT can be used instead. Mathematically, the multitap-windowed FFT is implemented as [26]
$$\displaystyle{ \hat{R}_{k}[m] = \frac{1} {N_{F}}\sum _{n=0}^{N_{F}-1}\left (\sum _{ p=0}^{P-1}\omega (n + pN_{ F})r(n + pN_{F} + kN_{F})\right )e^{-2\pi nm/N_{F} }, }$$
(36)
where p is the tap index and P is the total number of taps. In this case, the multitap-windowed energy detector is expressed as
$$\displaystyle{ \varLambda _{ME,l} = \frac{1} {N_{l}}\sum _{k=0}^{N_{l}-1}\sum _{ m}\vert \hat{R}_{k}[m]\vert ^{2}, }$$
(37)
where N l is the channel-specific sensing time. The sensing time differs across channels depending on the interfering powers in each channel. For a desired (P f , P d ) pair and \(\mathop{SNR}\nolimits\) sensitivity, N l can be computed as follows:
$$\displaystyle{ N_{l} =\mu \left [\left (1 + \frac{\sigma _{I,l}^{2}} {\sigma _{w}^{2}} \right )\frac{Q^{-1}(P_{f}) - Q^{-1}(P_{d})} {\mathop{SNR}\nolimits } - Q^{-1}(P_{ d})\right ]^{2}, }$$
(38)
where μ is a fitting factor that can be calculated beforehand. This expression can be derived from the detection performance of the multitap-windowed energy detector, which is shown to be [25]
$$\displaystyle{ P_{d} = Q\left ( \frac{1} {1 +\mathop{ SNR}\nolimits +\sigma _{I,l}^{2}/\sigma _{w}^{2}}\left [Q^{-1}(P_{ f})(1 +\sigma _{ I,l}^{2}/\sigma _{ w}^{2}) -\sqrt{\frac{N_{l } } {\mu }} \mathop{ SNR}\nolimits \right ]\right ). }$$
(39)
Once the PSD is finely estimated for each channel, the threshold is corrected before the multiband detection. Specifically, it is computed as
$$\displaystyle{ \lambda _{l} = \left (\sigma _{w}^{2} +\sigma _{ I,l}^{2}\right )\left (\sqrt{\mu /N_{ l}}Q^{-1}(P_{ f}) + 1\right ). }$$
(40)
Figure 18 illustrates the DSP used for the multitap-windowed energy detector.
Fig. 18

The multitap-windowed energy detector

Figure 19a shows the number of samples needed for (P f , P d ) = (0. 1, 0. 9) with variations on the interfere-to-noise-power ratio (INR), i.e., σ I, l 2σ w 2. Three detectors are shown: the multitap-windowed detector; the windowed detector, i.e., P = 1; and the conventional one, i.e., no windowing is used. It is assumed that \(\mathop{SNR}\nolimits = -5\) dB, and the adjacent interferes are one bin away from the band of interest. It is evident that the multitap-windowed detector provides a significant reduction in the sensing time. Figure 19b shows the false alarm probability with variations of the INR. It is clear that adapting the threshold maintains the false alarm to 0. 1 as desired.
Fig. 19

Performance of different energy-based detectors in the presence of adjacent interference

RF Front-End Nonlinearity

In wideband sensing, the received signal may contain multiple primary user signals. Even if all primary signals are transmitted with the same power, at the CR receiver, these signals can have various power levels, depending on the distance of these users to the CR receiver and channel fading. In the presence of strong signals, the receiver’s LNA may operate in a nonlinear region. Such nonlinearity introduces harmonics and intermodulation (IM) terms.

Figure 20 shows a receiver sensing a wideband spectrum that contains two strong signals, henceforth denoted as blockers, located at f b1 and f b2. The signal of interest is located at f 0 = 2f b2f b1. At the output of the LNA, the blockers introduce IM terms in the same band as the desired signal. These spurious terms remain after downconverting the wideband spectrum into baseband. Hence, the received samples, resulted from digitizing the baseband, are corrupted with the IM terms.
Fig. 20

LNA nonlinearity introduces IM terms

Modeling LNA Nonlinearities

Different IM terms are generated due to the nonlinearity of the LNA. However, not all these terms affect the received samples. For instance, even-order IM terms lie outside the frequency support of the signal of interest, and hence they can be filtered efficiently. Similarly, odd-order nonlinearities are typically dominated by third-order nonlinearities, making the impact of high odd-order terms, e.g., 5th order and higher, negligible. Hence, only third-order nonlinearities are considered in the subsequent analysis.

In the presence of nonlinearities, the received samples can be modeled as [19, 30]
$$\displaystyle{ r(n) =\beta _{1}x(n) +\beta _{3}x(n)\vert x(n)\vert ^{2} + w(n), }$$
(41)
where x(n) is the wideband signal and w(n) is an additive white Gaussian noise. Here, β 1 and β 3 are constants that are characteristics of the receiver front-end. Note that β 1 x(n) is the linear term, whereas β 3 x(n) | x(n) |2 is the third-order nonlinear term.
In the example of a single pair of blockers, it can be shown that the signal in the channel of interest is given by [30]
$$\displaystyle{ r_{d}(n) \approx \left (\beta _{1}x_{d}(n) + \frac{3} {2}\beta _{3}x_{b1}^{{\ast}}(n)x_{ b2}^{2}(n)\right )e^{2\pi f_{if}nT_{s} } + w(n), }$$
(42)
where x d (n) is the signal of interest, x b1(n) and x b2(n) are the two blockers, and f if = 2f 2f 1 is the intermediate frequency, where the signal of interest resides. Note that x d (n) is present in H 1 and absent in H 0, and the CR receiver must determine which hypothesis is true. The challenge here is that in the presence of strong blockers, the IM terms can be stronger than the signal of interest, making the detection difficult.

Mitigating LNA Nonlinearities

By viewing the IM terms as interference, one may follow the same approach used in Fig. 17, where the sensing time and the decision threshold are adapted. This requires estimating the blocker power [19]. The challenge here is that estimating the blocker, say in f b1, cannot be directly done by measuring the energy in that channel since it does not only contain the blocker signal but also the self-interference as shown in Fig. 20. Indeed, recall the received signal:
$$\displaystyle\begin{array}{rcl} r(n)& =& \mathop{\underbrace{\left (\beta _{1}x_{d}(n) + \frac{3} {2}x_{b1}^{{\ast}}(n)x_{ b2}^{2}(n)\right )e^{2\pi f_{if}nT_{s}}}}\limits _{ \text{signal at }f_{if}} +\mathop{\underbrace{ \bar{x}_{b1}(n)e^{2\pi f_{1}nT_{s}}}}\limits _{ \text{signal at }f_{1}} +\mathop{\underbrace{ \bar{x}_{b2}(n)e^{2\pi f_{2}nT_{s}}}}\limits _{ \text{signal at }f_{2}} \\ & & +w(n), {}\end{array}$$
(43)
where \(\bar{x}_{bi}(n)\) is a function of the i-th blocker signal x bi (n), and it can be shown that [19]
$$\displaystyle{ \bar{x}_{b1}(n) =\beta _{1}x_{b1}(n) + \frac{3} {2}\beta _{3}x_{b1}(n)\vert x_{b1}(n)\vert ^{2} + 3\beta _{ 3}x_{b1}(n)\vert x_{b2}(n)\vert ^{2}. }$$
(44)
The self-interference is hence defined as \(\phi _{i}(n) =\bar{ x}_{bi}(n) -\beta _{1}x_{bi}(n)\). It is observed that estimating the blocker power cannot be implemented using a time average of \(\bar{x}_{bi}(n)\) due to the presence of self-interference. Hence, a more advanced estimation is needed. Once estimated, the sensing time and the decision threshold can be adapted.

An alternative approach is to cancel the IM terms instead of estimating them since the latter approach typically requires increasing the sensing time to mitigate the presence of interference. Indeed, it is shown in (38) that N l σ I, l 2σ w 2, which shows that higher interference power requires longer sensing duration in order to mitigate it.

The cancellation scheme is as follows. An additional band-pass filtering stage is applied to the received samples to estimate the IM terms that fall in the channel of interest, i.e.,
$$\displaystyle{ \hat{x}_{b}(n) = \frac{1} {\beta _{1}^{3}}\bar{x}_{b1}(n)\bar{x}_{b2}^{2}(n). }$$
(45)
Then, this estimate is subtracted from the received signal in the band of interest, i.e.,
$$\displaystyle{ \hat{r}_{d}(n) = r_{d}(n) -\frac{3} {2}\theta \hat{x}_{b}(n), }$$
(46)
where θ is a parameter that can be optimized using an adaptive filter to minimize the IM term. The receiver architecture of this method is shown in Fig. 21.
Fig. 21

An architecture that cancels the IM terms

Figure 22a shows the detection performance of the energy and cyclostationarity detectors in the presence and absence of LNA nonlinearities. It is assumed that the signal-to-blocker ratio (SBR) is set to − 70 dB, \(\mathop{SNR}\nolimits = 3\) dB, and the thresholds are optimized to achieve P f = 0. 1. It is observed that the detection performance is degraded when the LNA operates in the nonlinear regime, making it critical to compensate for this impairment. Figure 22b shows the probability of detection with variations of the SBR, where the compensation algorithm presented in Fig. 21 is used with adaptive θ [19]. It is assumed that N = 500 and \(\mathop{SNR}\nolimits = 3\) dB. Using the cancellation algorithm, the performance significantly improves particularly when the blocker power is strong relative to the signal power .
Fig. 22

Effect of nonlinearities on detection performance in the presence and absence of the compensation algorithm

Summary

Spectrum sensing is an integral component of the cognitive radio system. To this end, modeling the different sensing techniques is critical to ensure reliable detection. While spectrum sensing has been largely studied using the classical detection modeling tools, there are key differences that are inherited to cognitive radio. In particular, spectrum sensing should be robust in negative SNR regimes, where measurements have shown that the detection performance may deviate from that predicted by the theoretical expressions. Indeed, in negative SNRs, noise power estimation becomes difficult and synchronization leads to frequency offsets. In addition, it is shown that converting the wideband sensing problem into several narrowband sensing problems requires additional care due to the adjacent interfering power resulted from the presence of strong signals and the IM terms resulted from the RF front-end nonlinearities.

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Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Electrical EngineeringUniversity of California, Los Angeles (UCLA)Los AngelesUSA

Section editors and affiliations

  • Wei Zhang
    • 1
  1. 1.School of Electrical Engineering and TelecommunicationsThe University of New South WalesSydneyAustralia

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