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HigherOrder Cyclostationarity Detection for Spectrum Sensing
EURASIP Journal on Wireless Communications and Networking volume 2010, Article number: 721695 (2010)
Abstract
Recent years have shown a growing interest in the concept of Cognitive Radios (CRs), able to access portions of the electromagnetic spectrum in an opportunistic operating way. Such systems require efficient detectors able to work in low SignaltoNoise Ratio (SNR) environments, with little or no information about the signals they are trying to detect. Energy detectors are widely used to perform such blind detection tasks, but quickly reach the socalled SNR wall below which detection becomes impossible Tandra (2005). Cyclostationarity detectors are an interesting alternative to energy detectors, as they exploit hidden periodicities present in manmade signals, but absent in noise. Such detectors use quadratic transformations of the signals to extract the hidden sinewaves. While most of the literature focuses on the secondorder transformations of the signals, we investigate the potential of higherorder transformations of the signals. Using the theory of HigherOrder Cyclostationarity (HOCS), we derive a fourthorder detector that performs similarly to the secondorder ones to detect linearly modulated signals, at SNR around 0 dB, which may be used if the signals of interest do not exhibit secondorder cyclostationarity. More generally this paper reviews the relevant aspects of the cyclostationary and HOCS theory, and shows their potential for spectrum sensing.
1. Introduction
Many studies have shown that the static frequency allocation for wireless communication systems is responsible for the inefficient use of the spectrum [1]. This is so because the systems are not continuously transmitting. Cognitive Radios (CRs) networks try to make use of the gaps that can be found in the spectrum at a given time. This opportunistic behavior categorizes CR as secondary users of a given frequency band, by contrast with the systems that were permanently assigned this band (primary users) [2]. For the CR concept to be viable, it is required that it does not interfere with the primary user services. It means that the system must be able to detect primary user signals in low signaltonoise ratio (SNR) environments fast enough. Efforts are being made to improve the performance of the detectors [3].
A radiometer (also called energy detector) can be used to detect completely unknown signals in a determined frequency band [4]. It is historically the oldest and simplest detector, and it achieves good performance when the SNR is strong enough. Unfortunately, since it is based on an estimation of the inband noise power spectral density (PSD), it is affected by the noise level uncertainty (due to measurement errors or a changing environment), especially at low SNR [5], where it reaches an absolute performance limit called the SNR wall. Another type of detector is based on the spectral redundancy present in almost every manmade signal. It is called a cyclic feature detector and will be the kind of detector of interest in this paper.
Cyclic feature detectors make use of the cyclostationarity theory, which can be divided in two categories: the secondorder cyclostationarity (SOCS) introduced by Gardner in [6–8] and the higherorder cyclostationarity (HOCS) introduced by Gardner and Spooner in [9, 10]. The SOCS uses quadratic nonlinearities to extract sinewaves from a signal, whereas the HOCS is based on thorder nonlinearities. The idea behind this theory is that manmade signals possess hidden periodicities such as the carrier frequency, the symbol rate or the chip rate, that can be regenerated by a sinewave extraction operation which produces features at frequencies that depend on these hidden periodicities (hence called cyclic features and cycle frequencies resp.). Since the SOCS is based on quadratic nonlinearities, two frequency parameters are used for the sinewave extraction function. The result is called the spectral correlation density (SCD), and can be represented in a bifrequency plane. The SCD can be seen as a generalization of the PSD, as it is equal to the PSD when the cycle frequency is equal to zero. Therefore, the SOCS cyclic feature detectors act like energy detectors, but at cycle frequencies different from zero. The advantage of these detectors comes from the absence of features (at least asymptotically) when the input signal is stationary (such as white noise), since no hidden frequencies are present, or when the input signal exhibits cyclostationarity at cycle frequencies different than the one of interest. The HOCS cyclicfeature detectors are based on the same principles, but the equivalent of the SCD is a dimensional space (). Like SOCS detectors, HOCS detectors have originally been introduced in the literature to blindly estimate the signal frequency parameters.
It has been shown that the secondorder cyclostationarity detectors perform better than the energy detectors in low SNR environments [7], and this has recently triggered a lot of research on the use of cyclostationarity detectors for spectrum sensing in the context of cognitive radios [11, 12]. However the secondorder detectors suffer from a higher computational complexity that has just become manageable. First fieldprogrammable gate array (FPGA) implementations are presented in [13, 14].
Higherorder detectors are generally even more complex, and since the variance of the features estimators increases when the order rises, most research results concern secondorder detectors. We will nevertheless demonstrate that it is possible to derive fourthorder detectors that bear comparable performances to secondorder ones to detect linearly modulated baseband signals at SNR around 0 dB. The paper will include a mathematical analysis of the detection algorithm, the effects of each of its parameters and its computational complexity. Performance will be assessed through simulations and compared with the secondorder detector.
After introducing the system model in Section 2, we will briefly review the basic notions of cyclostationarity theory in Section 3 in order to understand how secondorder detectors work and identify their limitations. Afterwards, we will move on to HOCS theory, and present its most relevant aspects in Section 4, which will be used to characterize the linearly modulated signals in Section 5 and to derive an algorithm that may be used for signal detection of linearly modulated signals in Section 6. We will conclude by a comparison of the new detector performance with secondorder detector and energy detector performances in Section 7.
2. System Model
This paper focuses on the detection of linearly modulated signals, like pulse amplitude modulation (PAM) or quadrature amplitude modulation (QAM) signals. The baseband transmitted signal is usually expressed as
where is the sequence of information symbols transmitted at the rate and is the pulse shaping filter (typically a squareroot Nyquist filter). After basebandtoradio frequency (RF) conversion, the RF transmitted signal is given by:
where and is the carrier frequency. In the PAM case, the symbols are real and only the cosine is modulated. In the QAM case, the symbols are complex and both the cosine and sine are modulated. A QAM signal can be seen as two uncorrelated PAM signals modulated in quadrature.
For the sake of clarity, we assume that the signal propagates through an ideal channel. Our results can nevertheless be extended to the case of multipath channels, if we consider a new pulse shape that is equal to the convolution of squareroot Nyquist filter with the channel impulse response. However, this would make the new pulse random. Simulations have shown that both secondorder and fourthorder detectors are affected in the same way by a multipath channel (equivalent degradation of performances). Therefore it does not seem critical to introduce multipath channels in order to compare the two, and it allows us to work with a constant pulse shape. Additive white Gaussian noise (AWGN) of onesided PSD equal to corrupts the signal at the receiver. Some amount of noise uncertainty can be added to . The detection of the signal at the receiver can be either done directly in the RF domain or in the baseband domain after RFtobaseband conversion.
3. SecondOrder Cyclostationarity
Two approaches are used to introduce the notion of cyclostationarity [8]. While the first approach introduces the temporal features of cyclostationary signals, the second approach is more intuitive and is based on a graphical representation of spectral redundancy. Both approaches lead to the same conclusion. This section reviews the main results of the secondorder cyclostationarity theory, which will be generalized to higherorder cyclostationarity in the next sections.
3.1. Temporal Redundancy
A widesense cyclostationary signal exhibits a periodic autocorrelation function [6, 7]
where denotes the statistical expectation operator. Since is periodic, it can be decomposed in a Fourier series
where the sum is over integer multiples of the fundamental frequencies. The coefficient is called the cyclic autocorrelation function, and represents the Fourier coefficient of the series given by
When the signal is cycloergodic, the expectation in the definition of the autocorrelation can be replaced by a time average so that
The cyclic autocorrelation is therefore intuitively obtained by extracting the frequency sinewave from the timedelay product . The SCD is defined as the Fourier transform of over . We notice that the only cyclic frequencies for which the SCD will not be null are the ones corresponding to the Fourier coefficients.
3.2. Spectral Redundancy
Let be the Fourier transform of . The SCD measures the degree of spectral redundancy between the frequencies and ( being called the cyclic frequency). It can be mathematically expressed as the correlation between two frequency bins centered on and when their width tends toward zero [6, 7]
where denotes the shorttime Fourier transform of the signal
Since the SCD depends on and , it can be graphed as a surface over the bifrequency plane . When , the SCD reduces to the PSD.
3.3. Baseband and RF SecondOrder Features
The performance of the cyclic feature detectors will first depend on the strength of the features they are trying to estimate. The two most common features exploited to detect the linearly modulated signals are linked with the symbol rate and the carrier frequency.

(i)
The symbol rate feature is usually exploited after RFtobaseband conversion at the receiver. As its name suggests it, it originates from the symbol rate at the transmitter. Since this is a discrete signal, its frequency spectrum is periodic, with a period equal to the inverse of the sample rate (which is equal to the symbol rate before RF conversion). If there is some excess bandwidth in the system, or in other words, if the pulse shaping filter does not totally cut off the frequency components larger than half the inverse of the symbol rate, some frequencies will be correlated, as shown in Figure 1.

(ii)
The doubledcarrier frequency feature is directly exploited in the RF domain. It is based on the symmetry of the RF spectrum, and it is much stronger than the symbol rate feature (it is as strong as the PSD). Since it depends on the symmetry of the spectrum of the baseband signal, it only exists if the modulation used has no quadrature components. If a real PAM scheme is used, the carrier feature exists, as illustrated in the left part of Figure 2. If a complex QAM scheme is used, the carrier feature vanishes, as illustrated in the right part of Figure 2.
Since complex modulations are quite common, it would not be possible to implement a cyclic feature detector for CRs based on the doubledcarrier frequency feature. On the other hand, the symbol rate feature solely depends on the pulse shaping filter. Provided that there is some excess bandwidth, the symbol rate feature will exist, whatever the modulation. Unfortunately, that feature is relatively small and depends on the amount of excess bandwidth. We can therefore ask ourselves if it would not be possible to find greater features using a fourthorder detector.
4. HigherOrder Cyclostationarity
The higherorder cyclostationarity (HOCS) theory is a generalization of the secondorder cyclostationarity theory, which only deals with secondorder moments, to thorder moments [9, 10]. It makes use of the fractionoftime (FOT) probability framework (based on time averages of the signals) which is closely related to the theory of highorder statistics (based on statistical expectations of the signals), by ways of statistical moments and cumulants. This section reviews the fundamentals of the HOCS theory and highlights the metrics that can be used for spectrum sensing.
4.1. LagProduct
We must always keep in mind that the goal of the HOCS theory is to extract sinewaves components from a signal, in which they are hidden by random phenomena. To extract, or regenerate, these frequencies, a nonlinear operation must be called upon. The secondorder theory uses the timedelay product which will be transformed in the autocorrelation after averaging. A natural and intuitive generalization of this operation to the thorder is called the lagproduct and can be expressed as [9]:
where the vector is composed of the individual delays (). The notation indicates an optional conjugation of the signal .
4.2. Temporal Moment Function and Cyclic Temporal Moment Function
If the signal possesses a thorder sinewave of frequency , then the averaging of the lagproduct, multiplied by a complex exponential of frequency , must be different from zero [9]:
Obviously, is a generalization of the cyclic autocorrelation function described in (5). It is called the thorder cyclic temporal moment function (CTMF). The sum of the CTMF (multiplied by the corresponding complex exponentials) over frequency is called the temporal moment function (TMF) and is a generalization of the autocorrelation function described in (3):
Each term of the sum in (12) is called an impure thorder sinewave. This is so because the CTMF may contain products of lowerorder sinewaves whose various orders sum to . In order to extract the pure thorder sinewave from the lagproduct, it is necessary to subtract the lowerorder products. The pure thorder sinewave counterpart of the CTMF, denoted by , is called the cyclic temporal cumulant function (CTCF). The pure thorder sine wave counterpart of the TMF, denoted by , is called the temporal cumulant function (TCF).
4.3. Temporal Cumulant Function and Cyclic Temporal Cumulant Function
The CTMF and TMF have been computed by using the FOT probability framework. In order to compute the CTCF and TCF, it is interesting to make use of the equivalence between the FOT probability framework and the highorder statistics theory. More specifically, the paper [9] demonstrates that the TMF of a signal can be seen as the thorder moment of the signal, and that the TCF of a signal can be seen as the thorder cumulant of the signal (hence their names). By using the conventional relations between the moments and the cumulants found in the highorder statistics theory, the TCF takes therefore the form:
where denotes the set of partitions of the index set (10), is the number of elements in the partition , and is the TMF of the thelement of order of the partition .
The CTCFs are the Fourier coefficients of the TCF and can be expressed in terms of the CTMFs:
where denotes the set of vectors of cycle frequencies for the partition that sum to (), and is the CTMF of the thelement of order of the partition at the cycle frequency .
The CTCF is periodic in : ( is the dimension vector composed of ones, meaning that is added to all elements of ). Therefore, it is not absolutely integrable in . To circumvent this problem, one dimension is fixed (e.g., ), and the CTCF becomes: . This function is called reduced dimensionCTCF (RDCTCF). It is the key metric of the ensuing algorithms for HOCS detectors. It should be noted that the equivalent exists for the CTMF and is called the RDCTMF (). However the RDCTMF is generally not absolutely integrable.
4.4. Cyclic Polyspectrum
The need for integrability comes from the desire to compute the Fourier transform of the RDCTCF, which gives the cyclic polyspectrum (CP). The CP is a generalization of the SCD plane for cyclostationnary signals. However it is not necessary to compute the CP of a signal for sensing applications since detection statistics can be directly derived from a single slice of the RDCTCF. For this reason, and the computational complexity gain, we will put the spectral parameters aside and devote our attention to the RDCTCF.
5. FourthOrder Features of Linearly Modulated Signals
We have previously talked about the secondorder cyclic features for communication signals, and we saw that the carrier frequency features tend to vanish from the SCD plane if the modulation is complex. We also asked ourselves if a fourthorder transformation of the signal may suppress the destructive interferences of quadrature components of a signal. We now have to gauge the potential of these fourthorder features. In this section, we compute the RDCTCF of the baseband and RF linearly modulated signals and identify the interesting features that can be used for signal detection.
5.1. Baseband Signals
The TCF of the baseband signal (1) has been computed in paper [10]. The mathematical derivation results in:
in which is the thorder cumulant of the symbol sequence :
where is the set of partitions of the set , is the number of elements in the partition , and is the order of the thelement in the partition (). is the thorder moment of the symbol sequence :
The expression of the moment can be understood this way: given a particular type of modulation, do the symbol variables elevated to the power (with optional conjugation specified by the operator ) gives a constant result? The answer to this question is helpful in assessing if a given signal may exhibit thorder features and what kind of conjugation must be used in the lagproduct (10). The appendix illustrates this result for the binary PAM and the quaternary QAM constellations (see also [10, 15]).
Computing the Fourier transform of the TCF and canceling reveals the RDCTCF in the form of:
where the cycle frequencies are integer multiples of the symbol rate ( with integer). The RDCTCF of the baseband signal is nonzero only for harmonics of the symbol rate. The amplitude of the features tend to zero as the harmonic number increases.
5.2. RF Signals
The RDCTCF of the RF signal specified by (2) can be inferred from the RDCTCF of the baseband signal by noting that the RF signal is obtained by modulating two independent PAM signals in quadrature. We need to calculate the CTCFs of PAM, sine and cosine signals, and to combine them using the following rules:

(i)
The cumulant of the sum is equal to the sum of the cumulants if the signals are independent. Therefore, if where and are two independent random signals, we have:
(19)and, after Fourier transform, we obtain:
(20) 
(ii)
The moment of the product is equal to the product of the moments if the signals are independent. Therefore, if where and are two independent random signals, we have:
(21)and, after Fourier transform, we obtain:
(22)
Equation (22) means that we have to multiply all CTMFs of and which sum to . If one of the signals is nonrandom ( in our case), the CTMF of the random signal can be replaced by its CTCF:
The CTCFs of the baseband PAM signals can be computed using (18). The only difference with a QAM signal resides in the cumulant of the symbol sequence , which must be computed for PAM symbols through (16) and (17) (see the binary PAM case in the appendix).
The CTMF of the sine and cosine signals can easily be determined from the expression of their lagproducts:
where . The lagproduct can be decomposed into a sum of cosine signals at various frequencies using Simpson formulas:
for the second order, and:
for the fourth order. It is clear that the CTMF of sine or cosine signals is made of Dirac's deltas at cycle frequencies , , and .
Since the real and imaginary parts of are two statistically independent PAM signals, the CTCF of is the sum of two CTCFs of modulated PAM signals in quadrature. The CTCFs of and are equal and denoted by in our next results. We can finally write:
For , we observe the destructive interference between the components of and at twice the carrier frequency, as was introduced in Section 3.
For , we also observe that the components of and at twice the carrier frequency cancel out, just as they do for the second order. There only remain the features at zero and four times the carrier frequency:
Since is a sum of cosines that depend on and (the notation indicates an optional sign change according to the expressions (25)(26)), the features are six times smaller than the features (at least when is null) and are therefore less suited for sensing scenarios.
5.3. Baseband and RF FourthOrder Features
We have to choose between baseband or RF signals and decide on the cycle frequency that will be used by the detector. We have seen that baseband QAM signals have features at the cycle frequencies that are multiples of the symbol rate (), whereas RF signals have additional features at cycle frequencies that depend on the carrier frequency (, , ). It has been shown that these additional features are small and that the strongest feature for both baseband and RF signals is obtained when the cycle frequency is equal to zero. Since noise signals do not have any fourthorder feature (the fourthorder cumulant of a Gaussian random variable is equal to zero), even when . Note that is a degenerated cycle frequency, which is present even in stationary signals. However, since it gives the strongest 4thorder feature, it is the frequency that will be preferred for our sensing scenario, even if the denomination "cyclicfeature detector" becomes inappropriate in this case.
Simulations made with baseband or RF signals for have shown that the two detectors exhibit similar performances. From now on, we will focus on the fourthorder feature detection for baseband signals and let aside the fourthorder feature detection for RF signals, as it enables a significant reduction of the received signal sampling frequency. The feature obtained in this situation is illustrated in Figure 3.
6. FourthOrder Feature Detectors
6.1. RDCTCF Estimator
In order to estimate the RDCTCF of the baseband QAM signal, we would have to use (14). Luckily, the signal is complex and the second order features disappear if we do not use any conjugation in the lag product (see the quaternary QAM example in the appendix). Therefore the RDCTCF is equal to the RDCTMF:
In practice, the RDCTCF is estimated based on a size finite observation window of the received sequence obtained after sampling the received signal.
with and are the elements of the discrete lagvector of size .
6.2. Noise Mean and Variance
When there is only noise in the system, the mean of the RDCTCF is equal to since the fourthorder cumulant of a Gaussian random variable is null. On the other hand, the variance of the RDCTCF is a function of the lagvector given by:
in which is the variance of AWGN noise samples at the input of the RDCTCF estimator. Simulations illustrated in Figure 4 confirm the result (31). Every discrete lagvector for which two or more values , are identical should be avoided, since it increases the noise variance. However, to afford the luxury of choosing lag values that are different from zero, we would have to increase the sampling rate at the receiver, which in turn would increase the noise power. Simulations have shown that it is better to use the lowest sampling rate that still satisfies Shannon's theorem, and set all lag values equal to zero. The RDCTCF variance also quite naturally decreases as the observation window is increased.
6.3. Detector
The detector has to decide between two hypotheses: hypothesis implying that no signal is present, hypothesis implying that the linearly modulated signal is present. The absolute value of the feature (here the RDCTCF) is compared to a threshold to make a decision:
The threshold is usually fixed to meet a target probability of false alarm (decide if ). In order to compute the threshold level as a function of the probability of false alarm, we must know the distribution of the RDCTCF. We already know its mean and variance values and using the centrallimit theorem, we assume that the output distribution is Gaussian (see also [16]). As a consequence, the absolute value of the RDCTCF takes the form of a Rayleigh distribution and the threshold level can be found using:
where is the probability of false alarm.
7. Detector Comparison
We will now briefly review the principles of all detectors previously mentioned in this paper, and compare their performance and computational complexity. We assume that secondorder and fourthorder detectors work only at a single location of the feature they exploit (the secondorder detector works at most favorable frequency, the fourthorder detector works at the most favorable value of the discrete lagvector ). MonteCarlo simulations were used, each of which used 5000 iterations.
7.1. Energy Detector
This is the most widely used detector in wireless communication systems. It averages the square modulus of the received sequence over time:
Its advantages are its simplicity and its ability to perform blind detection (since it does not require any information about the signal it is trying to detect). Unfortunately, it has been demonstrated that it cannot be used in lowSNR environments due to its sensitivity to noise uncertainty [6].
7.2. SecondOrder Detector
This detector computes an estimation of the SCD by averaging, over time and frequency domains, the cyclic periodogram of the signal spectrum computed for a finite time window at time :
where is the number of time windows and is the number of frequency bins. It is a much more complex and less efficient detector, which requires some characteristics of the signal in order to work (e.g., the symbol rate must be known in advance). Its advantage resides in the absence of features (at least asymptotically) when the input signal is a white noise, which results in the output mean of the detector always being equal to zero in presence of noise, therefore shielding the detector from noise uncertainty effects. Its computational complexity evolves as if the FFTs used to evaluate the cyclic periodogram [6] have a length of 1024 samples, and the total number of samples is equal to .
7.3. FourthOrder Detector
This detector averages the lagproduct of the received sequence over time:
This detector is simpler to implement than the previous one (no Fourier transform of the signal is required since we work in the time domain), which results in a computational complexity evolving as , the total number of samples. It benefits from the same immunity to noise uncertainty, and is therefore suited for operations at low SNR.
7.4. Performance Comparison
We may now take a look at the performance of the different detectors. Figure 5 illustrates the probability of missed detection (decide if ) curves as a function of the SNR for the three detectors under consideration. The threshold has been set in the three cases to achieve a target probability of false alarm equal to . These curves have been obtained without adding any noise uncertainty to the signal. In such conditions, the energy detector is the optimal detector for blind detection, and can be considered as a reference. It appears that the secondorder detector and the fourthorder detector, have similar performances when the SNR is around zero dB: for the same complexity, (that leads to an observation time ten times longer for the fourthorder detector), both detectors exhibit the same probability of missed detection (roughly 1 percent) at an SNR of −0.8 dB. However, when we consider an SNR of −4 dB, the fourthorder detector requires much more samples, which makes it more complex than the secondorder. Besides, the detectiontime constraints that are part of the cognitive radios reglementation would not be met if the observation time is too long.
If we add some amount of noise uncertainty, the energy detector cannot perform reliable detections and must be discarded, whereas the cyclic feature detectors remain unaffected. In order to verify this assumption, we computed the receiver operating characteristics (ROC) curves of the fourthorder detector for two situations, one without any noise uncertainty, and one with dB of noise uncertainty. The results are illustrated in Figure 6. We observe that the energy detector, which had the best ROC curve in the first case is a lot more affected by the noise uncertainty than the fourthorder detector. ROC curves for the secondorder detector can be found in [7], and show the same immunity to noise uncertainty than the fourthorder.
8. Conclusion
This paper has started from the need for robust detectors able to work in low SNR environments. A brief review of the secondorder cyclostationarity and secondorder cyclic feature detectors has exposed the advantages and drawbacks of such detectors, and explained the intuition that lead to the study of higherorder cyclostationarity (HOCS). The main guideline is to identify features of sufficient strength and to design a detector able to extract it from the signal. The most relevant aspects of HOCS theory have then been analyzed and we have derived a new fourthorder detector that can be used for the detection of linearly modulated signals. Simulation results have shown that fourthorder cyclic feature detectors may be used as a substitute for secondorder detectors at SNR around zero dB, which could be needed if the received signals do not exhibit secondorder cyclostationarity.
Appendices
A. Cumulants of the Binary PAM
This section computes the second and fourthorder cumulants of a binary PAM sequence. The symbols take the values .
A.1. SecondOrder Cumulant
There are possible partitions of the set : and . Since the binary PAM constellation is symmetric, only the first partition has a chance to give a product of moments different from . We will limit our investigations to the first partition.
The partition gives for its single element, so that .
A.2. FourthOrder Cumulant
There are possible partitions of the set , but only the ones that group by two or four have a chance to give a product of moments different from , which reduces the number of interesting partitions to four: ; ; and .
The first partition gives for its single element, and the three last partitions ; and give for their two elements, so that .
B. Cumulants of the Quaternary QAM
This section computes the second and fourthorder cumulants of a 4QAM sequence. The symbols take the values .
B.1. SecondOrder Cumulants
There are possible partitions of the set : and . Since the 4QAM constellation is symmetric, only the first partition has a chance to give a product of moments different from . We limit therefore our investigations to the first partition.
Different results are obtained according to the number of conjugations in the lagproduct (10):
(i)When no conjugation or two conjugations are used in the lagproduct, the partition gives for its single element, so that .
(ii)When one conjugation is used in the lagproduct, the partition gives for its single element, so that .
B.2. FourthOrder Cumulant
There are possible partitions of the set , but only the ones that group by two or four have a chance to give a product of moments different from , which reduces the number of interesting partitions to four: ; ; and .
Different results are obtained according to the number of conjugations in the lagproduct (10):

(i)
When no conjugation or four conjugations are used in the lagproduct, the first partition gives for its single element, and the three last partitions ; and give for their two elements, so that .

(ii)
When two conjugations are used in the lagproduct, arbitrary placed for this example on the second and fourth element of the lagproduct, the partition gives for its single element, the two partitions and give for their two elements, and the partition gives for its two elements, so that .

(iii)
When one or three conjugations are used in the lagproduct, the partition gives for its single element, and the three last partitions ; and give for at least one of their two elements, so that .
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Renard, J., VerlantChenet, J., Dricot, JM. et al. HigherOrder Cyclostationarity Detection for Spectrum Sensing. J Wireless Com Network 2010, 721695 (2010). https://doi.org/10.1155/2010/721695
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DOI: https://doi.org/10.1155/2010/721695