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 Signal-to-Noise 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 so-called 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 man-made signals, but absent in noise. Such detectors use quadratic transformations of the signals to extract the hidden sine-waves. While most of the literature focuses on the second-order transformations of the signals, we investigate the potential of higher-order transformations of the signals. Using the theory of Higher-Order Cyclostationarity (HOCS), we derive a fourth-order detector that performs similarly to the second-order ones to detect linearly modulated signals, at SNR around 0 dB, which may be used if the signals of interest do not exhibit second-order cyclostationarity. More generally this paper reviews the relevant aspects of the cyclostationary and HOCS theory, and shows their potential for spectrum sensing.
th-order nonlinearities. The idea behind this theory is that man-made signals possess hidden periodicities such as the carrier frequency, the symbol rate or the chip rate, that can be regenerated by a sine-wave 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 sine-wave 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 cyclic-feature 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.
is the sequence of information symbols transmitted at the rate 
and 
is the pulse shaping filter (typically a square-root Nyquist filter). After baseband-to-radio frequency (RF) conversion, the RF transmitted signal is given by:
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.
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 RF-to-baseband conversion.
exhibits a periodic autocorrelation function [
denotes the statistical expectation operator. Since 
is periodic, it can be decomposed in a Fourier series
is called the cyclic autocorrelation function, and represents the Fourier coefficient of the series given by
sine-wave from the time-delay 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.
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 [
denotes the short-time Fourier transform of the signal
and 
, it can be graphed as a surface over the bifrequency plane 
. When 
, the SCD reduces to the PSD.
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 
and 
where 
is the symbol rate. The symbol rate feature exists for baseband PAM/QAM signals if there is some excess bandwidth in the system.
and 
where 
is the carrier frequency. The doubled-carrier frequency feature exists for RF PAM signals as the baseband frequency spectrum exhibits a correlation between negative and positive frequencies. In the absence of any filtering, this correlation produces a symmetric frequency spectrum (left part). The doubled-carrier frequency feature vanishes for RF QAM signals as the baseband frequency spectrum is uncorrelated (right part).
th-order moments [
which will be transformed in the autocorrelation after averaging. A natural and intuitive generalization of this operation to the 
th-order is called the lag-product and can be expressed as [
is composed of the individual delays 
(
). The notation 
indicates an optional conjugation of the signal 
.
th-order sine-wave of frequency 
, then the averaging of the lag-product, multiplied by a complex exponential of frequency 
, must be different from zero [
is a generalization of the cyclic autocorrelation function described in (5). It is called the 
th-order 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):
th-order sine-wave. This is so because the CTMF may contain products of lower-order sine-waves whose various orders sum to 
. In order to extract the pure 
th-order sine-wave from the lag-product, it is necessary to subtract the lower-order products. The pure 
th-order sine-wave counter-part of the CTMF, denoted by 
, is called the cyclic temporal cumulant function (CTCF). The pure 
th-order sine wave counter-part of the TMF, denoted by 
, is called the temporal cumulant function (TCF).
th-order moment of the signal, and that the TCF of a signal can be seen as the 
th-order cumulant of the signal (hence their names). By using the conventional relations between the moments and the cumulants found in the high-order statistics theory, the TCF takes therefore the form:
denotes the set of partitions of the index set 
(10), 
is the number of elements in the partition 
, and 
is the TMF of the 
th-element of order 
of the partition 
.
denotes the set of vectors of cycle frequencies for the partition 
that sum to 
(
), and 
is the CTMF of the 
th-element of order 
of the partition 
at the cycle frequency 
.
: 
(
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 dimension-CTCF (RD-CTCF). 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 RD-CTMF (
). However the RD-CTMF is generally not absolutely integrable.
is the 
th-order cumulant of the symbol sequence 
:
is the set of partitions of the set 
, 
is the number of elements in the partition 
, and 
is the order of the 
th-element in the partition 
(
). 
is the 
th-order moment of the symbol sequence 
:
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 
th-order features and what kind of conjugation must be used in the lag-product (10). The appendix illustrates this result for the binary PAM and the quaternary QAM constellations (see also [
reveals the RD-CTCF in the form of:
with 
integer). The RD-CTCF 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.
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:
where 
and 
are two independent random signals, we have:
where 
and 
are two independent random signals, we have:
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:
, which must be computed for PAM symbols through (16) and (17) (see the binary PAM case in the appendix).
. The lag-product can be decomposed into a sum of cosine signals at various frequencies using Simpson formulas:
, 
, and 
.
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:
, we observe the destructive interference between the components of 
and 
at twice the carrier frequency, as was introduced in Section 3.
, 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:
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.
), 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 fourth-order feature (the fourth-order 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 4th-order feature, it is the frequency that will be preferred for our sensing scenario, even if the denomination "cyclic-feature detector" becomes inappropriate in this case.
have shown that the two detectors exhibit similar performances. From now on, we will focus on the fourth-order feature detection for baseband signals and let aside the fourth-order 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 
and 
for the cycle frequency 
. The values of 
and 
have been fixed to 
and 
respectively. The system parameters are: 
MHz symbol frequency, 
MHz sample frequency, normalized square-root Nyquist pulse shaping filter of 
roll-off factor.
finite observation window of the received sequence 
obtained after sampling the received signal.
and 
are the elements of the discrete lag-vector 
of size 
.
since the fourth-order cumulant of a Gaussian random variable is null. On the other hand, the variance of the RD-CTCF is a function of the lag-vector given by:
is the variance of AWGN noise samples at the input of the RD-CTCF estimator. Simulations illustrated in Figure 
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 RD-CTCF variance also quite naturally decreases as the observation window 
is increased.
and 
for the cycle frequency 
. The values of 
and 
have been fixed to 
and 
, respectively. The system parameters are: 
MHz symbol frequency, 
MHz sample frequency, 
. The number of noise realizations is 
.
implying that no signal is present, hypothesis 
implying that the linearly modulated signal is present. The absolute value of the feature (here the RD-CTCF) is compared to a threshold 
to make a decision:
if 
). In order to compute the threshold level as a function of the probability of false alarm, we must know the distribution of the RD-CTCF. We already know its mean and variance values and using the central-limit theorem, we assume that the output distribution is Gaussian (see also [
is the probability of false alarm.
). Monte-Carlo simulations were used, each of which used 5000 iterations.
computed for a finite time window at time 
:
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 [
.
, the total number of samples. It benefits from the same immunity to noise uncertainty, and is therefore suited for operations at low SNR.
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 second-order detector and the fourth-order 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 fourth-order 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 fourth-order detector requires much more samples, which makes it more complex than the second-order. Besides, the detection-time constraints that are part of the cognitive radios reglementation would not be met if the observation time is too long.
). The system parameters are: baseband QPSK signal with 
MHz symbol frequency, 
MHz sample frequency, square-root Nyquist pulse shaping filter of 
roll-off factor. No noise uncertainty added. The second-order detector is set to detect the symbol-rate feature (cf = 20 MHz), and the fourth-order detector works with the feature at four-times the carrier frequency, which is equal to zero in the present situation (cf = 0 MHz).Two observation times are considered for the three detectors: 50 and 500 
s. An observation time of 10 ms has been added for the fourth-order detector
dB of noise uncertainty. The results are illustrated in Figure 
dB uncertainty). The system parameters are: 
MHz symbol frequency, 
MHz sample frequency, square-root Nyquist pulse shaping filter of 
roll-off factor, SNR = 
dB.
.
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.
gives 
for its single element, so that 
.
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 
.
gives 
for its single element, and the three last partitions 
; 
and 
give 
for their two elements, so that 
.
.
possible partitions of the set 
: 
and 
. Since the 4-QAM 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.
gives 
for its single element, so that 
.
gives 
for its single element, so that 
.
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 
.
gives 
for its single element, and the three last partitions 
; 
and 
give 
for their two elements, so that 
.
gives 
for its single element, the two partitions 
and 
give 
for their two elements, and the partition 
gives 
for its two elements, so that 
.
gives 
for its single element, and the three last partitions 
; 
and 
give 
for at least one of their two elements, so that 
.