We have previously talked about the second-order 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 fourth-order transformation of the signal may suppress the destructive interferences of quadrature components of a signal. We now have to gauge the potential of these fourth-order features. In this section, we compute the RD-CTCF 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 . The mathematical derivation results in:
in which is the th-order 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 th-element in the partition (). is the th-order 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 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 [10, 15]).
Computing the Fourier transform of the TCF and canceling reveals the RD-CTCF in the form of:
where the cycle frequencies are integer multiples of the symbol rate ( 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.
5.2. RF Signals
The RD-CTCF of the RF signal specified by (2) can be inferred from the RD-CTCF 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:
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:
and, after Fourier transform, we obtain:
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:
and, after Fourier transform, we obtain:
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 lag-products:
where . The lag-product 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 Fourth-Order 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 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.
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 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 3.