I recently came across a published paper with the title Cyclostationary Correntropy: Definition and Application, by Aluisio Fontes et al. It is published in a journal called Expert Systems with Applications (Elsevier). Actually, it wasn’t the first time I’d seen this work by these authors. I had reviewed a similar paper in 2015 for a different journal.
I was surprised to see the paper published because I had a lot of criticisms of the original paper, and the other reviewers agreed since the paper was rejected. So I did my job, as did the other reviewers, and we tried to keep a flawed paper from entering the literature, where it would stay forever causing problems for readers.
The editor(s) of the journal Expert Systems with Applications did not ask me to review the paper, so I couldn’t give them the benefit of the work I already put into the manuscript, and apparently the editor(s) did not themselves see sufficient flaws in the paper to merit rejection.
It stings, of course, when you submit a paper that you think is good, and it is rejected. But it also stings when a paper you’ve carefully reviewed, and rejected, is published anyway.
Fortunately I have the CSP Blog, so I’m going on another rant. After all, I already did this the conventional rant-free way.
The paper is about a particular nonlinear transformation of a signal, and how to represent and exploit a representation of that transformation. Sounds familiar, right?
Let’s ignore Section 2.1, which reviews cyclostationarity, and begin with Section 2.2, which reviews the correntropy function. Equations (9)–(11) are reproduced here:
I see no problems here. Equation (11) follows from the definition of correntropy and the series expansion of . The critical question is about the expectations in the sum over . The authors suggest that these expectations could be periodic or polyperiodic functions of time when is cyclostationary, and so one may extract the Fourier coefficients from such periodic functions by the usual formula, which appears in (18),
where denotes our familiar infinite-time averaging operation.
So far, so good. Then the authors want to represent as a Taylor series, as before, and this is where things go wrong. We should see the following
which is a sum of a bunch of different cyclic temporal moment functions.
But the authors combine the complex exponential with the function and then attempt to expand that single exponential function. They write Equation (20)
(where they went back to the formulation of the cyclic component as the Fourier coefficient of the time-varying expected value). Here I see a sign error on the last term in the bracket, and an extra/mysterious term inside the bracket.
They then invoke cycloergodicity and return to the infinite-time averaging operation on the signal itself to claim (21):
So now the infinite-time averaging operation includes time averages of powers of .
In (22) through (25), the authors attempt to deal with this infinite sum. They lump all the bad terms involving powers of and products of and the signal into a function called , which is not explicitly written out. They get to (24),
Then they say
“… assuming as it tends to infinite [sic] in (24)”
(which is not true), and get (25)
which is wrong. Here we know that the term in (24) leads to , the term is a phase-shifted version of that term, namely , and the cross term gives rise to .
Moreover, the first term in (25) is simply the infinite-time average of a complex sine wave, which is unity for and zero for all other . Yet the authors state
“Besides, Eq. (25) shows that a sinusoisal [sic] function is always responsible for phase shifting the response regardless the [sic] stochastic process type. This property is due to the term .”
But that term is either () or (all other ), so can’t shift the phase of anything.
So I claim the theoretical development here is fundamentally flawed and nonsensical. The problem is that a lot of the mathematical development is hidden in the mysterious function. In my work on higher-order cyclostationarity, I take the opposite approach: What can we say about the utility of just the unique statistical information associated with each order ? I try to separate all the contributions to the signal’s probability density functions due to different moment/cumulant orders. These authors’ mission is to intentionally mix them all together. Why not just expand (A) above, and identify the various terms using previously defined functions? The various cyclic moments just fall right out of that expression.
The simulations section is also seriously flawed. First, why not compare their correntropy structure with something optimal or close, such as the single-cycle or multicycle detectors? Second, the frequency axes on the figures are quite hard to understand in light of the values in Table 1. The axes are labeled with normalized frequency (), the table tells us the carrier is MHz and the sampling rate is MHz, so the doubled-carrier feature for their BPSK signal should be at . Instead it is shown at 8e-7 (Fig 2). And the symbol rate shows up at 1e-7 but should be .
I also don’t believe Steps 3 and 4 of the algorithm in Section 4.1 can work in general. This is like the time-smoothing method, but they’ve left off the complex phase factor that accounts for the relative delay between the successive blocks. It will work provided the cycle frequencies that are exhibited by the signal in the data are equal to , where is the block length, but otherwise it won’t. In the real world, the cycle frequencies are almost never that convenient.
In Figure 6, where is the evidence that higher-order information is coming into view? The appearance of the small harmonics of 1e-7 is not explained. For BPSK, we’d expect to see evidence of the quadrupled carrier (which is quite strong), as well as the doubled carrier, but we see in the four plots mostly just the typical second-order cycle-frequency pattern for textbook BPSK.
Finally, the probabilities of detection for the author’s correntropy method are achieved at SNRs of more than dB less than those for the competing cyclostationary method. The author’s show that the conventional spectral correlation function for the noisy BPSK signal is completely obliterated (Figure 3). So we are to believe that the noise absolutely destroys the second-order cyclostationarity, but magnificently preserves the higher-order cyclostationarity, so that excellent detection results are obtained. Hmmm….
I have more objections, but I’ve run out of energy for this. Let me know if you disagree with me about this paper, or if I’ve went wrong somewhere in analyzing it.