I recently came across the 2014 paper in the title of this post. I mentioned it briefly in the post on the periodogram. But I’m going to talk about it a bit more here because this is the kind of thing that makes things a bit harder for people trying to learn about cyclostationarity, which eventually leads to the need for something like the CSP Blog.
The idea behind the paper is that it would be nice to avoid the need for prior knowledge of cycle frequencies when using cycle detectors or the like. If you could just compute the entire spectral correlation function, then collapse it by integrating (summing) over frequency , then you’d have a one-dimensional function of cycle frequency and you could then process that function inexpensively to perform detection and classification tasks.
The author sets up an expression for one of the terms of the incoherent suboptimal multicycle detector, which has an idealized counterpart that is equal to the sum (integral) of the magnitude-squared spectral correlation function. Something like this:
The author then tries to simplify this kind of expression, and that is where things start to go very wrong. The root of it all is the author’s equation (3),
where is the Fourier transform of the given data restricted to a time interval centered at and having width . In other words, the author starts off by stating that the spectral correlation function is the limit of the cyclic periodogram as the amount of data increases without bound. Exactly what we show is not true in the periodogram post. The limit does not exist.
The (idealized) multi-cycle detection statistic of interest to the author is
This can be re-expressed as
The author then tries to convert the sum over cycle frequency into an integral, but does not use impulse functions to do so, ending up with
But we know that the spectral correlation function is continuous in spectral frequency and discrete in cycle frequency . In many cases of practical interest, there are only a finite number of cycle frequencies for which the spectral correlation function is not identically zero (think of square-root raised-cosine PSK, QAM, PAM), so that in (6) the integral over cycle frequency is zero. In other cases, the number of cycle frequencies is infinite, but countable (think rectangular-pulse BPSK), which still leads to a zero-valued integral over cycle frequency. In general, the spectral correlation function is non-zero only on a set of measure zero in the bi-frequency plane, and so such integrals as the author writes in (6) above are always zero. So (6) is a mistaken representation of the desired quantity.
A double-integral representation of is possible by using impulse functions (Dirac delta functions).
Then the author tries to substitute (2) above into (4), which is a mistake because (2) is not true, but he also does it incorrectly. In particular, he writes
which he then simplifies to
by using a change of variables for . But this is an incorrect substitution. The author’s (3) (my (2) above) must be substituted for each appearing in the expression, leading to two limits (over two time variables and ). In other words, he should have ended up with a double limit.
In reality, each spectral correlation function is itself correctly represented by a double limit involving the data observation interval length and a smoothing or averaging parameter. In the case of the frequency-smoothing method, it is the width of the smoothing function . So, we would end up with four limits in (9) if we did things properly.
But the problems continue. The author then defines as the (scaled) periodogram,
He ends up with
(the conjugation is superfluous since is non-negative).
So the quantity , which was defined as the integral of the squares of all non-zero spectral correlation functions, is now reduced (incorrectly!) to the convolution of the periodogram with itself. All the cyclostationarity of the signal–no matter how rich or how poor–can be extracted by computing only the periodogram!
There are lots of other problems with the paper, including in the simulations section, but they don’t really matter because the theoretical development is seriously mathematically flawed. And most of the problem arises from trying to take the limit of the cyclic periodogram.
Comments and corrections are welcome!