# Cumulant (4, 2) is a Good Discriminator?

Let’s talk about another published paper on signal detection involving cyclostationarity and/or cumulants. This one is called “Energy-Efficient Processor for Blind Signal Classification in Cognitive Radio Networks,” (The Literature [R69]), and is authored by UCLA researchers E. Rebeiz and four colleagues.

My focus on this paper it its idea that broad signal-type classes, such as direct-sequence spread-spectrum (DSSS), QAM, and OFDM can be reliably distinguished by the use of a single number: the fourth-order cumulant with two conjugated terms. This kind of cumulant is referred to as the $(4, 2)$ cumulant here at the CSP Blog, and in the paper, because the order is $n=4$ and the number of conjugated terms is $m=2$.

The referenced paper claims that multi-carrier modulation (OFDM is the main type of practical multi-carrier modulation [MCM]) can be distinguished from single-carrier modulation using the $(4,2)$ cumulant. Typical single-carrier modulation types are our old friends the digital QAM/PSK/CPM signals, DSSS signals (non-frequency-hopped), ATSC DTV, AM, FM, and many others.

The authors of the paper aren’t interested in using cyclic cumulants. The cumulant they are talking about is the conventional stationary-signal cumulant. That’s why they call it $(4, 2)$ rather than, say, $(4, 2, 0)$ where the third value is the harmonic number $k$. The $(4,2)$ cumulant for a zero-mean complex-valued stationary signal $x(t)$ and all lags equal to zero is defined by

$\displaystyle C_x (\mathbf{0};4,2) = R_x(\mathbf{0}; 4,2) - 2R_x^2(\mathbf{0}; 2, 1) - R_x(\mathbf{0}; 2,0)R_x(\mathbf{0}; 2,2) \hfill (A)$

Equation (A) here is the theoretical counterpart to the $C(4,2)$ estimator in the paper, which is Equation (1).

There is a large number of very different signals in the non-MCM class, so it seemed to me upon first reading the paper that it might be hard to find one statistic that could distinguish all of these non-MCM signals from all of the MCM signals. So I devised an experiment to see for myself.

In the experiment, I first applied the authors’ $C(4,2)$ to a set of captured waveforms, each of which is downconverted to complex baseband before processing, and each of which has inband SNR greater than $20$ dB. The captured signals include several OFDM signals (LTE, satellite radio) and two non-MCM signals (CDMA, WCDMA).

The authors are trying to minimize computational cost, so they want to use the minimum number of samples to estimate $C(4,2)$; they recommend $90$ samples. (This appears to be independent of the sampling bandwidth, something I don’t understand.) So I applied the cumulant estimator to $128$ samples of the various captured signals.

I then created a set of simulated signals that included OFDM, DSSS, and a couple textbook DQAMs (QPSK and 64QAM). The results are shown here:

The lower plot indicates that the $C(4,2)$ cumulant is centered at value near zero for the simulated OFDM signals, and is large and negative for the DSSS and QPSK signals. The $C(4,2)$ value for the 64QAM signal is somewhere inbetween. If the threshold could be moved to around $-0.4$ or so, then the MCM and non-MCM simulated signals could be distinguished with some error due to the overlap in the distributions for 64QAM and OFDM.

However, the upper plot shows the opposite for captured signals. The $C(4,2)$ value for white Gaussian noise is centered near zero, as it should be, but the distributions for the various captured MCM and non-MCM  signals all overlap with each other and with the distribution for WGN. There is no threshold that could be used to distinguish between MCM and non-MCM captured signals here.

I also redid the experiment with a much larger block length of 4096 samples:

We see mostly the same behavior as for the short block length of $90$ samples. The various distributions are more compact, but the conclusion is the same: simulated signals might be accurately categorized as either MCM or non-MCM through the use of a thresholded $C(4,2)$ value, but the captured signals cannot be. In fact, the $C(4,2)$ value for one of the OFDM signals is actually smaller than the others, the opposite of the simulated case.

So this appears to be another case of Textbook Signals Ruining Everything. I’ve also applied this experimental approach to captured ATSC-DTV and broadcast FM (both non-MCM signals) for a block length of $4096$ samples. For the DTV signal, the values were tightly clustered around $-0.3$ and for the FM signal around $-1.0$.