Let’s look into the statistical properties of a class of textbook signals that encompasses digital quadrature amplitude modulation (QAM), phase-shift keying (PSK), and pulse-amplitude modulation (PAM). I’ll call the class simply digital QAM (DQAM), and all of its members have an analytical-signal mathematical representation of the form
In this model, is the symbol index, is the symbol rate, is the carrier frequency (sometimes called the frequency offset), is the symbol-clock phase, and is the carrier phase. The finite-energy function is the pulse function (sometimes called the pulse-shaping function). Finally, the random variable is called the symbol, and has a discrete distribution that is called the constellation.
Model (1) is a textbook signal when the sequence of symbols is independent and identically distributed (IID). This condition rules out real-world communication aids such as periodically transmitted bursts of known symbols, adaptive modulation (where the constellation may change in response to the vagaries of the propagation channel), some forms of coding, etc. Also, when the pulse function is a rectangle (with width ), the signal is even less realistic, and therefore more textbook.
We will look at the moments and cumulants of this general model in this post. Although the model is textbook, we could use it as a building block to form more realistic, less textbooky, signal models. Then we could find the cyclostationarity of those models by applying signal-processing transformation rules that define how the cumulants of the output of a signal processor relate to those for the input.
The original RF signal is the real part of the analytical signal, when the parameter is actually the RF carrier frequency. Let’s distinguish the carrier offset in (1) from the RF carrier frequency by calling the latter . Then the RF signal is given by
When the RF signal is downconverted (shifted in frequency), perhaps in multiple stages involving several shifts that sum to near , and the signal is otherwise undisturbed, then the model (1) holds for the result, where is the difference between the actual RF carrier and the effective (total) frequency shift. So, if the reception of the signal, and its downconversion, is of high quality, then the carrier offset frequency is small compared to the bandwidth of . We still refer to it as a carrier frequency sometimes, even though it is really an offset frequency. Ideal or ‘perfect’ downconversion corresponds to .
To recap, the complex signal representation (positive-frequency signal representation) for the RF signal and the complex signal representation for the baseband signal differ only by the value of the carrier phase and by the value of the carrier frequency. For the RF signal, the carrier frequency is , and it is very large with respect to the bandwidth of , and for the baseband (complex envelope) signal, it is small compared to the bandwidth of , and is ideally zero.
So this means we can analyze the complex-envelope representation, and the results will easily translate to the analytic-signal representation, which is essentially the RF signal.
Cyclic Cumulants of DQAM
Let’s first assume perfect downconversion, that is, let’s set , , and to obtain the complex-envelope (baseband) model given by
It can be shown that the th-order cyclic temporal cumulant function (cyclic cumulant) for the DQAM model (3) with IID symbols is given by (My Papers [6,13])
In (4), the key parameter is the cumulant for the symbol variable, which we call . This is the cumulant for of order using conjugations. Such a cumulant can be computed by hand or numerically provided that the probability mass function for the discrete random variable is known. But that probability mass function is just the constellation with equal probability given to each constellation point. So, we can compute those values. Let’s first look at some constellations:
For plotting purposes, the shown constellations are normalized so that the maximum of the real and imaginary parts are equal to . For computations of moments and cumulants, discussed below, the constellations are normalized so that their variance is .
By applying the usual cumulant machinery, involving lower-order moments and the partitions of the set , we can compute the exact values of the cumulants . And of course we can compute the moments too. Here are the cumulants and moments for the real-valued constellations above (BPSK, 4ASK (4-level PAM), and 8ASK),
and the cumulants for some of the complex-valued constellations are shown in the following table:
All of the complex-valued constellations considered here exhibit no cyclostationarity for order with or . The corresponding modulated signals possess only three second-order cycle frequencies for , which are , if we assume a typical bandwidth-efficient pulse like square-root raised-cosine. That is, if I had bothered to create columns for , the entries for would be zeros, and the entries for would be .
The reduced-dimension cyclic temporal cumulant function is obtained from the cyclic cumulant by setting one of the lag values to zero, say . For clarity, this reduced-dimension function for digital QAM/PSK is given by
for cycle frequencies .
Cyclic Polyspectra of DQAM
The cyclic polyspectrum is the dimensional Fourier transform of the reduced-dimension cyclic cumulant function (5). (There is a CSP Blog post on the cyclic polyspectrum.) The cyclic polyspectrum for DQAM/PSK is given by
for cycle frequencies .
To obtain the exact expressions for the cyclic cumulant and polyspectrum for the more general complex representation (1) that includes imperfect downconversion and unknown delay (that is, , , and can all be different from zero), we can apply the input-output relationships for basic signal processing operations discussed in this post. The residual carrier offset is just the multiplication of the baseband signal by a complex sine wave, and the delay is just a, well, delay, which is easily represented by a simple linear time-invariant system. This exercise leads to the following conclusions regarding the general form for the cycle frequencies for DQAM/PSK:
If we let range over all the integers, we obtain the largest possible set of cycle frequencies for DQAM. This set is actually achieved by the textbook rectangular-pulse BPSK signal, which has infinite bandwidth, and so has an infinite number of cycle frequencies for each combination, provided is even.
In practice, the constellation for and the pulse function limit the number of cycle frequencies, and determine which ones are actually exhibited by a signal. For example, the odd-order moments and cumulants of the symmetric constellations above are zero, so the cyclic cumulants for DQAM/PSK are zero for all odd orders . Real-world signals typically use a bandlimited pulse function , such as the square-root raised-cosine pulse, and this severely limits the range of . For , we have .
Specialization to Second Order ()
For , the reduced-dimension cyclic temporal cumulant function reduces to a lag-shifted version of the usual non-conjugate cyclic autocorrelation function,
and the cyclic polyspectrum is a frequency-shifted version of the non-conjugate spectral correlation function,
Let and rewrite (9),
Making the same substitution as before, , we obtain
which is also consistent with a frequency-shifted version of the conjugate spectral correlation function for DQAM.
These cyclic polyspectrum formulas provide insight into the maximum cycle frequencies that can exist for DQAM. The cyclic polyspectrum is seen to be the product of shifted pulse-function transforms. When the transform is bandlimited (zero outside of some finite frequency interval), then there is a largest that can result in overlap between the factors in the transform-product.
For example, when considering the square-root raised-cosine pulse type, we know that it has maximum width of for roll-off factor of , and minimum width of for roll-off factor of . The cycle frequencies are . For roll-off of , then, only can produce overlap between the two shifted pulse transforms in the cyclic polyspectrum (spectral correlation) formula. That is, DQAM with roll-off of has no non-trivial non-conjugate cycle frequencies. On the other hand, for roll-off of , will result in overlap. Similar remarks can be made about the conjugate spectral correlation function (cyclic polyspectrum for above).
Here let’s show some estimated cyclic cumulants and cyclic polyspectra for various orders and numbers of conjugated factors for the DQAM/PSK constellations that we graphed above.
We start with the basics–the power spectra. The FSM is used to estimate the power spectra for simulated versions of the DQAM/PSK signals. All signals employ symbols that are independent and identically distributed (IID), and the pulse functions are square-root raised-cosine with roll-off of (excess bandwidth of %), and samples. The PSD estimates are shown below:
The thing to notice is that when all the signals have the same average power, the same pulse function, and IID symbols, their power spectra are identical. From the point of view of, say, modulation recognition, this is not a good outcome. The signals cannot be distinguished from each other.
Here are plots of the estimated non-conjugate spectral correlation function for the symbol-rate cycle frequency:
As expected, these spectral correlation functions are identical. Now consider the conjugate spectral correlation for :
Here we see that the three real-valued constellations, BPSK, 4ASK, and 8ASK, have non-zero spectral correlation, while all the complex-valued constellations do not. Finally, then, there is some basis for distinguishing some of the signals from others based only on second-order statistics. Yet distinguishing between BPSK, 4ASK, and 8ASK remains impossible at second order.
Enter the cumulants. From the table above, we know that starting with order , the cyclic cumulants begin to take on different values for different constellations. Here are the cyclic cumulant magnitudes for :
Not all the modulations have distinct cyclic cumulants here. In particular, QPSK, 8PSK, and 16PSK all have identical cumulants (see the table above). But many do have distinguishable cumulant values.
Here are the estimated cyclic cumulants for and :
Finally, just for fun, here are several sets of estimated sixth-order cyclic cumulants:
The jpeg and MATLAB .fig files are posted in the Downloads page.
What’s important about the cyclic cumulants for DQAM/PSK is that there is always some order for which they differ for any two distinct constellations. This idea has its roots in the fact that the sets of th-order probability density functions for the two signals must differ for some , else the two signals are actually the same random process.
Now, an individual selection of may yield identical cyclic cumulants for multiple even for distinct signal constellations. However, when we look at the sets of cyclic cumulants over a number of , , and , we can still obtain distinguishing features. For example, for QPSK and 8PSK, the cyclic cumulants are identical. As are the cyclic cumulants. But the cyclic cumulants are not identical, and QPSK can be distinguished from 8PSK by the presence or absence of the cyclic cumulant for .
Now there are two (at least) pieces of bad news. The first is that distinguishability for a pair of constellations may require a very large value of indeed. Estimating moments and cumulants for large is a computationally challenging task. The second is that the differences between the cumulants may not be all that large when they do occur for a relatively small value of , meaning distinguishability will be difficult except for in very high SNR situations.
In My Papers [25,26,28] I lay out one approach to exploiting the fact that sets of cyclic cumulants differ to create interference-tolerant and quite general modulation recognition algorithms. When combined with blind processing that is based on the strip spectral correlation analyzer, we’re on our way toward true RF scene analysis.
Application to More Realistic Signal Models
This post has considered textbook signals of the DQAM and PSK type. Are there really any out there? (I asked here if you want to add your opinion or experience.) There are some real-world effects that plague DQAM signals that could be addressed using the basic constellation-based formalism described here. Any transmitter imperfection that results in a constant distortion of the constellation simply results in a new constellation, and that new constellation has its own values. For example, a power imbalance in the inphase and quadrature channels of the transmitter will stretch or compress the constellation points along either the or dimension in the constellation plots at the top of this post.
Some realistic signal models may combine textbook DQAM with slotting or framing or departures from IID symbols, and these signals may be mathematically representable in terms of signal processing operations involving the DQAM textbook signal and one or more other signals. The post on signal processing operations might then be helpful.