When we considered complex-valued signals and second-order statistics, we ended up with two kinds of parameters: non-conjugate and conjugate. So we have the non-conjugate autocorrelation, which is the expected value of the normal second-order lag product in which only one of the factors is conjugated (consistent with the normal definition of variance for complex-valued random variables),
and the conjugate autocorrelation, which is the expected value of the second-order lag product in which neither factor is conjugated
The complex-valued Fourier-series amplitudes of these functions of time are the non-conjugate and conjugate cyclic autocorrelation functions, respectively.
The Fourier transforms of the non-conjugate and conjugate cyclic autocorrelation functions are the non-conjugate and conjugate spectral correlation functions, respectively.
I never explained the fundamental reason why both the non-conjugate and conjugate functions are needed. In this post, I rectify that omission. The reason for the many different choices of conjugated factors in higher-order cyclic moments and cumulants is also provided. These choices of conjugation configurations, or conjugation patterns, also appear in the more conventional theory of higher-order statistics as applied to stationary signals.
The reason we need to consider various numbers of conjugated and non-conjugated terms in our moments and cumulants is that we desire to represent all of our signals and data as complex-valued random processes (complex signals). I’ll repeat this later, but a powerful reason to use complex-valued discrete-time processes is that they can represent real sampled RF signals using a sampling rate that is independent of the carrier frequency of the signal. Only the occupied bandwidth of the RF signal matters when representing an RF signal in terms of a lowpass (frequency content [Fourier transform, PSD] near zero frequency) complex-valued process.
Let’s take this step by step.
Suppose we have a simple real-valued radio-frequency signal, which is just a real-valued message modulated by (multiplied by) a real-valued sine wave,
(This is usually referred to as an amplitude-modulated (AM) signal.) The message signal is real, and so it has a symmetric PSD. Here is a numerical example:
I’ve denoted the width of the PSD for as . Throughout this post we assume that the carrier frequency is much greater than , so that the symmetric PSD of the real signal contains two well-separated bumps:
This means that each bump could be completely separated from the other using linear time-invariant filters. Now let’s look at the real-valued signal in more detail, with an eye toward expressing it in terms of simpler complex-valued components.
Recall Euler’s Formula is given by
Using this in our real-valued radio-frequency signal gives us
Using elementary Fourier transform analysis, the signal corresponds to the positive-frequency bump in the PSD of and corresponds to the negative-frequency bump. So suppose we have just one or the other of and . Can we recover ? Sure, recalling that is real here, we can just take the real part of ,
where is an operator that returns the real part of its argument. This must mean that all the statistical information in is available in . An advantage of working with signals like is that they can be frequency shifted to zero frequency, then sampled at a rate equal to (using the basic sampling theorem). The basic sampling rate for is , which can be much much greater than . (See also the Signal Processing ToolKit post on analytic signals and the complex envelope.)
Now, let’s look at the statistics of . In particular, let’s look at the expected value of the second-order lag product, which is the autocorrelation function,
We can express this expected value in terms of and ,
Or, in terms of the autocorrelation for the message signal ,
Now, if is cyclostationary (second-order) with cycle frequencies , then will have cycle frequencies
Middle two terms:
For example, if the real signal is a pulse-amplitude modulated (PAM) signal, it will have cycle frequencies , where is the symbol rate of the PAM signal. So the middle two terms above give us those cycle frequencies, the first term gives us (which includes itself when ), and the last term gives us . BPSK is such a PAM signal.
Quadrature Modulated Signals
The more general case is defined by adding signals in phase quadrature,
where is the inphase component and is the quadrature component of . Again, through the use of Euler’s Formula, we can express this real-valued radio-frequency signal as
As before, we can represent the real signal by the complex envelope signal provided the carrier is much larger than the bandwidths of the inphase and quadrature components. If we want to examine the complete picture of the second-order statistics of , we can do that by looking at the lag product
By multiplying the two terms, we find additive terms corresponding to all of the four possible conjugation configurations:
So to obtain all the statistical information for the original signal , we need to obtain the statistical information from each of the distinct conjugation configurations corresponding to the complex signal .
Now, some of the configurations are redundant. For instance, the lag product is just the complex conjugate of the lag product , so you can determine everything about one of these lag products from the other.
This means that in the end, for second-order, we always need to consider the “no conjugations” case as well as the “one conjugation” case , which leads us back to the conjugate and non-conjugate cyclic autocorrelation and spectral correlation functions, respectively.
Many signals possess significant features in both the non-conjugate and conjugate spectral correlation planes. By “significant” I mean that the energy in the spectral correlation function for a particular cycle frequency is of the same order of magnitude as the signal power, or perhaps one order of magnitude less, but no smaller. Examples include the canonical rectangular-pulse BPSK signal, OOK, DSSS BPSK, DSSS SQPSK, FSK, and GFSK. Many others possess only non-conjugate features, such as all QAM/PSK with greater than two points in the constellation. A few possess only conjugate features, such as SQPSK with square-root raised-cosine pulses, AM, and GMSK.
th-Order Moments and Cumulants
When we look at th-order lag products, as we did in the posts on higher-order moments and cumulants, various conjugation configurations come out of the complex-valued signal representation. In general, they are all needed to determine the full suite of cycle frequencies, cyclic moments, and cyclic cumulants for the original real-valued signal. Symmetries again apply, however, so that we can capture the statistical information by using a minimum set of configurations. When all the delays are equal, this becomes particularly easy. For example, for , we may consider only the cases of no conjugations, one conjugation, and two conjugations. The case of three conjugations is covered by the case of one conjugation, and the case of four is covered by the case of none.
Typical Complex-Signal Model
The typical model for a complex-valued communication signal is
where is the complex envelope of the transmitted signal, is called the carrier offset frequency, is some amplitude factor, and is the residual carrier phase.
The carrier offset frequency is the result of imperfectly downconverting (frequency shifting) the RF signal to zero frequency. Typically, is small compared to the bandwidth of . To understand the statistics of , then, it is required to look at the moments and cumulants of with all possible distinct conjugation configurations.
The next step in such modeling is to add noise, interference, and a propagation channel (such as discrete multipath).
After that, transmitter and receiver impairments can be considered, but we’re getting ahead of ourselves…
And so that’s it. The various conjugation configurations are required to fully study the statistical structure of communication signals that have been (perhaps imperfectly) converted to complex baseband (zero frequency). And it is desired to work with complex baseband signals because they require small sampling rates compared to the RF or even the intermediate-frequency (IF) signals. In the end, we can blame the existence of the various conjugation configurations on the desire to work with complex numbers.