In the real world, the electromagnetic field is a multi-dimensional time-varying real-valued function (volts/meter or newtons/coulomb). But in mathematical physics and signal processing, we often use complex-valued representations of the field, or of quantities derived from it, to facilitate our mathematics or make the signal processing more compact and efficient.
So throughout the CSP Blog I’ve focused almost exclusively on complex-valued signals and data. However, there is a considerable older literature that uses real-valued signals, such as The Literature [R1, R151]. You can use either real-valued or complex-valued signal representations and data, as you prefer, but there are advantages and disadvantages to each choice. Moreover, an author might not be perfectly clear about which one is used, especially when presenting a spectral correlation surface (as opposed to a sequence of equations, where things are often more clear).
An example is the following sequence of four surfaces taken from [R151]:
In this post, I show my own surfaces for real- and complex-valued representations of these common PSK signals. In a previous post, I explained mathematically how the complex-valued representation relates to the real-valued representation. In a future Signal Processing ToolKit post, I’ll go over all the steps involved in obtaining the complex-valued representation of a signal from a real-valued one.
Review of Surfaces for Complex-Valued PSK Signals
Let’s focus on the four simple common signals from Figure 1: BPSK, QPSK, SQPSK, and MSK.
BPSK and QPSK are pulse-amplitude-modulated signals, which just means each transmitted symbol multiplies a particular time-shifted pulse ,
In the case of BPSK, the values of are constrained to lie in the set , and in the case of QPSK they lie in the set .
SQPSK and MSK are not simple pulse-amplitude-modulated signals. Instead, they are the sum of two such pulse-amplitude-modulated signals, which are independent, and one is delayed by half the symbol duration relative to the other. Each of the involved signals inside SQPSK and MSK is a binary signal like BPSK. In the case of SQPSK, the pulse function is typically a square-root raised-cosine pulse, but it can be rectangular too. In the case of MSK, the pulse is defined to be half a period of a cosine wave. That is, MSK is a special case of SQPSK. It is also a special case of continuous-phase modulation (My Papers ).
We’ll focus on rectangular pulses for BPSK, QPSK, and SQPSK in this post because that is what is used in Figure 1.
You can see the spectral correlation surfaces and cyclic autocorrelation surfaces for the four signals in the spectral correlation gallery and the cyclic correlation gallery, respectively, but I’m going to reproduce them here for convenience.
First the four pairs of spectral correlation surfaces for complex-valued data:
The conjugate spectral correlation surface for QPSK is missing because it is identically equal to zero. Here are the corresponding cyclic correlation surfaces:
Note that the spectral correlation and cyclic correlation surfaces I just showed are estimated from simulated data, not numerically evaluated theoretical formulas like those in Figure 1. Also, I do not include negative non-conjugate cycle frequencies for reasons of symmetry–that makes the non-conjugate plots a little cleaner.
The carrier offset for the four simulated signals is not zero, but it is small relative to the bandwidths of the signals, and also small relative to the sampling rate. So these signals are close to “complex baseband” signals. To consider real-valued versions, we cannot simply take the real part of such complex-valued signals–frequency components will be mixed up. Recall that if we have a passband (RF or IF) signal, to get at the complex baseband signal, we have to isolate either the positive-frequency portion of the signal or the negative-frequency portion, then frequency shift the isolated part to zero or near-zero frequency. To reverse that process, we have to shift the complex baseband signal up to something like a quarter of the sampling rate, then restore the negative (or positive, depending) frequencies by taking the real part. This implies that we have to have an oversampled complex baseband signal in the first place, so that we may need an initial preprocessing step that increases the sampling rate should the complex-valued signal be critically sampled.
The preceding means that the PSDs for the complex-valued and real-valued versions of the exact same signal will look different. For example, the PSD (non-conjugate spectral correlation function for cycle frequency equal to zero) for the real-valued signal (properly obtained as above) will always be symmetric around , but that for the complex-valued signal need not be (e.g., non-zero carrier frequency offset).
Surfaces for Real-Valued PSK Signals
Here are my estimated spectral correlation plots for the real-valued versions of the four signals above:
As with the complex-valued signals, the power spectrum for these real-valued versions corresponds to , and we can clearly see each signal possesses two power-spectrum peaks, one in the positive-frequency interval and one in the negative. For BPSK, SQPSK, and MSK, we see two groups of identical carrier-related features (look along ), one for positive cycle frequency and one for negative. These correspond to features near the doubled carrier and the negative of the doubled carrier. When we convert the real-valued signal to complex-valued, we keep one or the other (depending on whether we zero out the positive frequencies or the negative ones), and we shift the signal down to near zero frequency, so in the end we just see one of the doubled-carrier groups, and the new doubled-carrier frequency (also referred to as an offset because it is ideally equal to zero, but can be offset from that) is very much smaller than the doubled-carrier frequency in the real-valued case.
For completeness, here are the cyclic autocorrelation surfaces for the four signals:
So, watch out when you go looking around the CSP literature–any displayed surfaces will have clues about whether the data of interest is modeled as real or complex.