The cyclic autocorrelation function (CAF) for rectangular-pulse BPSK can be derived as a relatively simple closed-form expression (see My Papers  for example or The Literature [R1]). It can be estimated in a variety of ways, which we will discuss in future posts. The non-conjugate cycle frequencies for the signal are harmonics of the bit rate, , and the conjugate cycle frequencies are the non-conjugate cycle frequencies offset by the doubled carrier, or .
Recall that our simulated rectangular-pulse BPSK signal has samples per bit, or a bit rate of , and a carrier offset of , all in normalized units (meaning the sampling rate is unity). We’ve previously selected a sampling rate of MHz to provide a little physical realism; let’s do that here too. This choice means the bit rate is kHz and the carrier offset frequency is kHz. From these numbers, we see that the non-conjugate cycle frequencies are kHz, and that the conjugate cycle frequencies are kHz, or kHz.
The blindly estimated CAF for our noisy rectangular-pulse BPSK signal is shown here:
The non-conjugate CAF for is the conventional autocorrelation function. Here we can see it is triangular in shape, with an additional inflation of the peak for , due to the presence of noise, which has non-zero power. For comparison, here is the numerically evaluated theoretical formula for the CAF for rectangular-pulse BPSK:
The match is excellent.
The theoretical formulas for the cyclic autocorrelation and spectral correlation function for BPSK signals (and other digital QAM/PSK signals) can be found in several places. One of the first places it was published is in the book Statistical Spectral Analysis by W. A. Gardner (The Literature [R1], Chapter 12). The formulas for both the th-order reduced-dimension cyclic cumulant and the th-order cyclic polyspectrum for PAM/PSK/QAM can be found in My Papers  and my dissertation. The -th order cyclic cumulant reduces to a lag-shifted (asymmetric) version of the cyclic autocorrelation for .
Interpreting the CAF
The non-conjugate CAF can be interpreted as the correlation between the lag product and the complex-valued sine wave . From that point of view, the CAF is the complex amplitude of the additive sine wave that is present in the lag product. That is, that sine wave has amplitude and phase . Note that the amplitude and phase depend on the delay . When the CAF is zero, there is no finite-strength additive sine-wave component in the lag product for the value of chosen; otherwise there is.
The conjugate CAF can be interpreted as the correlation between the lag product and the complex-valued sine wave .
Some signals may have only non-zero non-conjugate CAF values (such as large-alphabet digital QAM), some only non-zero conjugate CAF values (such as analog amplitude modulation, discounting the ever-present non-conjugate cycle frequency of zero), and some have both (such as our favorite signal, the rectangular-pulse BPSK signal).
Comparison with a Bandwidth-Efficient BPSK Signal
The rectangular-pulse signal has infinite bandwidth, and therefore it possesses an infinite number of cycle frequencies. However, the strength of the cyclic autocorrelation for most of those cycle frequencies is very small, so that in practical terms, the rectangular-pulse signal possesses ten or so significant features. That’s actually quite a lot compared to practical real-world signals like BPSK with square-root raised-cosine (SRRC) pulses. Here is the estimated CAF for a BPSK with SRRC pulses and a pulse roll-off factor of 0.3 (the excess bandwidth is 30%):
The non-conjugate CAF possesses exactly three cycle frequencies, (only two of those are shown in the plot), and the conjugate CAF also possesses three cycle frequencies . Note also that the width of the CAF is much larger than for the rectangular-pulse signal (the axes are different between the plots for the two signals). This is a direct consequence of the fact that the rectangular-pulse signal is strictly time-limited whereas the SRRC-pulse signal is strictly bandlimited.