BPSK Cycle Frequency

The BPSK cycle frequency arises from the signal's second-order statistics. When a BPSK signal is squared, the phase transitions are removed, revealing a spectral line at twice the carrier frequency offset plus the symbol rate. This is mathematically expressed as α = 2f_c + R_s, where α is the cyclic frequency, f_c is the carrier offset, and R_s is the symbol rate. This specific frequency is where the Spectral Correlation Function (SCF) exhibits its maximum non-zero density.

The strongest cyclostationary signature for BPSK is observed at the cycle frequency α = 2f_c. This is the most prominent feature for detection and coarse estimation. A secondary, weaker feature exists at α = 2f_c ± R_s, which is directly tied to the symbol rate. The Cyclic Domain Profile (CDP) will show a clear, isolated peak at these locations when projected from the SCF, making it a robust feature for automatic modulation classification.

The cyclostationary feature at the BPSK cycle frequency is highly robust to stationary noise and interference. Because additive white Gaussian noise (AWGN) is a stationary process, its spectral correlation is zero for all non-zero cyclic frequencies. This means the Cyclic Autocorrelation Function (CAF) at α = 2f_c will theoretically converge to zero for noise, providing a significant processing gain. This property makes cyclic feature detection far superior to energy detection in low-SNR environments.

Once the carrier-related cycle frequency (2f_c) is identified, the symbol rate can be blindly estimated. By searching for the secondary cycle frequencies at α = 2f_c ± R_s, the symbol rate R_s is directly measured as the difference between the primary and secondary peaks. This non-data-aided estimation is crucial for automatic modulation classification and for configuring demodulators for unknown intercepted signals.

A critical differentiator between BPSK and QPSK lies in their cycle frequencies. While a BPSK signal exhibits strong cyclostationarity at α = 2f_c, a QPSK signal does not. QPSK's second-order statistics are stationary; its cyclostationary features appear only in its fourth-order cumulants. Therefore, the presence of a strong peak at the BPSK cycle frequency in the SCF is a definitive, high-confidence discriminator for BPSK modulation over QPSK.

The exact shape and bandwidth of the cyclostationary feature are influenced by the pulse-shaping filter. A root-raised cosine filter, commonly used in digital communications, limits the signal's bandwidth and smooths the spectral correlation. The cycle frequency location remains unchanged, but the feature's support in the spectral frequency domain is constrained to the signal's occupied bandwidth, creating a distinct pattern in the Spectral Coherence map that can further identify the transmitter's configuration.