Cyclic Prefix (CP) Correlation

The core mechanism computes the correlation between the CP and its copy at the end of the OFDM symbol. By multiplying the received signal with a delayed conjugate of itself at a lag equal to the useful symbol length (Tu), the CP samples correlate while noise and data remain uncorrelated. The correlation window slides sample-by-sample, and the peak magnitude indicates the symbol boundary. This lag product is defined as R[n] = r[n] · r*[n + N_FFT], where N_FFT is the FFT size.

Raw correlation samples are noisy. To improve reliability, the lag product is passed through a moving average filter of length equal to the CP duration (N_CP). This integrates the correlated energy over the guard interval, producing a triangular-shaped correlation plateau in the magnitude profile. The length of this plateau directly reveals the CP duration, enabling blind discrimination between normal CP and extended CP modes in LTE and 5G NR signals.

The phase angle of the correlation peak provides a direct estimate of the fractional carrier frequency offset (CFO). Since the CP is a copy of the end of the symbol, any frequency offset causes a phase rotation between the two correlated segments. The CFO is calculated as: Δf = (1 / 2π · Tu) · arg(R_peak), where Tu is the useful symbol duration. This yields a fractional CFO estimate within ±0.5 subcarrier spacing, which can be combined with the Schmidl-Cox algorithm for integer CFO resolution.

The CP correlation can be derived from a maximum likelihood estimation framework. Van de Beek et al. formulated the joint ML estimator for symbol timing and CFO by modeling the received signal as a Gaussian process with cyclostationary properties. The log-likelihood function consists of a weighted correlation term and an energy correction term. This ML approach provides the theoretical foundation for CP-based synchronization and defines the optimal detection statistic under AWGN conditions.

In multipath channels, the CP correlation plateau becomes distorted by inter-symbol interference (ISI) from previous symbols. However, as long as the channel delay spread is shorter than the CP, the correlation structure remains detectable. The plateau edges may be smeared, but the center of mass of the correlation profile still provides a reliable timing estimate. Advanced techniques apply threshold detection or slope analysis to locate the plateau boundaries in dispersive channels.

CP correlation is computationally lightweight compared to cyclostationary feature detection or matched filtering. It requires only one complex multiply per sample for the lag product and a running sum for the moving average. This makes it ideal for real-time FPGA implementation and low-power spectrum sensors. The algorithm can be further optimized using recursive averaging to avoid redundant additions, reducing the per-sample operations to O(1) complexity.