FAM Algorithm

The FAM algorithm uses a bank of bandpass filters implemented via a short-time FFT to decompose the input signal into narrowband frequency channels.

• Decimation: Each channel is downsampled to reduce the computational load for subsequent processing.
• Parallel Processing: The architecture is inherently parallel, making it suitable for FPGA and GPU implementation.
• Resolution Trade-off: The channel bandwidth directly controls the spectral frequency resolution (Δf) of the final SCF estimate.

FAM computes the spectral correlation function through a cascade of two distinct FFT operations.

• First FFT (Channelizer): A windowed FFT computes the complex envelope of each frequency channel over time.
• Second FFT (Strip Correlation): A second FFT is applied along the time axis of the product of two frequency-shifted channel outputs.
• Result: This dual-FFT structure efficiently maps the signal onto the cyclic frequency (α) and spectral frequency (f) plane.

The primary output of the FAM algorithm is a high-resolution estimate of the cyclic spectrum, also known as the spectral correlation density.

• Detection of Hidden Periodicity: It reveals the correlation between spectral components separated by a cyclic frequency α, which is invisible to standard power spectral density analysis.
• Modulation Fingerprinting: Different modulation schemes (BPSK, QPSK, 16-QAM) produce distinct, unique patterns of spectral correlation peaks.
• Noise Robustness: Stationary noise has no spectral correlation (α ≠ 0), so the cyclic spectrum naturally suppresses noise and interference.

FAM achieves significant computational savings over direct spectral correlation estimators by trading off temporal resolution for frequency resolution.

• Complexity Reduction: Reduces the complexity from O(N²) for a direct cyclic periodogram to approximately O(N log N) by leveraging the FFT.
• Strip Spectral Correlation: The algorithm computes the SCF along narrow frequency strips, limiting the range of cyclic frequencies evaluated.
• Parameter Tuning: Engineers can balance cycle frequency resolution (Δα) and spectral frequency resolution (Δf) by adjusting the channelizer size (N') and the number of temporal samples (N).

A critical post-processing step extracts alpha profiles from the computed SCF for use as feature vectors in modulation classifiers.

• Definition: An alpha profile is a one-dimensional slice of the SCF at a fixed spectral frequency f, plotting correlation magnitude versus cyclic frequency α.
• Peak Detection: The locations and amplitudes of peaks in the alpha profile directly correspond to the signal's symbol rate, carrier offset, and modulation-specific periodicities.
• Feature Engineering: These profiles serve as compact, highly discriminative inputs to neural networks or support vector machines for automatic modulation classification.

FAM enables the estimation of a signal's physical-layer parameters without any prior knowledge of the transmission scheme.

• Symbol Rate Estimation: The cyclic frequency α at which the first strong spectral correlation peak appears corresponds directly to the symbol rate.
• Carrier Frequency Offset: The shift of spectral correlation features along the spectral frequency axis reveals the difference between the transmitter and receiver oscillators.
• OFDM Detection: The cyclic prefix in OFDM signals induces a specific cyclostationary signature at α = 1/Ts (where Ts is the symbol duration), allowing for blind identification and parameter estimation.

The Spectral Correlation Function is the fundamental two-dimensional transform that the FAM algorithm is designed to estimate efficiently. It measures the correlation between frequency-shifted versions of a signal—specifically, the correlation between the signal's spectral component at frequency f + α/2 and the conjugate of its component at f - α/2. This reveals hidden periodicities in the signal's spectral content that are invisible to standard power spectral density analysis. The SCF is non-zero only at specific cyclic frequencies (α) for cyclostationary signals, creating a unique signature for each modulation type.

The cyclic frequency, denoted by α, is the parameter that quantifies the periodicity of a signal's statistical moments. For a digitally modulated signal, significant cyclic frequencies occur at integer multiples of the symbol rate, the carrier frequency offset, and combinations thereof. The FAM algorithm's output is indexed by both spectral frequency f and cyclic frequency α, creating an alpha profile that serves as a unique fingerprint. Key cyclic features include:

• α = 0: Corresponds to the standard power spectral density
• α = symbol rate: Reveals the modulation's pulse-shaping characteristics
• α = 2× carrier offset: Indicates residual carrier components

The cyclic periodogram is the simplest and most direct estimator of the spectral correlation function, computed as the product of two frequency-shifted, finite-time Fourier transforms. It is the time- and frequency-smoothed version of this raw periodogram that the FAM algorithm approximates. While computationally straightforward, the raw cyclic periodogram is an inconsistent estimator—its variance does not decrease with increasing data length. The FAM algorithm overcomes this limitation through channelized time-frequency smoothing, trading resolution for statistical reliability and making it suitable for real-world signal classification tasks.

An alpha profile is a one-dimensional slice of the spectral correlation function at a fixed spectral frequency f, showing the magnitude of correlation across all cyclic frequencies α. This compact representation is often used as a cyclic feature vector for modulation classification. For the FAM algorithm, extracting an alpha profile involves selecting a single frequency bin from the channelizer output and plotting the correlation magnitude across all computed cycle frequencies. Key peaks in the alpha profile directly correspond to the signal's symbol rate, carrier offset, and their harmonics, enabling blind parameter extraction without prior knowledge of the transmission scheme.

Second-order cyclostationarity is the property of a signal whose autocorrelation function is periodic in time—the mathematical foundation upon which the FAM algorithm operates. Most man-made communication signals exhibit this property due to periodic processing operations like modulation, pulse shaping, and coding. The FAM algorithm exploits second-order statistics exclusively, making it computationally efficient compared to higher-order methods. It is particularly effective for:

• Linearly modulated signals (QAM, PSK, ASK)
• OFDM signals with cyclic prefixes
• Signals with periodic preambles or pilot tones