FAM Algorithm

FAM operates by first computing a sliding Short-Time Fourier Transform (STFT) on the input complex baseband signal. This channelization step decimates the wideband signal into N' narrowband, critically sampled frequency bins. By performing the subsequent cross-correlation on these decimated bins rather than the full-bandwidth signal, the algorithm achieves a computational savings factor proportional to the number of channels. The decimation factor is a critical design parameter balancing cycle frequency resolution against computational load.

The core of FAM is a two-stage process:

• Stage 1 (Channelization): A sliding FFT computes the complex demodulates for each frequency bin, effectively isolating narrowband components.
• Stage 2 (Correlation): The complex demodulates from two frequency-separated bins are pointwise multiplied and then smoothed via a second FFT. This product directly estimates the spectral correlation density at a specific pair of spectral frequencies (f) and cyclic frequency (alpha). This architecture maps the SCF estimation problem onto a sequence of FFT operations, exploiting the efficiency of the radix-2 butterfly.

For a signal with N samples and N' frequency channels, the FAM algorithm reduces the complexity of SCF estimation from O(N²) for direct methods to approximately O(N * N' * log N'). This makes it the preferred algorithm for real-time or near-real-time cyclostationary analysis on software-defined radios (SDRs). The trade-off is a reduction in cyclic frequency resolution, as the channelization process inherently limits the maximum resolvable alpha to the channel bandwidth.

FAM provides independent control over spectral frequency (f) resolution and cyclic frequency (alpha) resolution through two separate smoothing parameters:

• Channel bandwidth (Δf): Determined by the STFT window length and overlap, controlling the resolution along the spectral frequency axis.
• Correlation smoothing length (Δt): Determined by the second FFT length, controlling the resolution along the cyclic frequency axis. This decoupled control allows the analyst to tailor the bifrequency resolution to the specific signal of interest, such as resolving closely spaced BPSK cycle frequencies.

The direct output of the FAM algorithm is a discrete estimate of the Spectral Correlation Function (SCF)—a two-dimensional complex-valued matrix indexed by spectral frequency (f) and cyclic frequency (alpha). Key post-processing steps include:

• Normalization: Computing the Spectral Coherence by magnitude-normalizing the SCF to produce a scale-invariant feature bounded between 0 and 1.
• Projection: Collapsing the 2D SCF along the f-axis to generate a Cyclic Domain Profile (CDP), a compact 1D feature vector ideal for machine learning-based modulation recognition and emitter identification.

When implementing FAM on real IQ data, engineers must address:

• Channel overlap: Using overlapping STFT windows (e.g., 50% overlap) to prevent scalloping loss and ensure accurate correlation between adjacent bins.
• Data tapering: Applying a window function (e.g., Hamming) to each STFT segment to suppress spectral leakage, which can create false cyclic features.
• Strip vs. Full FAM: The SSCA (Strip Spectral Correlation Analyzer) is a closely related time-smoothing alternative that computes the complex demodulate of the entire signal before correlation, offering a different complexity-resolution trade-off for specific use cases.