FFT Size Detection

The FFT size manifests as a distinct cyclic period in the signal's autocorrelation function. By computing the autocorrelation of the received IQ samples and searching for peaks at specific lag values, the useful symbol duration (Tu) is revealed. The FFT size is directly proportional to this period given the system's base sampling rate.

• Mechanism: The repetition of the time-domain waveform every Tu seconds creates a cyclostationary signature.
• Key Metric: The lag corresponding to the maximum autocorrelation peak after CP removal indicates the FFT size.
• Example: An LTE 20 MHz signal with a 2048-point FFT exhibits a strong autocorrelation peak at a lag of 2048 samples at the appropriate sampling rate.

The Spectral Correlation Density function reveals the FFT size by identifying the frequency separation between correlated spectral components. OFDM signals exhibit spectral correlation at cyclic frequencies that are integer multiples of the subcarrier spacing. The FFT size is derived from the ratio of the sampling rate to the estimated subcarrier spacing.

• Process: Compute the cyclic autocorrelation and transform it to the frequency-frequency domain.
• Feature: Peaks in the SCD appear at cyclic frequencies α = k * Δf, where Δf is the subcarrier spacing.
• Robustness: SCD-based methods are highly resilient to noise and interference, functioning reliably at low Signal-to-Noise Ratios (SNR).

A Maximum Likelihood (ML) estimator can jointly estimate the useful symbol duration and cyclic prefix length by exploiting the redundancy introduced by the CP. The log-likelihood function is computed over a range of possible FFT sizes and CP ratios, and the combination that maximizes this function is selected.

• Input: A window of received IQ samples.
• Optimization: The algorithm iterates over candidate FFT sizes (e.g., 128, 256, 512, 1024, 2048) and CP lengths.
• Output: The FFT size and CP length that maximize the correlation between the CP and the end of the symbol.

Convolutional neural networks can be trained to directly regress the FFT size from raw IQ samples or spectrograms. The network learns to detect the periodic temporal patterns and frequency-domain comb structures characteristic of specific FFT sizes without explicit feature engineering.

• Architecture: Typically a 1D-CNN processing time-domain IQ samples or a 2D-CNN processing spectrograms.
• Training Data: Generated by synthesizing OFDM signals with varying FFT sizes, CP lengths, and channel impairments.
• Advantage: Generalizes well to non-standard or proprietary OFDM variants where analytical models may fail.

The FFT size can be indirectly determined by first estimating the subcarrier spacing (Δf). A bank of adaptive notch or comb filters is applied to the signal's power spectrum. The filter spacing that maximally nulls the signal energy corresponds to the subcarrier spacing. The FFT size is then calculated as N_FFT = Fs / Δf, where Fs is the known or estimated sampling rate.

• Method: Sweep a comb filter across the frequency band and measure the residual power.
• Detection: The deepest null indicates the correct subcarrier spacing.
• Application: Useful when the signal bandwidth is known but the FFT size is not.

Many OFDM standards embed known pilot subcarriers at fixed intervals in the time-frequency grid. By correlating the received signal with a bank of candidate pilot patterns, the FFT size is identified as the pattern that yields the highest correlation peak. This is a data-aided technique that requires knowledge of the standard's pilot structure.

• LTE Example: Cell-specific Reference Signals (CRS) are embedded every 6 subcarriers in frequency.
• Process: Demodulate the signal with candidate FFT sizes and correlate the extracted resource grid with the known pilot sequence.
• Constraint: Requires a priori knowledge of the specific wireless standard being analyzed.