Diagonal Slice Spectrum

The primary advantage of the diagonal slice is the drastic reduction in computational complexity. While a full bispectrum requires O(N²) operations for an N-point FFT, the diagonal slice reduces this to O(N) or O(N log N). This makes real-time implementation feasible on FPGAs and embedded processors where full 2D polyspectral processing would be prohibitive. The slice captures the bispectral values where f1 = f2, effectively tracing the main symmetry axis of the bifrequency plane.

The diagonal slice specifically retains information about self-coupling and harmonic generation within a signal. When a transmitter's non-linear power amplifier generates harmonics, the phase relationship between a fundamental frequency and its first harmonic manifests along the diagonal axis. This preserves the critical quadratic phase coupling signature that distinguishes one physical device from another, as these non-linear interactions are direct products of unique hardware impairment profiles.

Like the full bispectrum, the diagonal slice theoretically suppresses Gaussian noise. Since the bispectrum of a Gaussian process is identically zero, any energy appearing on the diagonal slice must originate from non-Gaussian signal components or non-linear interactions. This property allows the diagonal slice to extract hardware-induced fingerprints that are buried below the noise floor in traditional power spectrum analysis, making it highly robust for low-SNR environments.

The diagonal slice is often further compressed into a scalar or low-dimensional feature vector through integration. The integrated diagonal bispectrum sums the slice values across frequency, producing a compact representation of total non-linear energy distribution. This integrated value, or a set of segmented integrations, serves as a highly discriminative input to support vector machines (SVMs) or neural network classifiers for rapid emitter identification without requiring full 2D convolutional processing.

The diagonal slice is closely related to the bicepstrum, which is the inverse Fourier transform of the logarithm of the bispectrum. By operating on the diagonal slice, one can efficiently compute a 1D bicepstral representation that separates the system's transfer function from the input excitation. This is particularly useful for blind channel estimation and isolating the transmitter's inherent non-linear kernel from the effects of the propagation channel.

The diagonal slice is not a lossless representation. It discards cross-frequency coupling information where f1 ≠ f2, which can encode intermodulation products between unrelated spectral components. For complex modulations like OFDM with dense subcarrier interactions, the off-diagonal bispectral content may carry significant device-specific signatures. In such cases, radial integration or the full bispectrum may be required to capture all discriminative non-linear interactions.