fft

FFT's operation is defined as an efficient algorithm for computing the DFT. Remove the DFT — the mathematical transform that decomposes signals into frequency coefficients — and FFT has nothing to compute. The dependency is constitutive: FFT is the 'how' of DFT computation, DFT is the 'what'.

DERIVED_FROM test (historical): The Fourier transform concept (continuous, 1822) predates the FFT algorithm (Cooley-Tukey, 1965). The FFT is an efficient computational method for computing the discrete Fourier transform — it derives from and extends the Fourier transform concept.

FFT is an algorithm whose entire operational purpose is to compute the discrete Fourier transform efficiently. Remove DFT and FFT has no mechanism or purpose — present-tense removal test (Law 8) passes. DFT (epoch 0.06) predates FFT (0.83).

FFT (Fast Fourier Transform) is a specific efficient algorithm for computing the DFT. A competent speaker would call FFT a DFT computation method. It is a kind of DFT algorithm, not merely dependent on it.

FFT is a specific signal processing algorithm for computing the discrete Fourier transform efficiently. A competent DSP practitioner would classify FFT as a signal processing operation. Specific to general per Law 9.

zero-padding (appending zeros to a signal) was specifically designed to increase the apparent frequency resolution of FFT output. Its purpose is to serve FFT by interpolating the frequency domain — zero-padding SERVES fft.

STFT computes FFT on sliding window segments of the signal. Remove FFT and STFT loses its computational core — it stops operating. Removal test (Law 8) passes.

Spectral gating computes magnitude spectra per frame to estimate noise floor and attenuate bins; FFT is the computational core that produces those magnitude spectra. Remove FFT and spectral gating has no spectral representation to threshold — it stops operating. Pinned sense: spectral gating as per-band threshold attenuation in the frequency domain.

Remove FFT (or any DFT equivalent) — the spectral flanger stops operating. It requires frequency-domain decomposition via FFT to compute the modulated phase ramp across bins; without frequency-domain transform capability, the effect cannot function.

Overlap-save needs FFT to operate: its core mechanism is computing the FFT of overlapping signal blocks, multiplying in frequency domain, then IFFT-ing back. Remove FFT and overlap-save ceases to function. This passes the removal test (Law 8).

Remove FFT — cepstral-peak-prominence stops operating. CPP is computed as the peak of the cepstrum, which requires FFT → log-spectrum → IFFT pipeline. Without DFT/FFT, no cepstrum can be computed.

Cepstral-peak-picking DEPENDS_ON fft — computing the real cepstrum requires taking the FFT of the signal, then the log, then the inverse FFT. Remove FFT and the entire cepstral computation collapses: there is no way to obtain the cepstrum without it.

Phase unwrapping operates on phase spectra obtained via FFT; remove FFT and there is no phase data to unwrap, so the unwrapping procedure ceases to operate.

The cepstrum algorithm requires FFT as its core computational step: it computes the FFT of the log-power spectrum to produce the quefrency domain representation. Remove FFT and the cepstrum cannot operate. Which-came-first: FFT (epoch 0.69) predates cepstrum (epoch 0.88).

Phase vocoder computes spectra via FFT — remove FFT and the phase vocoder cannot compute the analysis/synthesis spectra it needs. The which-came-first test: FFT existed long before phase vocoder, but this is a present-tense dependency, not historical.