stft

STFT (1940s, Dennis Gabor) derives from the Fourier Transform (late 1800s, Fourier): it applies Fourier analysis to short time windows, extending the basic transform to time-varying signals. Which-came-first test: Fourier Transform (c. 1822) predates STFT by over a century.

STFT was developed within and derives from the field of signal processing. It addresses a specific signal-processing need — analyzing non-stationary signals — by extending Fourier methods to the time-varying domain.

STFT IS a sliding-window Fourier transform. Remove the Fourier transform and the core computation vanishes — STFT cannot operate without it. Removal test passes: X stops operating when Y is removed. DEPENDS_ON holds.

STFT is a specific kind of time-frequency representation: it represents a signal in joint time-frequency space via a sliding-window Fourier analysis. A competent speaker calls STFT 'a time-frequency representation.' Per Law 9.

STFT is a specific kind of time-frequency analysis technique: it analyzes signals in joint time-frequency space via sliding-window Fourier analysis. A competent speaker would call STFT 'a time-frequency analysis method.' INSTANCE_OF specific→general.

Removal test: remove windowing-function now and STFT stops operating as a practical technique — without windowing you get the classical (non-time-localized) Fourier transform, not the short-time variant. STFT requires windowing to segment signals and localize analysis in time.

STFT concept DERIVED_FROM spectrum: spectrum (frequency-domain representation) existed first as a fundamental signal analysis concept. STFT extends the spectrum concept to non-stationary signals by windowing, making it older/simpler and the direct ancestor. Which existed first? Spectrum.

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.

STFT computes Fourier transforms of windowed segments of a signal. Remove windowing and STFT has no mechanism to create overlapping time-localized segments — it collapses to plain Fourier transform which is a different concept. Windowing is constitutive to STFT's operation.

Which existed first? STFT (Short-Time Fourier Transform) predates mel-spectrogram. STFT was developed in the 1940s as a signal analysis method; mel-spectrogram came later when the mel-frequency scale was applied to spectrogram computation. STFT fed into mel-spectrogram.

Mel-spectrogram needs STFT to operate now: it is computed by applying STFT to get the time-frequency representation, then projecting onto mel bins. Remove STFT and the mel-spectrogram cannot be computed — the removal test passes.

STFT existed first and fed into the spectrogram concept: the spectrogram is computed as the squared magnitude of the STFT output. Spectrogram is a direct derivative of STFT — take the STFT and compute |X|^2.

Griffin-Lim operates by alternating between inverse-STFT and forward-STFT. Remove STFT and the algorithm stops operating entirely — it cannot function without the STFT/ISTFT pair. DEPENDS_ON direction correct.

Phase-vocoder operates via STFT: it computes short-time Fourier transforms, manipulates the spectra, and reconstructs via inverse STFT. Remove STFT and the phase-vocoder has no mechanism — ceases to operate. Present-tense dependency.

Spectrogram computation requires STFT as its core operation — a spectrogram is the magnitude of the STFT over time. Remove STFT as a concept and the spectrogram ceases to be computable. The removal test passes.

MFC computation requires STFT as the foundation — the mel-filterbank operates on the STFT magnitude spectrum. Remove STFT and MFC has no input to warp. The removal test passes.

MFCC computation requires STFT — the mel-filterbank operates on the STFT magnitude spectrum. Remove STFT and MFCC has no spectral decomposition to work from. The removal test passes.

Removal test: remove STFT now and spectral-subtraction stops operating — the technique computes STFT of the noisy signal, estimates noise from silent segments, then subtracts. Without STFT there is no time-frequency decomposition to work with.

Which-came-first test: STFT (1940s-60s) predates MFCC (1980s). MFCC computation takes the STFT as its first step — the historical lineage runs STFT → magnitude spectrum → log → mel filterbank → DCT → MFCC.

Mel-frequency spectrum is computed by first obtaining the STFT magnitude spectrum, then warping it to the mel scale. Removal test: remove stft and there is no spectral representation to warp. Constitutive dependency.

Cepstral analysis DEPENDS_ON STFT for operation. The standard pipeline: STFT → power spectrum → log → DCT → cepstral coefficients. Remove STFT and the analysis pipeline collapses at the first step. The removal test passes.

Harmonic-percussive-separation needs STFT to operate now: the technique decomposes audio via time-frequency representations (spectrograms), and removing the STFT means no time-frequency representation is available — the method cannot run. The removal test (Law 8): remove STFT, HPS stops operating, not merely being sayable.

Griffin-Lim operates on the magnitude spectrum of the STFT. Its entire iterative procedure — compute STFT, iterate magnitude/phase, inverse-STFT — requires the STFT framework. Remove STFT and Griffin-Lim stops operating.

Mask-based enhancement operates on time-frequency representations to estimate masks and apply them to magnitude spectra. Remove STFT and the mask estimation and application cannot operate — the technique stops functioning. Operational dependency per Law 8b, not merely sayable. Remove-Y-now-X-stops-operating test satisfied.

Phase vocoder was developed as a specific application/derivative of STFT-based analysis-synthesis, building directly on STFT frames.

Phase retrieval algorithms operate on magnitude spectrograms produced by STFT; without STFT analysis there is no input representation for retrieval.

Harmonic-percussive-source-separation operates on STFT magnitude spectrograms to build harmonic and percussive masks. Remove STFT and the method has no time-frequency representation to filter and ceases to operate.