discrete-fourier-transform

The discrete Fourier transform was derived from the continuous Fourier transform by extending it to discrete sequences. FT existed first as a mathematical tool (1822); DFT was derived from it as the discrete analog. Which came first test: FT predates DFT and fed into DFT's formulation.

The discrete Fourier transform is a specific signal processing technique for converting time-domain signals to frequency-domain representation. A competent speaker calls DFT a type of signal processing operation. Per Law 9: specific→general INSTANCE_OF. Nearest kind is signal-processing.

DFT is a specific computational method for producing a spectral representation. Which came first? The general concept of representing signals in the frequency domain (spectral representation) predates the specific algorithm (DFT). spectral-representation=0.03, DFT=0.06 → spectral-representation is older and more general.

The cepstrum computation fundamentally requires the discrete Fourier transform: cepstrum = IDFT(log(|DFT(x)|²)). Remove DFT and the cepstrum cannot be computed. The DFT concept predates cepstrum (1820s vs 1963).

The cepstrum-coefficient is computed via the inverse DFT of the log-power spectrum: c[n] = IDFT(log(|X[k]|²)). The DFT concept (1820s) predates cepstral coefficients (1963, Bogert). Which-came-first test passes: DFT existed first and fed into cepstrum-coefficient.

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).

Cepstral analysis computes the cepstrum via FFT (DFT) of the log power spectrum. Remove DFT and the cepstrum computation stops working entirely. Removal test passes: cepstral analysis needs DFT as its computational engine.

Windowing is applied before DFT to reduce spectral leakage from finite signal truncation. Remove DFT and windowing as a DSP technique has no target — you always window a DFT. Direction: windowing depends on discrete-fourier-transform.

Spectral leakage is a direct consequence of the DFT's finite-length sampling of signals. Remove the discrete Fourier transform and spectral leakage ceases to exist — it is an artifact intrinsic to the DFT's mathematical structure. Epochs confirm DFT (0.06) predates spectral-leakage (0.31).