spectral-decomposition

Fourier analysis (1822) predates modern spectral decomposition techniques (mid-20th century). Spectral decomposition — representing a signal as a sum of frequency components — is built on the Fourier transform. Which-came-first: fourier-transform is older and fed into spectral-decomposition as the foundational mathematical tool.

Spectral decomposition IS a form of spectral representation — it represents a signal as a decomposition into frequency components. This is the nearest kind for spectral-decomposition.

Spectral decomposition (Fourier analysis, 1822; mid-20th century techniques) predates cepstral-envelope (1980s). Cepstral-envelope computation requires spectral decomposition: the cepstrum is the inverse FFT of the log power spectrum, and the power spectrum itself comes from spectral decomposition via FFT. Which-came-first (Law 7) passes cleanly.

Spectral decomposition predates MFCC (1980s). MFCC computation begins with STFT, which is a form of spectral decomposition — it decomposes the signal into frequency components over time. Without spectral decomposition, MFCC coefficients cannot be computed. Law 7 passes.

Spectral decomposition (1822 Fourier → mid-20th century techniques) predates the cepstrum (1976, Bogert et al.). The cepstrum is computed as IFFT(log|FFT(x)|) — the FFT is spectral decomposition, and the cepstrum is derived from the spectral representation. Which-came-first (Law 7) passes.

Spectral decomposition (Fourier analysis, 1822) predates spectral-envelope extraction techniques (1970s). The spectral envelope is the smooth envelope of the power spectrum, which itself comes from spectral decomposition. Without spectral decomposition, there is no spectrum to extract an envelope from. Law 7 passes.

Mel-frequency-spectrogram (1980s) derives from spectral decomposition: its computation begins with STFT, a time-frequency spectral decomposition, followed by mel-scale warping. Spectral decomposition (Fourier analysis, 1822) predates and enables the mel-spectrogram. The which-came-first test passes.

Spectral-decomposition (FFT-based decomposition of signals into frequency components) existed first as the foundational technique; the cepstral-transform builds directly on this by taking the log magnitude of the spectrum and inverting it to produce the cepstrum. The transform's first and critical step is spectral decomposition.

Spectral subtraction needs spectral decomposition (FFT) to operate now — it transforms the signal to the frequency domain, estimates and subtracts noise, then reconstructs. Remove spectral decomposition and spectral subtraction stops working.