spectrum

Spectrum is built and maintained for the sake of analysis — its designed purpose is to decompose complex signals into constituent components, making structure visible. Law 8d: for whose sake? Analysis. The note shows purpose by design: spectrum analyzers, spectrograms, and spectral decomposition methods are all instruments and methods created specifically to serve analytical work.

Spectrum analysis OPERATES by measuring frequency components of a signal. Remove measurement and spectral analysis ceases to function — you cannot decompose or measure frequencies without measurement. Per Law 8 removal test: remove measurement and spectrum stops operating as a concept/method.

A spectrum is the representation of a signal in the frequency domain, computed via the Fourier transform. Remove the Fourier transform concept and the spectrum ceases to be computable — it has no operational mechanism. The spectrum depends on the Fourier transform for its very existence. Object-level operational dependency. Law 8.

Power spectrum is a specific kind of spectrum — it describes the distribution of power across frequency. Law 9: 'X is a specific kind of Y' is INSTANCE_OF. A competent speaker calls a power spectrum 'a spectrum.'

Magnitude spectrum maps each frequency to its amplitude — it is a specific kind of spectrum. The filer pins the sense: magnitude (amplitude vs. frequency), not phase spectrum or complex spectrum. Law 9: magnitude-spectrum is a specific case of the general spectrum.

The cepstrum concept is derived from the spectrum: quefrency-domain analysis operates on the Fourier transform of the log-power spectrum. The spectrum existed first and directly fed into the construction of the cepstrum concept. Which-came-first test passes.

The spectral-envelope is derived from the raw spectrum: it extracts the smooth, coarse shape from the fine spectral detail. The spectrum concept existed first and the envelope is a transformation applied to it. Which-came-first test passes.

Which-came-first: the spectrum concept predates and feeds into spectral-peak. A spectral peak is a feature identified within a spectrum — you need the spectrum concept first to identify peaks within it. The spectrum existed first and directly fed into constructing the spectral-peak concept.

Spectral tilt computes the slope of spectral energy across frequency bands. It requires a spectrum as input — remove spectrum and you cannot compute the tilt measure. This is a present-tense operating dependency per Law 8, not meta-level.

Spectral bandwidth measures the spread of spectral energy around the centroid. It requires a spectrum as input — remove spectrum and spectral bandwidth cannot be computed. Valid present-tense DEPENDS_ON per Law 8.

Spectrogram needs spectrum to operate now: it is a time-varying representation of spectral content. Remove the concept of spectrum and the spectrogram loses its data domain entirely — it stops functioning as a representation. The removal test passes.

Mel-spectrogram needs spectrum as its data foundation: it transforms spectral magnitudes using the mel scale. Remove spectrum and there is no underlying representation for the mel transform to work on. Removal test passes.

Cepstral coefficients are computed directly from the spectrum: take the log of the power spectrum, then apply the inverse Fourier transform. Which existed first? Spectrum (as a frequency-domain representation) predates cepstral coefficients (coined 1963). Spectrum is the direct input domain from which cepstral coefficients are derived.

Freq-warping operates on spectra: it remaps the frequency axis of a spectral representation. Remove spectrum from the pipeline and freq-warping has no data to transform — it stops operating entirely. Real object-level removal test.

The mel-filterbank concept DERIVED_FROM spectrum: it takes the spectral representation as its input and organizes frequency bins along the mel scale. Spectrum existed first as the frequency-domain representation, and the mel-filterbank was built on top of it as a perceptual weighting mechanism.

MFCC computation genuinely needs spectrum to operate: the mel-filterbank operates on the power spectrum to produce mel-spectrogram before DCT. Remove the spectrum concept and MFCC has nothing to operate on.

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.

Chroma extraction operates on the magnitude spectrum — removing spectrum and chroma has no spectral data to quantize into pitch-class bins. Removal test passes.

Spectral peaks are local maxima IN a magnitude spectrum. Remove spectrum and spectral-peaks has nothing to operate on. Removal test passes.

Cepstral analysis operates on the spectrum (computes the spectrum of the log spectrum). Remove spectrum and cepstral-analysis cannot operate. Removal test passes.

spectrum (frequency-domain representation) existed first as a fundamental concept. STFT extends spectrum to non-stationary signals by adding windowing. Historical test: the spectrum concept predates STFT and fed into it.

spectral-centroid computes the weighted mean frequency of the spectrum. It directly operates on the power spectrum — remove spectrum and centroid has no input. Genuine operational dependency.

spectral-contrast measures the energy distribution across frequency bands from the spectrum. Remove spectrum and the contrast computation has no data. Genuine operational dependency.

Spectrum (the concept of decomposing a signal into frequency components) predates and fed into phase-estimation as a concept. Spectrum existed as the Fourier concept from the 19th century; phase-estimation algorithms emerged later to recover the phase component that spectrum provides magnitude for. Which-came-first test passes.

Spectrum (frequency decomposition of signals) predates the concept of spectral peaks. The spectral-peaks concept — identifying local maxima in a frequency representation — derives from having a spectrum to examine. Which-came-first test passes. Note: DEPENDS_ON stft was already struck as too narrow; this DERIVED_FROM to the broader spectrum concept is correct.

The mel-frequency spectrum is derived from the general spectrum concept: it applies a mel-scale frequency warping to the linear frequency spectrum. The mel scale existed as a perceptual model after the spectrum concept, and the mel-frequency spectrum as a concept is obtained by transforming the spectrum through the mel-scale filterbank. Which existed first? The linear spectrum concept predates the mel-warping transformation.

Which came first? The concept of spectrum predates spectral-whiteness. Spectral-whiteness was derived by applying the notion of 'whiteness' (flat spectral density) to the spectrum concept. Spectrum fed into the development of spectral-whiteness as a specific characteristic of spectral distributions.

Spectral-flatness needs spectrum to operate now: it measures the flatness of a spectral density distribution. Remove spectrum and spectral-flatness has no data to analyze — the removal test (Law 8) passes.

Spectral whiteness quantifies how flat a spectral density distribution is. It operates by analyzing spectral data — remove spectrum and there is no spectral density to characterize. The removal test passes: spectral whiteness needs spectrum as its data domain.

Spectral slope measures the rate of change of spectral magnitude with respect to frequency. It is computed directly from the spectrum — taking the first derivative of the spectral envelope. Remove the concept of spectrum and spectral slope has no data to analyze. Present-tense operational dependency.

Spectral energy is the integrated power across a frequency band or the entire spectrum. It is computed from the spectral magnitude values. Remove spectrum and spectral energy has no values to integrate. Present-tense operational dependency.

The cepstrum is computed as the inverse Fourier transform of the logarithm of the magnitude spectrum. Remove the concept of spectrum and the cepstrum cannot be computed — the entire derivation chain collapses. The removal test passes.

Spectral decay measures the rate at which spectral energy decreases over time. It operates on spectral data — remove the concept of spectrum and there is no spectral energy to measure, no spectral domain in which decay is defined. The removal test passes: no spectrum concept = spectral decay is undefinable.