power-spectrum

A power spectrum is a specific kind of spectral density — it quantifies the distribution of signal power across frequency. A competent speaker calls it 'a type of spectral density.' Nearest kind is spectral-density.

A power spectrum is computed to measure the distribution of signal power across frequency bins. Its designed purpose is to serve measurement of power content — it is an instrument of spectral measurement, not a standalone physical description.

Power spectrum as a signal-processing concept derives from the Fourier transform: the PSD is computed by taking the Fourier transform of the autocorrelation function (Wiener-Khinchin theorem). Fourier transform existed first (1822) and fed into the power spectrum concept (20th century). Historical chain is clear and directionally correct per Law 7.

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.'

The power spectrum is computed as the squared magnitude of the Fourier transform. Remove fourier-transform and the power spectrum has no operational basis — it cannot be computed or defined without it. Removal test passes: X without Y ceases to operate.

A power spectrum is a specific type of spectral representation that shows the distribution of signal power across frequency. 'Power spectrum is a spectral representation' — a competent speaker would call X a Y. Files against nearest kind per Law 11.

Power spectrum (mid-20th century) predates cepstrum (1960s, Bogert et al.). Cepstrum's core operation — taking the log of a power spectrum and applying inverse FT — directly uses the power spectrum as input. Which-came-first test: power spectrum came first and fed into cepstrum's design.

Spectral-centroid is computed as the weighted mean frequency of the power spectrum: Σf·P(f)/ΣP(f). Remove the power-spectrum concept and the spectral-centroid cannot be computed — it is literally the center-of-mass of the power spectrum. The removal test passes: no power-spectrum, no spectral-centroid.

Cepstrum-coefficients are computed from the power spectrum (c[n]=IDFT(log|X[k]|²)). Remove the power-spectrum concept and cepstrum-coefficients cannot be computed — the removal test passes as present-tense operational necessity.

The cepstral-domain concept was derived from the power spectrum: the construction procedure takes the power spectrum → log → IDFT. Which came first? Power spectrum (Fourier analysis concept, predates cepstrum by decades). Historical derivation.

Spectral-flatness is computed directly FROM a power spectrum — it is the ratio of geometric to arithmetic mean of spectral power values. Remove the power spectrum and spectral-flatness has no input to operate on; the removal test passes.

Spectral-flux measures the rate of change of the power spectrum over time. It operates on the power spectrum — remove it and spectral-flux has no input to operate on. Removal test passes.

Spectral contrast quantifies the difference between peaks and valleys in a spectral representation — this computation fundamentally requires the power spectral density as input. Remove the power spectrum and spectral contrast has no data to operate on. This is a present-tense operational dependency: spectral contrast cannot function without the power spectrum as its computational substrate.

Spectral rolloff computes the frequency below which X% of total spectral energy is contained — this requires a power spectrum as input to accumulate magnitudes. Remove the power spectrum and spectral rolloff cannot operate. Genuine DEPENDS_ON per Law 8's removal test.

The mel-spectrum is computed by applying the mel filterbank to a power spectrum. Remove the power spectrum and there is no input to transform — the mel-spectrum computation stops entirely. Present-tense dependency confirmed.

DERIVED_FROM test: the power spectrum concept existed first and fed into spectral flux. Spectral flux measures the rate of change of the power spectrum over time — it is a derived metric built on the power spectrum foundation. The note pins the sense: spectral flux as a specific measure DERIVED FROM the broader power spectrum concept.

Power spectrum existed first and fed into spectral energy. Spectral energy is computed by integrating a power spectrum — the power spectrum is the source from which spectral energy is derived. Which-came-first test (Law 7): power spectrum concept predates spectral energy as a named scalar quantity.

Cepstral coefficients are computed from the power spectrum: the cepstrum is the inverse Fourier transform of the log power spectrum. Remove the concept of power spectrum and the cepstral computation collapses — the log magnitude is the power spectrum, which is the essential input to the cepstral transform.

Spectral subtraction estimates the noise power spectrum and subtracts it from the noisy signal's power spectrum. Remove power spectrum and the subtraction operation has no data to work on — the entire method is defined in terms of power spectral density manipulation.

Spectral decay measures the rate at which spectral energy decreases over time. It operates on the power spectrum — remove power spectrum and there is no spectral energy measure on which decay can be computed. The removal test passes.

Spectral flatness is computed from the power spectrum: the measure is the ratio of geometric mean to arithmetic mean of the spectral power values. Remove the power spectrum and the flatness calculation has no input and stops operating.