magnitude-spectrum

The magnitude spectrum is obtained by applying a frequency-domain transform (Fourier, Laplace, or Z-transform) to a signal. Remove the Fourier transform concept and the magnitude spectrum cannot be computed — the amplitude values at each frequency are the modulus of the complex transform output. This is an operational dependency.

The magnitude spectrum is designed and maintained for the sake of analysis — it maps each frequency to its amplitude to enable inspection of a signal's spectral content. Remove analysis as the purpose and the magnitude spectrum has no reason to exist; it is a tool built to serve analytical inquiry into signals.

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.

Magnitude spectrum is computed from the Fourier transform — the FT existed first as a mathematical tool (1822), and the magnitude spectrum representation (magnitude vs. frequency) was derived from it by taking the absolute value of complex FT coefficients. Which-came-first test: FT predates magnitude spectrum.

Magnitude-spectrum is computed specifically to serve signal-processing: it represents signal power distribution across frequency for diagnostic and analytical purposes. For whose sake? The spectrum points at signal-processing as its master domain.

magnitude-spectrum is a specific kind of spectral-representation: it represents signal energy across frequency using raw (or log) magnitude values. A competent speaker would call it 'a spectral representation'.

Spectral centroid is computed from the magnitude spectrum: it is the center of mass of the spectrum, calculated as the weighted average of frequencies using magnitudes as weights. Magnitude spectrum existed first as a representation; spectral centroid is derived from it. Which came first? The magnitude spectrum representation predates the centroid statistic.

Spectral centroid is computed as the center of mass of the magnitude spectrum — a weighted average of frequencies using magnitudes as weights. Remove the magnitude spectrum and spectral centroid computation stops operating entirely. Law 8 present-tense removal test passes.

The cepstrum is computed from the magnitude spectrum: take the log of the magnitude spectrum, then apply the inverse Fourier transform. The magnitude spectrum existed first as a concept; the cepstrum was derived from it.

Phase retrieval recovers phase from magnitude-only spectral data. Remove the magnitude spectrum and phase retrieval has no input to operate on. The removal test is satisfied: without magnitude spectra, phase retrieval cannot function.

Envelope extraction needs magnitude-spectrum to operate: it computes the slow-varying amplitude envelope from spectral magnitude data. Remove magnitude-spectrum and envelope extraction has no data domain. Removal test passes.

The spectrogram operates by computing magnitude spectra over successive time frames. Remove magnitude-spectrum as the underlying construct and the spectrogram has no mechanism to compute — it ceases to operate. Present-tense dependency.

The mel-spectrogram operates by applying a mel-scale filterbank to the magnitude spectrum of a signal. Remove magnitude-spectrum and the mel-spectrogram has no input — it ceases to operate. Present-tense dependency.

Spectral subtraction estimates noise power from the magnitude spectrum and subtracts it. Remove magnitude-spectrum and spectral subtraction has no input to operate on. The removal test passes.

Pitch-class profiles are computed by folding the magnitude spectrum into 12 frequency bins. Remove the magnitude spectrum and pitch-class profile computation collapses — the removal test passes at the object level.

Log-magnitude-spectrum is derived from magnitude-spectrum by applying a log transformation. Which existed first? magnitude-spectrum predates and feeds into log-magnitude-spectrum. The which-came-first test passes.

Mel-frequency spectrum is computed by mapping the magnitude spectrum onto the mel scale. Remove the magnitude spectrum concept and mel-frequency spectrum computation collapses — there is nothing to map. Removal test passes (Law 8b). The log-magnitude variant similarly depends on magnitude-spectrum.

Spectral bandwidth measures the spread of frequencies around the spectral centroid, computed from the magnitude spectrum. Remove magnitude spectrum and spectral bandwidth computation collapses — there is no spectral distribution to measure. Removal test passes (Law 8b).

Spectral rolloff computes the frequency below which a certain percentage of magnitude spectrum energy resides. Remove magnitude spectrum and spectral rolloff computation collapses. Removal test passes (Law 8b).

Log-magnitude-spectrum is computed by applying a logarithmic transform to magnitude-spectrum values. Remove magnitude-spectrum as a concept and the input for the log transform disappears — the operation stops. Real operational dependency (Law 8b).

Which-came-first test: magnitude-spectrum (1960s) predates spectral-flatness (1970s, Widmer). Spectral-flatness operates by computing the ratio of geometric mean to arithmetic mean of the power spectrum values. The magnitude spectrum is the input data that feeds into this computation.

Which-came-first test: magnitude spectrum (19th century, post-Fourier) predates spectral entropy (1970s). Spectral entropy is computed by normalizing the magnitude (or power) spectrum into a probability distribution and calculating its Shannon entropy. The magnitude spectrum is the direct input.

Spectral entropy needs magnitude-spectrum to operate: it normalizes the magnitude spectrum into a probability distribution then computes Shannon entropy. Remove magnitude-spectrum and spectral entropy has no input to normalize. Both DERIVED_FROM (historical) and DEPENDS_ON (operational) coexist.

Spectral flux computes the L1 or L2 norm of the difference between consecutive magnitude spectra. Remove magnitude-spectrum and spectral flux has no input data to operate on. The removal test passes.

Spectral energy is computed as the sum of squared magnitude spectrum values across frequency bins. Remove magnitude-spectrum and spectral energy has no input data. The removal test passes.

Chroma features are computed by folding the magnitude spectrum into 12 pitch classes. Remove magnitude-spectrum and chroma extraction has no spectral input to fold. The removal test passes.

Spectral contrast computes energy differences between peaks and valleys across frequency bands in the magnitude spectrum. Remove magnitude-spectrum and spectral contrast has no input data to operate on.

Spectral flatness (Wiener entropy) computes the ratio of geometric mean to arithmetic mean of the magnitude spectrum. Remove magnitude-spectrum and spectral flatness has no input data. The removal test passes.

Spectral slope is computed by fitting a line to the magnitude spectrum across frequency bins using least-squares regression. Remove magnitude-spectrum and spectral slope has no data to operate on — the removal test passes. This is an object-level operational dependency.

chroma computation requires magnitude spectrum as input — chroma bins accumulate energy from magnitude spectrum frequency bins into pitch class bins. Remove magnitude-spectrum and chroma has no data to bin. Removal test passes.

Phase-estimation genuinely needs magnitude spectrum to operate: iterative phase retrieval algorithms (e.g., Gerchberg-Saxton, Fienup) use magnitude spectrum as the constraint in Fourier space. Remove magnitude spectrum and phase estimation algorithms have no data constraint. Removal test (Law 8) passes.

Griffin-Lim operates on the magnitude spectrum as its input — it iteratively estimates phase from magnitude alone. Remove the magnitude spectrum and the algorithm has no input to work from. Present-tense operational dependency.