Spectral Descriptors

Spectral Descriptors explicitly characterize how a declared nonnegative spectral quantity—such as signal magnitude, squared magnitude, power, power spectral density, energy-like weight, or periodic structure—is organized with respect to frequency over a declared Behavioral Signal support. These descriptors rely on the semantic context of the input signal, the declared frequency domain, the spectral estimator used, scaling, normalization, the chosen descriptor functional, parameterization, units, and interpretive limits to acquire scientific meaning.

Discrete Fourier Transform (DFT) or Fast Fourier Transform (FFT) coefficients, magnitude spectra, power spectra, periodograms, power spectral density (PSD) estimates, and spectrograms are spectral representations or estimates of a signal’s frequency content. However, these spectral objects are not automatically complete Spectral Descriptors by themselves. Spectral frequency is not behavioral frequency by definition; rather, spectral frequency is a coordinate in the frequency domain that relates to the signal’s temporal sampling and support but does not directly equate to behavioral events or phenomena without interpretation. A spectral descriptor only attains scientific relevance when all aspects of its context—signal semantics, support, frequency coordinates, spectral estimator, scaling, normalization, descriptor functional, parameters, units, and interpretive boundaries—are explicitly declared and understood.

Meaning and Spectral Representation

A Spectral Descriptor is an identifiable characterization of one or more properties of a declared spectral representation over a declared frequency domain. These properties include, but are not limited to, total or band-specific content, dominant frequency, spectral location, spread, roll-off, shape, concentration, entropy, flatness, slope, harmonic structure, or cepstral structure, provided they are scientifically meaningful in the given context. The identity of a Spectral Descriptor must include the specific spectral quantity to which the descriptor functional is applied, not merely the descriptor name, to preserve clarity and scientific validity.

Complex Fourier coefficients, magnitude spectra, squared-magnitude or power spectra, periodograms, power spectral density estimates, normalized spectral distributions, log spectra, and Spectral Descriptors all represent distinct spectral objects:

• Complex Fourier coefficients retain phase information and thus contain different information than nonnegative magnitude or power representations.
• Magnitude spectra contain amplitude information without phase.
• Power or squared-magnitude spectra represent energy or power content per frequency bin.
• Periodograms estimate power spectra from finite-length signals under specific windowing.
• Power spectral density (PSD) estimates spectral power distribution per unit frequency (density semantics).
• Normalized spectral distributions adjust for total power or energy, losing absolute content information.
• Log spectra transform spectral magnitudes or powers into logarithmic scale, altering arithmetic meanings.
• Spectral Descriptors are functionals or summary statistics derived from these spectral objects, reflecting scientifically interpretable characteristics.

A descriptor computed from one spectral representation should not be treated as numerically or scientifically equivalent to the same-named functional computed from another without explicit justification.

Support-level Spectral Descriptors characterize frequency organization aggregated across the declared support interval. If frequency organization changes materially within that support, a single global spectral representation can mix several temporal regimes, resulting in descriptors that summarize a composite rather than a homogeneous state. This global characterization is distinct from explicitly time-localized frequency characterization, where spectral structure evolution through time is part of the target property. A single global spectral descriptor does not prove that oscillatory or frequency structure was constant throughout the support.

Frequency Axis, Sampling, and Observable Spectral Structure

Frequency-coordinate semantics must be explicit and consistent. Common frequency units and conventions include:

• Hertz (Hz): cycles per unit time, typically seconds.
• Cycles per unit time: alternative to hertz, depending on time unit.
• Cycles per sample: normalized frequency relative to sample intervals.
• Angular frequency: radians per second or per sample.
• Normalized frequency: frequency normalized by the Nyquist frequency or sampling frequency.
• Positive and negative frequencies: for complex-valued spectral representations.
• One-sided versus two-sided frequency axes: one-sided spectra show only nonnegative frequencies (common for real signals), two-sided spectra include negative frequencies.

Sampling-time units and normalization references must be explicit to avoid confusion. Do not mix descriptors expressed in hertz, normalized-to-Nyquist units, relative harmonic number, cadence-normalized frequency, or other reference-dependent frequencies under one semantic identity.

The uniformly sampled discrete Fourier transform (DFT) and its physical frequency mapping are precisely:

Here, N is the number of uniformly spaced samples in the analyzed finite sequence, n is the sample index, x_n is the sample value, k is the DFT-bin index, i is the imaginary unit, X_k is the complex DFT coefficient, f_s is the sampling frequency in declared units, and f_k is the corresponding DFT-bin frequency under this conventional grid. The DFT is a representation-building transform, not a Spectral Descriptor by itself.

Observable frequency range and aliasing considerations apply for uniformly sampled signals. For a real-valued signal sampled at frequency f_s, frequencies above the Nyquist limit (f_s / 2) can alias into lower observed frequencies if acquisition or prior resampling does not adequately control out-of-band content. Post-acquisition analysis cannot recover frequency information that was never observed or that was irreversibly aliased. Spectral peaks should not be interpreted outside the scientifically supported frequency range.

Support duration, sample count, DFT-bin spacing, zero padding, and practical frequency resolution interact as follows: under uniform sampling, the nominal bin spacing depends on sampling frequency and transform length, but the ability to resolve nearby spectral components depends also on observed duration, window characteristics, signal-to-noise ratio, and spectral estimator choice. Zero padding increases frequency grid density but does not increase true resolving power, which depends on the duration and nature of the observed data.

Spectral leakage and windowing arise because finite support implicitly truncates the signal. Applying a taper or window modifies the spectral response by changing main-lobe width, side-lobe behavior, amplitude calibration, and effective weighting of samples. The declared Descriptor Support is distinct from the window or taper applied within that support. No window universally optimizes frequency resolution, leakage suppression, amplitude estimation, and variance simultaneously; trade-offs are inherent.

Spectral Estimation, Scaling, and Preprocessing Dependence

Periodogram and Welch-type power spectral estimation are representative but not exclusive estimator families. A periodogram estimates spectral power or power density from a finite support under a declared window and scaling. Welch-type estimation divides the evidence into possibly overlapping subsegments, computes modified periodograms for each, and averages them. This averaging trades frequency resolution, variance, robustness, and effective use of evidence according to segment length, overlap, window, FFT length, and averaging rule. No estimator default can be presented as a universal Behavioral Signal Processing standard.

Spectral estimation for irregularly sampled evidence requires distinct methodology. Ordinary uniform-grid FFT frequency semantics should not be imposed on irregular timestamps without justified resampling or an estimator designed for irregular sampling, such as Lomb–Scargle-type approaches when their assumptions fit the scientific question. Whether timing was resampled, interpolated, weighted, or analyzed natively must be preserved because these choices fundamentally change the spectral object.

Spectral scaling and units must be carefully distinguished. Density-like quantities are expressed per unit frequency, while spectrum-like quantities represent integrated power over frequency bins. Magnitude spectra differ fundamentally from squared magnitude or power spectra. One-sided and two-sided conventions redistribute spectral weight differently for real signals. Descriptor definitions involving sums, integrals, ratios, centroids, entropy, or flatness must explicitly identify the spectral representation and scaling from which values are computed.

Preprocessing such as DC content removal, mean removal, detrending, filtering, normalization, and others are scientifically consequential for Spectral Descriptors. Removing a mean or trend reduces low-frequency content but may also remove the property of interest. Filtering can alter band power, peak location, centroid, entropy, and spectral shape. Amplitude normalization preserves relative spectral shape but destroys absolute power information. The analyzed signal state must be preserved and not treated as invisible or irrelevant.

Support-level stationarity and regime mixing affect interpretability. Periodogram- or PSD-derived support summaries can be descriptive even without strict stochastic stationarity, but frequency-domain meaning weakens when a support combines distinct regimes, strong trends, isolated transients, or changing frequencies. Shortening support improves localization only by changing the evidence scope and may reduce frequency discrimination. Explicit time-localized spectral evolution requires a different characterization.

Spectral Content, Frequency Bands, and Peaks

Total spectral content and integrated spectral power are meaningful only under a declared spectral representation and units. A sum over squared-magnitude bins, an integral of a power spectral density, and a physical signal-energy or power quantity coincide only under appropriate scaling and signal definitions. Arbitrary squared spectral magnitude should not be labeled as physical energy or power merely because of mathematical resemblance.

Discrete band content from a density-like spectral estimate is defined as:

Here, a and b are declared lower and upper band boundaries with an explicit edge convention, K_{a,b} is the set of frequency bins admitted to that band, k is a frequency-bin index, P_k is a nonnegative density-like spectral estimate at bin k, Δf_k is the represented frequency width associated with that bin, and B_{a,b} is the integrated band content in the corresponding non-density units. Exact numerical integration can require edge interpolation or other quadrature when band boundaries do not coincide with bin centers or edges.

Absolute band content, relative band content, band fractions, and band ratios are distinct descriptors. Relative band content requires a declared normalization denominator and frequency domain; band ratios require both numerator and denominator bands and can become unstable or undefined near zero denominators. Normalization improves comparability but removes absolute amplitude information and makes the descriptor dependent on the chosen reference band or total range.

Frequency bands can be externally standardized for a modality, study-defined, participant-specific, task-normalized, peak-centered, harmonically defined, or data-derived. Band edges learned or tuned from analyzed outcomes are not equivalent to prespecified scientific bands and can introduce circularity or evaluation leakage. Conventional modality-specific bands should not be generalized to unrelated signals simply because numerical frequency values are available.

Peak frequency, dominant frequency, peak spectral magnitude or power, top-k peaks, and peak-to-background or peak-to-total relationships require explicit definitions of peak search range, local-maximum rule, interpolation, minimum prominence or separation, treatment of DC, ties, and edge peaks. The highest spectral bin is not automatically a fundamental frequency, and narrow artifacts or interference components can dominate a peak descriptor without representing the intended behavior.

Spectral Location, Spread, Shape, and Concentration

The spectral centroid and p-order spectral spread are defined exactly as:

Here, K is the declared frequency-bin set used by the descriptor, k is bin index, f_k is bin frequency, q_k is the declared nonnegative spectral weight for bin k, Q is total weight Q = Σ_{k∈K} q_k with Q > 0, f_c is the spectral centroid, p is a declared positive spread order, and σ_{f,p} is the p-order spectral spread about the centroid. Magnitude-weighted and power-weighted versions differ and must be declared explicitly.

Spectral median frequency and roll-off (spectral-edge) frequency are cumulative spectral-location descriptors. The median frequency is defined by a 50% cumulative criterion under a chosen spectral weight, while roll-off uses another declared fraction ρ. Frequency domain, cumulative-weight representation, fraction, interpolation rule, and zero total spectral weight handling must be specified. Roll-off is not an observed maximum frequency.

Higher spectral moments and shape descriptors include frequency-weighted variance-like spread, skewness, and kurtosis when the nonnegative spectral weight can be normalized meaningfully over frequency. Weight representation, frequency range, normalization, and estimator conventions must be declared. Kurtosis is distinguished from excess kurtosis. Spectral skewness characterizes asymmetry across frequency, not temporal waveform asymmetry.

Spectral slope, spectral decrease, and other trend-of-spectrum descriptors are distinct and require explicit definition. Whether the fit or difference uses linear frequency, log frequency, linear spectral level, log power, decibels, selected bins, excluded DC, or another axis transform must be declared. Slope values lack meaning without these axis and range conventions. Spectral decrease is not a synonym for least-squares spectral slope.

Spectral flatness, crest (peak-to-average concentration), and related concentration descriptors compare geometric and arithmetic spectral means under a declared nonnegative representation and usually require a positive numerical floor when zeros occur. Crest emphasizes concentration at the largest spectral component. Interpretations such as "noise-like" or "tone-like" are modality- and model-dependent and should not be generalized automatically to arbitrary behavioral signals.

Normalized discrete spectral entropy is defined exactly as:

Here, K is the declared set of included spectral bins, M is the number of bins in K with M > 1, k is bin index, q_k is the declared nonnegative spectral weight, Q is total weight Q = Σ_{k∈K} q_k with Q > 0, p_k is the normalized spectral weight, and H_s is the entropy normalized by log(M) using the same logarithm base in numerator and denominator. Zero contribution is adopted for bins with p_k = 0. This entropy is of a discretized spectral-weight distribution, not entropy rate, sample entropy, permutation entropy, or a universal measure of behavioral complexity.

Harmonic, Cepstral, and Modality-Specific Spectral Structure

Fundamental-frequency and harmonic descriptors are valid only when the signal support plausibly contains periodic or approximately periodic structure and when the descriptor includes an explicit fundamental-frequency definition. Spectral peak frequency must be distinguished from fundamental frequency, harmonic index from absolute frequency, and a harmonic series from any set of regularly spaced peaks. An absent or ambiguous fundamental frequency should produce an explicit undefined or uncertain state rather than a forced estimate.

Harmonic power ratio, harmonic-to-nonharmonic content, inharmonicity, harmonic spacing error, and related descriptors conceptually characterize conformity to a declared harmonic structure. These require a fundamental or harmonic model, frequency tolerance, harmonic count or range, weighting, and treatment of broadband or unresolved components. Such descriptors should not be applied to clearly aperiodic evidence simply because spectral peaks exist.

Cepstral descriptors characterize the spectrum after a declared transformation of a log-magnitude or log-power spectral representation into a quefrency- or coefficient-domain object. The cepstrum differs from the original spectrum, and generic cepstral coefficients differ from modality-specific constructs such as mel-frequency cepstral coefficients (MFCCs). MFCCs depend additionally on perceptual frequency mapping, filter-bank design, logarithm, discrete cosine transform convention, coefficient selection, and audio-oriented assumptions.

Absolute, normalized, perceptually warped, task-relative, cadence-relative, and participant-relative spectral coordinates are distinct choices. Frequency normalization can aid comparison but changes the descriptor’s reference. Perceptual scales, chroma, tonal descriptors, and audio-specific spectral contrast are scientifically appropriate for acoustic evidence but should not be imposed on movement, physiological, or other modalities without a substantive mapping giving those coordinates meaning.

Multichannel and vector-valued spectral characterization differ fundamentally. Channel-wise spectra, spectra of vector magnitude, spectra after coordinate rotation, and spectra of projected components are different signal objects and yield different Spectral Descriptors. Channel identity, coordinate frame, units, transformations, and whether channels were combined before or after spectral estimation must be preserved. Multiple single-channel spectra should not be interpreted as cross-signal coherence or coupling without explicit cross-spectral analysis.

Spectral Uncertainty, Sensitivity, and Verification

Spectral Descriptor uncertainty and invalid states arise from finite support, spectral-estimator variance, short observation, weak peaks, close components, leakage, aliasing, irregular timing, missing samples, uncertain sampling rate, clipping, quantization, detrending, window choice, band-edge placement, zero or near-zero total spectral weight, and model-dependent harmonic assumptions. It is essential to distinguish an uncertain peak from no peak, an undefined band ratio from zero band ratio, unresolved frequency from zero frequency, and invalid spectral descriptor from behavioral absence.

Sensitivity and robustness analyses compare scientifically plausible support lengths, preprocessing states, windows/tapers, detrending rules, spectral estimators, Welch segment lengths and overlaps, averaging rules, FFT lengths, one-/two-sided conventions, density-versus-spectrum scaling, frequency ranges, band boundaries, peak rules, spectral weights, entropy normalization, numerical floors, slope axis transforms, and harmonic tolerances. Material dependence on defensible alternatives should remain visible rather than being hidden by selecting configurations that produce the desired behavioral association.

Spectral Content, Frequency Bands, and Peaks

Spectral Location, Spread, Shape, and Concentration

Harmonic, Cepstral, and Modality-Specific Spectral Structure

Spectral Uncertainty, Sensitivity, and Verification

Spectral Provenance and Scientific Interpretation

Verification and provenance are a coherent responsibility in Spectral Descriptor analysis. Verification includes confirming source signal identity and version, support, sampling times and frequency, valid frequency range, anti-aliasing or resampling history, preprocessing and detrending state, spectral-estimator identity and version, window or taper used, segment length and overlap, FFT length, one-/two-sided convention, frequency grid, magnitude/power/PSD scaling and units, averaging rule, normalization, band definitions, peak-search rules, centroid/spread weights, roll-off fraction, entropy normalization and log base, flatness floor, slope axes and range, harmonic model, cepstral configuration, invalid states, implementation version, numerical tolerances, uncertainty, and sensitivity results.

Verification benefits from synthetic signals with known frequencies, known mixtures, broadband examples, and cross-implementation checks to establish arithmetic conformance to spectral definitions. Behavioral interpretation depends critically on signal semantics, support, modality, and scientific claim, and cannot be substituted by computational verification alone.