Source: https://jmrplens.github.io/phonometry/signals/spectra/ # Signals and spectra Band levels answer *how much*; this section answers *what is in the signal*. Where the rest of the core works in fractional octave bands, these pages work with the fine-grained estimators of classical signal analysis: spectral densities, correlation functions, delays and envelopes. They share one discipline, taken from Bendat & Piersol: every estimate is **calibrated** (the same dB SPL / dBFS reference frames as the rest of the library) and carries its **statistical quality**, so a spectrum is not just a curve but a curve with a confidence interval. Eight pages sit under that discipline, in six families: the frequency-domain estimators ([spectral analysis](https://jmrplens.github.io/phonometry/signals/spectral-analysis/), [multiple and partial coherence](https://jmrplens.github.io/phonometry/signals/miso-coherence/)), the time-frequency middle ground ([spectrograms](https://jmrplens.github.io/phonometry/signals/time-frequency/)), the spectrum-shape methods ([cepstrum](https://jmrplens.github.io/phonometry/signals/cepstrum-echoes/)), the period-domain methods ([synchronous averaging](https://jmrplens.github.io/phonometry/signals/synchronous-averaging/)), the time-domain estimators ([correlation and delay](https://jmrplens.github.io/phonometry/signals/correlation-delay/)), and the stimulus and system toolbox ([test signals](https://jmrplens.github.io/phonometry/signals/test-signals/), [system measurement](https://jmrplens.github.io/phonometry/signals/system-measurement/)). They differ in what they estimate and share two decisions. The first is the **segment length**. It fixes the resolution bandwidth of a PSD, the cell shape of a spectrogram, the number of averages behind every confidence interval, the longest delay a generalized cross-correlation can see and the degrees of freedom a conditioned MISO estimate has left, so choosing it once and keeping it is what makes a PSD, a coherence and a delay computed from the same record mutually consistent. The second is **stationarity**. Every average on these pages, and every error formula quoted beside it, assumes the process did not drift while it was being recorded, which is exactly what the [data qualification](https://jmrplens.github.io/phonometry/metrology/data-qualification/) tests decide; when a record fails them, the honest tools are the short-time views rather than the averaged ones. [Calibrated spectral analysis](https://jmrplens.github.io/phonometry/signals/spectral-analysis/) is where the frequency-domain family starts. The Welch power and cross-spectral density estimators report their effective number of averages, normalized random errors and chi-square confidence intervals; the coherent output spectrum splits a measured output into the part explained by an input and the part that is noise, with a spectral signal-to-noise ratio; fractional-octave smoothing bridges back to the banded world; and the colored-noise generators synthesize white, pink, red, blue and violet test signals with an exact power-law slope. [Multiple and partial coherence](https://jmrplens.github.io/phonometry/signals/miso-coherence/) carries that same cross-spectral machinery to several correlated sources at once: from multiple inputs and one output it separates the coherence a source genuinely contributes from the part it merely shares with another, and its partial coherent output spectra say which source dominates each band. [Time-frequency analysis](https://jmrplens.github.io/phonometry/signals/time-frequency/) is the view in between: the calibrated STFT spectrogram shows what happens *when* - a passing siren, an impact, a run-up - with every cell reading an absolute level in the same scaling as the Welch estimators, and the zoom FFT computes the spectrum of a narrow band on an arbitrarily fine grid to separate tones closer than a practical FFT bin. [Cepstrum, echoes and the envelope spectrum](https://jmrplens.github.io/phonometry/signals/cepstrum-echoes/) works on the *shape* of the spectrum. The power, real and complex cepstrum collapse periodic spectral ripple onto quefrency spikes, echo detection reads a reflection's delay and coefficient off the cepstral peak, liftering splits a log spectrum into smooth envelope and fine structure, and the envelope spectrum turns amplitude modulations into discrete lines at the modulation frequency. [Time synchronous averaging](https://jmrplens.github.io/phonometry/signals/synchronous-averaging/) extracts a repetitive waveform of known period from asynchronous noise by ensemble-averaging successive periods: the residual noise falls as the square root of the number of averages, and choosing that number to place a comb node on an interfering order rejects it far better than the habitual power of two. [Correlation, time delay and envelope](https://jmrplens.github.io/phonometry/signals/correlation-delay/) is where its time-domain counterpart starts. Auto- and cross-correlation come with the Bendat & Piersol normalizations and random errors; time-delay estimation offers the direct correlator, the cross-spectrum phase slope and the Knapp & Carter generalized cross-correlation weightings (Roth, SCOT, PHAT, maximum likelihood); impulse responses can be delayed and aligned with sub-sample precision; and the Hilbert transform yields the envelope with instantaneous phase and frequency. [Test signals and sample-rate tools](https://jmrplens.github.io/phonometry/signals/test-signals/) is the toolbox underneath the rest: tone bursts with the exact gating of IEC 60268-1 (zero-crossing start, integral full periods, repetitive trains) that exercise detector ballistics, polyphase resampling behind an explicit anti-alias specification whose designed filter travels with the result and which every cross-rate comparison needs, and band-limited fractional delay with a linear or circular boundary, whose kernel is shared by the impulse-response alignment of [Correlation, time delay and envelope](https://jmrplens.github.io/phonometry/signals/correlation-delay/) and the non-integer period alignment of [Time synchronous averaging](https://jmrplens.github.io/phonometry/signals/synchronous-averaging/). [System measurement](https://jmrplens.github.io/phonometry/signals/system-measurement/) turns the toolbox toward measuring systems themselves: complementary Golay pairs deconvolve a time-invariant system with zero correlation noise, the Mueller & Massarani shaped sweeps put the excitation energy where a target spectrum asks for it while keeping a swept sine's crest factor, and the regularized spectral inversion converts a measured response into a safe equalizer with an analytic bound on flatness and out-of-band gain. These estimators feed the rest of the library: transfer functions and distortion analysis build on the cross-spectral machinery, room impulse response work leans on delay estimation and alignment, and the [uncertainty pages](https://jmrplens.github.io/phonometry/metrology/) supply the error-analysis vocabulary the estimates are stated in. ## Pages in this section - [Calibrated spectral analysis](https://jmrplens.github.io/phonometry/signals/spectral-analysis/): Welch PSD/CSD with chi-square confidence intervals, the coherent output spectrum and spectral SNR, 1/n-octave smoothing and exact-slope colored-noise generators. - [Multiple and partial coherence](https://jmrplens.github.io/phonometry/signals/miso-coherence/): the Bendat & Piersol multiple-input/output coherence functions for multiple correlated sources and one output, with the Gaussian-elimination conditioning that tells a genuine cause from a source that merely correlates with it, and the partial coherent output spectra that say which source dominates each band. - [Time-frequency analysis](https://jmrplens.github.io/phonometry/signals/time-frequency/): the calibrated STFT spectrogram in absolute units (dB SPL for pascals) with the time-versus-frequency resolution trade-off, and the zoom FFT that resolves tones closer than a practical FFT bin. - [Cepstrum, echoes and the envelope spectrum](https://jmrplens.github.io/phonometry/signals/cepstrum-echoes/): the power/real/complex cepstrum with quefrency analysis, echo detection with the reflection coefficient read off the peak, lowpass/highpass liftering, the homomorphic round trip and the envelope spectrum of amplitude modulations. - [Time synchronous averaging](https://jmrplens.github.io/phonometry/signals/synchronous-averaging/): extraction of a periodic waveform of known period by time domain averaging, the comb filter that describes the operation in the frequency domain, the square-root noise-reduction law, and the choice of the number of averages that places a comb node on an interfering order (McFadden 1987). - [Correlation, time delay and envelope](https://jmrplens.github.io/phonometry/signals/correlation-delay/): correlation estimates with their random errors, time-delay estimation by direct correlation, phase slope and GCC weightings, sub-sample impulse-response alignment, and the Hilbert envelope. - [Test signals and sample-rate tools](https://jmrplens.github.io/phonometry/signals/test-signals/): IEC 60268-1 tone bursts with exact gating, resampling with a stated anti-alias specification, and band-limited fractional delay. - [System measurement](https://jmrplens.github.io/phonometry/signals/system-measurement/): complementary Golay pairs with exactly noise-free deconvolution to an impulse response, sweeps that follow an arbitrary target magnitude spectrum by group-delay shaping, and the Kirkeby-regularized inversion of a measured response. ## See also Pages elsewhere on the site that this section leans on: - [Machine fault frequencies](https://jmrplens.github.io/phonometry/vibration/machinery/machine-diagnostics/): the kinematic fault-frequency families of rotating machinery (Norton & Karczub Section 8.4) drawn on top of a measured envelope spectrum: bearing BPFO, BPFI, BSF and cage frequencies, gear-mesh sidebands, induction-motor slip, pole-pass and rotor-slot harmonics, and blade-passing tones. ## What this section does not cover These are Bendat & Piersol's textbook estimators, not a certification method: no page here carries clause numbers or acceptance limits, and no result is a compliance verdict. Three capabilities a reader looks for are genuinely absent. **Multiple arrivals are not separated.** `time_delay` and `echo_detection` report the single largest peak, so a record with a direct path plus several reflections needs manual peak-picking or repeated calls on narrowed bands. **There is no multi-sensor geometry.** Delay estimation is pairwise; there is no built-in TDOA solver, no beamformer and no source localisation, and a multiple-output system needs one `miso_coherence` call per output. **There are no perceptual features.** The cepstrum is the plain linear-frequency one, with no mel warping and no MFCC variant. Two estimators that share this section's Welch core are documented where they are used instead: the transfer function and ordinary coherence on [Electroacoustics](https://jmrplens.github.io/phonometry/devices/electroacoustics/electroacoustics/), and the two-microphone intensity probe on [Sound intensity](https://jmrplens.github.io/phonometry/devices/emission/intensity/).