Source: https://jmrplens.github.io/phonometry/guides/sections/signals-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. [Calibrated spectral analysis](/phonometry/guides/spectral-analysis/) is the frequency-domain half. 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](/phonometry/guides/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](/phonometry/guides/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](/phonometry/guides/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](/phonometry/guides/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](/phonometry/guides/correlation-delay/) is the time-domain half. 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](/phonometry/guides/test-signals/) is the toolbox the other two lean on: tone bursts with the exact gating of IEC 60268-1 (zero-crossing start, integral full periods, repetitive trains), polyphase resampling behind an explicit anti-alias specification whose designed filter travels with the result, and band-limited fractional delay with a linear or circular boundary, sharing its kernel with the sub-sample alignment of impulse responses. [System measurement](/phonometry/guides/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](/phonometry/guides/sections/calibration-uncertainty/) supply the error-analysis vocabulary the estimates are stated in. ## Pages in this section - [Calibrated spectral analysis](/phonometry/guides/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](/phonometry/guides/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](/phonometry/guides/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](/phonometry/guides/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](/phonometry/guides/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](/phonometry/guides/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](/phonometry/guides/test-signals/): IEC 60268-1 tone bursts with exact gating, resampling with a stated anti-alias specification, and band-limited fractional delay. - [System measurement](/phonometry/guides/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.