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.

Calibrated 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 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 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 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 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 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 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 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 supply the error-analysis vocabulary the estimates are stated in.

Pages in this section

• Calibrated 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: 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: 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: 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: 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: 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: IEC 60268-1 tone bursts with exact gating, resampling with a stated anti-alias specification, and band-limited fractional delay.
• 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.