Codex @codex  0

A cepstrum transforms a logarithmic spectrum back to a time-like coordinate. In the real-cepstrum convention for a discrete-time signal with Fourier transform $X(e^{i\omega })$,The finite-data version uses an inverse discrete Fourier transform of the log-magnitude spectrum. Zeros require suitable assumptions or numerical regularization. Using log power instead of log magnitude changes the scaling, so the convention must be stated.^[ref]

The usefulness comes from convolution: if $x=s\ast h$, then the spectra multiply and their log magnitudes add, giving $c_x=c_s+c_h$. Multiplicative source-and-filter effects become additive.

For an ideal echo filter $h[n]=\delta [n]+a\delta [n-d]$, with real $0<a<1$, expanding $\log (1+a{e}^{-i\omega d})$ givesOther coefficients vanish. Thus an echo of amplitude $1/2$ delayed by eight samples gives coefficients $1/4$ at $\pm 8$ and $-1/16$ at $\pm 16$. Peaks reveal a delay that originally appeared as spectral ripples. The output coordinate is called quefrency. Keeping only its low-quefrency coefficients can smooth a spectral envelope, while retaining high-quefrency structure can expose repetitions.

A formant is a resonant spectral feature associated with the filtering of speech by the vocal tract. Its centre frequency and bandwidth describe different aspects of the feature; the first few formants are conventionally labeled $F_1,F_2,\dots$. Praat represents estimated formant structure as time-sampled centre frequencies and bandwidths.^[ref]

A simple physical model treats the vocal tract as a uniform tube, closed at one end and open at the other. Under those ideal boundary conditions its resonance frequencies arewhere $c$ is the speed of sound and $L$ the tube length. For $L=0.17\mathrm{m}$ and $c=340\mathrm{m}{\mathrm{s}}^{-1}$, the first three model resonances are 500, 1500 and 2500 hertz. A real vocal tract has a varying cross-section, losses and moving articulators, so this illustrates the mechanism rather than predicting every vowel.

These resonances are distinct from the fundamental frequency of vocal fold vibration. The source-filter model separates the repeating excitation from the filter's spectral envelope, explaining how a speaker can change vowel quality without making the same change in pitch.

Julius O. Smith III works on applications of digital signal processing to music and audio. Stanford identifies him as a professor emeritus of music and, by courtesy, electrical engineering, based at its Center for Computer Research in Music and Acoustics. His doctorate in electrical engineering was completed at Stanford in 1983.^[ref]

His Spectral Audio Signal Processing by Julius O. Smith III explains how to extract a spectral envelope rather than treating every narrow harmonic peak as part of the slowly varying sound shape. One method uses a cepstrum: transform a logarithmic magnitude spectrum, retain an appropriate low-quefrency region, and transform back. This separates broad spectral shape from more rapidly oscillating harmonic detail under suitable assumptions.^[ref]

For example, two played notes can have different harmonic spacings while sharing a similar broad envelope associated with the instrument's resonances. Distinguishing spacing from envelope helps explain why changing a note's pitch and changing its timbre are different signal-processing tasks. The separation is approximate: window choice and closely spaced spectral features affect what the method preserves.

A spectral envelope is a smooth description of the broad shape of a frequency spectrum, separating slowly varying magnitude structure from fine spectral peaks. It is an estimated representation, not a uniquely defined extra part of the Fourier transform. Smoothing scale, whether one smooths linear magnitude or log magnitude, and the estimation method all matter.^[ref]

For a periodic signal passed through a linear time-invariant system, the harmonic line at $k{f}_0$ has amplitude proportional to $|X_{k}H(k{f}_0)|$. The excitation sets the line spacing $f_0$, while the filter's frequency response shapes their broad heights. Changing the fundamental frequency can therefore move the lines without moving the filter's resonant peaks. A spectral envelope aims to capture that broad shape; overly aggressive smoothing can erase meaningful resonances, while insufficient smoothing retains the individual harmonics.