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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.

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.

Quefrency is the independent coordinate of a cepstrum. When a cepstrum is obtained by an inverse Fourier transform of a logarithmic magnitude spectrum expressed as a function of frequency in hertz, its quefrency coordinate has units of time. A repeated pattern across frequency can consequently appear at the corresponding reciprocal spacing on the quefrency axis.^[ref]

For an $N$-point discrete Fourier transform with sampling rate $f_s$, bin spacing is $\Delta f=f_s/N$. The inverse-transform coordinate at index $n$ has quefrencyWith $f_s=8000\mathrm{Hz}$, index 40 represents $5\mathrm{ms}$. An ideal echo delayed by 40 samples generates spectral ripples spaced by $1/(0.005\mathrm{s})=200\mathrm{Hz}$ and a cepstrum contribution at that $5\mathrm{ms}$ delay, with further contributions at its multiples. The cepstrum's real-valued convention can also produce mirrored peaks at negative quefrencies.

A $5\mathrm{ms}$ quefrency peak is therefore not a $5\mathrm{ms}$-period oscillation of the original waveform by definition. Depending on the signal, it might reflect an echo or a fundamental frequency near $200\mathrm{Hz}$. Interpretation requires a model of how the spectral repetition arose.