Quefrency ($q$)

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.