Cepstrum ($c_x={\mathcal{F}}^{-1}(\log |\mathcal{F}x|)$)

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.

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.