Let $\omega= e^{-2\pi i/N}$ and let $$\mathbf{W}= \left[\omega^{jk}\right]_{j,k=0,\cdots,N-1}$$

Then we can take a vector $\mathbf{x}\in \mathbb{R}^N$ and transform it via

$$\mathbf{X} = \mathbf{W}\mathbf{x}$$

Source: https://en.wikipedia.org/wiki/DFT_matrix

I then have

$$\mathbf{X}_{dB} = \left [ 20\log_{10}|X_{k}| \right ]_{k=1}^{N}$$

The cepstrum can be found via

$$\mathbf{c} = \frac{1}{N} \mathbf{W}^{*}\mathbf{X}_{dB}$$

I could have made mistakes, so feel free to correct me.

I understand what a signal is and I understand what the spectrum is. Can someone explain intuitively what $\mathbf{X}_{dB}$ and $\mathbf{c}$ are? In particular, how do we interpet the meaning of the components of $\mathbf{c}$? e.g. what does $c[0]$ mean or $c[1]$?


1 Answer 1


The cepstrum is commonly referred to as a "spectrum of a spectrum." By performing $\mathbf{X}=\mathbf{W}\mathbf{x}$, you are calculating the complex result of the discrete Fourier transform of your input $\mathbf{x}$. Taking the squared absolute value of $\mathbf{X}$ renders the energy spectral density, which simply expresses the energy of $\mathbf{x}$ at each frequency of the spectrum and disregards any information about phase.

$$ E_{\mathbf{x}} =|\mathbf{X}|^2$$

Recalling the logarithmic property,

$$c \log x=\log x^c,$$

it is clear that $\mathbf{X}_{dB}$ is the decibel representation of the energy spectral density:

$$\mathbf{X}_{dB}=20 \log_{10}|\mathbf{X}|=10 \log_{10}|\mathbf{X}|^2=10 \log_{10}E_{\mathbf{x}}.$$

Note that $\mathbf{X}_{dB}$ is a purely real quantity. Finally, the cepstrum, $\mathbf{c}$, is found by taking the discrete Fourier transform of $\mathbf{X}_{dB}$, again exposing a complex spectrum that now describes the periodic components of the energy spectral density. A "peak" in the cepstrum at anywhere other than DC (i.e. $k\ne0$) means that the energy spectral density has periodic components (e.g. harmonics). The frequency spacing of these harmonics in $E_\mathbf{x}$ would correspond to the location of the peak in the cepstrum, also known as quefrency. The value at DC, $c[0]$, would describe the broadband energy of $\mathbf{x}$, or how much the entire spectrum is positively offset for all frequencies.

The reason for the $\log_{10}$ scaling in this is to reduce undesired artifacts from showing up in the cepstrum, which can be observed if you take the Fourier transform of a periodic pulse train.


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