# Interpreting the cepstrum of a signal

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

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

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 $$c[1]$$?

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.