9.3.5 The Cepstrum: Inverse Discrete Fourier Transform

While it would be possible to use the mel spectrum by itself as a feature representation for phone detection, the spectrum also has some problems, as we will see. For this reason, the next step in MFCC feature extraction is the computation of the cepstrum. The

cepstrum has a number of useful processing advantages and also significantly improves phone recognition performance.

One way to think about the cepstrum is as a useful way of separating the source and filter. Recall from Sec. ?? that the speech waveform is created when a glottal source waveform of a particular fundamental frequency is passed through the vocal tract, which because of its shape has a particular filtering characteristic. But many characteristics of the glottal source (its fundamental frequency, the details of the glottal pulse, etc) are not important for distinguishing different phones. Instead, the most useful information for phone detection is the filter, i.e. the exact position of the vocal tract. If we knew the shape of the vocal tract, we would know which phone was being produced. This suggests that useful features for phone detection would find a way to deconvolve (separate) the source and filter and show us only the vocal tract filter. It turns out that the cepstrum is one way to do this.

For simplicity, let's ignore the pre-emphasis and mel-warping that are part of the definition of MFCCs, and look just at the basic definition of the cepstrum. The cepstrum can be thought of as the spectrum of the log of the spectrum. This may sound confusing. But let's begin with the easy part: the log of the spectrum. That is, the cepstrum begins with a standard magnitude spectrum, such as the one for a vowel shown

in Fig. 9.14(a) from Taylor (2008). We then take the log, i.e. replace each amplitude value in the magnitude spectrum with its log. as shown in Fig. 9.14(b).

The next step is to visualize the log spectrum as if itself were a waveform. In other words, consider the log spectrum in Fig. 9.14(b). Let's imagine removing the axis labels that tell us that this is a spectrum (frequency on the x-axis) and imagine that we are dealing with just a normal speech signal with time on the x-axis. Now what can we say about the spectrum of this 'pseudo-signal'? Notice that there is a high-frequency repetitive component in this wave: small waves that repeat about 8 times in each 1000 along the x-axis, for a frequency of about 120 Hz. This high-frequency component is caused by the fundamental frequency of the signal, and represents the little peaks in the spectrum at each harmonic of the signal. In addition, there are some lower frequency components in this 'pseudo-signal'; for example the envelope or formant structure has about four large peaks in the window, for a much lower frequency.

Fig. 9.14(c) shows the cepstrum: the spectrum that we have been describing of the log spectrum. This cepstrum (the word cepstrum is formed by reversing the first letters of spectrum) is shown with samples along the x-axis. This is because by taking the spectrum of the log spectrum, we have left the frequency domain of the spectrum, and gone back to the time domain. It turns out that the correct unit of a cepstrum is the sample.

Examining this cepstrum, we see that there is indeed a large peak around 120, corresponding to the F0 and representing the glottal pulse. There are other various components at lower values on the x-axis. These represent the vocal tract filter (the position of the tongue and the other articulators). Thus if we are interested in detecting phones, we can make use of just the lower cepstral values. If we are interested in detecting pitch, we can use the higher cepstral values.

For the purposes of MFCC extraction, we generally just take the first 12 cepstral values. These 12 coefficients will represent information solely about the vocal tract filter, cleanly separated from information about the glottal source.

It turns out that cepstral coefficients have the extremely useful property that the variance of the different coefficients tends to be uncorrelated. This is not true for the spectrum, where spectral coefficients at different frequency bands are correlated. The fact that cepstral features are uncorrelated means, as we will see in the next section, that the Gaussian acoustic model (the Gaussian Mixture Model, or GMM) doesn't have to represent the covariance between all the MFCC features, which hugely reduces the number of parameters.

For those who have had signal processing, the cepstrum is more formally defined as the inverse DFT of the log magnitude of the DFT of a signal, hence for a windowed frame of speech x[n]:

$$ c[n]=\sum_{n=0}^{N-1}log\left(\left|\sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi}{N}kn}\right|\right)e^{j\frac{2\pi}{N}kn} $$