9.3.2 Windowing

Recall that the goal of feature extraction is to provide spectral features that can help us build phone or subphone classifiers. We therefore don’t want to extract our spectral features from an entire utterance or conversation, because the spectrum changes very quickly. Technically, we say that speech is a non-stationary signal, meaning that its statistical properties are not constant across time. Instead, we want to extract spectral features from a small window of speech that characterizes a particular subphone and for which we can make the (rough) assumption that the signal is stationary (i.e. its statistical properties are constant within this region).

We'll do this by using a window which is non-zero inside some region and zero elsewhere, running this window across the speech signal, and extracting the waveform inside this window.

We can characterize such a windowing process by three parameters: how wide is the window (in milliseconds), what is the offset between successive windows, and what is the shape of the window. We call the speech extracted from each window a frame, and we call the number of milliseconds in the frame the frame size and the number of milliseconds between the left edges of successive windows the frame shift.

The extraction of the signal takes place by multiplying the value of the signal at

time n, s[n], with the value of the window at time n, w[n]:

$$ y[n]=w[n]s[n] $$

Figure 9.10 suggests that these window shapes are rectangular, since the extracted windowed signal looks just like the original signal. Indeed the simplest window is the rectangular window. The rectangular window can cause problems, however, because it abruptly cuts of the signal at its boundaries. These discontinuities create problems when we do Fourier analysis. For this reason, a more common window used in MFCC extraction is the Hamming window, which shrinks the values of the signal toward zero at the window boundaries, avoiding discontinuities. Fig. 9.11 shows both of these windows; the equations are as follows (assuming a window that is L frames long):

rectangular

$$ w[n]~=~\left\{\begin{array}{l l}{1}&{\quad0\leq n\leq L-1}\\ {0}&{\quad\mathrm{o t h e r w i s e}}\\ \end{array}\right. $$

hamming

$$ w[n]=\left\{\begin{array}{l l}{0.54-0.46\cos(\frac{2\pi n}{L})}&{\quad0\leq n\leq L-1}\\ {0}&{\quad\mathrm{o t h e r w i s e}}\end{array}\right. $$