9.3.6 Deltas and Energy

The extraction of the cepstrum via the Inverse DFT from the previous section results in 12 cepstral coefficients for each frame. We next add a thirteenth feature: the energy

from the frame. Energy correlates with phone identity and so is a useful cue for phone detection (vowels and sibilants have more energy than stops, etc). The energy in a frame is the sum over time of the power of the samples in the frame; thus for a signal x in a window from time sample $t_{1}$ to time sample $t_{2}$, the energy is:

$$ Energy=\sum_{t=t_{1}}^{t_{2}}x^{2}[t] $$

Another important fact about the speech signal is that it is not constant from frame to frame. This change, such as the slope of a formant at its transitions, or the nature of the change from a stop closure to stop burst, can provide a useful cue for phone identity. For this reason we also add features related to the change in cepstral features over time.

We do this by adding for each of the 13 features (12 cepstral features plus energy) a $ \Delta $ or velocity feature, and a $ \Delta $ double $ \Delta $ or acceleration feature. Each of the 13 delta features represents the change between frames in the corresponding cepstral/energy feature, while each of the 13 double delta features represents the change between frames in the corresponding delta features.

A simple way to compute deltas would be just to compute the difference between frames; thus the delta value $ d(t) $ for a particular cepstral value $ c(t) $ at time t can be estimated as:

$$ d(t)=\frac{c(t+1)-c(t-1)}{2} $$

Instead of this simple estimate, however, it is more common to make more sophisticated estimates of the slope, using a wider context of frames.