Linear Predictive Coding (or “LPC”) is a method of predicting a sample of a speech signal based on several previous samples. Similar to the method employed by the cepstrum, we can use the LPC coefficients to separate a speech signal into two parts: the transfer function (which contains the vocal quality) and the excitation (which contains the pitch and the sound). The method of looking at speech as two parts which can be separated is known as the Source Filter Model of Speech.

We can predict that the nth sample in a sequence of speech samples is represented by the weighted sum of the p previous samples:

The number of samples (p) is referred to as the “order” of the LPC. As p approaches infinity, we should be able to predict the nth sample exactly. However, p is usually on the order of ten to twenty, where it can provide an accurate enough representation with a limited cost of computation. The weights on the previous samples (ak) are chosen in order to minimize the squared error between the real sample and its predicted value. Thus, we want the error signal e(n), which is sometimes referred to as the LPC residual, to be as small as possible:

We can take the z-transform of the above equation:

Thus, we can represent the error signal E(z) as the product of our original speech signal S(z) and the transfer function A(z). A(z) represents an all-zero digital filter, where the ak coefficients correspond to the zeros in the filter’s z-plane. Similarly, we can represent our original speech signal S(z) as the product of the error signal E(z) and the transfer function 1 / A(z):

The transfer function 1/A(z) represents an all-pole digital filter, where the ak coefficients correspond to the poles in the filter’s z-plane. Note that the roots of the A(z) polynomial must all lie within the unit circle to ensure stability of this filter.

The spectrum of the error signal E(z) will have a different structure depending on whether the sound it comes from is voiced or unvoiced. Voiced sounds are produced by vibrations of the vocal cords. Their spectrum is periodic with some fundamental frequency (which corresponds to the pitch). Examples of voiced sounds include all of the vowels. Unvoiced signals, however, do not have a fundamental frequency or a harmonic structure. Instead, they are just white noise.