Speex is based on CELP, which stands for Code Excited Linear Prediction. This section attempts to introduce the principles behind CELP, so if you are already familiar with CELP, you can safely skip to section 7. The CELP technique is based on three ideas:
- The use of a linear prediction (LP) model to model the vocal tract
- The use of (adaptive and fixed) codebook entries as input (excitation) of the LP model
- The search performed in closed-loop in a ``perceptually weighted domain''
This section describes the basic ideas behind CELP. Note that it's still incomplete.
Linear Prediction (LPC)
Linear prediction is at the base of many speech coding techniques, including CELP. The idea behind it is to predict the signal 
using a linear combination of its past samples:
where 
is the linear prediction of . The prediction error is thus given by:
The goal of the LPC analysis is to find the best prediction coefficients 
which minimize the quadratic error function:
That can be done by making all derivatives 
equal to zero:
The filter coefficients are computed using the Levinson-Durbin algorithm, which starts from the auto-correlation 
of the signal .
For an order 
filter, we have:
The filter coefficients are found by solving the system 
. What the Levinson-Durbin algorithm does here is making the solution to the problem 
instead of 
by exploiting the fact that matrix 
is toeplitz hermitian. Also, it can be proven that all the roots of 
are within the unit circle, which means that 
is always stable. This is in theory; in practice because of finite precision, there are two commonly used techniques to make sure we have a stable filter. First, we multiply 
by a number slightly above one (such as 1.0001), which is equivalent to adding noise to the signal. Also, we can apply a window to the auto-correlation, which is equivalent to filtering in the frequency domain, reducing sharp resonances.
The linear prediction model represents each speech sample as a linear combination of past samples, plus an error signal called the excitation (or residual).
In the z-domain, this can be expressed as
where is defined as
We usually refer to as the analysis filter and as the synthesis filter. The whole process is called short-term prediction as it predicts the signal using a prediction using only the past samples, where is usually around 10.
Because LPC coefficients have very little robustness to quantization, they are converted to Line Spectral Pair (LSP) coefficients which have a much better behaviour with quantization, one of them being that it's easy to keep the filter stable.
Pitch Prediction
During voiced segments, the speech signal is periodic, so it is possible to take advantage of that property by approximating the excitation signal 
by a gain times the past of the excitation:
where 
is the pitch period, 
is the pitch gain. We call that long-term prediction since the excitation is predicted from 
with 
.
Innovation Codebook
The final excitation will be the sum of the pitch prediction and an innovation signal 
taken from a fixed codebook, hence the name Code Excited Linear Prediction. The final excitation is given by:
The quantization of is where most of the bits in a CELP codec are allocated. It represents the information that couldn't be obtained either from linear prediction or pitch prediction. In the z-domain we can represent the final signal 
as
Analysis-by-Synthesis and Error Weighting
Most (if not all) modern audio codecs attempt to ``shape'' the noise so that it appears mostly in the frequency regions where the ear cannot detect it. For example, the ear is more tolerant to noise in parts of the spectrum that are louder and vice versa. That's why instead of minimizing the simple quadratic error
where 
is the encoder signal, we minimize the error for the perceptually weighted signal
where 
is the weighting filter, usually of the form
|
(1) |
with control parameters 
. If the noise is white in the perceptually weighted domain, then in the signal domain its spectral shape will be of the form
If a filter has (complex) poles at 
in the 
-plane, the filter 
will have its poles at 
, making it a flatter version of .
Analysis-by-synthesis refers to the fact that when trying to find the best pitch parameters (, ) and innovation signal , we do not work by making the excitation as close as the original one (which would be simpler), but apply the synthesis (and weighting) filter and try making 
as close to the original as possible.
参考资料：
1 百科总结： https://zh.wikipedia.org/wiki/%E7%A0%81%E6%BF%80%E5%8A%B1%E7%BA%BF%E6%80%A7%E9%A2%84%E6%B5%8B
2 详细介绍： http://ntools.net/arc/Documents/speex/manual/node8.html