Linear prediction and its implementation in Matlab

Linear prediction analysis is one of the core technologies in modern speech signal processing. It has made great contributions to the rapid development of modern speech signal processing. It has been widely used in speech analysis, synthesis, coding, recognition, etc., and is still one of the most effective speech analysis technologies. For example, the G.729 CS-ACELP speech coding algorithm used in VoIP and H.323 online multimedia communication systems is a high-quality speech coding standard based on conjugate algebraic code excited linear prediction.

1 Basic principles of linear prediction

The most commonly used model in speech signal processing is the full-pole model. The basic concept of linear prediction is that a unique set of predictor coefficients can be determined by minimizing the sum of squares of the differences between the actual speech samples and the linear prediction samples, that is, approximating the minimum mean square error. If a random process is simulated by the output generated by a p-order full-pole system excited by white noise, let the transfer function of this system be:

Where: p is the predictor order; G is the channel filter gain. Therefore, the relationship between the speech sample s(n) and the excitation signal e(n) can be expressed by the following difference equation:

That is, there is a correlation between speech samples, and the past sample values ​​can be used to predict future sample values. For voiced sounds, the excitation e(n) is a unit impulse repeated at the pitch period; for unvoiced sounds, e(n) is a stationary white noise.

In the process of model parameter estimation, the following system is called a linear predictor:

Where: ai is called the linear prediction coefficient. Therefore, the system function of the p-order linear predictor has the following form:

A(z) in equation (1) is called the inverse filter, and the base transfer function is:

The linear prediction equation can be obtained as follows: The short-term average prediction error within a certain frame is defined as:

From this, the prediction coefficients can be calculated.

Due to the short-term stability of speech signals, they need to be processed in frames (10 to 30 ms). For a speech segment s(n) with N sample points selected from the window at time n, it is recorded as Φn(j, i):

2 Basic Principles of Linear Prediction Analysis in G.729

Using 10th order linear prediction (LP) for short-term analysis, the LP synthesis filter is defined as:

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3 Implementing Linear Prediction on Matlab

3.1 Windowing and autocorrelation function calculation

G.729 speech coding algorithms are mostly implemented in C language, which makes the program very lengthy and has certain limitations. Since Matlab has powerful functions in data analysis, automatic control, digital signal processing, and drawing, for intuitive expression, Matlab is used to implement the linear prediction analysis algorithm.

Figure 1 shows a schematic diagram of a hybrid window, which consists of two parts. The first half is a 1/2 Hamming window; the second half is a 1/4 cosine function:

Where: 5 ms (40 samples) come from the next frame, 15 ms (120 samples) come from the previous frame, and the windowing process of the current frame 10 ms (80 samples) is shown in Figure 1.

The windowed image is shown in Figure 2.

3.2 Determination of LP coefficients

To solve the linear prediction filter coefficients ai, the most classic Levinson-Durbin algorithm will be used. The steps of this algorithm are:

The superscripts in brackets in the above formulas represent the divisors of the predictor. Steps (1) to (4) can be recursively solved for I=1, 2, ..., P, where: E is the minimum mean square prediction error; R is the autocorrelation coefficient; k is the reflection coefficient, which ranges from [-1, 1]; aij is the jth coefficient of the i-th order predictor. Through calculation, it is found that in fact, in the calculation process, although the goal is to calculate all the coefficients of a p-order linear predictor, in fact, in the recursive process, only all the coefficients with divisors lower than the p-order linear predictor are calculated, and the minimum prediction error energy is calculated at the same time (the range of the reflection coefficient in the recursive formula is a necessary and sufficient condition to ensure the stability of the system H(z), that is, all the roots of the polynomial A(z) fall within the unit circle).

From the derivation, we know that the physical meaning of φn(j, i) can be understood as the short-time autocorrelation function of sn. Therefore, it reflects the actual situation of the speech waveform, that is, different waveforms have different values. However, the value of ai is determined by φn(j, i) and changes with the change of φn(j, i), so it can also be said that ai reflects the actual situation of the speech waveform.

Its Matlab description is as follows:

4 Conclusion

Linear prediction is widely used in speech processing, and using Matlab to implement it can intuitively know the analysis results, laying the foundation for the next step of implementing the algorithm on DSP.