Self-Implemented Linear Prediction Coefficient Calculation

Linear prediction analysis is a fundamental technique in signal processing and speech analysis, based on the principle of predicting current signal samples from past observations. This method computes a set of linear prediction coefficients (LPC) that characterize the short-term correlations within signals. The implementation follows this computational workflow.

### 1. Autocorrelation Function Computation The program first calculates the autocorrelation function of the input signal, which measures signal similarity across different time lags. The implementation typically extracts the first N autocorrelation values where N represents the prediction order. In MATLAB code, this involves using xcorr() function or manual convolution with zero-padding for finite-length signals.

### 2. Levinson-Durbin Recursive Algorithm The core LPC computation employs the Levinson-Durbin algorithm - an efficient recursive method that progressively solves for optimal prediction coefficients while minimizing prediction error. Key computational steps include: Initializing prediction error energy and reflection coefficients Iteratively computing LPC coefficients through order-update recursions Updating error terms at each stage The algorithm's MATLAB implementation typically involves nested loops handling the recursion with O(N²) complexity, using intermediate arrays for partial correlation storage.

### 3. Prediction Error Analysis The derived LPC coefficients enable signal prediction and error evaluation. Small prediction errors indicate effective modeling of short-term signal characteristics. The implementation includes error calculation through convolution between LPC coefficients and signal history, with residual analysis validating model accuracy.

### 4. Application Extensions This methodology finds extensive applications in speech coding (LPC vocoders), audio compression, and signal modeling. Understanding LPC computation facilitates advanced research in speech synthesis, noise reduction, and pattern recognition. The MATLAB program encapsulates this complete pipeline, requiring only input signals and prediction order parameters to generate LPC coefficients for subsequent analysis and applications.