Linear Predictive Coding (LPC) Speech Coding and Synthesis

Course: Digital Audio Analysis and Processing

Programme: MSc in Music and Acoustic Engineering, Politecnico di Milano

Work Team: Di Lorenzo Giuliano, Longhi Filippo

Overview

This project implements a speech coding and synthesis system based on Linear Predictive Coding (LPC), inspired by the speech synthesis approach used in the Texas Instruments Speak & Spell educational toy.

The goal of the project is to analyze speech signals, extract a compact LPC representation, and reconstruct the original audio using the estimated vocal tract model and excitation signal. LPC allows efficient speech representation by modeling the speech production mechanism through a prediction filter and an excitation source.

The implementation was developed in MATLAB2024b and includes both encoding and decoding stages.

Methodology

The LPC system is divided into two main stages:

1. Encoding
2. Decoding

The speech signal is processed frame-by-frame, assuming local stationarity of speech. Each frame is analyzed to estimate the LPC parameters that describe the spectral envelope of the signal.

LPC Analysis

The speech frame is modeled as:

$$ s(n) \approx \sum_{k=1}^{p} a_k s(n-k) $$

where $s(n)$ is the speech signal, $a_k$ are the LPC coefficients, and $p$ is the prediction order.

The LPC coefficients are obtained from the autocorrelation of each frame using the Levinson-Durbin recursion.

Different prediction orders are used depending on the speech characteristics: $p = 10$ for voiced frames, $p = 4$ for unvoiced frames.

Voiced and unvoiced classification allows the system to select an appropriate excitation model during synthesis.

Encoding Process

The encoding stage performs the following operations:

1. Frame extraction: the signal is divided into short frames using a Hamming window, which allows independent analysis of each speech segment.

2. LPC coefficient computation: for every frame, the autocorrelation function is computed, LPC coefficients are estimated using the Levinson-Durbin algorithm, and the prediction error is obtained through the whitening filter. The prediction error represents the excitation information required for reconstruction.

3. Pitch estimation: for voiced frames, the pitch period is estimated from the prediction error using autocorrelation analysis. The pitch information is then used to generate the excitation signal during decoding.

Decoding Process

The reconstruction stage recreates the speech signal from the stored LPC parameters.

Two excitation models are used:

1. Voiced frames: the excitation signal is generated as an impulse train

   • Impulse spacing corresponds to the estimated pitch period.
   • The amplitude is controlled by the estimated gain.

2. Unvoiced frames: the excitation signal is modeled as white noise

   • Zero-mean Gaussian noise
   • Variance controlled by the LPC gain estimation

Speech Reconstruction

The excitation signal is passed through the LPC synthesis filter:

$$ H(z)=\frac{1}{1-\sum_{k=1}^{p}a_kz^{-k}} $$

The reconstructed frames are then combined to obtain the final synthesized speech signal.

Analysis Results

The project includes analysis plots comparing:

• Speech spectrum and LPC shaping filter response
• Prediction error in the time domain
• Prediction error spectrum

These plots allow evaluation of how accurately the LPC model represents voiced and unvoiced speech components.

Synthesis Results

The LPC model successfully reproduces the general characteristics of the original speech signal.

Observations:

• Voiced sounds maintain their harmonic structure thanks to the pitch-based excitation model.
• Unvoiced sounds are reconstructed using noise excitation and maintain their spectral behavior.
• The reconstructed signal preserves the temporal evolution of the original speech.

Some differences remain, particularly in spectral detail and consonant clarity, due to the simplified excitation model and LPC approximation.