What Is Linear Predictive Coding?

Linear predictive coding (LPC) is a signal processing technique that models a time-domain signal by predicting each sample as a weighted linear combination of a fixed number of previous samples, then representing the signal through the set of predictor coefficients rather than the raw samples. Applied to speech, the predictor models the vocal tract as an all-pole filter excited by either a periodic impulse train (for voiced sounds) or random noise (for unvoiced sounds). Because the predictor captures most of the signal's power, the residual after subtracting the prediction is small and can be encoded or discarded, enabling substantial compression. LPC was developed in the late 1960s and early 1970s by Atal, Itakura, Saito, and Schroeder, and remains a central technique in speech coding, speech synthesis, and speaker recognition.

LPC draws on the theory of autoregressive models from statistics and on the acoustics of the human vocal tract. The connection between autoregressive prediction and resonance modeling means that LPC coefficients encode the spectral envelope of the signal, particularly the formant frequencies that characterize vowel identity. This interpretation makes LPC valuable not just for compression but for acoustic feature extraction in speech analysis.

Autoregressive Modeling

The mathematical core of LPC is the p-th order autoregressive (AR) model: the predicted sample at time n is the sum of the product of the ith predictor coefficient ai with the sample at time n minus i, summed from i equal 1 to p. The optimal coefficients in the minimum mean-squared-error sense satisfy the Yule-Walker equations, a symmetric Toeplitz linear system that the Levinson-Durbin recursion solves in O(p^2) operations. The order p is typically 10 to 16 for telephone-bandwidth speech, providing one coefficient per kilohertz of signal bandwidth plus a few additional orders for the glottal pulse and lip radiation. The Stanford CCRMA treatment of LPC as all-pole resonance modeling gives a detailed derivation of how the AR model connects to the physical acoustics of tube resonances.

Analysis-Synthesis Framework

In the LPC vocoder, the speech signal is analyzed in short frames of typically 20 to 30 milliseconds. For each frame, the predictor coefficients are computed, the residual signal is analyzed to determine pitch period and voicing decision, and a compressed representation is transmitted or stored. At the synthesis end, the decoder reconstructs an excitation signal from the voicing and pitch parameters, passes it through the all-pole filter defined by the predictor coefficients, and recovers a perceptually acceptable speech signal. The MATLAB DSP System Toolbox tutorial on LPC analysis and synthesis demonstrates this pipeline step by step. This framework underlies the earliest digital voice coders used in secure telephone equipment and remains the conceptual basis for more complex modern codecs.

Extensions and Modern Codecs

Code-excited linear prediction (CELP), introduced by Atal and Schroeder in 1985, extended the basic LPC vocoder by replacing the simple voiced/unvoiced excitation model with a codebook of stored excitation vectors, selected by closed-loop analysis to minimize the perceptually weighted error. CELP achieved dramatically better quality at rates of 4 to 16 kbit/s and spawned the family of codecs standardized in ITU-T G.728, G.729, and G.723.1, which are widely deployed in VoIP and wireless systems. The ACM paper on matrix quantization and LPC-based vocoder systems illustrates how LPC remains central even in low-resource speech recognition pipelines. Mixed excitation LPC and algebraic CELP variants continue to appear in standards for mobile telephony and voice over IP.

Applications

Linear predictive coding has applications across a range of fields, including:

• Speech compression in mobile telephony and VoIP codecs
• Speaker verification and speech recognition feature extraction
• Text-to-speech synthesis systems
• Secure voice encryption for military and government communications
• Pitch estimation and prosody analysis in speech science