Linear Predictive Coding (LPC)

LPC models a discrete-time signal, s(n), as a linear combination of its p previous samples plus an excitation (or error) term, e(n). This is expressed by the linear prediction equation:

s(n) ≈ a₁s(n-1) + a₂s(n-2) + ... + aₚs(n-p) + e(n)

• a₁...aₚ: The LPC coefficients or predictor coefficients that define the filter.
• p: The prediction order, typically 10-16 for speech. Higher orders model more spectral detail.
• e(n): The residual signal or prediction error. For voiced speech, this approximates the glottal pulse; for unvoiced, it resembles white noise. The goal is to compute coefficients that minimize the energy of e(n).

The LPC model corresponds to an all-pole or autoregressive (AR) filter. The derived coefficients define a filter whose transfer function is:

H(z) = G / (1 - Σ aₖz⁻ᵏ)

• This filter's frequency response models the spectral envelope of the original signal, capturing the formants (resonant frequencies) in speech.
• The gain, G, is set to match the energy of the original signal.
• The all-pole assumption is highly effective for modeling the resonant structure of the human vocal tract but cannot directly represent spectral nulls (zeros).

LPC coefficients are calculated by solving the Yule-Walker equations, which arise from minimizing the prediction error. This involves estimating the signal's autocorrelation. Key algorithms include:

• Levinson-Durbin Recursion: An efficient O(p²) algorithm that solves the Toeplitz system of normal equations derived from autocorrelation. It also yields reflection coefficients (k-parameters), which have desirable quantization properties.
• Covariance Method: Uses a windowed segment of the signal without assuming it is zero outside the window. Can be more accurate for short, stable signal segments. These methods ensure stable synthesis filters when the reflection coefficients have a magnitude less than 1.

LPC's efficiency and accuracy in modeling speech have made it a cornerstone of low-bitrate audio coding and analysis:

• Speech Coding & Compression: Foundational to Code-Excited Linear Prediction (CELP) used in standards like GSM (13 kbps) and VoIP (G.729). It separates the slowly varying filter (vocal tract) from the rapidly changing excitation.
• Speech Synthesis & Vocoding: Used in early text-to-speech systems. By transmitting or storing only the LPC coefficients and a simplified excitation signal, intelligible speech can be resynthesized.
• Speech Recognition: LPC-derived features (or related cepstral coefficients like LPC-Cepstrum) serve as compact, informative feature vectors for phonetic classification.
• Pitch & Formant Analysis: The residual signal is used for pitch detection, while the filter's poles indicate formant frequencies.

LPC coefficients are often transformed into other domains for increased robustness and utility:

• Line Spectral Frequencies (LSFs): A transformed representation of LPC coefficients with better quantization and interpolation properties. LSFs are ordered frequencies between 0 and the Nyquist rate, crucial for modern speech codecs.
• LPC-Cepstrum: The cepstral coefficients derived from the LPC coefficients. They provide a smoothed, orthogonal representation of the log spectrum and are closely related to Mel-Frequency Cepstral Coefficients (MFCCs).
• Reflection Coefficients (RCs) / PARCOR Coefficients: Direct byproducts of the Levinson-Durbin recursion. Their magnitude is bounded (|k| < 1), ensuring filter stability, making them suitable for quantization.

While powerful, LPC has specific constraints that engineers must account for:

• All-Pole Limitation: Cannot perfectly model signals with spectral zeros (e.g., nasals in speech, certain musical instruments), leading to approximation error.
• Window Length & Stationarity: Assumes the signal is stationary within the analysis window (typically 20-30 ms for speech). Too long a window violates this; too short provides insufficient data for reliable coefficient estimation.
• Noise Sensitivity: Performance degrades with background noise, as the model attempts to fit the noise spectrum as well.
• Synthesis Quality: The simplistic excitation model (impulse train + white noise) in basic vocoders produces the characteristic "buzzy" or robotic quality of early synthetic speech, addressed by more advanced models like CELP.

Mel-Frequency Cepstral Coefficients (MFCC) are the dominant feature representation for speech and speaker recognition, providing a compact, perceptually relevant spectral envelope. Unlike LPC's all-pole model, MFCCs are derived by:

• Applying a Short-Time Fourier Transform (STFT) to windowed audio frames.
• Mapping the power spectrum to the mel scale, which approximates human auditory perception.
• Taking the logarithm of the mel powers and applying a Discrete Cosine Transform (DCT) to decorrelate the coefficients.

MFCCs are robust to additive noise and filter characteristics, making them a standard baseline. LPC, in contrast, provides a more parametric model of the vocal tract, useful for synthesis and low-bitrate coding.

The Short-Time Fourier Transform (STFT) is the fundamental time-frequency analysis tool that underpins most audio features. It computes the Fourier transform of short, overlapping windowed segments of a signal, producing a spectrogram. This is a prerequisite step for features like MFCCs and is conceptually contrasted with LPC:

• STFT: Provides a direct, non-parametric view of frequency content over time.
• LPC: Provides a parametric, all-pole model that estimates the spectral envelope by predicting future samples as a linear combination of past samples.

While STFT reveals harmonic structure, LPC excels at modeling formants (resonant frequencies of the vocal tract) by factoring out the source excitation.

Voice Activity Detection (VAD) is a critical preprocessing step that identifies segments of an audio signal containing human speech versus silence or background noise. It is often applied before feature extraction techniques like LPC to conserve computational resources and improve model accuracy. Common VAD methods utilize:

• Energy-based thresholds on the time-domain signal.
• Spectral features like zero-crossing rate or band energy ratios.
• Machine learning classifiers trained on features such as MFCCs or LPC coefficients themselves.

In a pipeline, VAD gates the application of LPC analysis, ensuring the linear predictive model is only fit to frames where speech is present, leading to more stable parameter estimates.

Linear Predictive Coding is a specific type of autoregressive (AR) model. An AR model of order p predicts the current value of a time series based on a linear combination of its p previous values, plus a white noise innovation term. In LPC:

• The time series is the audio signal sample.
• The LPC coefficients are the weights of the AR model.
• The prediction error (or residual) is the white noise excitation.

This formal statistical foundation connects LPC to broader time-series analysis. The Levinson-Durbin recursion is the standard efficient algorithm for solving the Yule-Walker equations to estimate the AR coefficients, which are the LPC parameters.

Code-Excited Linear Prediction (CELP) is a seminal speech coding algorithm that builds directly upon LPC. It is the basis for many modern voice codecs (e.g., ITU-T G.728, G.729). CELP enhances basic LPC by:

• Using a more sophisticated model for the excitation signal. Instead of a simple pulse or noise, it uses a codebook of possible excitation vectors.
• Performing analysis-by-synthesis: searching the codebook to find the excitation that, when passed through the LPC synthesis filter, produces output closest to the original speech.

This hybrid approach allows CELP to achieve high-quality speech at bitrates as low as 4.8 kbps, demonstrating LPC's core role in compression.

Formants are the resonant frequencies of the vocal tract, visible as peaks in the speech spectrum. Formant analysis is a primary application of LPC. The LPC-derived spectral envelope makes formant locations (F1, F2, F3) easy to estimate:

• The roots of the LPC polynomial in the z-domain correspond to the formant frequencies and bandwidths.
• This provides a parametric, compact description crucial for phonetics, speech therapy, and speaker characterization.

Compared to picking peaks from an STFT spectrogram, LPC-based formant tracking is more robust and provides direct bandwidth information. It is a clear example of LPC's strength in parametric spectral estimation for structured signals like speech.