Linear Prediction Coefficients

Description

Linear Prediction (LP) coefficients, also referred to as Linear Predictive Coding (LPC) coefficients, are a generic descriptive term for the short-term filter coefficients used in speech coding. In the context of 3GPP, these coefficients are central to the operation of speech codecs such as Adaptive Multi-Rate (AMR), Adaptive Multi-Rate Wideband (AMR-WB), and Enhanced Voice Services (EVS). The LP coefficients define the transfer function of the short-term synthesis filter that models the spectral envelope of the speech signal. They are derived from the input speech signal using linear prediction analysis, typically employing Levinson-Durbin recursion to solve the Yule-Walker equations. The order of the LP filter varies by codec: for example, AMR narrowband uses an order of 10, while AMR-WB uses an order of 16 to capture wider bandwidth characteristics.

The LP coefficients are computed on a frame-by-frame basis, usually every 20 ms, and are quantized for transmission. Because direct quantization of LP coefficients is inefficient and sensitive to quantization errors, they are often transformed into other representations such as Line Spectral Frequencies (LSFs), Immittance Spectral Frequencies (ISFs), or Line Spectral Pairs (LSPs) before quantization. These transformed representations have better quantization properties and ensure filter stability. The quantized coefficients are then transmitted to the decoder, which uses them to reconstruct the short-term filter for speech synthesis.

In the decoder, the LP coefficients are used to filter the excitation signal (which may be generated from an adaptive codebook and a fixed codebook) to produce the synthesized speech. The short-term filter shapes the spectrum of the excitation to match the original speech spectrum, thereby reproducing the formant structure. The accuracy of the LP coefficients directly affects the quality and intelligibility of the reconstructed speech. Therefore, robust quantization and interpolation techniques are employed to minimize distortion, especially in error-prone channels.

LP coefficients are also used in other parts of the codec, such as in the computation of the perceptual weighting filter, which shapes the quantization noise to be less audible. The weighting filter is derived from the LP coefficients by applying a bandwidth expansion factor. This technique is common in Code-Excited Linear Prediction (CELP) codecs. Furthermore, LP coefficients are used in the adaptive codebook search to compute the target signal for pitch prediction.

The term "LP coefficients" is generic and applies across multiple 3GPP specifications. For instance, TS 26.090 specifies the AMR codec, TS 26.190 specifies the AMR-WB codec, TS 26.290 specifies the Extended Adaptive Multi-Rate Wideband (AMR-WB+) codec, and TS 46.060 specifies the Enhanced Full Rate (EFR) codec. These specifications detail the exact algorithms for computing, quantizing, and applying LP coefficients. The concept has remained stable across releases from Rel-8 to Rel-19, reflecting its foundational role in speech coding.

Purpose & Motivation

The primary purpose of LP coefficients is to enable efficient digital compression of speech signals by modeling the human vocal tract as a time-varying linear filter. Speech production can be approximated by an excitation source (glottal pulses or noise) filtered by the vocal tract, which imposes a spectral envelope characterized by formants. By estimating the LP coefficients, the codec captures this spectral envelope with a small number of parameters, allowing the excitation to be encoded separately with fewer bits. This separation of spectral and excitation information is the basis of low-bit-rate speech coding.

Before the widespread adoption of linear prediction, speech coding often relied on waveform coding techniques such as PCM or ADPCM, which required high bit rates (64 kbps or more) to achieve acceptable quality. Linear prediction, combined with analysis-by-synthesis techniques like CELP, reduced the required bit rate to 4.75–12.2 kbps while maintaining good quality. This was crucial for mobile communications, where bandwidth is limited and efficient spectrum usage is paramount.

The LP coefficients solve the problem of representing the short-term spectral characteristics of speech in a compact and quantizable form. They provide a mathematically tractable model that can be estimated from the speech signal using well-known algorithms (e.g., Levinson-Durbin). The transformation to LSFs or ISFs further improves quantization efficiency and robustness, making LP coefficients a cornerstone of all modern 3GPP speech codecs. Their continued use from Rel-8 through Rel-19 underscores their enduring effectiveness and the lack of a need for fundamental change.