Short term synthesis filter

Description

The short term synthesis filter is a core component of linear predictive coding (LPC) based speech codecs, including those specified in 3GPP TS 26.090 (AMR), TS 26.190 (AMR-WB), TS 26.290 (AMR-WB+), and TS 46.060 (GSM EFR). It operates at the decoder, taking the excitation signal — which represents the residual after removing short-term correlations at the encoder — and filtering it to reconstruct the spectral envelope of the original speech. This spectral envelope corresponds to the resonances of the vocal tract, often called formants.

In LPC-based speech coding, the encoder analyzes short segments of speech (typically 20 ms frames) to estimate a set of linear prediction coefficients (LPCs) that describe the short-term correlation of the signal. These coefficients are quantized and transmitted to the decoder. The decoder uses them to configure the short term synthesis filter, which is an all-pole filter of the form 1/A(z), where A(z) is the LPC polynomial. The excitation signal, which may be a combination of pitch contributions and noise-like components, is passed through this filter to produce the synthesized speech.

The filter is called 'short term' because it models correlations within a short time window, typically corresponding to one pitch period or less. This contrasts with the long-term synthesis filter, which models pitch periodicity. Together, the short-term and long-term synthesis filters form the synthesis filter cascade in codecs such as AMR and AMR-WB. The short-term filter is responsible for the overall spectral shape, while the long-term filter adds harmonic structure.

Stability of the short term synthesis filter is critical. Because the filter is all-pole, its coefficients must correspond to a stable filter — that is, all poles must lie inside the unit circle in the z-plane. Encoders ensure stability by using algorithms such as Levinson-Durbin recursion and by bandwidth expansion or other stabilization techniques. If the filter were unstable, the synthesized speech would diverge and produce loud artifacts. The decoder may also apply interpolation of LPC coefficients between frames to ensure smooth transitions and avoid abrupt spectral changes.

In practical implementations, the short term synthesis filter is applied sample-by-sample or block-by-block, depending on the codec. It is computationally efficient, typically requiring only a few multiply-accumulate operations per sample. Its parameters are transmitted at a low bit rate (e.g., 2-4 kbps for the LPC coefficients in AMR), making it a key enabler of low-bit-rate speech compression. The filter's accuracy directly affects speech quality, and modern codecs use techniques such as line spectral pairs (LSPs) for efficient quantization and robust interpolation.

Purpose & Motivation

The short term synthesis filter exists to reconstruct the spectral envelope of speech from a compact set of parameters, enabling low-bit-rate speech coding without sacrificing intelligibility. In the source-filter model of speech production, the vocal tract acts as a filter that shapes the excitation from the vocal cords. By modeling this filter with a small number of LPC coefficients, codecs can represent speech efficiently, transmitting only the filter parameters and a simplified excitation signal.

Before LPC-based codecs, waveform codecs such as PCM and ADPCM required high bit rates (64 kbps or 32 kbps) to achieve acceptable quality. The short term synthesis filter, as part of the LPC framework, allowed bit rates to drop to 12.2 kbps (GSM EFR), 7.95 kbps (AMR), and even lower, while maintaining good quality. This was essential for mobile networks, where spectrum is scarce and capacity is paramount.

The filter also solves the problem of spectral mismatch: without it, the excitation signal alone would sound flat and unnatural. By reintroducing short-term correlation, the filter restores the formant structure that gives speech its characteristic timbre. This makes synthesized speech sound like the original speaker rather than a robotic or muffled version. The concept has been refined over decades and remains fundamental to modern codecs, including those used in VoLTE and VoNR.