Line Spectral Frequencies

Description

Line Spectral Frequencies (LSF), also known as Line Spectral Pairs (LSP), are a mathematical representation of the Linear Predictive Coding (LPC) coefficients used in speech coding. In 3GPP speech codecs, such as the Adaptive Multi-Rate (AMR) and Enhanced Voice Services (EVS) codecs, LSF are used to represent the short-term spectral envelope of the speech signal. The LSF are derived from the LPC coefficients through a transformation that ensures the resulting frequencies are ordered and lie within the range (0, π). This property makes LSF highly suitable for quantization, as it allows for efficient vector quantization and ensures stability of the synthesis filter.

The LSF are computed by converting the LPC polynomial into two auxiliary polynomials, P(z) and Q(z), whose roots lie on the unit circle in the z-plane. The angles of these roots are the LSF. The LSF have several advantageous properties: they are ordered (i.e., the frequencies are monotonically increasing), they are robust to quantization errors, and they provide a direct link to the formant frequencies of the speech signal. These properties make LSF ideal for use in low-bit-rate speech codecs, where efficient quantization is critical.

In 3GPP codecs, the LSF are typically quantized using predictive or multi-stage vector quantization. For example, in the AMR codec, the LSF are quantized using a split vector quantizer with 3 or 4 sub-vectors, depending on the bit rate. The quantized LSF are then transmitted to the decoder, which reconstructs the LPC coefficients and uses them to filter the excitation signal to synthesize the speech. The accuracy of the LSF quantization directly affects the quality of the reconstructed speech, so careful design of the quantizer is essential.

The LSF representation is also used in other 3GPP codecs, such as the Adaptive Multi-Rate Wideband (AMR-WB) and the EVS codec. In EVS, the LSF quantization is more sophisticated, employing a combination of predictive and non-predictive quantization to achieve high quality at low bit rates. The LSF are also used in the speech quality assessment and in the design of codec modes for different network conditions.

Overall, Line Spectral Frequencies are a fundamental component of 3GPP speech codecs, enabling efficient and robust representation of the spectral envelope. Their mathematical properties make them ideal for quantization, and their use is widespread in mobile telephony and Voice over LTE (VoLTE) services.

Purpose & Motivation

Line Spectral Frequencies exist to provide a robust and efficient representation of the spectral envelope of speech signals for use in speech coding. In the context of 3GPP, they are essential for codecs like AMR and EVS, which are used in mobile networks to compress voice for transmission over bandwidth-limited channels. The LSF representation solves the problem of quantizing LPC coefficients in a way that guarantees filter stability and allows for efficient vector quantization.

Before LSF, other representations such as reflection coefficients or log area ratios were used, but they had drawbacks in terms of quantization efficiency and stability. LSF offer a better alternative because they are ordered and bounded, which simplifies quantization and ensures that the reconstructed synthesis filter is stable. This is crucial for maintaining speech quality, especially at low bit rates where quantization errors can be significant.

The LSF concept was introduced in the early days of speech coding and has been adopted in many standards, including 3GPP codecs. In 3GPP, the term 'Line Spectral Frequencies' is used interchangeably with 'Line Spectral Pairs', and the definition simply refers to the latter. The LSF have been part of 3GPP specifications since at least Release 8, and they continue to be used in the latest codecs like EVS. Their importance lies in enabling high-quality voice communication over mobile networks, which is a fundamental service.