Immittance Spectral Pair

Description

Immittance Spectral Pairs (ISP) are a representation of Linear Predictive Coding (LPC) parameters used in speech coding, defined in 3GPP specifications 26.190 (AMR-WB) and 26.290 (AMR-WB+). The ISP are obtained by decomposing the LPC inverse filter transfer function A(z) into two transfer functions: one with even symmetry and one with odd symmetry. The roots of these polynomials on the z-unit circle are the Immittance Spectral Pairs, which are also referred to as Immittance Spectral Frequencies (ISF).

The mathematical derivation starts with the LPC polynomial A(z) of order p, which represents the short-term spectral envelope of the speech signal. Two auxiliary polynomials are formed: P(z) = A(z) + z^-(p+1) A(z^-1) and Q(z) = A(z) - z^-(p+1) A(z^-1). P(z) has even symmetry, and Q(z) has odd symmetry. Due to these symmetry properties, the roots of P(z) and Q(z) lie on the unit circle and interlace with each other. The angular positions of these roots, expressed as frequencies, are the ISP. For an LPC order p, there are p ISP values, which fully describe the spectral envelope.

In the AMR-WB codec, LPC analysis is performed once per 20 ms frame, and the resulting ISP are quantized and transmitted to the decoder. The ISP representation is preferred over raw LPC coefficients because it has better quantization properties: the values are more evenly distributed, less sensitive to quantization errors, and allow efficient vector quantization with a small number of bits. The ISP also have a natural ordering property (sorted in ascending frequency), which simplifies quantization and interpolation. In AMR-WB, the ISP are quantized using split vector quantization, and the quantized values are interpolated between frames to provide smooth spectral transitions.

The ISP representation is also used in the EVS codec (26.445) and other 3GPP speech codecs. In the decoder, the quantized ISP are converted back to LPC coefficients, which are then used to filter the excitation signal to reconstruct the speech. The stability of the synthesis filter is guaranteed if the ISP remain ordered and within the unit circle, which is ensured by the quantization scheme. The ISP are also used in speech enhancement and noise suppression algorithms within the codec.

In summary, Immittance Spectral Pairs are a mathematically elegant and practically efficient representation of the short-term spectral envelope of speech. They are central to the operation of wideband speech codecs in 3GPP systems, enabling high-quality voice services over mobile networks. The term is used interchangeably with Immittance Spectral Frequencies, and both refer to the same underlying concept.

Purpose & Motivation

The Immittance Spectral Pair representation exists to provide a robust and efficient way to encode the short-term spectral envelope of speech in LPC-based codecs. Raw LPC coefficients are difficult to quantize because small errors can cause the synthesis filter to become unstable, and their dynamic range varies widely. ISP solve this problem by transforming the LPC coefficients into a set of frequencies that are inherently ordered and bounded, making quantization more reliable and efficient.

The use of ISP also enables efficient interpolation between frames, which is important for smooth transitions and high speech quality at low bit rates. In the context of 3GPP codecs such as AMR-WB and EVS, ISP allow the codec to achieve transparent quality at bit rates as low as 6.6 kbps, which is essential for spectral efficiency in mobile networks. The concept was introduced in Release 8 as part of the AMR-WB codec specification, and it has been carried forward in subsequent releases without fundamental changes to its definition.

Historically, the ISP representation was developed in the speech coding research community as an alternative to other LPC parameterizations such as Line Spectral Frequencies (LSF) and reflection coefficients. ISP are closely related to LSF but are derived from a slightly different polynomial decomposition. The 3GPP adopted ISP for AMR-WB because of its excellent quantization properties and its ability to support wideband speech (50-7000 Hz) with low complexity. The term has remained stable across releases, reflecting its mature and well-understood nature.