Inverse filter

Description

The inverse filter is a key component in speech coding, particularly in linear predictive coding (LPC) analysis. Its primary function is to remove the short-term correlation from the speech signal, effectively whitening the signal to produce a residual that is easier to encode. The filter models the inverse frequency response of the vocal tract, which means it acts as an all-zero filter that flattens the spectral envelope of the speech signal. This is achieved by applying the inverse of the vocal tract's transfer function, which is typically represented by an all-pole filter in the synthesis model.

In the architecture of a speech codec, such as AMR, EVS, or other 3GPP codecs, the inverse filter is used at the encoder. The encoder first estimates the short-term prediction coefficients (LPC coefficients) that characterize the vocal tract. These coefficients define the synthesis filter, which is an all-pole filter. The inverse filter is then the inverse of this synthesis filter, i.e., an all-zero filter with coefficients derived from the LPC coefficients. When the speech signal is passed through the inverse filter, the output is the prediction residual (also called excitation signal), which has a relatively flat spectrum. This residual is then quantized and transmitted along with the LPC coefficients.

The inverse filter is mathematically expressed as A(z) = 1 - sum_{i=1}^{p} a_i z^{-i}, where a_i are the LPC coefficients and p is the prediction order. This filter is applied to the input speech to compute the residual. The design of the inverse filter is crucial for efficient compression: a well-designed filter removes redundancy, allowing the residual to be encoded with fewer bits. The filter is typically implemented as a finite impulse response (FIR) filter, and its coefficients are quantized and transmitted to the decoder, where they are used to construct the synthesis filter for reconstruction.

In 3GPP specifications, the inverse filter is defined in the context of speech codecs such as those in TS 26.090 (AMR), 26.190 (AMR-WB), 26.290 (EVS), and 46.060 (AMR-NB). These specifications detail the exact computation of the LPC coefficients and the inverse filtering process. The inverse filter is also used in other speech processing tasks, such as speech enhancement and recognition, but its role in coding is paramount for achieving low bit rates while maintaining speech quality.

Overall, the inverse filter is a fundamental signal processing block that enables efficient speech compression by decorrelating the speech signal. It is a cornerstone of modern speech codecs and is essential for the operation of mobile communication systems where bandwidth is limited.

Purpose & Motivation

The inverse filter exists to facilitate efficient speech compression by removing the short-term correlation inherent in speech signals. Speech is produced by the vocal tract, which acts as an acoustic filter with a specific frequency response. This response introduces correlation between adjacent samples, making the signal redundant. By applying an inverse filter that models the inverse of the vocal tract response, the encoder can obtain a residual signal with a flat spectrum, which is much easier to encode at low bit rates. This is the basis of linear predictive coding, a technique that has been used since the 1960s and is still central to all modern speech codecs.

Before the widespread adoption of LPC, speech coding relied on waveform coding techniques like PCM and ADPCM, which required higher bit rates for acceptable quality. The inverse filter, combined with LPC analysis, allowed a significant reduction in bit rate by exploiting the source model of speech production. In 3GPP, the inverse filter was standardized as part of the AMR codec in Release 8 (though AMR itself was introduced earlier, the definition in the glossary appeared in Rel-8). It continues to be a key element in subsequent codecs like AMR-WB and EVS, ensuring interoperability and high-quality speech at low data rates. The concept is fundamental to the success of mobile telephony, enabling efficient use of spectrum.