Autoregressive Modeling of Temporal/Spectral Envelopes With Finite-Length Discrete Trigonometric Transforms

The theory of autoregressive (AR) modeling, also known as linear prediction, has been established by the Fourier analysis of infinite discrete-time sequences or continuous-time signals. Nevertheless, for various finite-length discrete trigonometric transforms (DTTs), including the discrete cosine and sine transforms of different types, the theory is not well established. Several DTTs have been used in current audio coding, and the AR modeling method can be applied to reduce coding artifacts or exploit data redundancies. This paper systematically develops the AR modeling fundamentals of temporal and spectral envelopes for the sixteen members of the DTTs. This paper first considers the AR modeling in the generalized discrete Fourier transforms (GDFTs). Then, we derive the modeling to all the DTTs by introducing the analytic transforms which convert the real-valued vectors into complex-valued ones. Through the process, we build the compact matrix representations for the AR modeling of the DTTs in both time domain and DTT domain. These compact forms also illustrate that the AR modeling for the envelopes can be performed through the Hilbert envelope and the power envelope. These compact forms can be used to develop new coding technologies or examine the possible defects in the existing AR modeling methods for DTTs, We apply the forms to analyze the current temporal noise shaping (TNS) tool in MPEG-2/4 advanced audio coding (AAC).

I. Introduction

Autoregressive (AR) modeling is widely used for power spectrum estimation [1], [2]. An order- AR model, also referred to as linear prediction model, is defined by , where is the data sequence and is a random sequence of variance . The power spectral density, or power spectrum, of the order- AR model is given by $$\eqalignno{P_{X}^{\rm AR} (f) & = {\mbi \sigma}_{e}^{2} \cdot \left\vert {1 + \sum_{k = 1}^{P} {a_{k} \exp (- j2\pi fk)}} \right\vert ^{- 2}\cr & & {\hbox {for}}\ 0 \leq f \leq 1. }$$ From the aspect of filtering, an AR model describes a data sequence generated from an all-pole filter fed by a white excitation sequence. As a source-filter model with low dimension parameters, the AR modeling method has been successfully utilized to model the acoustical system of speech production, which comprises the vocal tract and the glottal excitation, and has become an essential method in many speech applications, such as speech coding, speech recognition, pitch tracking, and formant estimation.