有限長度的時態 /光譜的離散三角變換(Finite-Length Discrete Trigonometric Transforms) 之回歸模型(AR Modeling)

Abstract

回歸模型（AR Modeling）理論，也稱為線性預測，已設立的傅里葉分析無限離散時間序列或連續時間信號。儘管如此，由於各種有限長度離散三角變換 (Discrete trigonometric transforms, DTTs)，包括離散餘弦和正弦變換不同類型的，這個理論是不完備。目前的音頻編碼中採用了DTTs，而AR建模方法可用於減少Audio Artifacts或利用編碼數據冗餘(Redundancy)。本研究將以兩年時間對DTTs 的16個成員有系統的建立時間和光譜Envelope的理論基礎，並實驗對音訊壓縮方法的改進效果。 : (一) 本研究首先對廣義離散傅里葉變換（GDFTs）建立AR Modeling的理論。 (二) 本研究再透過Analytic Transform 來建立DTTs時態 /光譜Envelope的AR Modeling的理論。 (三) 本研究將提出緊湊的形式(Compact Form)和Hilbert Envelope，來開發新的編碼技術，或檢查現存壓縮方法使用DTT的可能漏洞。 (四) 本研究將對現有壓縮標準中用到DTT的模組提出改進方法。

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 project will systematically develops the AR modeling fundamentals of temporal and spectral envelopes for the sixteen members of the DTTs. This project will conduct in two years the the basic theorems and consider the applying to the current audio coding standard for the possible improvement (1) This project considers first the AR modeling through the generalized DTT forms which is the generalized discrete Fourier transforms (GDFTs). (2) We will derive the modeling to all the DTTs by introducing the analytic transforms which convert the real-value vectors into complex-value ones. Through the process, we want to build the compact matrix representations for the AR modeling of the DTTs in both time domain and DTT domain. (3) This project want to show that the AR modeling for the envelops can be investigated through the Hilbert envelope and the power envelope. (4) We adopt these compact forms to review the DTTs in current audio coding and develop new coding technologies for the possible improvement.