MDCT Sinusoidal Analysis for Audio Signals Analysis and Processing

Abstract

The Modified Discrete Cosine Transform (MDCT) is widely used in audio signals compression, but mostly limited to representing audio signals. This is because the MDCT is a real transform: Phase information is missing and spectral power varies frame to frame even for pure sine waves. We have a key observation concerning the structure of the MDCT spectrum of a sine wave: Across frames, the complete spectrum changes substantially, but if separated into even and odd subspectra, neither changes except scaling. Inspired by this observation, we find that the MDCT spectrum of a sine wave can be represented as an envelope factor times a phase-modulation factor. The first one is shift-invariant and depends only on the sine wave's amplitude and frequency, thus stays constant over frames. The second one has the form of sinθ for all odd bins and cosθ for all even bins, leading to subspectra's constant shapes. But this θ depends on the start point of a transform frame, therefore, changes at each new frame, and then changes the whole spectrum. We apply this formulation of the MDCT spectral structure to frequency estimation in the MDCT domain, both for pure sine waves and sine waves with noises. Compared to existing methods, ours are more accurate and more general (not limited to the sine window). We also apply the spectral structure to stereo coding. A pure tone or tone-dominant stereo signal may have very different left and right MDCT spectra, but their subspectra have similar shapes. One ratio for even bins and one ratio for odd bins will be enough to reconstruct the right from the left, saving half bitrate. This scheme is simple and at the same time more efficient than the traditional Intensity Stereo (IS).