Audio Signal Processing and Coding

6.8 DISCRETE FOURIER AND DISCRETE COSINE TRANSFORM

This section offers abbreviated filter bank interpretations of the discrete Fourier transform (DFT) and the discrete cosine transform (DCT). These classical block transforms were often used to achieve high-resolution frequency analysis in the early experimental transform-based audio coders (Chapter 7) that preceded the adaptive spectral entropy coding (ASPEC), and ultimately, the MPEG-1 algorithms, layers I–III (Chapter 10). For example, the FFT realization of the DFT plays an important role in layer III of MPEG-1 (MP3). The FFT is embedded in efficient realizations of both MP3 hybrid filter bank stages (pseudo-QMF and MDCT), as well as in the spectral estimation blocks of the psychoacoustic models 1 and 2 recommended in the MPEG-1 standard [ISOI92]. It can be seen that block transforms are a special case of the more general uniform-band analysis-synthesis filter bank of Figure 6.1. For example, consider the unitary DFT and its inverse [Akan92], which can be written as, respectively,

where W = e^jπ/M. If the analysis filters in Eq. (6.1) all have the same length and L = 2M, then the filter bank could be interpreted as taking contiguous L sample blocks of the input and applying to each block the transform in Eq. (6.39a). Although the ...