Integer Approximate Cosine/Sine-Modulated Filter Banks

Digital audio signals to be processed are integer-valued. But the perfect reconstruction cosine/sine-modulated filter banks and cosine-modulated QMF banks are real-valued transforms which map integer signal into real-valued spectral coefficients. Although their fast algorithms reduce the computational complexity, due to floating-point finite-length representation and corresponding rounding-off errors they cannot be used for lossless audio coding. Actually, almost all modern perceptual audio coding schemes developed so far operate in floating-point arithmetic and therefore, are lossy in nature. However, some audio coding applications require completely lossless preservation of the audio signal. An enabling technology for transform-based lossless audio coding is the integer transform. Integer transform is a transform which maps integers to integers by a reversible (invertible) way so that it preserves all mathematical properties of the original real-valued transform such as perfect reconstruction, energy compaction property and fast algorithm. Indeed, the integer modified discrete cosine transform (IntMDCT) or integer modulated lapped transform (IntMLT) enabled to design and implement this innovative coding technology for scalable lossy to lossless audio coding. The local and global methods to integer approximation of perfect reconstruction cosine/sine-modulated filter banks and cosine-modulated QMF banks are discussed in detail. They are based on computational methods of linear algebra, matrix theory and matrix computations, and in particular, on the matrix decompositions. In fact, the scalar and block matrix decompositions are powerful mathematical tools to construct the reversible (invertible) integer transforms.