Date of Award
Doctor of Philosophy (PhD)
Electrical and Computer Engineering
During the past decade, discrete block transforms have evoked considerable interest in the signal processing area. Discrete block transform is realized by first dividing the sample sequence of a signal to be processed into a series of blocks. The transform operation is then carried out by taking the inner products between the finite-length signal and a set of basis functions. This produces a set of coefficients which may then be further processed in compression, quantization, and encoding. Generally, block transforms can be divided into two types. One is called "nonlapped transform"; the other, "lapped transform". In the first system, the blocks are contiguous and no overlap (conjunction) is involved. The second system has overlapped operations: i.e., the data blocks may be overlapped.
In signal or image compression, it is well known that the non-lapped transforms produce an artifact called the "blocking effect" in the reconstructed signal or image. The blocking effect is partially due to the sharp cutoff of the data block. In lapped transforms where the data blocks overlap, the gradual decay of the basis functions in the overlapped region helps to reduce this blocking effect.
In this dissertation, we use a bell function to establish the discrete local sine (DLS) transform and the discrete local cosine (DLC) transform. The basis functions of this general type have a smooth cutoff and are shown to have orthogonality properties. The orthonormality and lapped orthogonality of DLS/DLC transforms indicate that DLS and DLC belong to the family of the lapped block transform system. We derive sufficient and necessary conditions for perfect reconstruction for lapped block transforms based on DLS/DLC basis functions, although, in general, lapped transform matrices are not unitary. These conditions are useful for designing optimum lapped block transforms in special applications. For example, in image and speech coding, we can design an optimal lapped transform system with the perfect signal reconstruction property.
Other properties of the transforms are also discussed. These include scaling-in-time, shifting-in-time, difference-in-time, the uniqueness property, and the convolution property. Fast algorithms for implementing DLS and DLC are developed. Based on Given's rotations and butterfly operations, we decompose an Mth-order DLS or DLC with length M + L⁻ into a set of sparse matrices, where L is the length of the overlapped region. Because of the partially recursive nature of the structure, DLS and DLC fast algorithms can be implemented with parallel processors.
As examples, we consider two applications using lapped transforms. The first application is in acoustic echo cancellation. We use block transforms to implement a subband acoustic echo canceller. However, due to the frequency aliasing problem in a filter bank system , the direct application of block transforms in an echo canceller does not function very well. We propose an improved method. By changing the subsampling rate in block transforms, echo residuals can be reduced significantly. Moreover, we develop an optimum lapped transform using a criterion of maximum energy concentration. With the optimum designed lapped transform, an obvious performance improvement can be observed from echo suppression. Image compression , a tool for efficiently encoding a picture (two dimensional data), is the second application of lapped orthogonal transforms discussed in this thesis. We develop a new sub-optimal lapped transform based on DLS and DLC for image coding. The results are compared with those obtained from using the lapped orthogonal transform (LOT) and modulated lapped transform (MLT). Computer simulation results show significant improvement in the reduction of block effects.
Li, Jingyun, "Lapped transforms based on DLS and DLC basis functions and applications" (1997). Open Access Dissertations and Theses. Paper 3395.