Finite Precision Error Modeling and Analysis

Abstract

Discrete wavelet transforms (DWTs) have excellent energy compaction characteristics and are able to provide near perfect reconstruction (PR). They are ideal for signal/image analysis and encoding. Hardware implementation of DWT is fast and area efficient in fixed-point arithmetic. DWT encoding has been drawing much attention because of its ability to decompose signals into a hierarchical structure that is suitable for adaptive processing in the transform domain. In existing architectural designs for the DWT, little consideration has been given to word size and precision. Present chapter addresses this problem, showing how the word size requirements can be calculated for a specific problem (based on the range of input data and wavelet used). A simplified, statistical model is proposed. As the depth of the DWT filtering increases, the data word length requirement increases. It is important to investigate how this can affect the potential of the resulting hardware implementation of DWT. The issue has been analyzed for both pyramid structure DWT and parallel filter DWT. The organization of this chapter is as follows. Section 3.1 presents background material related to subject. Section 3.2 presents in brief the computational complexity of DWT. Section 3.3 presents finite precision modeling of two-channel PR filter bank in moderate detail, including modeling of quantized coefficient filters. Section 3.4 presents the proposed statistical modeling of DWT to study the effects of finite word length implementation. This includes construction of new DWT filters to accommodate round-off errors followed by corresponding mathematical derivation.