A deterministic analysis of limit-cycle oscillations, which occur in fixed-point implementations of recursive digital filters due to roundoff and truncation quantization after multiplication, is performed. Amplitude bounds, based upon a correlated (nonstochastic) signal approach using Lyapunov's direct method, as well as a general matrix formulation for zero-input limit cycles, are derived and tested for the two-pole filter. The limit cycles are represented on a successive-value phase-plane-type diagram from which certain symmetry properties are derived. The results are extended to include limit cycles under input-signal conditions.

Limit-cycle oscillations in digital filters

Wavelet transforms (WT) have growing impact on signal processing theory and practice. This is because of two reasons: (a) unifying role of wavelet transform and (b) their successes in wide variety of applications. Digital filter banks, the basis of wavelet-based algorithms, have become standard signal processing operators. Filter banks are the fundamental tools required for processing of real signals using digital signal processors (DSP) [133,139]. Vaidyanathan in his book [134] has discussed connection between theory of filter bank and DSP. The purpose of this book is to look at wavelet-related issues from a signal processing perspective. This book focuses on and around new implementation techniques of discrete wavelet transform (DWT) and their applications in denoising and classification. On this account, it is required to introduce the wavelet theory in brief. The organization of this chapter is as follows: Section 1.1 introduces the subject in brief. Section 1.2 presents historical review of multiresolution analysis and wavelet transform. Various kinds of wavelet transform applied to signal processing applications viz. continuous wavelet transform (CWT) and DWT (one dimension and two dimensions) are discussed in brief. Section 1.3 reviews implementation issues and applications of DWT from signal processing viewpoint. Section 1.4 concludes this chapter by outlining major contribution of the book.

Efficient Algorithms for Discrete Wavelet Transform

When a digital filter is realized with floating-point or fixed-point arithmetics, errors and constraints due to finite word length are unavoidable In this paper, we show how these errors can be mechanically analysed using the HOL theorem prover We first model the ideal real filter specification and the corresponding floating-point and fixed-point implementations as predicates in higher-order logic We use valuation functions to find the real values of the floating-point and fixed-point filter outputs and define the error as the difference between these values and the corresponding output of the ideal real specification Fundamental analysis lemmas have been established to derive expressions for the accumulation of roundoff error in parametric Lth-order digital filters, for each of the three canonical forms of realization: direct, parallel, and cascade The HOL formalization and proofs are found to be in a good agreement with existing theoretical paper-and-pencil counterparts

Error analysis of digital filters using theorem proving

Equation (17) is an explicit formula for the elements of the function matrix P = g (A). It does not (directly) require the knowledge of the similarity transformation matrices T and T-' in (3). Thus, the elements of P can be computed directly from the knowledge of the companion matrix A, its eigenvalues, and the function g. Now, it will be shown that (17) has also a remarkable recursive property (24), which allows one to compute the function matrix P using somewhat less than 6n2 multiplications and n2 divisions. The preceding statement assumes, of course, that all eigenvalues of A are known, and all g(Ai) (i= 1, n) are also known. It is also interesting to note that the evaluation of P using the similarity transformation (3) would require about n3 multiplications and n3 additions more than is required using (24), which is derived below. From (8) and (14) it is easy to see that qi,j-1 = qi.j& + am-,+ 1 (18) These values are now used to obtain the remaining two elements of the matrix P, using the recursive formula (24). Note that the subscript i in (24) must vary faster than the subscript j. Also note that j = n-l n-2. .. Upon completion of these simple computations, the matrix P has the following form, And, of course, it can be easily checked that P 2 = A. The evaluation of g = exp (A) can be performed in a similar manner. ACKNOWLEDGMENT and The thor wishes to thank Dr.

C. Weinstein

Papers

Correction to "A comparison of roundoff noise in floating point and fixed point digital filter realizations"