Radomir S. Stankovic

The Haar wavelet transform: its status and achievements

Abstract This paper is a brief survey of basic definitions of the Haar wavelet transform. Different generalizations of this transform are also presented. Sign version of the transform is shown. Efficient symbolic calculation of Haar spectrum is discussed. Some applications of Haar wavelet transform are also mentioned.

Spectral Transform Decision Diagrams

This chapter proposes spectral decision diagrams (STDDs), that are graphical representations of spectral transforms of switching functions and integer-valued functions. Binary decision diagrams (BDDs) and functional decision diagrams (FDDs) are graphical representations for switching functions and their Reed-Muller transforms, respectively. Multi-terminal decision diagrams (MTBDDs), arithmetic transform decision diagrams (ACDDs), and Walsh transform decision diagrams (WDDs) are graphical representations for integer-valued functions, their arithmetic transforms, and their Walsh transforms, respectively. This chapter shows that an STDD represents a function and its spectral transform at the same time. As for n-bit adders, ACDDs and WDDs require O(n) nodes while MTBDDs require O(2 n ) nodes. As for n-bit multipliers, ACDDs and WDDs require O(n 2) nodes while MTBDDs require O(4 n ) nodes.

Reduction of sizes of decision diagrams by autocorrelation functions

This paper discusses optimization of decisions diagrams (DDs) by total autocorrelation functions. We present an efficient algorithm for construction of linearly transformed binary decision diagrams (LT-BDDs) and Linearly transformed multiterminal BDDs (LT-MTBDDs) for systems of Boolean functions, based on linearization of these functions by the corresponding autocorrelation functions. Then, we present a method for reduction of sizes of DDs by a level-by-level reduction of the width of DDs using the total autocorrelation functions. The approach provides for a simple procedure for minimization of LT-BDDs and LT-MTBDDs and upper bounds on their sizes. Experimental results for benchmarks illustrate that the proposed method on average is very efficient.

Circuit design from Kronecker Galois field decision diagrams for multiple-valued functions

In this paper we define the Kronecker Galois field decision diagrams (KGFDDs), a generalization of Kronecker decision diagrams (KDDs) to the representation of multiple-valued (MV) functions. Starting from the multi-place decision diagrams (MDDs) and Galois field decision diagrams (GFDDs) we give a generalization that allows more compact representation with respect to the nodes needed. Based on KGFDDs we present a new method for circuit design for MV circuits. In contrast to previously presented approaches the resulting circuits have only logarithmic (instead of linear) depth.

Functional decision diagrams for multiple-valued functions

We first give a uniform interpretation of the binary and functional decision diagrams for binary switching functions. Based upon that, the concept of functional decision diagrams is extended to MV functions relative to the Galois field and Reed-Muller-Fourier representations. The application of the introduced decision diagrams in the realization of functions and the calculation of Galois field and Reed-Muller-Fourier coefficients is considered.

Optimization of GF(4) expressions using the extended dual polarity property

A method for optimization of Fixed Polarity Reed-Muller expressions (FPRM) using the dual polarity property has been presented in [7]. In [2], this method has been extended to optimization of Kronecker expressions by introducing the notion of extended dual polarity property. In this paper, we propose a generalization of this method to optimization of Fixed polarity Galois field (GF) expressions for quaternary functions. The proposed method exploits a simple relationship between fixed polarity GF expressions for dual polarities.

Efficient calculation of fixed-polarity polynomial expressions for multiple-valued logic functions

This paper presents a tabular technique for the calculation of fixed-polarity polynomial expressions for multiple-valued (MV) functions. The technique is derived from a generalization of the corresponding methods for fixed-polarity Reed-Muller (FPRM) expressions for switching functions. All useful features of tabular techniques for FPRMs (e.g. simplicity of the involved operations and high possibilities for parallelization of the calculation procedure) are preserved. The method can be extended to the calculation of coefficients in Kronecker expressions for MV functions.

Calculation of Reed-Muller-Fourier coefficients of multiple-valued functions through multiple-place decision diagrams

We extend the method for the calculation of Walsh transform of binary switching functions through the binary decision diagrams to the calculation of Reed-Muller-Fourier transform of p-valued through multiple-place decision diagrams functions through multiple-place decision diagrams. The calculation of Reed-Muller coefficients of binary switching functions is involved as a special case for p=2.<<ETX>>

Decision diagram method for calculation of pruned Walsh transform

Discrete Walsh transform is an orthogonal transform often used in spectral methods for different applications in signal processing and logic design. FFT-like algorithms make it possible to efficiently calculate the discrete Walsh spectrum. However, for their exponential complexity, these algorithms are practically unsuitable for large functions. For this reason, a Binary Decision Diagram (BDD) based recursive method for Walsh spectrum calculation has been introduced in Clarke et al. (1993). A disadvantage of this algorithm is that the resulting Multi-Terminal Binary Decision Diagram (MTBDD) representing the Walsh spectrum for f can be large for some functions. Another disadvantage turns out if particular Walsh coefficients are to be computed separately. The algorithm always calculates the entire spectrum and, therefore, it is rather inefficient for applications where a subset of Walsh spectral coefficients, i.e., the pruned Walsh spectrum, is required. In this paper, we propose another BDD-based method for Walsh spectrum calculation adapted for application where the pruned Walsh spectrum is needed. The method takes advantage of the property that for most switching functions, the size of a BDD for f is usually quite a bit smaller than the size of the MTBDD for the Walsh spectrum. In our method, a MTBDD representing the Walsh spectrum is not: constructed. Instead, two additional fields are assigned to each node in the BDD for the processed function f. These fields are used to store the results of intermediate calculations. Pairs of spectral coefficients are calculated and stored in the fields assigned to the root node. Therefore, the calculation complexity of the proposed algorithm is proportional to the size of the BDD for f whose spectrum is calculated. Experimental results demonstrate the efficiency of the approach.

Reading the Sampling Theorem in Multiple-Valued Logic: A Journey from the (Shannong) Sampling Theorem to the Shannon Decomposition Rule

Signals described by functions of continuous and discrete variables can be uniformly studied in a group theoretic framework. This paper presents a consideration which shows that in the case of multiple-valued (MV) functions, the notion of bandwidth relates to the concept of essential variables. Sampling conditions convert into requirements for periodicity and regularity in the truth-vectors of MV functions. Due to that, by starting from the sampling theorem, we derive generalized Shannon decomposition rules for MV functions that include the classical Shannon decomposition rule in binary-valued logic as a particular case. It follows from these considerations that the sampling theorem provides a regular way for the decomposition of a MV function into subfunctions of smaller numbers of variables. In circuit synthesis, this allows decomposition of a network to realize a function into subnetworks realizing subfunctions depending on subsets of variables, where the cardinality of the subsets is determined by the bandwidth selected. As there are no convergence problems, the sampling theorem for discrete functions can be formulated in terms of a class of Fourier-like transforms with certain properties provided. Due to that, different decompositions of a given function can be determined by selecting various Fourier-like transforms.