a Abstract—Symbolic Circuit Analysis (SCA) is a technique used to generate the symbolic expression of a network. It has become a well-established technique in circuit analysis and design. The symbolic expression of networks offers excellent way to perform frequency response analysis, sensitivity computation, stability measurements, performance optimization, and fault diagnosis. Many approaches have been proposed in the area of SCA offering different features and capabilities. Numerical Interpolation methods are very common in this context, especially by using the Fast Fourier Transform (FFT). The aim of this paper is to present a method for SCA that depends on the use of Wavelet Transform (WT) as a mathematical tool to generate the symbolic expression for large circuits with minimizing the analysis time by reducing the number of computations. Keywords—Numerical Interpolation, Sparse Matrices, Symbolic Analysis, Wavelet Transform. I. INTRODUCTION CA refers to the calculation of network functions where all or part of the circuit parameters are represented by symbols. The applicability of symbolic simulation techniques to the analysis and synthesis of analog integrated circuits has been known for a long time [1,2,3]. SCA has been developed to help designers get a better understanding of circuit behaviours using the symbolic expressions for the circuit performances. This technique is quite mature in analysis of linear circuits [4,5]. Given the exponential increase in complexity and the time required to do SCA with the circuit size, finding a method that can handle large circuits keeping both the complexity and time as minimum as possible is a challenging factor [5]. SCA methods (in a close connection with numerical methods) can be divided mainly into two categories. These are the topological and the numerical methods [6]. Each one of these methods has its own advantages and limitations. For instance, in topological methods the number of elements represented as symbols is large but the circuits that can be handled is small. On the other hand, in numerical methods, fairly large networks can be handled but the number of symbolic variables is limited. The numerical interpolation method constitutes a very efficient technique for the calculation of network function coefficients with only the complex frequency in symbolic form [6]. The direct Ali Al-Ataby was with the Department of Computer Engineering, College of Engineering, Nahrain University, PO Box 64040, Baghdad, Iraq. He is now with the Department of Electronics and Electrical Engineering, Liverpool University, Liverpool, UK (email: AliAlAtaby@liverpool.eng.uk.com). Fawzi Al-Naima is with the Department of Computer Engineering, College of Engineering, Nahrain University, PO Box 64040, Baghdad, Iraq. (Corresponding author, email: fawzi_cmc@yahoo.com). application of numerical interpolation method can be used to solve problems of system matrix size of around 30 and about 10 elements only represented as variables beside the complex frequency “s” [7,8]. Modified versions of numerical interpolation method are available and proved to be efficient; however, it still suffers from serious limitation in practice, which is the rapidly increasing amount of calculation as the number of symbols to be handled increases. This naturally leads to escalation of computer CPU time and memory requirements, and hence, the famous overflow problem. Looking to the issue from linear system window, it is so obvious that the size of the matrices under processing needs to be reduced. This can be done naturally by two methods: approximation (omission of insignificant terms in the system matrix) [5] and compression (introducing sparsity to increase the number of zeros in the system matrix). Following this strategy, larger circuits can be analysed with less computation efforts. This paper follows the second method, i.e. compressing the system matrix by introducing sparsity. To achieve this aim, a clever and promising mathematical transform will be used. This transform is the Discrete Wavelet Transform (DWT). To simulate the application to SCA, a program was written and tested using MATLAB. The next sections will describe the numerical interpolation traditional method using FFT, then the DWT will be introduced and its use as a method to compress system matrices will be illustrated. Some experimental results will be shown, and the paper will be concluded with comments on the proposed method. II. NUMERICAL INTERPOLATION METHOD FOR SYMBOLIC ANALYSIS Numerical interpolation methods are based on the theory and implementation of numerical methods for generating symbolic functions of networks. They seem to have a lower computational cost than other well-known symbolic analysis algorithms such as parameter extraction method. The following discussion will introduce the idea of using interpolation in finding network transfer functions using the Discrete Fourier Transform (DFT) in interpolation [8, 9,10, 11]. A. Polynomial Interpolation First, N+1 points will be found by evaluating the function: )] ( det[ ) ( x A x P N = (1) at x 0 , x 1 , ..., x N , where N is the maximum power of x. Now, there are N+1 distinct points (x i , y i =P N (x i )), i=0, 1, ..., N. Both Symbolic Analysis of Large Circuits Using Discrete Wavelet Transform Ali Al-Ataby , Fawzi Al-Naima S World Academy of Science, Engineering and Technology International Journal of Electronics and Communication Engineering Vol:3, No:4, 2009 920 International Scholarly and Scientific Research & Innovation 3(4) 2009 ISNI:0000000091950263 Open Science Index, Electronics and Communication Engineering Vol:3, No:4, 2009 publications.waset.org/9149/pdf