Fast fourier transform method for the design and analysis of F.I.R. digital filters.

Abstract

I feel somewhat daunted at the thought of coming in to set the scene for other star performers in the field of signal processing. However, I shall attempt to indicate some of the reasons for the importance of signal processing, particularly in communications and also the scope of this study. Sigp.al processing methods and techniques form the basis of very important developments in physics, electrical and electronic engineering, particularly in communications and radar and sonar, instrumentation and industrial process control. During the last decade digital methods for signal processing have become more significant in that now they not only replace more classical analogue techniques in many areas, but they are also being applied to new areas. There are several reasons for this development : the consistently high level of digital techniques permits better signal processing and analysis. There is greater flexibility within applications, and there is an increasing availability of general purpose computers and minicomputers, at a decreasing cost. Of all the methods and techniques used for digital signal processing, digital filtering is the most important. In the past it has been limited to theoretical research, but recently has been used in many important practical applications for processing 1-D and 2-D signals. This fact may be attributed to the availability of efficient and relatively simple design methods simply fast technological advances in large and very LSI circuits for multipliers , memories, adders, with an increase in maximum working frequency and new devices, such as charge coupled devices and advances in microcomputer hardware and software, particularly with the introduction of microprocessors and minicomputers. This thesis attempts to which can be used to design, put together, a technique which can be used to design, analyse and implement Finite Impulse Response digital filters. It is divided into four main chapters. The first chapter is essentially tutorial in character and presents an introduction to the discrete fourier transform (DFT), and with the relevant theory which leads to the well known Fast Fourier Transform (FFT) algorithm. A BASIC program design to implement the FFT is then derived and improvements made in order to make it compatible with similar designs in the field. From this mathematical basis in the second chapter, the extension of the FFT in Spectral analysis by the Periodogra method is illustrated and a program developed to enable frequency Spectron analysis to be made. Material derived in this section is based on material developed in the first chapter. In the third chapter, criteria for the practical design of 1-D Fill digital filters are derived and :filter impulse response coefficients and frequency responses presented. The implementation to the approximation problem is then presented and useful software designed and documented. The various merits and de-merits of the approach is thoroughly analysed and assessed against other techniques in the field. The fourth and last chapter is dedicated to the performing of the actual filtering operation by convolution in the frequency domain, using material derived from other sections and conclusions drawn on typical results obtained0 All chapters terminate with a detailed discussion of results, and typical examples and illustrations given, together with analysis and comments. Programme software developed in the various chapters was designed to operate on a PDP 11 multipurpose computer operating under a multiuser· RT-11 operating system, and using multiuser- BASIC interpreter as the programming language. Appendices A,B and C give relevant information regarding system operation and program access.