The evaluation of fourier transforms by the sampling method

Abstract

The sampling method is a well known method for evaluating Fourier transforms. Since the conditions imposed by the sampling theorem cannot, generally, be satisfied exactly when one is dealing with experimental responses and signals, a number of authors have used weighted sampling methods which reduce, more or less, the error due to frequency folding. These methods have been commonly considered to be based on approximations of the given function by elementary functions (staircase function, trapezoidal approximation, etc.). Up to now, no general theory regarding the selection of the sampling rate, and that of the appropriate weight function, had been given. In this paper it is shown that if the frequency response (or spectrum) to be evaluated approaches a high frequency asymptote of the form βω − k , there is an optimum weight function which can be easily selected. Essentially, different weight functions have to be used for the real and the imaginary parts of the spectrum, since they have a different asymptotic behavior. These weight functions (or multiplicative correction functions) are closely related to the additive correction functions which are also considered in this paper. Sometimes the latter lead to more accurate results than the former, as is illustrated in the applications to some test systems which will be discussed. The sampling method is most favorable from the point of view of numerical evalua on. A time saving scheme is indicated that is suitable for hand computation using simple aids and a desk calculator. When a digital computer is used, the fast Fourier transform technique can be easily adapted to the modified sampling method.