The look-up table for deriving the Fourier transforms of cosine-pulses

This paper presents a new and easy method of obtaining the Fourier transforms of the nth order cosine-pulses which have uniform amplitudes. This new method focuses on deriving formulas which are recursively related to their orders, and thus making it also applicable to numerical solutions. Concerning the procedures needed to obtain the analytical solutions, this new method proves to be simpler than conventional methods, because the results consist of a sum of two functions which can be easily calculated recursively. It must be noted that the formula can be represented as a complete recursion by separating the coefficients in the manner originated by the authors. The resulting equation is the sum of the original “sinc” functions shifted by some symmetrical factors and then multiplied by several constants. The constants are easily determined by the binomial coefficients and the shifting factors from the corresponding exponential differences in the expansion of (a+b) ⁿ. Furthermore, a lookup table is obtained, making it possible to get all the coefficients and factors needed for the Fourier transform of the cosine-pulses of any order