13 - Fourier Transform

Summary

This chapter discusses the transition between Fourier series and Fourier Transform, which is the tool for spectrum analysis. Generally, the use of linearly independent base functions allows a wide range of linear regression models that work in a least square sense such that the total error squared is minimized in finding the coefficients of the base functions. A special case is sinusoidal functions based on a fundamental frequency and all its harmonics up to infinity. This leads to the Fourier series for periodic functions. In this chapter, we start from the original Fourier series expression and convert the sinusoidal base functions to exponential functions. We can then consider the limit when the length of the function and the period of the original function approach infinity (so that the fundamental frequency approaches 0, including aperiodic functions), leading to the Fourier integral and Fourier Transform. We can then define the inverse Fourier Transform and establish the relationship between the coefficients of Fourier series and the discrete form Fourier Transform. All these are preparations for the fast Fourier Transform (FFT), an efficient algorithm of computation of the discrete Fourier Transform that is widely used in data analysis for oceanography and other applications.