This chapter introduces the continuous-time Fourier transform (CTFT) and its properties. Many examples are presented to illustrate the properties. The inverse CTFT is derived. As one example of its application, the impulse response of the ideal lowpass filter is obtained. The derivative properties of the CTFT are used to derive many Fourier transform pairs. One result is that the normalized Gaussian signal is its own Fourier transform, and constitutes an eigenfunction of the Fourier transform operator. Many such eigenfunctions are presented. The relation between the smoothness of a signal in the time domain and its decay rate in the frequency domain is studied. Smooth signals have rapidly decaying Fourier transforms. Spline signals are introduced, which have provable smoothness properties in the time domain. For causal signals it is proved that the real and imaginary parts of the CTFT are related to each other. This is called the Hilbert transform, Poisson’’s transform, or the Kramers–Kronig transform. It is also shown that Mother Nature “computes” a Fourier transform when a plane wave is propagating across an aperture and impinging on a distant screen – a well-known result in optics, crystallography, and quantum physics.

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.