Continuous -time Fourier Transform

The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the discrete summation of the Fourier series transforms into a continuous integral known as the Fourier transform.

The transition from the Fourier series to the Fourier transform is pivotal for analyzing nonperiodic functions. The Fourier series decomposes a periodic function x(t) into a sum of sines and cosines, expressed as:

Where x_n are the Fourier coefficients and ω₀​ is the fundamental angular frequency. As the period of the function extends to infinity, the fundamental frequency ω₀ tends to zero, and the summation over discrete frequencies nω₀ evolves into an integral over a continuous frequency variable ω:

This integral defines the Fourier transform X(ω), representing the original function x(t) in the frequency domain.

The initial skepticism about representing any periodic function with sinusoids led to the establishment of the Dirichlet conditions. These conditions provide criteria under which a periodic function can be expanded in terms of sinusoids. Specifically, a function x(t) can be represented by a Fourier series if the function has finite discontinuities, finite number of maxima and minima, and it is absolutely integrable over the period.

In practical applications, particularly in image processing, the Fourier transform plays a crucial role. It aids in enhancing images and filtering out noise, thereby making details more distinct and sharper. By transforming an image into the frequency domain, various filtering techniques can be applied to emphasize certain features or reduce noise, and then the inverse Fourier transform is used to convert the processed image back to the spatial domain. This approach is foundational in modern image analysis, enabling advanced techniques in medical imaging, remote sensing, and digital photography.