17.1 : 連続時間フーリエ変換

The Fourier series is an effective tool for representing periodic functions like a train of square waves.

Consider a pulse-train waveform consisting of a sequence of rectangular pulses. If the period of these pulses is finite, the waveforms can be represented by a Fourier series.

Now, if the period of the pulse-train increases, the frequency of the obtained line spectra decreases.

But what if the period goes to infinity, resulting in a single pulse? In this case, the summation in the Fourier series evolves into a continuous integral, known as the Fourier transform.

According to Dirichlet conditions, a periodic function can be expanded in terms of

sinusoids if it has a finite number of discontinuities, maxima, and minima and is

integrable. If a function fails these conditions, it cannot be represented by a Fourier series.

Fourier transform is commonly used in image processing, where it helps enhance images and filter out noise, making the details clearer and sharper.