How the Fourier Transform Works

In my last book: "How the Fourier Series Works" we studied Fourier's original theory, published in 1822. However, the Fourier Series can only model repeating signals. It took the further work of the German mathematician, Peter Gustav Lejeune Dirichlet, to expand the capabilities of the Fourier Series so that it could model non-repeating signals.
In this book, we'll follow the journey of the development of the Fourier Transform from Fourier's original idea, through Dirichlet's changes, and into the digital age of sampled signals, all the way to the Fast Fourier Transform that is used so widely today.