Unit 4: Fourier Transform#

This chapter continues our coverage of Fourier Analysis with an introduction to the Fourier Transform.

• Fourier Series is used when we are dealing with signals that are periodic in time. It is based on harmonics of the fundamental frequency ${\mathrm{\Omega }}_0$ of the periodic signal where the period $T=2\pi /{\mathrm{\Omega }}_0$.

• The line spectrum occur at integer multiples of the fundamental frequency $k{\omega }_0$ and is a discrete (or sampled) function of frequency.

• As the period $T$ is increased, the distance between harmonics decreases because ${\mathrm{\Omega }}_0$ reduces.

• In the limit $T\to \mathrm{\infty }$, the signal becomes aperiodic and $k{\mathrm{\Omega }}_0\to \omega$ which is a continuous function of frequency.

This is the basis of the Fourier Transform which is very important as the basis for data transmission, signal filtering, and the determination of system frequency response.