Fourier Series | CT Fourier Transform |

Introduction | Examples | Properties | FT of periodic signals |

Recall: Fourier series representation of a periodic signal $\tilde{x(t)}$ with time period $'T'$ is given by:-

(1)Suppose, $x(t)$ is not periodic.Is there a representation for $x(t)$ as a linear combination of complex exponentials?

The main idea is to think of $x(t)$ as the limit of $\tilde{x}(t)$ when $T \rightarrow \infty$ i.e $$\lim_{T \to \infty}\tilde{x}(t)$$.

Summary:-

1. F.S representation applies to periodic signals i.e A signal contains only frequencies which are integer multiples of a fundamental frequency.

2. F.T representation applies to Non-periodic (and periodic) signals i.e The signal may contain a continuum of frequencies $X(j\omega)$ refers to the F.T,where $\omega$ is a continuously changing variable.

So,the Analysis and Synthesis Equations respectively are given by:-

(3)