Introduction

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

(1)
\begin{align} \tilde{x}(t)=\sum_{k=-\infty}^{\infty}\tilde{X}[k]e^{jkw_0t} \end{align}
(2)
\begin{align} \tilde{X}[k]=\frac{1}{T}\int_{0}^{T}\tilde{x}(t)e^{-jkw_0t}dt \end{align}

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)
\begin{align} X(j\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt \end{align}
(4)
\begin{align} x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\omega)e^{j\omega t}d\omega \end{align}