A representation for aperiodic signals

We have already seen that a broad class of functions (which satisfy the Dirichlet's conditions) can be written in the form of Fourier series .That is ,for a periodic function x(t) satisfying the Dirichlet's conditions , we may say

                                              where       

Although this covers a broad class of functions, it puts a serious restriction on the function. That is, Periodicity . So the next question that naturally pops up in one's mind is , "Can we extend our idea of the Fourier series so as to include non periodic functions ?" . This precisely is our goal for this part , the basic inspiration being, an aperiodic signal may be looked at as a periodic signal with an infinite period. Note what follows is not a mathematically rigourous excercise, but will help develop an intuition for the Fourier Transform for aperiodic signals.