3.7 How to find the CTFT Fourier transform of a periodic signal?

Given \(x\left ( t\right ) \), a periodic signal, such as \(\cos \Omega _{0}t\), we want to find its \(X\left ( \Omega \right ) \), then this is the algorithm

1.

Find \(c\left ( k\right ) \), the Fourier series coefficients of \(x\left ( t\right ) \) from \(c\left ( k\right ) =\frac{1}{2T}{\int \limits _{T}}x\left ( t\right ) e^{-j\Omega t}dt\) where \(T\) is the periodic of \(x\left ( t\right ) \)

2.

Now \(X\left ( \Omega \right ) =2\pi{\sum \limits _{n=-\infty }^{\infty }}c\left ( n\right ) \delta \left ( \Omega -n\Omega _{0}\right ) \)

For example \(\cos \Omega _{0} t = \frac{e^{j\Omega _{0} t}+e^{-j\Omega _{0} t}}{2}\), hence \(c(-1) = \frac{1}{2}\) and \(c(1)= \frac{1}{2}\), therefore \begin{align*} X(\Omega ) & = 2\pi \left ( \frac{1}{2} \delta \left ( \Omega + \Omega _{0} \right ) + \frac{1}{2} \delta \left ( \Omega -\Omega _{0}\right ) \right ) \\ & = \pi \left ( \delta \left ( \Omega + \Omega _{0} \right ) +\delta \left ( \Omega - \Omega _{0} \right ) \right ) \end{align*}

Therefore, we see that all periodic functions in time, will have fourier transform which is a pulse train.