Fourier Tansform

Question

What will be the Fourier transform of the wave shown below:

Waveform

Basically i want to know the Fourier Transform of a square wave of varying width!

Answer 1

You seem to be asking for the Fourier Transform of $$ f(t)=\sum_k a_kp(t-kT_s) $$ where the $a_k$ are either $0$ or $1$, and: $$ p(t)=\left\{ \begin{array}{cc} 1,& t\in(0,T_s]\\ 0,& \mbox{otherwise} \end{array} \right. $$ Then, by linearity of the FT and the appropriate translation theorem: $$ F(\omega)=(\mathcal{F}f)(\omega)=\sum_k a_k e^{i\omega kT_s}P(\omega)=P(\omega)\sum_k a_k e^{i\omega kT_s} $$ where $P=\mathcal{F}p$.

Note: the sign in the exponents in the sum may be negative if you are using a definition of the forward direction of the FT in the opposite sense to what I have taken.