# Spectrum of windowed version of original continuous signal

Suppose we have the complex signal $x(t)= \exp(j\omega_0 t)$. Using the properties of Fourier transform we can prove its CTFT is Dirac $\delta$ function.

If any one ask me about the spectrum of $x(t)$ "Does $x(t)$ has continuous spectrum or discrete spectrum", my answer will be "The spectrum of $x(t)$ is discrete".

Now, if I apply a rectangular window to the complex exponential $x(t)$ in the time domain and then take CTFT I will end up with $\mathrm{sinc}$ function. Now the spectrum of the windowed complex exponential is continuous and not discrete.Is this interpretation true?

• I'd suggest asking a specific question, and also editing the title to summarize your question. – MBaz Feb 12 '17 at 1:51
• This wasn't on hold when I started answering, and so I gave it an attempt. After trying to answer it, I agree now that you should refine your title and question. I hope my attempt to answer will help you in that regard. I think I have a pretty good idea of what you are asking now, so if you want me to help clean up the question, I probably can. I don't want to do too much without permission from you (the OP) though, because it would require quite a bit of alteration. – hops Feb 12 '17 at 17:16

It is true that a complex-valued signal $x(t) = \exp\left(j \omega_0 t\right)$ and the Dirac delta function form a CTFT pair. I can agree with you that the spectrum of this signal is discrete (nonzero for a finite number of frequencies, in this case a single frequency).
After applying a rectangular window in time to the complex-valued signal $x(t)$, the CTFT is a frequency-translated sinc function centered at $\omega_0$ with a lobe width inversely proportional to the size of the time domain window. This shows us that the result of truncating a time domain signal with infinite support in time and a discrete spectrum in frequency can lead to a new time domain signal with finite support and a continuous spectrum. So, the answer is yes, the interpretation given in the question is true.
• If I'm understanding your question, then yes. The sinc-like response of the DTFT is a direct result of applying a rectangular window to the complex exponential in the time domain and sampling it. If you apply this window and then take the CTFT, you do end up with a sinc response (not sinc-like). When the spectrum of this signal with infinite bandwidth becomes $2\pi$-periodic after downsampling, you obtain the sinc-like response alluded to in the answer. – hops Feb 12 '17 at 16:53