# Bandwidth of a trimmed sinus wave

I generate a sinus which duration is 15ms and frequeuncy of 13KHz. Is its bandwidth is (1/15ms) ? Or in other words, BW = 66.6Hz? I ask that because I have two freqs in my system: 13K & 13.1K. I know that they are seperated by 100Hz. And as the tone I generate have a lower duration I think it spreads into its friend area.

• It depends on how you define bandwidth. There is no frequency $f_c$ above which there is zero frequency content, due to the finite duration of the sinusoid, i.e. theoretically the signal has infinite bandwidth. But of course its power decays away from the center frequency. Jul 9 '15 at 16:26
• Let's say -6db. I just need an estimation for that. In other words, is the energy of the 13K will affect seriously the 13.1K? Jul 9 '15 at 16:28
• See my answer below. Jul 9 '15 at 16:45

The spectrum of your windowed sinusoid is just a shifted sinc function (assuming a rectangular window). If $T=0.015s$ is the length of the signal then the magnitude of the spectrum at positive frequencies is given by
$$S(f)=T|\text{sinc}((f-f_0)T)|,\quad f>0$$
with $f_0$ the frequency of the sinusoid, and where I use the definition $\text{sinc}(x)=\sin(\pi x)/(\pi x)$. So with $\Delta f=100Hz$ (the distance to the other sinusoid), you get a relative attenuation of
$$|\text{sinc}(\Delta f\cdot T)|=0.21221\doteq 13.5\,\text{dB}$$
So at the frequency of the neighboring sinusoid, the spectrum of the other (finite length) sinusoid is about $13.5\,\text{dB}$ lower.