Fourier Transform of a full rectified sine wave

[Dextrine]

Homework Statement

Derive the FT for a full-wave rectified sine wave, i.e., |sin(wt)|

Homework Equations

$$1/(√2π)\int_{a}^{b} |Sin[wt]| {e}^{-i w t}dt$$

The Attempt at a Solution

I'm not entirely sure how to start doing this problem. What I tried doing was noticing that both of these equations are even, thus so is their product. So we can change the limits to go from 0 to infinity and multiply the result by two. this doesn't really help much however. I also tried writing the exponential as sin and cosine but that didn't change anything either...

 

[MaxwellsCat]

Well first notice that the absolute value gives you slightly different versions of the exponential functions that make up $|\!\sin{ωt}|$, right?

Are you supposed to derive the expression you have listed or are you supposed to just find the Fourier transform of $|\!\sin{ωt}|$? You say both of these equations, which two?

 

[Dextrine]

Pretty sure we just have to calculate the Fourier transform and show our work. However, I had yet to try to make the absolute value into exponential form, i will try that and report back if I encounter any more issues. thanks for the push in the right direction.

 

[vela]

You might want to express the signal as the convolution of a single pulse with a train of Dirac delta functions and then use the convolution theorem.

 

[Dextrine]

So I managed to get the problem to something that at least seems more manageable yet I am stuck again. I rewrote $$|Sin[wt]|$$ as $$ \sqrt{Sin^2[wt]}$$ and rewrote the exponential in Trig formand now I have
$$\int_{-inf}^{inf}{\sqrt{Sin^2[wt]}Cos[wt]}$$
the imaginary term canceled out to zero since it is an odd function.

But now I am having difficulty with THIS integral... Is there some sort of special technique to it? Am I at least on the right track?

 

[MaxwellsCat]

So your substitution is technically correct, but I would try writing it as an exponential function - generally MUCH easier to integrate. I'm not sure that your integral is correct either, though I don't have your working.

Are you familiar with how to write sin as an exponential?

 

[Dextrine]

I did try going that route but also got stuck. Here is what I tried when going the exponential route.
$${\frac{1}{2\sqrt{2 pi}}}\int_{-inf}^{inf}|(e^{2iwt}-1)|e^{-iwt}dt$$

which again leaves me with the same problem of dealing with the magnitude.

 

[Dextrine]

I wrote the absolute value of sin[wt] as $$\frac{1}{2}|(e^{2iwt}-1)|$$

 

[MaxwellsCat]

Maybe try and look up how to evaluate integrals with absolute values inside - aside from that you're there. Also keep in mind what a Fourier transform does - it takes a function from position/time-space to frequency-space. That should tell you a little bit about what it should look like - think about what |sin(ωt)| looks like in frequency-space.

 

[vela]

One mistake you should fix before getting too far is that the frequency of the sine function shouldn't be the same variable as the frequency that appears in the complex exponential.

 

[jssamp]

Are you trying to find the Fourier series representation or the Fourier transform of a periodic signal? In either case you don't need to deal with the absolute value. A rectified sine wave is a periodic signal with a period equal to half of the full sinusoid, I would write the sine in exponential form and integrate over one period to find the series coefficients. Then if you want the FT you combine this result and the FT of a frequency shifted impulse in the summation for the time representation of the FS.