Fourier Transforms: cos(pi*t)[u(t+0.5)-u(t+0.5)], etc

[mikalcarbine257]

Hello forum,

I am having a little difficult with Fourier transforms. I understand the concept completely and the proof but implementing it (with look up tables) seems to be causing problems with me.

I have two problems in particular:

The image doesn't seem to be posting in the thread so check here: http://img227.imageshack.us/my.php?image=fouriersz7.jpg

Sorry for the crappy graphs, best I could do in paint haha

I need to read these graphs and compute the fourier transform of them. My biggest problem is conveying the graph in terms of a function, here is what I have for a)

cos(pi*t)[u(t+0.5)-u(t+0.5)]

and for b)

e^(t)[u(t+1)-u(t)]+e^(-t)[u(t)-u(t-1)]

Do these seem right? My teacher gave us just the answers but I can't seem to get the transform right so I am thinking I am having problems with conveying the graph.

Any help is appreciated!

 

[Unco]

Re: Fourier Transforms

Hi Mikal,

If u is the Heaviside function, then your descriptions are absolutely correct. Show us your attempts and what you're expecting so we can sort out where you're tripping up (if at all).

 

[mikalcarbine257]

Re: Fourier Transforms

Yes, u is the heaviside function in this case, sorry about forgetting to mention that

Thanks for the reply, turns out I tried part b over again and was able to work it out

As for part a)

I start using the transforms for the functions cos(w*t) and u(t) from the lookup tables but it appears that my teacher used cos(w*t)<->pi*[D(w+w0)-D(w-w0)] (where D is the unit impulse function) and the pulse function ptau(t) <-> tau*sinc(tau*w/2*pi).

His answer is: 0.5{sinc[(w+pi)/(2*pi)]+sinc[(w-pi)/(2*pi)]}, is the above a correct assumption? If so the function would then be p1(t)*cos(pi*t)?

 

[Unco]

Re: Fourier Transforms

It is difficult without knowing what formulas you have to choose form (not to mention notation: what is p1(t)?). Rewriting part (a) in terms of rect(t):
\(\displaystyle f(t) = \cos{\pi t}\, \text{rect}(t)\)
and using the formula on the second page of here
\(\displaystyle A\cos{\left(\frac{\pi t}{2 \tau\right)}\text{rect}\left(\frac{t}{2\tau}\right) \mapsto \frac{A\pi}{\tau} \frac{\cos{w\tau}}{\left(\frac{\pi}{2\tau}\right)^2 - w^2}\)
gives
\(\displaystyle f(t) \mapsto 2\pi\frac{\cos{\frac{1}{2}w}}{\pi^2-w^2}\),
which is equal to your teacher's answer, using \(\displaystyle \text{sinc}{(x)} = \frac{\sin{(\pi x)}}{\pi x}\).