Fourier transform of rect function

[thomas49th]

Homework Statement

From the definition of the Fourier transform, find the Fourier transform of rect(t-5).

Homework Equations

$$G(w) = \int^{\infty}_{-\infty}g(t)e^{jwt}dt$$

The Attempt at a Solution

So, I sketched the function which has area 1 and centre at 5, with its lower bound @ 4.5 and upper at 5.5. SOOOO cannot I not just write$$G(w) = \int^{5.5}_{4.5}1e^{jwt}dt$$

for it's Fourier transform? Is that allowed. I know the rect function some how turns into sinc, but in this case how?

I can prove with limits -a and +a but when you have 2 positive or 2 negative limits you don't get the sin(-ax) = -sin(ax) which keeps the sine and gets rid of the cosine!

 

[lanedance]

why not try shifting the integal by using the transform s = t-5

 

[thomas49th]

I got the answer in the end by a clever little exponent trick. However the very next question on my question sheet is "Using the time shift property, compute again the Fourier transform of rect(t-5) and compare the 2 results"

The results being the one I compute for this one and the previous one (which I got sinc(w/2)e^(-jw5))

So basically I'm shifting the rect back around 0, so from -0.5 to 0.5. I believe the shift theorem says that a shift in the time domain translate to a shift in the phase... so a shift of 5 means I have to multiply the exponent value by 5.

Therefore I'm integrating (wrt t) between -0.5 and 0.5 e^(-5jwt)

which gives $$\frac{1}{-5jw}[e^{-5jwt}]^{0.5}_{-0.5}$$

Using euler to expand and sticking in the limits I get sinc(5/2 jwt)

Does that sound correct? I don't see how the 2 answers compare. Have I done it wrong? Can someone shed some light?

Thanks
Thomas

 

[lanedance]

have a look at this
http://en.wikipedia.org/wiki/Fourier_transform#Functional_relationships
F{f(t-a)} = e^(-i.2.pi.a)F(k)
which should line up better?

apologies for the abuse of notation

 

[thomas49th]

Ok so I have to multiply by e^-j2pi(a) where a is the distance of the rect function (centre) to the origin? I was using omega. What's going on?

Euuughhhh what does w (omega) equal again? Isn't it 2pi*f? I don't have an f? Maybe I'm just confusing myself...


After an hours break...

I think i have it:
Can I confirm that all you do is perform the Fourier transform THEN multiply by e^-jwa

and w = 2pi/t (which is 2pi f)

Right?

 

[vela]

After an hours break...

I think i have it:
Can I confirm that all you do is perform the Fourier transform THEN multiply by e^-jwa

Yes, that's right. Note you get this result simply by making the substitution t' = t-a and then integrating.

and w = 2pi/t (which is 2pi f)

Right?

Not quite or you're being sloppy. $\omega=2\pi f$ is correct, but $\omega=2\pi/t$ isn't (t is your integration variable).