How to Approach This Challenging Fourier Transform Problem?

[asdf12312]

Homework Statement

Homework Equations

here is list of Fourier transforms:
http://uspas.fnal.gov/materials/11ODU/FourierTransformPairs.pdf

The Attempt at a Solution

so I know the solution but I don't know how to get it. Here is what I think to do: the ramp function r(t) and the rect pτ(t). I know that r(t) is the integral of step u(t) which is integral of d(t), and I know the transform of u(t)..would that help? I guess I'm not really sure what to do. In the solution they use integral of pτ(t) minus delta function d(t)..not sure why.

 

[vela]

Well, you haven't really made much of an attempt here. All you've done is throw out some random facts that may or may not have to do with the problem.

You're given a graphical representation of x(t), and you want to find the Fourier transform
$$X(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty x(t) e^{-i\omega t}\,dt.$$ To evaluate the integral, you're going to need to figure out what to plug in for x(t). Start there.

 

[rude man]

Well, you haven't really made much of an attempt here. All you've done is throw out some random facts that may or may not have to do with the problem.

You're given a graphical representation of x(t), and you want to find the Fourier transform
$$X(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty x(t) e^{i\omega t}\,dt.$$ To evaluate the integral, you're going to need to figure out what to plug in for x(t). Start there.

I think you r integrand is in error ... the exponential?

 

[vela]

Fixed it.

 

[rude man]

The OP should also be forewarned that there are two versions of the Fourier transform pair.

vela gave one in his post #2. I gave the other in the duplicate post, to wit,
$$X(f) = \int_{-\infty}^\infty x(t) e^{-i\omega t}\,dt$$
where ω = 2πf.

I personally prefer mine since the annoying 1/2π factor is absent. In vela's way that fracton has to go in front of the transform but left out in the inverse transform. On the other hand, my way introduces f as the transform parameter while keeping ω in the exponential. Of course, you can get around this by substititing ω = 2πf in the exponential, but I never do. Pretty easy to remember ω = 2πf.

Pick your poison ...

 

[asdf12312]

well i could say x(t) is X+1 from -1<x<0 and 1 from 0<x<1 but i don't think that's doable with transforms. I am trying to follow how they got the solution, since x(t) has r(t) and u(t), could use integral ∫ of (u(t) - d(t)). but i think that, since the r(t) is only from -1<x<0 they used the rect function instead of u(t) (which is kinda same thing I guess). so I understand this much:

∫p1(λ+1/2) - d(λ-1) dλ

and transform of rect pτ(t) is τsinc(τω/2pi), transform of d(t-c) is e^-jωc.

so they should get something like ∫sinc(ω/2pi)e^jω/2 - ∫e^-jω which would be...

2/jω (sinc(ω/2pi)e^jω/2) + 1/jω (e^-jω ) ?

 

[rude man]

well i could say x(t) is X+1 from -1<x<0 and 1 from 0<x<1 ?

No you couldn't. Look again at what x(t) is from t = -1 to t=0.
Then use the integral vela or I gave you (depends on how your teach defines the Fourier transform) with suitable limits of integration.

 

[LCKurtz]

You are making it way too hard. Your function is zero except on $(-1,1)$. Just plug in the formulas for $x(t)$ and do the integration.

 

[asdf12312]

i am getting confused.. why is x(t) not t+1 from t=-1 to t=0 though? I guess the problem is I have no way to find transform of r(t+1), so instead I have to use integral of u(t) (or the rect p(t) which is easier). Is there another way to do it? I am just doing it the way they did it, to get the answer which is:

1/jω [ sinc(ω/2pi)e^jω/2 - e^-jω ]

just trying to figure out how they got that (how they could factor out 1/jω like that).

 

[rude man]

There is also the integration property: for the left side. the ramp is the integral of a constant from -1 to 0
The right side is the integral of the delta function delta(t) from 0 to +1
etc. There are various possibilities depending on where you want to start.

 

[LCKurtz]

i am getting confused.. why is x(t) not t+1 from t=-1 to t=0 though?

It is $t+1$ as you said, except you didn't use $t$ as the variable.

 

[rude man]

i am getting confused.. why is x(t) not t+1 from t=-1 to t=0 though? I guess the problem is I have no way to find transform of r(t+1), so instead I have to use integral of u(t) (or the rect p(t) which is easier). Is there another way to do it? I am just doing it the way they did it, to get the answer which is:

1/jω [ sinc(ω/2pi)e^jω/2 - e^-jω ]

just trying to figure out how they got that (how they could factor out 1/jω like that).

The answer I got is X(ω) = [1 - exp(jω) + jw exp(-jω)]/ω².

It sounds like they want you to use the transform of the triangle function for -1<t<0 and the rectangle for 0<t<1. I looked at that approach briefly & didn't see much advantage to it. Quite the opposite, actually. But I'm no Fourier expert & there may well be a trick to it that I don't see.

I suppose you could let sinc(ω/2pi) = sin(ω/2pi)/(ω/2pi)
then sin(ω/2pi) = [exp j(ω/2pi) - exp -j(ω/2pi)]/2j

which should eventually give the answer I got (assuming I got it right to begin with).

 

[vela]

well i could say x(t) is X+1 from -1<x<0 and 1 from 0<x<1 but i don't think that's doable with transforms. I am trying to follow how they got the solution, since x(t) has r(t) and u(t), could use integral ∫ of (u(t) - d(t)). but i think that, since the r(t) is only from -1<x<0 they used the rect function instead of u(t) (which is kinda same thing I guess). so I understand this much:

∫p1(λ+1/2) - d(λ-1) dλ

and transform of rect pτ(t) is τsinc(τω/2pi), transform of d(t-c) is e^-jωc.

I think you're saying that
$$x(t) = \int_{-\infty}^t [p_1(\lambda+1/2)-\delta(\lambda-1)]\,d\lambda,$$ where $p_1(t)$ denotes a rectangular pulse of unit width and centered at the origin. If so, that's fine.

so they should get something like ∫sinc(ω/2pi)e^jω/2 - ∫e^-jω which would be...

2/jω (sinc(ω/2pi)e^jω/2) + 1/jω (e^-jω ) ?

No, you don't just replace the integrand with its Fourier transform. Use the property in the table you linked to for the transform of
$$\int_{-\infty}^t f(\tau)\,d\tau.$$

 

[LCKurtz]

The answer I got is X(ω) = [1 - exp(jω) + jw exp(-jω)]/ω².

I agree with that answer (using your version of the FT integral).