Fourier Transform Time Shifting Property

[frenzal_dude]

Homework Statement

I tried to work out the FT of a sin function with a time delay using first mathematical manipulation, and then using the time shifting property.

However I get two very similar, but for some reason not identical answers.

Homework Equations

Please open the .jpg to see my working out.
The answers are identical except for the 'f' and 'f0' which are circled in red pen.

The Attempt at a Solution

Thanks for your help!

 

[vela]

It follows from the fact that $f(x)\delta(x-x_0) = f(x_0)\delta(x-x_0)$.

 

[frenzal_dude]

It follows from the fact that $f(x)\delta(x-x_0) = f(x_0)\delta(x-x_0)$.

could you please explain further?

 

[Hootenanny]

could you please explain further?

Are you familiar with the Dirac Delta function and its properties?

 

[frenzal_dude]

Are you familiar with the Dirac Delta function and its properties?

Yeh I am. And I understand the equation given. But where is that happening in my working out?

The exponential with f0 and t0 given is simply a complex number, and can't be considered a dirac delta function in the frequency domain right?

 

[vela]

What do you get if you apply that property to, say, $e^{j2\pi f t_0} \delta(f-f_0)$ (where $e^{j2\pi f t_0}$ is the function multiplying the delta function)?

 

[frenzal_dude]

What do you get if you apply that property to, say, $e^{j2\pi f t_0} \delta(f-f_0)$ (where $e^{j2\pi f t_0}$ is the function multiplying the delta function)?

I fully get it now!

Basically it doesn't matter if you use f or f0, the result is the same since you are multiplying with an impulse at f=f0!

Thanks so much for your help!