Fourier transform: signal with filter

[JustPeter]

Hi Guys,

I'm having trouble with the following:

A finite-time signal is the result of a filter G(t) applied to a signal. The filter is simply “on” (1) for t ∈ [0,T] and off (“0”) otherwise. If x(t) is the signal, and x(ω),its Fourier transform, compute the Fourier transform of the filtered signal. Next, take a simple sine for x(t), x(t) = sin(ω0t), and compute the Fourier transform for the finite-time signal. Write the result, it must involve the filter, and integrations should stretch [−∞,∞]

I don't really know what to do exactly, with the first problem.

I can try calculating the Fourier transform of the filter:

G(ω)= ∫₀^T e^-iωtdt = -1/(iω)⋅(e^-iωT-1)

The Fourier transform of the signal is: x(ω)

The convolution theorum says that the convolution of two functions is the product of the Fourier-transformed functions. Which makes: G(ω)x(ω).

But I have the idea that this isn't right. Could one of you guys assist me?

Peter

 

[BvU]

The convolution theorem also works the other way around: the Fourier transform of the product of a step function and some other function is the convolution of their Fourier transforms.

By the way: do use the template, don't erase it. It helps you order to your thinking and us to help you better

 

[JustPeter]

The convolution theorem also works the other way around: the Fourier transform of the product of a step function and some other function is the convolution of their Fourier transforms.

Ok. If I understand you correctly, you mean:

Fourier{x(t)g(t)} = 1/2π ⋅ X(ω)⊗G(ω) ==>

Writing -1/(iω)⋅(e^-iωT-1) to -1/iω⋅e^-iωT/2(e^-iωT/2-e^iωT/2) = T⋅e^-iωT/2⋅sinc(ωT/2)

Fourier{x(t)g(t)}=1/2π⋅∫_-∞^∞ X(w-w')⋅T⋅sinc(ω'T/2) dω' ??

By the way: do use the template, don't erase it. It helps you order to your thinking and us to help you better

Sorry, I will do that next time, thanks!

 

[BvU]

Looks reasonable (all the contributions are there -- didn't check the gory details. Most of the time I use a table like this)
I take it you mean $\ x(\omega-\omega_0) \$ ?

Now for the second part ...