Fourier transform: signal with filter

• JustPeter
In summary: 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.I think that might be what you are looking for. The Fourier transform of the product of a step function and some other function is the convolution of their Fourier transforms.
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)= ∫0T 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

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

BvU said:
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-eiωT/2) = T⋅e-iωT/2⋅sinc(ωT/2)

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

BvU said:
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!

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 ...

What is a Fourier Transform?

A Fourier Transform is a mathematical tool used to break down a signal into its individual frequency components. It converts a signal from the time domain to the frequency domain, allowing us to analyze the signal in terms of its frequency components.

What is a signal with filter?

A signal with filter refers to a signal that has been processed or altered by a filter, which is a device or algorithm that modifies the signal in some way. This can include removing certain frequencies, amplifying certain frequencies, or changing the shape of the signal.

How does a Fourier Transform work?

A Fourier Transform works by decomposing a signal into its individual frequency components using a complex mathematical formula. It breaks down the signal into a series of sine and cosine waves of varying amplitudes and frequencies, which can then be analyzed separately.

What are the applications of Fourier Transform?

The Fourier Transform has many applications in various fields, including signal processing, image processing, data compression, and physics. It is commonly used to analyze and filter signals in communication systems, medical imaging, and audio processing, among others.

What is the difference between a Fourier Transform and a Fourier series?

A Fourier Transform is used for continuous signals, while a Fourier series is used for periodic signals. The Fourier series decomposes a periodic signal into a series of harmonically related sinusoidal components, while the Fourier Transform can be used for both periodic and non-periodic signals.

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