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So I have a function $$f \in L^{2}(\mathbb{R} )$$ which can be reconstructed from its sample values if a sample rate is: $$\frac{1}{T}=2\Omega $$. The continuous function $$f \in L^{2}(\mathbb{R} );f:\mathbb{R} \rightarrow \mathbb{C},\enspace \mbox{be a 3$\Omega$ bandlimited}$$ $\hat{g}$ is the Fourier transform of the function $g \in L^{2}(\mathbb{R} )$ which is represented by the T-sampled version of $f$. We consider the T-sampled version of $f$ for $T \doteqdot \frac{1}{2\Omega } $

How should I sketch $|\hat{g}|$ if $|\hat{f}|$ has the following appearance: enter image description here

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  • $\begingroup$ Wait, $\hat g$ and $\hat f$are the Fourier transforms of what? $\endgroup$ – Marcus Müller Dec 16 '16 at 9:21
  • $\begingroup$ @MarcusMüller $$\hat{g}\enspace is \enspace the\enspace Fourier\enspace transform\enspace of\enspace the\enspace function\enspace g \in L^{2}(\mathbb{R} )$$ and $$\hat{f}\enspace is \enspace the\enspace Fourier\enspace transform\enspace of\enspace the\enspace function\enspace f \in L^{2}(\mathbb{R} )$$ $\endgroup$ – Jishan Dec 16 '16 at 9:34
  • $\begingroup$ You can write inline math simply by using one $ instead of two. Anyway, where does g come from? How does it relate to f? $\endgroup$ – Marcus Müller Dec 16 '16 at 9:36
  • $\begingroup$ You don't mention g anywhere in your question before you declare it to be the transform of f, which you know contradict in your comment... $\endgroup$ – Marcus Müller Dec 16 '16 at 9:37
  • $\begingroup$ @MarcusMüller updated the question $\endgroup$ – Jishan Dec 16 '16 at 9:56
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I'm answering the following problem:

Given a function $ f $ which has a Bandwidth (Half side) of $ 3 \Omega $ and the following Fourier Transform:

enter image description here

How would look like the Fourier Transform of $ g $ which is a samples version of $ f $ at rate $ T = \frac{1}{2 \Omega} $.

Sampling in the Time Domain creates Replications in the Fourier Domain.
The replications are $ \frac{1}{T} = 2 \Omega $ apart.

Now, just take the function you have in your drawing and add to it a replication of it centered at $ 2 \Omega $.

It means that the information marked as A and B will be added to C and D.

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  • $\begingroup$ could you show me the drawing please. I am still a bit confused $\endgroup$ – Jishan Dec 16 '16 at 13:18

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