4
$\begingroup$

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

$\endgroup$
5
  • $\begingroup$ Wait, $\hat g$ and $\hat f$are the Fourier transforms of what? $\endgroup$ Dec 16, 2016 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, 2016 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? $\endgroup$ Dec 16, 2016 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... $\endgroup$ Dec 16, 2016 at 9:37
  • $\begingroup$ @MarcusMüller updated the question $\endgroup$
    – Jishan
    Dec 16, 2016 at 9:56

1 Answer 1

6
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ could you show me the drawing please. I am still a bit confused $\endgroup$
    – Jishan
    Dec 16, 2016 at 13:18

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.