I have been trying hardly to check if these equalities are true or false. However, I have not been able to conclude anything. Could you help me, please?

$$y[2n]=h[n]\star x[2n] $$

$$\mathfrak{F}(x[-n])e^{-jp\pi Fm}=\mathfrak{F}(x[-n+m]) $$

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    $\begingroup$ Really ? At least be honest and tell us this is your homework. How do you want us to help here ? We don't even know what x, y,h are $\endgroup$ – MaximGi Feb 21 '16 at 16:26
  • $\begingroup$ Where I study we don't have to do homework, this is just a proposed problem that I want to know how to do. x[n] and y[n] are two not-specified sequences. $\endgroup$ – David103 Feb 21 '16 at 16:30
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    $\begingroup$ Please, allow me to rephrase : both properties you're showing are basics, and you should have, before posting this question, respectively look up "discrete convolution" and "Fourier's transform basic properties" on google and figured out what exactly was preventing you to understand those two equations. $\endgroup$ – MaximGi Feb 21 '16 at 16:35
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    $\begingroup$ I suppose $y[n]=h[n]\star x[n]$, otherwise it doesn't make much sense. It's common here to show your own efforts, so we can see where you're stuck and how we can best help you. $\endgroup$ – Matt L. Feb 21 '16 at 16:36
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    $\begingroup$ I'm with Matt L.: The first statement makes no sense unless we know that $y[n] = h[n] \star x[n]$. The second question is about a time shift, which you should be able to answer from a table of Fourier transform properties. $\endgroup$ – Peter K. Feb 21 '16 at 17:44

I'll try to get you started.

For the first part:

  1. Write out the convolution sum for $y[n]=h[n]\star x[n]$
  2. Replace $n$ by $2n$, which gives you an expression for $y[2n]$
  3. Write out the convolution sum for $h[n]\star x[2n]$
  4. Check if both expressions are equal. If you've done everything right, the answer should be 'no' (i.e., they're not equal). If you fail showing this, add your steps to your question and explain what the problem was.

For the second part:

  1. Write the Fourier transform of $x[-n]$ in terms of the Fourier transform of $x[n]$, and multiply by $e^{-j2\pi fm}$; that's the left-hand side of the equation.
  2. Compute the Fourier transform of the right-hand side. This answer will be helpful (replace $z$ by $e^{j2\pi f}$).
  3. Like step 4 above, just that in this case the answer should be 'yes' (i.e., they're equal).

If you make the variable change $2n=m$ you can easily see that $$y[m]=h[m]*x[m]$$ that is obviously different from $$y[m]=h[m/2]*x[m]$$ therefore it's FALSE.

The second sentence could be explained by the transposition property $$x[-n]=X(-F)$$ and the property of frequency displacement $$X(F)\cdot e^{(-j·2·\pi·m·F)} \longleftrightarrow x[n-m]$$ After applying that you got that it's TRUE.

PS: I'm your classmate

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    $\begingroup$ A bit of $\LaTeX$ would make your answer much more readable. $\endgroup$ – Matt L. Feb 21 '16 at 17:27

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