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For what signal applies, that its autocorrelation function at point $0$ is zero, i.e. $R_f(0) = 0\ ?$

I know that autocorrelation is (RootMeanSquare)^2 and for $R_f(0)$ this equals: $$ R_f(0) = \frac{1}{T}\int_0^T f(t)^2 dt = \sum_{k=-\infty}^\infty \lvert c_k\rvert^2 $$

But I have no idea how to formulate that signal from equation.

Thank you very much for hint that will guide me to the right way of solving this problems.

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  • $\begingroup$ So if $R_f(0)$ equals $0$, can you say what that implies for $f(t)$? No? Do you understand the concept of integral as the area the graph of $[f(t)]^2$ and the $t$ axis? $\endgroup$ Nov 30, 2017 at 19:14
  • $\begingroup$ You really shouldn't ask two different questions in the same question. Imagine someone would like to answer one, but not the other - then they've got no chance to be accepted as answer! So, remove the second question from your question (ask it separately). $\endgroup$ Nov 30, 2017 at 20:12
  • $\begingroup$ @MarcusMüller done $\endgroup$
    – Filip CZ
    Nov 30, 2017 at 21:08
  • $\begingroup$ @DilipSarwate thank you for your reaction, but I wasn't sure where to fit zeros in the formula, because of 'T' and 't' in the equation. $\endgroup$
    – Filip CZ
    Nov 30, 2017 at 21:12
  • $\begingroup$ $R_f(0) = 0$ is given. $R_f(0) = \sum_{k=-\infty}^\infty |c_k|^2$ is what your book says. Put the two together to get $\sum_{k=-\infty}^\infty |c_k|^2 = 0$. So, the sum of all the terms for which $|c_k|^2$ is positive is cancelled out by sum of all the terms for which $|c_k|^2$ is negative. Can you proceed from here? $\endgroup$ Dec 1, 2017 at 22:04

1 Answer 1

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This (and your other question) is really a clear task to look up the very basic (and unambiguous) definition of the entity at hand (here: the autocorrelation, there length of a convolution) and simply apply that definition.

So: since you're asking for a hint:

Simply write down the formula for the autocorrelation of a signal. Calculate what it is at 0, set that to zero and get your signal.

You got the formula for autocorrelation at 0 in your question, but you copied it from somewhere and denied yourself the chance to actually see what's happening.

You also seem to be prone to using formulas without being sure what the symbols in these mean (your $t$ vs $T$ confusion). So, read the page leading up to that formula.

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  • $\begingroup$ The question asked, right up front, is "What signal has an autocorrelation value of $0$ at $t=0$ or $\tau=0$, i.e. $R_f(0) = 0$? $\endgroup$ Dec 1, 2017 at 0:11

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