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Given a signal $x(t) = \frac4{10}\cos(800πt) + \frac12\cos(820πt) + \frac1{10}\cos(880πt)$ and knowing that the sampling frequency is $4000$ Hz.

How many samples are needed at least to represent the signal in a DFT representation?

A solution I've found states that you need to take the difference between the 2 smallest frequencies in the signal which are in this case $800π/2π$ = $400$ and $820π/2π$ = $410$ and compute $410-400 = 10$

And then use $F_s/N \le 10 \Leftrightarrow 4000/N \le 10$ resulting in $ N \ge 400$

Is this correct?

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    $\begingroup$ It wouldn't be of great help if we told you whether or not that formula is correct if you have no idea why and when it can be applied. Have you tried to understand what's going on, and what determines the number of required samples? This is a homework style question, and you should show some effort trying to solve the problem yourself. $\endgroup$
    – Matt L.
    Jun 10 at 14:51
  • $\begingroup$ And how's "needed" defined? A valid but unlikely answer is 3. Other answers can be based on aliasing or "undersampling". You'd ask a better question by disputing your source, if the source hasn't defined it, but "just need answer"'s off topic. $\endgroup$
    – OverLordGoldDragon
    Jun 11 at 5:48

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