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My question is similar to this one and this one, but neither answers address my concern.

Suppose you have a signal x(n) = {-1,2,-3,2,-1} where we can assume x(0) = -3. So this is an even signal.

How do you find the phase of the DFT of x(n)?

user1068636
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  • What you show is not an even signal, did you mean x(n) = {1,2,-3,2,1}? – Dan Boschen Mar 21 '20 at 03:28
  • This post should help you: https://dsp.stackexchange.com/questions/38544/ffts-of-a-complex-signal-separating-the-real-and-imaginary-parts/38545#38545 – Dan Boschen Mar 21 '20 at 03:29
  • Sorry my mistake. Now it's even – user1068636 Mar 21 '20 at 03:30
  • @DanBoschen - I am not sure I see which part of the post is relevant to this question. So the answer to this question is $\pi$, but I'm not sure I see why. – user1068636 Mar 21 '20 at 03:33
  • Your signal is real and it is an even function. So the DFT is real. What is the phase of a real signal? – Dan Boschen Mar 21 '20 at 03:34
  • If you are saying the phase is zero, then why is the answer in textbook $\pi$ ? – user1068636 Mar 21 '20 at 03:35
  • the phase of a real signal can be $0$ or $\pi$, right? Solve for the DFT and then look at the sign if they are all negative then the phase is $\pi$ – Dan Boschen Mar 21 '20 at 03:37
  • @DanBoschen - I am following what you are saying. But is there some general formula for finding the phase of an arbitrary list of real numbers? – user1068636 Mar 21 '20 at 03:40
  • If the real number is positive the phase is $0$, if the real number is negative, the phase is $\pi$. It's that simple. – Dan Boschen Mar 21 '20 at 03:40
  • @DanBoschen - I can ask a new question if you want. But just curious if you know of an example of a discrete signal whose phase is not 0 or $\pi$ ? – user1068636 Mar 21 '20 at 03:43
  • Any complex or imaginary number, so j would be the simplest example. Also $5e^{j\pi/3}$ or 2 + j3 are other examples. – Dan Boschen Mar 21 '20 at 03:46
  • But if you are asking for an example of sequence that would have an DFT that is not real, then refer again to my other question that I linked - it has all the properties of the FT they are trying to make sure you know. (Sorry if I seem like I am being cagey--- we're not supposed to do your homework for you but happy to help where you are stuck) – Dan Boschen Mar 21 '20 at 03:50
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    No problem, this was very helpful – user1068636 Mar 21 '20 at 03:51

1 Answers1

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Here is a hint that will help you:

The DFT is cyclical in time and in frequency. For the sequence given by

$$x(n) = [-1,2,-3,2,-1]$$

With x(0) = -3 would be solved using the standard DFT equation that starts at n=0 using

$$x(n) = [-3, 2, -1,-1, 2]$$

From that you can solve for the DFT and then determine easily for each result what it's phase is.

Dan Boschen
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