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Consider the following two statements:

In time axis:

  1. A signal without a DC component is a signal which doesn't have the zero frequency (the DC frequency)
  2. A signal without a DC component is averaged to zero

My questions are:

  1. Why is it incorrect? It sounds like the definition to me.
  2. What is the explanation? It has something to do with the Fourier representation of the signal, I suspect.
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    $\begingroup$ Useful link here. $\endgroup$ – Gilles Mar 11 '18 at 14:26
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Statement 1, as I read the garbled syntax, is correct, assuming "without a" and "doesn't have" imply a "zero" (an answer) and not a "null" (no answer).

The zero bin, also known as the DC bin, is the straight up sum of the signal over the sample frame in an unnormalized DFT calculation. If the sum is zero, the average is zero. So statement 2 is correct.

The explanation comes straight from the unnormalized definition of the DFT:

$$ X[k] = \sum_{n=0}^{N-1} { x[n] e^{ -i2 \pi { k \over N } n } } $$

Substitute in k = 0:

$$ X[0] = \sum_{n=0}^{N-1} { x[n] } $$

What is the source of these two statements? Maybe there is a larger context to take into consideration.

Hope this helps,

Ced

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  • $\begingroup$ Thank you. It's taken from some exam (sorry for the poor translation). $\endgroup$ – deficiencyOn Mar 11 '18 at 14:49
  • $\begingroup$ @deficiencyOn, You're welcome. Tricky wording is a problem with exam questions, translated or not. BTW, DC stands for Direct Current, so is really only appropriate in electronic applications, although is used more broadly than that. The "time axis" part throws me too. The components and frequency are frequency domain concepts. $\endgroup$ – Cedron Dawg Mar 11 '18 at 14:59

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