Why doesn’t DFT[0] tell us anything about the frequency of the input?

[Ross C]

Question from sp-m1-4-discrete-fourier-transform:
“Why doesn’t DFT[0] tell us anything about the frequency of the input?”

My attempt at an answer:

Because it only reports back the sum of the magnitude samples. It doesn’t relate these to different phase angles (= horizontal positions on a cosine wave) because at k=0 the angle is always set to 0, and so all the samples are just multiplied by 1 before summing them.

[Catherine Lai]

Yes, that’s right!

We can interpret this as telling us the bias of the input amplitude in the time domain. That is, when reconstructing the original waveform should we shift all the cosines representing our frequency components up (positive bias), down (negative bias) or keep them centred at zero (zero bias).

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