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I'm trying to decipher what exactly this statement means. So if I have a signal which has no pure sinusoidal components, does that mean I can't decompose my signal completely into sine-waves which may be amplitude/phase/period shifted? And so I would conclude that there is also like a saw-tooth component or a square-function component mixed up in the signal as well?

Saying that the signal contains NO pure sinusoidal component makes me feel like its decomposition should not include ANY amplitude/phase/period shifted sine-waves.. but that seems a little extreme.

Could someone clarify this for me? Thanks.

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  • $\begingroup$ Some pathological function that was not integrable might not have a Fourier transform. Otherwise wouldn't the existance of an FT of any non-DC signal imply that it did contain at least 1 pure sinusoid? $\endgroup$
    – hotpaw2
    May 4, 2012 at 5:46

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It contains no spikes in its frequency spectrum. (What is the Fourier transform of a sinusoid?)

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As Emre said, what they probably mean that there are no individual (as in, stand alone), sinusoidal components. (Spikes in the spectrum), in the example they are using. (Perhaps it is a sum of many sinusouds).

I think you would be hard pressed to come across a signal that is not composed of complex exponentials (sines and cosines) at all. (I cannot imagine such a signal). Any realistic signal imagined can be decomposed into a summation of sines and cosines.

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  • $\begingroup$ I'm trying to recall my Signals class, about 40 year ago. It was drilled into us that any WAVEFORM can be replicated as a sum of sinusoids, but I don't know how that applies to a complex signal of "infinite" length. $\endgroup$
    – Daniel R Hicks
    May 4, 2012 at 12:49

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