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You know that a sine corresponds to a pulse by J.Fourier transform. The lower is the frequency, the closer is the pulse to the origin. A constant signal is a sine (or cosine, that may be important) of frequency 0. It is a pulse in the origin. This is ok, since we know very well that a pulse transforms to the white spectrum: it is a combination of all sine waves at once.

But, I see a contradiction here. I have just shown that a constant is also a sine of frequency 0 and, also, a combination of all frequencies. A sine of frequency 0 is a sum of all frequencies. How is this possible?

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Val
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You are confusing time domain and frequency domain. A constant time domain function (a sine with frequency 0, if you like) corresponds to a delta impulse at the origin (i.e., frequency zero!) in the frequency domain. A delta impulse at frequency zero is zero for all other frequencies $\omega\neq 0$. Consequently, a constant in the time domain is not a combination of all frequencies. What you were probably thinking of is a delta impulse in the time domain, which corresponds to a constant in the frequency domain (i.e., it contains all frequencies).

So, summarizing, a constant in one domain corresponds to a delta impulse in the other domain. A constant in the time domain does not contain any frequencies other than 0 (delta at zero in the frequency domain). A constant in the frequency domain (i.e. a combination of all frequencies) corresponds to a delta impulse in the time domain.

Matt L.
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  • Yes, I shamefully flipped the domains without ever noticing this. So spread-minded. – Val Mar 31 '14 at 10:43