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How can the Doppler effect cause broadening of a spectral line?

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How can the Doppler effect cause broadening of a spectral line?

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Take an amplitude modulated carrier wave with a fixed modulating frequency. Un-shifted the sidebands would remain constant in width. If the signal source was moving towards us the modulating frequency would also increase. The increased modulating frequency would move the sidebands further away from the carrier, increasing bandwidth or broadening the spectrum.

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