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A fairly expensive spectrum analyzer is needed to resolve the sidebands of an AM signal separately from their carrier wave.
If the analyzer is capable of doing so, and is used as required to do so, then barring abnormalities and as Neil states, a double-sideband AM signal will show a constant carrier amplitude on that analyzer display screen, as it is modulated.
During modulation, two additional r-f signals will be present in the output spectrum of the transmitter. One appears below the carrier frequency, and one above the carrier frequency.
The frequency separation of those sidebands from the carrier is a function of the modulating frequency. The amplitude of each of those sidebands with respect to the amplitude of the carrier is a function of the percentage of amplitude modulation.
An example of this spectrum display is shown in Figure 18 at the link below.
http://www.anritsu.com/en-GB/Media-Room/Newsletters/files/guidetospectrumanalyzers.pdf
