The statement that the amplitude of the signal has to be less than half the amplitude of the carrier wave is related to a concept called the Nyquist-Shannon sampling theorem or Nyquist criterion. This theorem is used in digital signal processing and telecommunications to determine the minimum sampling rate required to accurately reconstruct a continuous-time signal from its samples.
According to the Nyquist-Shannon sampling theorem, in order to faithfully reconstruct a signal, the sampling rate must be at least twice the maximum frequency component present in the signal. This is known as the Nyquist rate.
When modulating a signal onto a carrier wave, the process involves changing the amplitude, frequency, or phase of the carrier wave to represent the information in the signal. In amplitude modulation (AM), the amplitude of the carrier wave is varied in accordance with the instantaneous amplitude of the signal.
To ensure that the modulated signal can be accurately reconstructed at the receiver, the sampling rate used to capture the modulated signal needs to satisfy the Nyquist criterion. In the case of amplitude modulation, the highest frequency component in the modulated signal is equal to the sum of the carrier frequency and the highest frequency component in the baseband signal.
If the amplitude of the signal were greater than half the amplitude of the carrier wave, it would result in distortion and overlap of frequency components during the sampling process. This would violate the Nyquist criterion and lead to aliasing, where higher frequency components are incorrectly represented as lower frequency components in the sampled signal.
By keeping the signal amplitude below half the carrier amplitude, we ensure that the highest frequency component in the modulated signal is within the permissible range for accurate sampling and reconstruction. This restriction helps prevent aliasing and allows for faithful recovery of the original signal at the receiver.