Final answer:
The carrier c(t) of the given AM signal is 20cos(2000π t), and the modulating signal m(t) is determined as cos(200π t). The modulating signal represents the frequencies that are modulating the amplitude of the carrier wave.
Step-by-step explanation:
To determine the modulating signal m(t) and the carrier c(t) for the given AM signal s(t), we need to analyze the components of the signal. The general form of an amplitude modulated (AM) signal is:
s(t) = Ac cos(ωc t) + Ac m(t) cos(ωc t)
where:
- Ac is the amplitude of the carrier wave
- ωc is the angular frequency of the carrier wave
- m(t) is the modulating signal
Given the AM signal s(t) = 5cos(1800π t) + 20cos(2000π t) + 5cos(2200π t), we can identify the carrier wave as the term with the highest amplitude, which is 20cos(2000π t). Therefore: c(t) = 20cos(2000π t)
The sidebands, which are created by the modulation process and contain the information of m(t), are the terms with the lower amplitudes. These are the terms 5cos(1800π t) and 5cos(2200π t), equidistant in frequency from the carrier wave.
From this structure, we can deduce that the modulating signal m(t) has frequencies corresponding to the difference in angular frequency between the sidebands and the carrier (2200π - 2000π = 200π; 2000π - 1800π = 200π), which gives us: m(t) = Acos(200π t)
Since the sideband amplitudes are a quarter of the carrier wave's amplitude, the modulation index must be 0.5.
The modulating signal m(t) then becomes: m(t) = cos(200π t)
Hence, the modulating signal m(t) is a cos wave with an angular frequency of 200π radians per second, and the carrier c(t) is a cos wave with an amplitude of 20 and an angular frequency of 2000π radians per second.