In AM (amplitude modulation) radio, there are two very important waveforms—the signal, s(t), and the carrier. All of the information we desire to transmit, voice, music, and so on, is contained in the signal waveform, which is in essence transported by the carrier. Therefore, the Fourier transform of s(t) contains frequencies from about 50 Hz to 20,000 Hz. The carrier, c(t), is a sinusoid oscillating at a frequency much greater than those in s(t). For example, the FCC (Federal Communications Commission) rules and regulations have allocated the frequency range 540 kHz to 1.7 MHz for AM radio station carrier frequencies. Even the lowest possible carrier frequency allocation of 540 kHz is much greater than the audio frequencies in s(t). In fact, when a station broadcasts its call letters and frequency, they are telling you the carrier’s frequency, which the FCC assigned to that station!
In simple cases, the signal, s(t), is modified to produce a voltage of the form
v(t)=[A+s(t)] \cos \left(\omega_{c} t\right)where A is a constant and \omega_{c} is the carrier frequency in rad/s. This voltage, υ(t), with the signal “coded” within, is sent to the antenna and is broadcast to the public, whose radios “pick up” a faint replica of the waveform υ (t).
Let us plot the magnitude of the Fourier transform of both s(t) and υ (t) given that s(t) is
where f_{a} is 1000 Hz, the carrier frequency is 900 kHz, and the constant A is unity.
