Application of the Fourier Transform to the Amplitude Modulation

Besides its usefulness for circuit analysis, the Fourier transform is used extensively in a variety of fields such as optics, spectroscopy, acoustics, computer science, and electrical engineering. In electrical engineering, it is applied in communications systems and signal processing, where frequency response and frequency spectra are vital. Here we consider two simple applications: amplitude modulation (AM) and sampling.

Amplitude Modulation

Electromagnetic radiation or transmission of information through space has become an indispensable part of a modern technological society. However, transmission through space is only efficient and economical at radio frequencies (above $ 20 \mathrm{kHz} $ ). To transmit intelligent signals such as for speech and music-contained in the low-frequency range of $ 50 \mathrm{~Hz} $ to $ 20 \mathrm{kHz} $ is expensive; it requires a huge amount of power and large antennas.
A common method of transmitting low-frequency audio information is to transmit a high-frequency signal, called a carrier, which is controlled in some way to correspond to the audio information. Three characteristics (amplitude, frequency, or phase) of a carrier can be controlled so as to allow it to carry the intelligent signal, called the modulating signal. Here we will only consider the control of the carrier's amplitude. This is known as amplitude modulation.
Amplitude modulation (AM) is a process whereby the amplitude of the carrier is controlled by the modulating signal.
$ \mathrm{AM} $ is used in ordinary commercial radio bands and the video portion of commercial television.
Suppose the audio information, such as voice or music (or the modulating signal in general) to be transmitted is $$ m(t)=V_{m} \cos \omega_{m} t $$ while the high-frequency carrier is $$ c(t)=V_{c} \cos \omega_{c} t $$ where $ \omega_{c}>>\omega_{m} $. Then an AM signal $ f(t) $ is given by $$f(t)=V_{c}[1+m(t)] \cos \omega_{c} t$$ Figure $ 1 $ illustrates the modulating signal $ m(t) $, the carrier $ c(t) $, and the AM signal $ f(t) $. We can use the result in Eq. (A) together with the Fourier transform of the cosine function to determine the spectrum of the AM signal:
$$\begin{aligned}F(\omega)=& \mathcal{F}\left[V_{c} \cos \omega_{c} t\right]+\mathcal{F}\left[V_{c} m(t) \cos \omega_{c} t\right] \\=& V_{c} \pi\left[\delta\left(\omega-\omega_{c}\right)+\delta\left(\omega+\omega_{c}\right)\right] \\&+\frac{V_{c}}{2}\left[M\left(\omega-\omega_{c}\right)+M\left(\omega+\omega_{c}\right)\right]\end{aligned}$$
Fig. 1: Time domain and frequency display of: (a) modulating signal, (b) carrier signal, (c) AM signal.
where $ M(\omega) $ is the Fourier transform of the modulating signal $ m(t) $. Shown in Fig. $ 2 $ is the frequency spectrum of the AM signal. Figure $ 2 $ indicates that the AM signal consists of the carrier and two other sinusoids. The sinusoid with frequency $ \omega_{c}-\omega_{m} $ is known as the lower sideband, while the one with frequency $ \omega_{c}+\omega_{m} $ is known as the upper sideband.
Frequency spectrum
of AM signal.
Fig. 2: Frequency spectrum of AM signal.
Notice that we have assumed that the modulating signal is sinusoidal to make the analysis easy. In real life, $ m(t) $ is a nonsinusoidal, band-limited signal - its frequency spectrum is within the range between 0 and $ \omega_{u}=2 \pi f_{u} $ (i.e., the signal has an upper frequency limit). Typically, $ f_{u}=5 \mathrm{kHz} $ for AM radio. If the frequency spectrum of the modulating signal is as shown in Fig. 3(a), then the frequency spectrum of the AM signal is shown in Fig. 3(b). Thus, to avoid any interference, carriers for AM radio stations are spaced $ 10 \mathrm{kHz} $ apart.
Fig. 3: Frequency spectrum of: (a) modulating signal, (b) AM signal.
At the receiving end of the transmission, the audio information is recovered from the modulated carrier by a process known as demodulation.
Example 1: A music signal has frequency components from $ 15 \mathrm{~Hz} $ to $ 30 \mathrm{kHz} $. If this signal could be used to amplitude modulate a $ 1.2-\mathrm{MHz} $ carrier, find the range of frequencies for the lower and upper sidebands.
The lower sideband is the difference of the carrier and modulating frequencies. It will include the frequencies from $$1,200,000-30,000 \mathrm{~Hz}=1,170,000 \mathrm{~Hz}$$ to $$1,200,000-15 \mathrm{~Hz}=1,199,985 \mathrm{~Hz}$$ The upper sideband is the sum of the carrier and modulating frequencies. It will include the frequencies from $$1,200,000+15 \mathrm{~Hz}=1,200,015 \mathrm{~Hz}$$ to $$1,200,000+30,000 \mathrm{~Hz}=1,230,000 \mathrm{~Hz}$$

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