Question 24.9: Amplitude modulation...
Amplitude modulation
Amplitude modulation is a technique that allows audio signals to be transmitted as electromagnetic radio waves. The maximum frequency of audio signals is typically 10 kHz. If these signals were to be transmitted directly then it would be necessary to use a very large antenna. This can be seen by calculating the wavelength of an electromagnetic wave of frequency 10 kHz using the formula
c=f \lambdaHere c is the velocity of an electromagnetic wave in a vacuum \left(3 \times 10^8 \mathrm{~m} \mathrm{~s}^{-1}\right), f is the frequency of the wave and A is its wavelength, and hence \lambda=\frac{c}{f}=30000 \mathrm{~m}. It can be shown that an antenna must have dimensions of at least one-quarter of the wavelength of the signal being transmitted if it is to be reasonably efficient. Clearly a very large antenna would be needed to transmit a 10 kHz signal directly. The solution is to have a carrier signal of a much higher frequency than the audio signal which is usually termed the modulation signal. This allows the antenna to be a reasonable size as a higher frequency signal has a lower wavelength. The arrangement for mixing the two signals is shown in Figure 24.5.
Let us now derive an expression for the frequency spectrum of an amplitude-modulated signal given by
where \omega_c is the angular frequency of the carrier signal and x(r) is the modulation signal.
Now \phi(t)=x(t) \cos \omega_c t can be written as
Taking the Fourier transform and using the first shift theorem yields
\begin{aligned} \mathcal{F}\{\phi(t)\}=\Phi(\omega) & =\mathcal{F}\left\{\frac{x(t)\left(\mathrm{e}^{j \omega_t t}+\mathrm{e}^{-\mathrm{j} \omega_2 t}\right)}{2}\right\} \\ & =\mathcal{F}\left\{\frac{\mathrm{e}^{j \omega_e t} x(t)}{2}\right\}+\mathcal{F}\left\{\frac{\mathrm{e}^{-j \omega_2 t} x(t)}{2}\right\} \\ & =\frac{1}{2}\left(X\left(\omega-\omega_{\mathrm{e}}\right)+X\left(\omega+\omega_{\mathrm{e}}\right)\right) \end{aligned}where X(\omega)=\mathcal{F}\{x(t)\}, the frequency spectrum of the modulation signal.
Let us consider the case where the frequency spectrum, |X(\omega)|,, has the profile shown in Figure 24.6(a). The frequency spectrum of the amplitude-modulated signal, |\Phi(\omega)|, is shown in Figure 24.6(b). All of the frequencies of the amplitude-modulated signal are much higher than the frequencies of the modulation signal thus allowing a much smaller antenna to be used to transmit the signal. This method of amplitude modulation is known as suppressed carrier amplitude modulation because the carrier signal is modulated to its full depth and so the spectrum of the amplitude-modulated signal has no identifiable carrier component.
