Figure P .7 .21 shows a carrier of frequency ωc being amplitudemodulated by a sine wave of frequ

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Figure P .7 .21 shows a carrier of frequency ωc being...

Figure $P .7 .21$ shows a carrier of frequency $\omega_{c}$ being amplitudemodulated by a sine wave of frequency $\omega_{m}$, that is, $$E=E_{0}\left(1+a \cos \omega_{m} t\right) \cos \omega_{c} t$$ Show that this is equivalent to the superposition of three waves of frequencies $\omega_{c}, \omega_{c}+\omega_{m},$ and $\omega_{c}-\omega_{m} .$ When a number of modulating frequencies are present, we write $E$ as a Fourier series and sum over all values of $\omega_{m} .$ The terms $\omega_{c}+\omega_{m}$ constitute what is called the upper sideband, and all the $\omega_{c}-\omega_{m}$ terms form the lower sideband. What bandwidth would you need in order to transmit the complete audible range?

Figure $P .7 .21$ shows a carrier of frequency $\omega_{c}$ being amplitudemodulated by a sine wave of frequency $\omega_{m}$, that is,
$$E=E_{0}\left(1+a \cos \omega_{m} t\right) \cos \omega_{c} t$$
Show that this is equivalent to the superposition of three waves of frequencies $\omega_{c}, \omega_{c}+\omega_{m},$ and $\omega_{c}-\omega_{m} .$ When a number of modulating frequencies are present, we write $E$ as a Fourier series and sum over all values of $\omega_{m} .$ The terms $\omega_{c}+\omega_{m}$ constitute what is called the upper sideband, and all the $\omega_{c}-\omega_{m}$ terms form the lower sideband. What bandwidth would you need in order to transmit the complete audible range?

Key Concepts

Amplitude Modulation

Amplitude modulation is a technique used in communications where the amplitude of a high-frequency carrier wave is varied in proportion to the instantaneous amplitude of a lower-frequency information (modulating) signal. This modulation process embeds the information signal into the carrier, making it suitable for long-distance transmission.

Sidebands

Sidebands are the frequency components that appear on either side of the carrier frequency as a result of amplitude modulation. The upper sideband contains frequencies that are the sum of the carrier and modulating frequencies, while the lower sideband contains the difference. They carry the information signal and are essential in understanding the spectral composition of an AM signal.

Fourier Series Representation

Fourier series representation is a mathematical tool used to decompose any periodic signal into a sum of simple sinusoidal components (sines and cosines). In the context of modulation, this allows for representing a complex modulating signal as a sum of sine waves, each of which generates its own pair of sidebands when modulating the carrier.

Bandwidth in AM Transmissions

The bandwidth required for an amplitude modulated signal is determined by the frequency content of the modulating signal. When the modulating signal spans the complete audible range, typically from 20 Hz to 20 kHz, the resultant signal will have sidebands extending this range on either side of the carrier. Therefore, to transmit the complete audible range, a total bandwidth of 40 kHz is needed.

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