Analysis of an Amplitude-Modulated Wave

A portion of an amplitude-modulated output wave (of the type transmitted in radio-broadcast systems, page 489) is shown in Fig. 17(a). The modulating signal is a sinusoidal wave, such as a 1000-cycle tone.

Figure 17. An amplitude-modulated carrier wave a is composed of a carrier-frequency component b, a lower sideband c, and an upper sideband d. The modulating wave used was a single frequency of constant amplitude. (These curves are only approximate.)

The instantaneous magnitude of a sinusoidal current wave is

In amplitude modulation, the value of I_m of a carrier-frequency wave ω_c = 2πf_c is caused to vary in amplitude by a voice-frequency signal wave of frequency f_v in accordance with the relation

where I_c is the maximum value of the unmodulated carrier wave, m is the percentage modulation, and ω_v = 2πf_v.

Substituting equation 2 in equation 1 and writing ω_c for ω gives, as the instantaneous value of an amplitude-modulated wave such as Fig. 17 (a),

From trigonometry, sin ω_vt sin ω_ct = 0.5*cos (ω_c - ω_v) t - 0.5*cos (ω_c + ω_v)t, and if these substitutions are made equation 3 becomes

This is the equation for an amplitude-modulated wave such as shown in Fig. 17 (a), and the equation shows that this wave contains three components as follows: first, a carrier-frequency component, I_c sin ω_ct which is not affected by the modulation process and is shown by Fig.

17(b); second, the lower sideband,¹ 0.5*mI_ccos(ω_c - ω_v)t, of maximum amplitude mI_c/2 and frequency (ω_c/2π - ω_v/2π) or (f_c - f_v), and as shown by Fig. 17(c); third, the upper sideband,¹ mI_c/2*cos(ω_c + ω_v), of maximum amplitude mI_c/2 and frequency (ω_c/2π + ω_v/2π) or (f_c + f_v), and as shown by Fig. 17 (d). It is important to note that, in amplitude modulation, sum frequencies (f_c + f_v) and difference frequencies (f_c - f_v) are created.

The preceding analysis and Fig. 17 are for amplitude modulation by a single-frequency wave. In this instance the sidebands (more correctly, side frequencies) also will be single-frequency waves. For example, if the carrier frequency is 10,000 cycles, and the modulating frequency is 1000 cycles, the lower side frequency will be a single frequency of 9000 cycles, and the upper side frequency will be a single frequency of 11,000 cycles.

If the carrier frequency is 10,000 cycles and the modulating frequency is a telephone conversation that is contained within a band of from 200 to 3500 cycles, then the lower sideband will be a complex wave lying within the band 9800 to 6500 cycles, and the upper sideband will lie within the band 10,200 to 13,500 cycles. It is important to note that in amplitude modulation two sidebands are created and that each of these varies in accordance with the modulating signal and contains all the information of the telephone conversation to be transmitted. The component of carrier frequency is unchanged by the amplitude-modulation process and contains no information. The idea that the component of carrier frequency, or "carrier," as it is usually called, in some way "carries" the information to the distant station is incorrect. In fact, most carrier telephone systems and many radio systems suppress the carrier and one sideband and transmit only one sideband to the distant station.