Dr. Riki Morikawa, George Mason University
 

AM Modulation

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                Amplitude Modulation (AM) is an analog modulation technique in which information (e.g., music, voice, etc.) is impressed upon a carrier wave's amplitude.  We use carrier waves in order to separate communication channels from one another.  Your radio is a prime example. The channels on your AM radio range from 535 kHz to 1.7 MHz.  So when you dial in 535 kHz on your radio, you're actually tuning in the carrier wave's frequency and not the frequency of the actual information signal used to modulate the carrier wave.

                Let's consider the following example in order to illustrate the concept. Fig. 1 show our information wave.  Note that real information such as voice or music does not appear as a pure sine wave.  We choose a sine wave because it is simpler to demonstrate how AM works.  Our information wave, denoted m(t), has an amplitude of 6 volts, a period T = 0.01 seconds.  Since we know the period of the wave, we can easily determine the frequency:  f = 1/T = 100 Hz.  The information signal is represented mathematically below.

m(t) = A_m sin (2πf_mt ± φ)

where A_m=amplitude, f_m=message's frequency, t is time, φ is phase which equals 0 in our example.

Figure 1.  Information wave (a.k.a. message as a function of time, m(t))

A_m = 6v

f_m = 100 Hz

φ = 0

                Our carrier wave will also be represented as a sine wave, but with a much higher frequency, fig 2.  The carrier equation with a frequency fc = 1000 Hz = 1 kHz, T=0.001s,

c(t) = A_c sin (2πf_ct ± φ), where A_c=6, f_c=1 kHz, φ = 0

Figure 2.  Carrier wave, c(t)

                It's important that the ratio of the modulating wave's amplitude, A_m, to the carrier wave's amplitude, A_c, is between 0 and 1.  This ratio is called the modulation index.  We will see later how changing this ratio effects out modulated signal wave, s(t) (i.e., the modulated carrier wave).

modulation index, μ = A_m/A_c, where  0 ≤ μ ≤ 1

                Since we want to represent our message by modifying the carrier's amplitude, we simply add our message, m(t), to the A_c.  Therefore, our signal is:

signal wave, s(t) = [A_c + m(t)] sin (2πf_ct ± φ)

                By substituting m(t) into s(t), and with φ = 0:

                        s(t) = [A_c + m(t)] sin (2πf_ct)

                                =  [A_c + A_m sin (2πf_mt)] sin (2πf_ct)

                Manipulating the modulation index equation gives us:  A_m= μA_c , and substituting A_m into the above equation give us the following.

                        s(t) =  [A_c + μA_c sin (2πf_mt)] sin (2πf_ct)

                                =  A_c [1 + μ sin (2πf_mt)] sin (2πf_ct)

                This resultant equation of the modulated carrier signal, s(t), is graphed below in fig 3.

Figure 3.  Blue colored signal wave, s(t), which represents the modulate carrier.  The red wave represents m(t)

                Let's consider the modulation index μ when the ratio of A_m/A_c is greater than 1.  In this case, we'll make μ=10 shown in fig. 4.

Figure 4.  μ = 10

                Fig. 4 shows that we begin to see distortion when m(t) is at the most negative amplitude of its cycle.  This means that the signal is beginning to deteriorate.  In fig 5, we increase the modulation index μ to 80 and we see that our signal distortion is at a point where our information or message signal cannot be extracted.

Figure 5.  μ = 80.

Also see  Way to solve a modulation problem