# Amplitude Modulation

Cuthbert Nyack
This was the first type of modulation used for communicating signals from one point to another and is still the simplest to understand. The signal can be written as:- v = ac (1 + m cos wmt) cos wct
This represents a signal at frequency wc whose amplitude is modulated by another frequency wm.
m = am/ac is the modulation index.
To find the frequency spectrum of the am signal the above expression can be rewritten as a sum of signals of constant amplitude:-
v(t) = ac{cos wct + m/2(cos(wc + wm)t + cos(wc - wm)t)}
Above expression shows that the frequency spectrum consists of 3 components at frequencies wc, wc + wm and wc - wm.

The percent of the power transmitted which is in the carrier is given by Pc = 100/(1 + m * m/2) and varies from 100% for m = 0 to 66.66% for m = 1. This is considered to be one of the disadvantages of AM since the carrier is a sine wave and contains no information.

If the modulating index becomes greater than 1 then the expression for the AM signal amplitude (1 + m cos wmt) can go negative. In practical AM systems The upper amplitude of the envelope will be limited to zero and the modulating index to one. However this needs to be modified for modulation with many sines.
The applet below for Fn = 0 shows how an AM signal is related to the carrier and the modulating signals.
An AM signal can be described by any combination of sines and cosines. As mentioned above:-

v = ac (1 + m cos wmt) cos wct = ac{cos wct + m/2(cos(wc + wm)t + cos(wc - wm)t)}

The following can also be used:-

v = ac (1 + m sin wmt) sin wct = ac{sin wct + m/2(cos(wc + wm)t - cos(wc - wm)t)}

v = ac (1 + m cos wmt) sin wct = ac{sin wct + m/2(sin(wc + wm)t + sin(wc - wm)t)}

v = ac (1 + m sin wmt) cos wct = ac{cos wct + m/2(sin(wc + wm)t - sin(wc - wm)t)}

Fn = 1 to 4 shows these 4 ways of describing an AM signal. They differ in the phase of carrier and modulating signal but the appearance of the AM is the same in all cases.
The applet allows changes in the phase of the carrier and upper sideband when the phase is changed. Carrier phase changes only affect the AM when the AM is derived by summing a carrier and 2 sidebands but not when the AM is derived from multiplying the carrier with the offset modulating signal.
Fn = 5 shows modulation with 2 sinusoids. The spectrum now has 2 pairs of sidebands and a total modulation index can be calculated from the 2 modulation indices.

Fn = 6 to 11 shows AM signals with different non sinusoidal modulating signals. In each case the spectrum and total modulation index is obtained from the Fourier Series representation of the modulating signal. Fn = 9 to 11 shows cases where the average of the modulating signal is zero but the magnitude of the positive peak is different from the magnitude of the negative peak. In all these cases the appearance of the AM can be very different from what one might expect from the total modulation index.

Fn = 8 shows a modulating signal which can be changed by 'a'.
Fn = 12 shows how the AM is built up from the carrier and sidebands for the case where the spectrum has several sidebands. The number of sidebands summed can be changed by 'ns'.
Fn = 11,12 show a modulating signal which can be changed by 'b'.

Fn = 13 illustrates the phasor interpretation of the AM. Position of Phasor plot can be changed by changing Pt. In this plot, the vertical axis is real while the horizontal axis is imaginary. The resultant AM moves up and down along the real axis because the AM does not have any phase variation.

Fn = 14 shows a phasor interpretation of what happens when the carrier is shifted by 90º. The signal here is obtained by adding the shifted carrier and the 2 sidebands. Since the resultant has components along the Vertical and Horizontal axes, then the resulting signal has both amplitude and phase variations and is not an AM signal. The orange x shows the amplitude(length of yellow line) variation. The unmodulated carrier is shown as dark red and the difference in the zero crossings between the cyan and dark red curves is a measure of the phase variation.