A study of AM signals is a convenient tool for showing a damaging effect arising from phase dispersion.
Amplitude modulation, or AM, is probably the simplest method of getting a voice or some music onto a radio signal and then sending that signal off to some far distant place. Because of that simplicity, a study of AM signals is a convenient tool for showing a damaging effect arising from phase dispersion.
With AM radio, the audio signal usually gets reproduced pretty nicely at the receiving end, but not always. Sometimes a phrase like, “You give us 22 minutes and we’ll give you the news” can come out sounding like “Y’mph gvmmph ush tentee-two mnshunts …” and maybe you would wonder why.
Consider an AM signal source from which there is a carrier that, just for the sake of example, we can amplitude modulate using either the first or the second of two “audio” signals or by using those two audio signals at the same time. The governing equations would look like the following:
Signal = Carrier + Lower Sideband 1 + Upper Sideband 1 + Lower Sideband 2 + Upper Sideband 2
We let Fc be the frequency of the carrier, Fmod1 be the frequency of the first “audio” signal 1, and Fmod2 be the frequency of the second “audio” signal 2.
In radian frequency, we let:
Wc = 2 * pi * Fc
Wm1 = 2 * pi * Fmod1
Wm2 = 2 * pi * Fmod2
Carrier = K0 * Sin (Wc * t)
Lower Sideband 1 = LSB1 = K1 * sin ((Wc − Wm1) * t)
Upper Sideband 1 = USB1 = K1 * sin ((Wc + Wm1) * t)
Lower Sideband 2 = LSB2 = K2 * sin ((Wc − Wm2) * t)
Upper Sideband 2 = USB2 = K2 * sin ((Wc + Wm2) * t)
Normalizing to the carrier amplitude, we let K0 = 1. We then set that K1 and K2 will be less than one and just for the sake of making the figure below look nice, I arbitrarily chose that Fc = 10 MHz, Fmod1 = 1 MHz, Fmod2 = 2.5 MHz, K1 = 0.3, and K2 = 0.2.
On a spectrum analyzer, we would see the carrier at its particular frequency, Fc, plus, for each audio signal, there will be a pair of sidebands; the upper sideband at the frequency Fc + Fmod and the lower sideband at the frequency of Fc − Fmod. If we have a spectrum analyzer and a fast enough scope, we can obtain displays along the lines (no pun intended) of Figure 1.
Please note that the envelopes of the various waveforms are also illustrated, as well as the signals themselves.
Figure 1 This figure illustrates amplitude modulation with no phase dispersion.
Note that, for the absence of any distortion, the peaks of the carrier waveform follow envelopes which exactly track the modulating audio waveforms.
As our artfully-produced AM radio signal travels from New York City to Brisbane, as it goes up into the ionosphere and back down again, the transit time from transmission at the starting location to the point of reception may differ for the carrier versus the various sidebands. The result is phase shifting of the sidebands with respect to their carrier. The overall effect is called phase dispersion.
Introducing phase dispersion into our equations, we introduce phase angle changes in the sideband signals with respect to the carrier as follows:
LSB1 = K1 * sin ((Wc − Wm1) * t − DegL1 * pi / 180)
USB1 = K1 * sin ((Wc + Wm1) * t + DegU1 * pi / 180)
LSB2 = K2 * sin ((Wc − Wm2) * t − DegL2 * pi / 180)
USB2 = K2 * sin ((Wc + Wm2) * t + DegU2 * pi / 180)
In the real world, I have no idea how many degrees of phase shift might actually be involved, however, just for the sake of making another viewable picture, I chose the phase shifts quite arbitrarily.
I chose that DegL1 = −45°, DegU1 = +45°, DegL2 = −112.5°, and DegU2 = +112.5°.
The effect on the waveforms is very much in evidence.
Figure 2 The effect of the phase dispersion is evident.
The envelopes from Figure 1 are reproduced here to show that because of phase dispersion, the peaks of the signal no longer conform to the original modulation waveform envelopes, even though the carrier and the sidebands are still at their same frequencies.
This illustrates the distortion, which is why the announcer sometimes sounds like he’s suffering from a 30-second sneeze.
John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).