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All of the Apollo radios were full duplex, so these noises could be generated even when an astronaut was not talking or his VOX was tripped.
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Did you mean VOX was NOT tripped?
Yeah. Poor grammatical framing. I mean to say NOT (talking OR vox_tripped). The astronaut's transmitter stays on the air, and the ground continues to listen, even when the path between the microphone and transmitter is not active at the moment.
Not being a ham operator or an electronics expert, I have deferred to you for many of the electronics "anomalies" discussed in Apollo. Do the "chirps" exist because of the internals of each components of the system, then?
I can't say what the exact mechanism is in this exact case, but it is common in analog radio and audio systems to have nonlinear distortion. When you sum two signals A and B, the output of the system is supposed to be simply A+B. With nonlinear distortion, the output is something else; it may be "clipped" to some limit, or it may be something else (usually less) than the actual sum. There can be any number of causes: poor design, excessive signal amplitudes, component degradation or failure, etc. Even a loose or dirty connector. This could happen anywhere along the signal chain, including in the receiver. I haven't watched a lot of the video but it did seem that it happens when the astronauts are very close to the rover camera so it could be overloading of the LCRU VHF receiver.
If you work out the math (Fourier transforms), nonlinear distortion produces frequencies not present in the original signal. If you put frequencies F1 and F2 into the system, the output may contain not only the original frequencies F1 and F2, but also n*F1+m*F2. If n and m are nonzero, including negative values, you get "intermodulation distortion". This is intentionally done in 'mixers', devices widely used in radio receives and transmitters to shift the frequency of a signal.
Each PLSS generates two telemetry subcarriers: the LMP at 3.9 and 7.35 kHz and the CMP at 5.4 and 10.5 kHz. Remember that the CMP receives the LMP's signal on an FM link at 279 MHz, adds his own, and transmits the sum to the LCRU (or LM) for relay to earth. Intermodulation distortion could easily produce difference frequencies between the various telemetry carriers falling into the audible range (300 to 2300 Hz). As I said before, the effect seems to happen when the astronauts are very close to the TV camera and LCRU so the commander's AM transmitter could be overloading the LCRU's receiver. You probably wouldn't get this effect on earlier (pre-J) missions using the LM for communications relay since the LM's EVA relay antenna is mounted on the top of the LM, away from the astronauts on the surface.