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In the following diagrams, a microphone pinout is shown for several radios which have DTMF mics that work with them.

How is DTMF sent to the radio without a pin designated "DTMF"? Is it through the microphone audio, or is it through the up/down pins, which are used only when the radio is not transmitting? Or is there some other way to send DTMF?

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DTMF is an audio-frequency based signaling system that sends tone pairs as audible frequencies,. Most DTMF microphones send the tones as though they were ordinary microphone audio to be transmitted, so in the diagrams you have shown, the tones will use the same "MIC AUDIO" or "TX AUDIO" path as when you are talking into the mic.

(And if you're interested in what those audio frequencies are, Wikipedia's page on DTMF shows the full tone matrix and a few other details about the signaling system.)

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