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your radio is turning off due to when the starter is turning the voltage drops below the recommended voltage needed to run the radio without damaging the radio. Some radios have this and some don't. Your antenna problem could be several issues, bad coax, bad coil, radio not operating correctly. I would buy a new coax and temporary hook it up to see if that fixes it. SWR meters will say that your SWR is perfect when it is only reading the cable and not the antenna due to a short. Try your radio in a different truck if that is an option or try a magntetic antenna temporary to see if you have a antenna problem or a radio problem