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If an an over-the-air radio signal of a certain frequency $f_c=300$ MHz travels in my direction while I'm standing still, then I observe the signal travelling towards me at the speed to light and I measure no Doppler switch.

However, if the speed of light is always constant to the eyes of a the observer regardless if the observer is moving or not, I should also measure $300$x$10^6$ periods a second (and thus no Doppler shift) if I move towards the light source. But theory tells that, when moving towards the light source, positive Doppler should happen.

Where is the mistake in my understanding?

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if you move relativ to the sending source of the wave, your time scale is different, so the frequency you measure is different.

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