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I'm using an antenna that works in the range from 1 GHz to 3 GHz. I want to match impedances between a cable from the transmitter and the input amplifier and from the output amplifier and the cable to the receiver to get the maximum power and the lossless.

I found that there is a technique called Bode-Fano equations when you use a wide range of frequencies (1GHz to 3GHz) instead only one frequency, but I don't understand it how could I used it. Can you help me?

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The Bode-Fano equations are not what you are looking for.

From a brief read around the net, it seems they are related to the study of the trade off between bandwidth, size and efficiency.

This is an interesting topic, there seem to be some fundamental limits to the size, bandwidth and efficiency of small antennas. This study is mainly interesting and useful when considering small antennas ( \$d<\lambda/2\$, narrow bandwidth (under a few %), and low efficiency (well below 100%).

Your antenna is most likely the opposite of most of these - wide band, efficient and probably bigger than half a wavelength - so these might not be directly applicable here.

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