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Antennas are generally tested for a nominal gain on the axis, then their radiation pattern is measured in E and H planes (in more planes as needed), for impedance matching over a specified bandwidth, for polarization, etc. Such tests are made either on an open range or in an anechoic chamber.
A tested antenna may be tested as "active" if the test signal is input into it, and the test receiver is located at the opposite side of the range.
Or, the test signal is sent from the opposite side of the range, and received by the antenna under test; then you may call it "passive".
The test results should be the same for both methods.
The only exception may be if a high power should be input to a tested antenna; some antennas can transmit a limited power level.
passive test is as described above, tests of radiation patterns, gain/directivity/efficiency etc...
active test can mean many things, but more often than not involves test of the antenna integrated with the intended electronics and running tests within an anechoic chamber that emulates the intended RF environment. The end result is often a measure of throughput.
active tests often involve antennas in handset designs and could include evdo testing so on. another active test that is currently being studied for standards is active MIMO testing, where the MIMO antennas are tested with the associated electronics to determine impact on the throughput etc...
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