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The temperature is done indirectly. For antennas it is the actual temperature of the objects it is pointed at. For active devices you measure the noise figure and then calculate the tempeature. An alternate way is to use a heated resistor to generate the noise for the test but this is not accurate.
The use of temperature works best for low noise systems.
Nois Figure is the Signal to Noise ratio auf the Output of a two port divided by the Signal to Noise Ratio on the input. I.e. the noise figure tells you how much noise the four port added.
The (equivalent) noise temperature is related to the latter. Your two-port-box (an amplifier for example) adds a specific amount of noise to your signal. Now, you have to consider your two port to have a Noise Figure of 0 (i.e. the two-port is ideally noise free) and have a noisy resistor conencted the input port. Then the effective noise temperature is the absolut Room-Temperatur of the Resistor with which the resistor created just as much noise as the the unideal noisy two port.
In formulation: Teff=(F-1)T
T=Temperatur Factor (most of often the is set to 290K)
The Noise Factor of a two port network, F, is defined as the ratio of the total noise power density at the output port when the noise temperature of the source is the standard noise temperature (290 K), to that part of the noise power density at the output port which originates from the source at standard noise temperature.
Remember that all the powers are available powers!
The IRE definition states that the noise temperature of a two port network is the noise temperature which assigned to the input port of a noise free equivalent of the network yields the saime available output noise power density as the actual network with noise free source.
Noise Temperature may be defined also for a 1-port network (Noise Factor doesn't).
The relation beetween these two quantities is:
F = 1 + Te/T0
where T0=290 K.
You may measure directly noise temperature with a radiometer.
Sorry, I forgot to mention the antenna.
Regarding an antenna:
The antenna receives atmospheric noise and especially earth noise.
So, the equivalant Noise Temperatur is the temperture of a resistor connected to the receiver that creates just as much noise as the antenna does pointed in a specific direction.
So, to determine the noise temperture of your antenna you must actually measure the noise level. You must then know all the noise figure values of your receiver components.
You can send a signal to the antenna and then see when your receiver receives a S/N of zero. Then substract all the Noise Figures and Gains (also antenna-gain) of your receiver.