Antenna Noise Temperature, Directivity, Antenna Gain

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Antenna Noise Temperature: It is the measure of all the external noise collected by a receiving antenna. Measured in Kelvin (K). It varies with antenna diameter, elevation angle and antenna polarisation. The larger the antenna, the lower the noise temperature. Major noise sources are cosmic noise (caused due to sun, moon and starts) and ground noise; caused due to noise energy radiated from the soil. 

Antenna Directivity
Directivity is a fundamental antenna parameter. It is a measure of how 'directional' an antenna's radiation pattern is. An antenna that radiates equally in all directions would have effectively zero directionality, and the directivity of this type of antenna would be 1 (or 0 dB).
Silly side note: When directivity is specified for an antenna, what is meant is 'peak directivity'. Directivity is technically a function of angle, but the angular variation is described by its radiation pattern. 
Antenna Gain
The term Antenna Gain describes how much power is transmitted in the direction of peak radiation to the power that would be radiated in the same direction by an isotropic antenna. Antenna gain is more commonly quoted than directivity in an antenna's specification sheet because it takes into account the actual losses that occur.

A transmitting antenna with a gain of 3 dB means that the power received far from the antenna will be 3 dB higher (twice as much) than what would be received from a lossless isotropic antenna with the same input power. Note that a lossless antenna would be an antenna with an  antenna efficiency of 0 dB (or 100%). Similarly, a receive antenna with a gain of 3 dB in a particular direction would receive 3 dB more power than a lossless isotropic antenna. The gain figure is only applicable to a particular direction and frequency.

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