Editing Noise temperature (section)

== Effective noise temperature ==
{{Main|Friis formulas for noise}}
The noise temperature of an amplifier is commonly measured using the [[Y-factor]] method. If there are multiple amplifiers in cascade, the noise temperature of the cascade can be calculated using the [[Friis formulas for noise|Friis equation]]:<ref name="mcclaning">McClaning, Kevin, and Tom Vito. ''Radio Receiver Design.''  Atlanta, GA:  Noble Publishing Corporation, 2000.  {{ISBN|1-884932-07-X}}.</ref>

:<math>T_\text{E} = T_1 + \frac{T_2}{G_1} + \frac{T_3}{G_1 G_2} + \cdots</math>

where
*<math> T_\text{E}</math> = resulting noise temperature referred to the input
*<math> T_1</math> = noise temperature of the first component in the cascade
*<math> T_2</math> = noise temperature of the second component in the cascade
*<math> T_3</math> = noise temperature of the third component in the cascade
*<math> G_1</math> = power gain of the first component in the cascade
*<math> G_2</math> = power gain of the second component in the cascade

Therefore, the amplifier chain can be modelled as a [[black box]] having a gain of <math>G_1 \cdot G_2 \cdot G_3 \cdots</math> and a noise figure given by <math>NF = 10 \log_{10} (1 + T_\text{E}/290)</math>. In the usual case where the gains of the amplifier's stages are much greater than one, then it can be seen that the noise temperatures of the earlier stages have a much greater influence on the resulting noise temperature than those later in the chain. One can appreciate that the noise introduced by the first stage, for instance, is amplified by all of the stages whereas the noise introduced by later stages undergoes lesser amplification. Another way of looking at it is that the signal applied to a later stage already has a high noise level, due to amplification of noise by the previous stages, so that the noise contribution of that stage to that already amplified signal is of less significance.

This explains why the quality of a [[preamplifier]] or [[Low-noise amplifier|RF amplifier]] is of particular importance in an amplifier chain. In most cases only the noise figure of the first stage need be considered. However one must check that the noise figure of the second stage is not so high (or that the gain of the first stage is so low) that there is SNR degradation due to the second stage  anyway. That will be a concern if the noise figure of the first stage plus that stage's gain (in decibels) is not much greater than the noise figure of the second stage.

One corollary of the [[Friis formulas for noise|Friis equation]] is that an [[attenuator (electronics)|attenuator]] prior to the first amplifier will degrade the noise figure due to the amplifier. For instance, if stage 1 represents a 6&nbsp;dB attenuator so that <math>G_1 = \frac{1}{4}</math>, then <math>T_\text{E} = T_1 + 4 T_2 +  \cdots</math>. Effectively the noise temperature of the amplifier  <math>T_2</math> has been quadrupled, in addition to the (smaller) contribution due to the attenuator itself  <math>T_1</math> (usually room temperature if the attenuator is composed of [[resistor]]s). An antenna with poor [[antenna efficiency|efficiency]] is an example of this principle, where <math>G_1</math> would represent the antenna's efficiency.