SM 5 BSZ - On noise figures and noise temperatures.
(June 3 2001)

#### Noise figure and noise temperature of amplifiers.

An ideal amplifier will amplify whatever signal and noise present at the input without adding any noise of its own. When connected to a resistor that is held at the absolute zero temp 0K (0 degrees Kelvin) an ideal amplifier will give no output signal at all.

Real amplifiers add noise due to losses within the amplifier (if the lossy components are not kept at 0K) and the amplifying process itself is normally adding some noise too.

The noise figure of an amplifier is the amount of noise at the output divided by the amount of noise one would have had from an ideal amplifier with the same gain connected to the same source. Noise figures are expressed in dB and they depend on what the source is. If nothing else is explicitly stated the input is connected to a resistor at room temperature, 290K.

The noise temperature is often easier to work with. The noise temperature of a preamplifier is the temperature you would have on a resistor connected to an ideal amplifier to get the same output as you would get from from the real amplifier if connecting it to a resistor that is kept at 0K.

Temperatures correspond to power levels. When the temperature of a resistor is doubled the power output from it is doubled. (The voltage is proportional to the square root of the temperature)

Powers from uncorrelated sources are additive so noise temperatures are additive. The noise temperature at a certain point in the receiver chain is simply another way of expressing the power level of random noise.

An amplifier having a noise figure of 3dB has a noise temperature of 290K and it gives twice as much noise as an ideal amplifier with the same gain when connected to a room temperature resistor.

The noise temperature at the output of an amplifier is the sum of the noise temperature of the source and the noise temperature of the amplifier itself multiplied by the power gain of the amplifier. This rule is important. It is simple and applying it will give a direct estimate of how much noise a receiver system has at the output compared to an ideal receiver.

Tout = Noise temperature at amplifier output in degrees Kelvin.
G = Power gain in linear scale. A 20 dB amplifier has G=100.
Tampl = Noise temperature of amplifier.
Tsource = Noise temperature of source.

This formula can be applied to a chain of amplifiers. Tout from one amplifier will be Tsource for the next amplifier in a chain of amplifiers.

By setting up Tout for the last amplifier in a chain of amplifiers and then comparing it to the Tout one would have obtained with all Tampl = 0 one gets the total degradation caused by the noise contributions from all the amplifiers in the amplifier chain.

The total degradation is the ratio between temperatures (powers) and it is usually expressed in dB.

Noise figures, although expressed in dB, do not express the system degradation directly unless the antenna temperature happens to be 290K

#### Noise figure and noise temperature of attenuators.

When analysing a receiver to find out where the different noise contributions come from and by how much they contribute to the total noise floor at the loudspeaker output one has to take losses in filters, cables and attenuators into account.

An attenuator is an amplifier with G below 1 and with a temperature at the output that goes asymptotically towards the physical temperature. The same formula is valid for amplifiers and attenuators:

Tout = Gatt * ( Tatt  + Tsource )

as an example, consider a 30dB attenuator connected to a resistor at room temperature. It is obvious that Tout will be 290K since both the source and the attenuator are resistive losses at this temperature. The formula gives:

290 = 0.001 * ( Tatt + 290 )

or Tatt = 289710K.

The noise temperature of an attenuator may be much higher than the physical temperature. This may seem strange at a first glance, but it is a natural consequence of the noise temperature definition. The ideal amplifier in this case is the attenuator held at 0K. The resistor at the input defining noise temperature of the amplifier (attenuator) has to be really hot to make the noise temperature at the output 290K after 30dB noiseless attenuation!

With source and attenuator both at 290K, the output temperature will be 290K regardless of gain (attenuation). This leads to:

The noise figure can be calculated from Tout calculated with this value and Tout calculated with Tatt = 0.

Not very surprising, this ratio turns out to be 1/Gatt which means that the noise figure of an attenuator is the same as the attenuation in dB!

#### Amplifier with attenuator in front of it

The output temperature of an attenuator followed by an amplifier will be:

Tout  = Gampl * ( Tampl  + Gatt * ( Tatt  + Tsource ) )

The noise figure can be calculated from Tout calculated with this value and Tout calculated with Tampl = 0 and Tatt = 0.

Comparing the ratio to the expression for the noise figure of the preamp allone one finds that the noise figure of an attenuator preceding a preamplifier is the noise figure of the preamplifier plus the attenuation of the attenuator in dB.

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