Friis formula (Derivation and problems) and Equivalent Noise Temperature

Friis Formula Derivation & Problems

Friis originated the concept and terminology “noise figure” and “noise factor” while working at Bell Labs in Holmdel, New Jersey, in 1942. Fundamentally, Friis’s NFgen “noise-figure” is a function of two variables that is a dB metric that answers the question, “If I connect a particular block (like a single transistor amplifier, or an entire complex receiver chain) to a source (like an antenna) that is putting out noise power in addition to the signal, how much worse (lower in dB) will the block’s output SNR be, relative to the SNR going into the block? In other words, how much worse (in dB) will the output SNR be from this specific block compared to an identical but noiseless block? Friis’ NFgen “noise-figure” is a system-specific metric (i.e. it relies on the application and the incoming SNR) that allows different amplifiers and receiver configurations to be compared based on a measure that captures exactly how many dB the SNR drops by in that system with that incoming SNR.

The function of two variables Friis developed is,

f (Te,Ts) = 1 +  Te/Ts

such that, Friis’s Noise Factor = Fgen =f(Te,Ts)  = 1 + Te/Ts = SNRin/SNRout = (sin/(kTsB)) /  SNRout

Noise Factor Fgen is a ratio of power ratios, and represents the factor by which the SNR is reduced across a device. Since Fgen is a power ratio (not a voltage ratio), the conversion to dB is 10 logs (not 20 logs), and therefore Friis’s noise-figure NFgen can be stated these ways,

NFgen  = 10 log10(Fgen)                                 _

=  10 log10 (SNRin/SNRout)

=  SNRin-dB   SNRout-dB _

Solving for Te, when using Friis’s Fgen definition, we get,

Te  = (Fgen − 1) Ts

Friis’ f(Te, Ts) function enables engineers to calculate the effect (or degradation) of an arbitrary block on the signal-to-noise ratio (SNR) of a signal travelling through it. NFgen cannot be used on a receiver or an amplifier by itself because the SNR deterioration depends on what the amplifier or receiver is linked to, i.e. how noisy the source is.

To calculate “SNR reduction,” the “noise-figure” function must be a function of two variables, Ts and Te. Friis’ noise-figureNFgen evaluates how much a block affects the signal-to-noise ratio when utilised in the system to which it is linked based on these inputs (SNR). The second definition cannot be stated to be the same.

Equivalent Noise Temperature

The equivalent noise temperature of a system is defined as the temperature at which the noise resistor must be maintained to produce the same amount of noise power at the system output as the actual system when connected to the input of a noiseless version of the system.

The noise at the input of the amplifier input is given by

This is the noise contributed by the amplifier. This noise power can be alternatively represented by some fictitious temperature Teq such that

Thus the equivalent noise temperature of the amplifier is given by

This equation shows that Teq is just an alternative measure for F.

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