Is Noise Figure dependent on input noise power?

Question

I was reading about the Noise Figure on Wikipedia, where I saw the following definition:

The noise factor is thus the ratio of actual output noise to that which would remain if the device itself did not introduce noise, or the ratio of input SNR to output SNR.

where the noise figure is $10\log_{10} \text{(Noise Factor)}$.

If the Noise Figure is given for an amplifier as a constant (for example Noise-Figure=$5.5 \,\rm{dB}$, since amplifier noise is additive (i.e. its power adds up to the input noise power), then the amplifier noise must be dependent on input noise power, which contradicts my intuition. However, we know that Noise Figure is highly used in practice. Where is my mistake in understanding it?

Answer 1

Wikipedia has a pretty good description at Noise figure, I will summarize here.

Let's start with some definitions. Noise figure $NF$ is related to the noise factor $F$ by: $$\mathrm{NF}=10\log_{10}(F)$$

The noise factor is the ratio of the input signal-to-noise (SNR), $\mathrm{SNR}_i$, ratio to the output SNR, $\mathrm{SNR}_o$, of a system. $$F = \frac{\mathrm{SNR}_i}{\mathrm{SNR}_o} = \frac{S_i/N_i}{S_o/N_o}$$ where $S_i$ is the input signal power, $S_o$ is the output signal power, $N_i$ is the input noise power, and $N_o$ is the output noise power.

Image from Wikipedia.

After a linear device the output SNR is $$\mathrm{SNR}_o = \frac{S_o}{N_o} = \frac{S_iG}{N_iG+N_a}$$ where $G$ is the device gain and $N_a$ is the noise added by the device.

Substituting into the noise factor definition gives: $$F = 1 + \frac{N_a}{N_iG}$$ which as was observed a function of the input noise power. However (here quoting from Wikipedia):

In cascaded systems $N_i$ does not refer to the output noise of the previous component. An input termination at the standard noise temperature is still assumed for the individual component. This means that the additional noise power added by each component is independent of the other components.

The typical convention is to set the input noise temperature to value of 290K, approximately room temperature, and therefore the reference input noise power is $$N_i = kTB$$ where $k$ is the Boltzmann constant ($k \approx 1.38\times 10^{-23}$ J/K), $T = 290K$ is the reference noise temperature, and $B$ is the bandwidth of consideration. (Equivalent derivations can be done in terms of noise power density in which case the bandwidth can be neglected.)

Combining all of this: $$F = 1 + \frac{N_a}{kTBG}$$ which is now independent of the input signal and noise power.