Estimation of the contribution of the frontend to the total noise figure

In a radio receiver composed of two stages, the total noise factor \(F\) can be computed using Friis’s formula as\[F = F_1 + \frac{F_2 – 1}{G_1},\]where \(F_1\) is the noise factor of the first block, \(G_1\) is the gain of the first stage and \(F_2\) is the noise factor of the second stage. If \(G_1\) is large enough, then the contribution of the second factor is small and the total noise factor of the whole system is essentially the same as the noise factor of the first stage. This is the reason why a low noise amplifier is useful as a frontend, because it has a low noise factor \(F_1\) and high gain \(G_1\).

If \(F_2\) and \(G_1\) are known (perhaps only approximately), then it is easy to check if the contribution of the frontend to the total noise figure is large enough so that the total noise figure is determined by the noise figure such frontend alone. However, it may happen that one or both of \(F_2\) and \(G_1\) are not known. In email communication, Leif Åsbrink mentioned that there is an easy way of checking the contribution of the frontend without knowing these parameters. The method is to switch off the frontend and note the drop in the noise floor. He gave the following estimates: if the noise floor drops by more than 10dB, then the total noise figure is the same as the noise figure of the frontend up to 1dB; if the noise floor drops by more than 17dB, then the total noise figure is the same as the noise figure of the frontend up to 0.1dB. Here I present the maths behind these kind of estimates.

Let \(T_1, T_2\) be the noise temperatures of the first and second stages respectively, \(T_0 = 290\mathrm{K}\) the standard noise temperature and \(G_2\) the gain of the second stage. When the frontend is switched on, the noise temperature at the output of the receiver is\[G_2(G_1(T_0 + T_1) + T_2).\](This means that if this quantity is multiplied by the Boltzmann constant \(k_B\), one gets noise power density, and if this quantity is multiplied by \(Bk_B\), where \(B\) is the bandwidth, one gets noise power). Similarly, when the frontend is switched off, the noise temperature at the output is \[G_2(T_0+T_2).\]

Let \(C\) be the ratio between the noise power at the output when the frontend is on and the noise power at the output when the frontend is off. Since the \(Bk_B\) factors cancel out, \(C\) equals the ratio between the noise temperature at the output when the frontend is on and the noise temperature at the output when the frontend is off. Hence,\[C=\frac{G_1(T_0 + T_1) + T_2}{T_0 + T_2}.\]

By Friis’s formula, the equivalent noise at the input for the complete system is\[T_0 + T_1 + \frac{T_2}{G_1}.\]Similarly, the equivalent noise at the input just for the frontend is\[T_0 + T_1.\]We want to estimate the ratio between these two quantities (since it coincides with the ratio between noise factors). We proceed as follows:\[\frac{T_0+T_1+\frac{T_2}{G_1}}{T_0+T_1}=1+\frac{T_2}{G_1(T_0+T_1)}.\]Since \(G_1(T_0+T_1) = C(T_0+T_2)-T_2\), this equals\[1+\frac{T_2}{C(T_0+T_2)-T_2} \leq 1+\frac{1}{C-1}.\]Hence, we see that the ratio between noise factors is smaller or equal than \(1 + (C-1)^{-1}\). Note that this estimate is good whenever \(T_2\) is much larger than \(T_0\), which is the same as the noise figure of the second stage being larger than about 2dB or 3dB.

We have seen that the difference in noise figures is smaller than\[10\log_{10}(1 + (C-1)^{-1}).\] If the drop in the noise floor is 10dB, then \(C=10\), so we get a difference in noise figures of 0.46dB. If the drop in the noise floor is 17dB, then C is approximately 50, and we get a difference in noise figures of 0.088dB. This matches the estimates given by Leif.

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