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Given a passive network at room temperature, the noise figure is the ratio between the output noise power density pw and and the output noise power density due to the source pwS

F = pw/pwS = 0.5*k*T/0.5*k* T*g= 1/g

where g is the network gain. Now I would say that the formula holds also for temperatures T1 different from room temperature T0 but turns out it doesn't. How can I prove the formula becomes F = 1 + T1 *(-1 + 1/g)/T0?

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Have you considered thermal noise equations?

https://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noise

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