Derive an expression$/$formula for the overall noise factor of a cascade of $4-$stage systems, as shown in the figure below; show all steps not just using Friis' equation. Here each noise factor $F$ is computed with respect to Rs$,$ a standardized matched impedance$-$level of the cascade system, and Rs is also the input source's resistance to the cascade system. Furthermore, each power gain $G$ is the available power gain, the power available under this matched standardized Rs condition $($ typically Rs$=50$ ohm, Rs not shown in the figure below$).$ A standarized impedance Rs system means that all input impedances and all output impedances equal Rs ohm such a $50-$ohm PCB system.

Hint: the noise factor $F-=\frac{\text{t}\text{o}\text{t}\text{a}\text{l}\text{o}\text{u}\text{t}\text{p}\text{u}\text{t}\text{n}\text{o}\text{i}\text{s}\text{e}\text{p}\text{o}\text{w}\text{e}\text{r}}{\text{o}\text{u}\text{t}\text{p}\text{u}\text{t}\text{n}\text{o}\text{i}\text{s}\text{e}\text{d}\text{u}\text{e}to\text{i}\text{n}\text{p}\text{u}\text{t}\text{s}\text{o}\text{u}\text{r}\text{c}\text{e}(Rs)}.$ As an example, for the $2$nd stage, F$2$&G$2,$ we have:

$F_2=\frac{(V_{Rs}^2+V_{\text{i}\text{n}\text{p}\text{u}\text{t},2}^2)G_2}{V_{Rs}^2{G}_2}=1+\frac{V_{\text{i}\text{n}\text{p}\text{u}\text{t},2}^2}{V_{Rs}^2}$

so$,$

$\frac{V_{\text{i}\text{n}\text{p}\text{u}\text{t}\text{,}2}^2}{V_{Rs}^2}=F_2-1$

Where $V_{\text{i}\text{n}\text{p}\text{u}\text{t}\text{,}2}^2$ is the input$-$referred noise power $($voltage ${?}^2)$ of the $2$ nd stage, and $V^2$ Rs is the noise power of the input$-$source resistance under matched conditions. Use the inputreferred noise powers as above for other stages in your derivation.

Note: noise figure NF is $F$ in dB unit.

Calculate the overall noise factor of the cascaded amplifiers shown in the figure below:

$F=$ noise factor of each stage

G $=$ available gain of each stage