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# Demostración de pasar de Friis a Rappaport y viceversa

Si se considera la antena transmisora como isotrópica, la densidad de potencia en la antena receptora: ${\displaystyle \omega ={\frac {P_{t}}{A}}={\frac {P_{t}}{4\pi d^{2}}}}$

de ganancia de la antena transmirsora es Gt. Luego, la densidad de potencia ${\displaystyle \omega ={\frac {P_{t}G_{t}}{4\pi d^{2}}}}$

En condiciones ideales, el área efectiva de la antena receptora es Ar. Luego, la potencia recibida ${\displaystyle P_{r}=\omega A_{r}=({\frac {P_{t}G_{t}}{4\pi d^{2}}})A_{r}}$

La ganancia de la antena transmisora es ${\displaystyle G_{t}={\frac {4\pi A_{t}}{\lambda ^{2}}}}$

Entonces,

${\displaystyle P_{r}=P_{t}({\frac {4\pi A_{t}}{4\pi d^{2}\lambda ^{2}}})A_{r}}$

${\displaystyle {\frac {P_{r}}{P_{t}}}=({\frac {G_{t}G_{r}\lambda ^{2}}{(4\pi )^{2}d^{2}}})}$

La ganancia de la antena receptora es: ${\displaystyle G_{r}={\frac {4\pi A_{r}}{(\lambda ^{2}}}}$ ${\displaystyle {\frac {P_{r}}{P_{t}}}=({\frac {G_{t}G_{r}\lambda ^{2}}{(4\pi )^{2}d^{2}}})}$ ${\displaystyle P_{r}=P_{t}({\frac {A_{t}A_{r}\lambda ^{2}}{(4\pi )^{2}d^{2}}})}$ ${\displaystyle G={\frac {4\pi A_{\varrho }}{\lambda ^{2}}}}$

${\displaystyle P_{r}=P_{t}({\frac {({\frac {4\pi A_{\varrho }}{\lambda ^{2}}})\lambda ^{2}}{(4\pi )^{2}d^{2}}})}$

${\displaystyle P_{r}=P_{t}({\frac {4\pi A_{\varrho }}{(4\pi )^{2}d^{2}A_{i}so}})}$

${\displaystyle P_{r}=P_{t}({\frac {4\pi A_{\varrho }}{(4\pi )^{2}d^{2}({\frac {\lambda ^{2}}{4\pi }})}})}$

${\displaystyle P_{r}=P_{t}({\frac {A_{t}A_{r}}{d^{2}\lambda ^{2}}})}$

${\displaystyle {\frac {P_{r}}{P_{t}}}=({\frac {A_{t}A_{r}}{d^{2}\lambda ^{2}}})}$