In the Friis transmission equation,

$$\frac{P_r}{P_t} = G_t G_r \left( \frac{\lambda}{4 \pi d} \right)^2,$$

it seems that if the frequency is halved while the other parameter values are fixed, the received power \${P_r}\$ will quadruple since the wavelength \$\lambda\$ is doubled. I mean, for example, if I work on a 2 GHz system and if I switch my signal to 1 GHz, is the power level of my received signal \${P_r}\$ going to become four times higher? Or do I miss something?

(I assumed the bandwidth of the antennas and other systems is suitable for both frequencies.)

  • \$\begingroup\$ Have you ever derived Friis transmission equation by yourself? \$\endgroup\$ Aug 27, 2023 at 11:25
  • \$\begingroup\$ Yes with a book help \$\endgroup\$
    – adba
    Aug 28, 2023 at 15:17

1 Answer 1


That's right, but you will also have to scale your antennas by the same amount, because \$G_{t,r}\$ for any kind of aperture antenna is quadratic with the ratio of length of antenna to wavelength. Halving your \$f\$ doubled your \$\lambda\$ and hence divided your \$G_t\$ and \$G_r\$ by a factor of 4. So, to keep \$G_{t,r}\$ constant while doubling the wavelength, you need to double the sizes of your antennas as well. Often, that's not very attractive…

There's no free lunch! If you just reduce the frequency of your transmission without changing your antennas, you win exactly nothing (best case, really, usually, because antenna systems are typically optimized for one wavelength).

  • \$\begingroup\$ If I had a log-periodic antenna having a constant gain in a scope of 1 and 4 GHz, I wouldn't have to change my antenna right? \$\endgroup\$
    – adba
    Aug 27, 2023 at 11:58
  • 1
    \$\begingroup\$ yeah, if you have an antenna that is already large enough for your lowest frequency, you would get that. But outside of measurement antennas, that's rarely the case; in practice, just for mechanical stability reasons, you'd want the smallest antenna that gives you the gain you need. \$\endgroup\$ Aug 27, 2023 at 12:02
  • \$\begingroup\$ Thank you I got it \$\endgroup\$
    – adba
    Aug 27, 2023 at 12:14

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