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Minimum detectable signal of base station in free space

Physics Asked by Quentin James on February 18, 2021

I am studying for tomorrow’s exam (Communication Systems, part of Computer Science course), and I’ve run into this question which involves some radio propagation physics. That’s why I decided to post my question here.

"A base station, having a radio coverage radius of 1.2km, with channels in the 1710MHz to 1785MHz band and 1805MHz to 1880MHz band for the uplink and downlink respectively, is installed on a high tower.
Using Friis’ free space model, find the minimum detectable signal if the output power of the base station is 10W and the antenna gain is 5dB. Assume that the mobile station antenna is isotropic."

I am using this version of the Friis’ free space model:
$P_{received (dBm)} = P_{transmitted(dBm)} + G_{transmitted(dBi)} + G_{received(dBi)} – 20log_{10}(4pi f/c) – 20log_{10}(d)$

… where $f$ is in Hz and $d$ is in m

I am not sure what to input as $f$, since the question specifies 2 different frequency bands. Intuitively, I used $d = 1200$m and the set $f$ as the highest possible frequency (1880MHz) since the equation gives a worse $P_{received}$ as $f$ increases. My thought process is, the signal strength is worst at the highest possible frequency, so this should be the minimum detectable signal (not sure if this is true). What should $f$‘s value be in the equation?

Any help is greatly appreciated.

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