Final answer:
To express the free space loss using kilometers and megahertz, the equation becomes LᵀB = 20 log(f MHz) + 20 log(d km) + 32.44 dB. This considers the changes in measurement units for distance and frequency while keeping the form of the original equation the same.
Step-by-step explanation:
To rewrite the equation for free space loss using kilometers (km) instead of meters (m) and megahertz (MHz) instead of hertz (Hz), you should convert the units within the original equation: LᵀB = 20 log(f) + 20 log(d) - 147.56 dB. Remember that 1 km = 1,000 m and 1 MHz = 1,000,000 Hz. So we can make the following substitutions into the formula:
- For distance (d) in km: log(d km) = log(d × 1,000 m)
- For frequency (f) in MHz: log(f MHz) = log(f × 1,000,000 Hz)
Using properties of logarithms, we can rewrite the equation as:
LᵀB = 20 log(f MHz × 10⁶) + 20 log(d km × 10³) - 147.56 dB
LᵀB = 20(log(f MHz) + log(10⁶)) + 20(log(d km) + log(10³)) - 147.56 dB
LᵀB = 20 log(f MHz) + 120 + 20 log(d km) + 60 - 147.56 dB
LᵀB = 20 log(f MHz) + 20 log(d km) + 32.44 dB
We simply collect the constants 120 and 60, subtract the original constant, and we are left with a new constant of 32.44 dB when dealing with frequencies in MHz and distances in km.