Calculate Transmission Signal Power Loss in Decibels

The power loss or gain in a transmission medium is often measured in decibels (dB). The formula to calculate power loss in dB is:

$ L_{dB} = 10 \log_{10} \left( \frac{P_{out}}{P_{in}} \right) $

Where:

• $L_{dB}$ is the power loss in decibels.
• $P_{out}$ is the output power.
• $P_{in}$ is the input power.

Attenuation Calculation

In this scenario, the signal power is reduced to one-half of its original value. Therefore:

• $P_{out} = P_2$
• $P_{in} = P_1$
• We are given that $P_2 = \frac{1}{2} P_1$.

Substituting these values into the formula:

$ L_{dB} = 10 \log_{10} \left( \frac{P_2}{P_1} \right) $

$ L_{dB} = 10 \log_{10} \left( \frac{1/2 P_1}{P_1} \right) $

$ L_{dB} = 10 \log_{10} \left( \frac{1}{2} \right) $

$ L_{dB} = 10 \log_{10} (0.5) $

Using the logarithm value, $\log_{10} (0.5) \approx -0.30103$:

$ L_{dB} \approx 10 \times (-0.30103) $

$ L_{dB} \approx -3.0103 \text{ dB} $

Rounding to the nearest whole number, the power loss is approximately -3 dB.