Signal Attenuation (Decibel Formula)
Signal attenuation measures how much a signal weakens over distance.
Learn the decibel formula for power and voltage with examples.
The Formulas
Voltage: dB = 20 × log₁₀(V_out / V_in)
Signal attenuation is the reduction in signal strength as it travels through a medium such as a cable, fiber optic line, or the atmosphere. It is measured in decibels (dB), a logarithmic unit named after Alexander Graham Bell.
A negative dB value means the signal is weaker at the output (attenuation). A positive dB value means the signal is stronger (amplification or gain). The logarithmic scale is used because signal levels can span many orders of magnitude.
Key reference points: -3 dB means the power is halved, -10 dB means it drops to 1/10, and -20 dB means it drops to 1/100. For voltage, -6 dB approximately halves the voltage, and -20 dB reduces it to 1/10. The factor of 20 (instead of 10) in the voltage formula accounts for the fact that power is proportional to voltage squared.
Variables
| Symbol | Meaning |
|---|---|
| dB | Attenuation or gain in decibels |
| P_in | Input power (in watts or milliwatts) |
| P_out | Output power (same units as P_in) |
| V_in | Input voltage |
| V_out | Output voltage |
Example 1
A fiber optic cable receives 5 mW of light and delivers 0.5 mW at the other end. What is the attenuation?
Apply the power formula: dB = 10 × log₁₀(P_out / P_in)
dB = 10 × log₁₀(0.5 / 5) = 10 × log₁₀(0.1)
dB = 10 × (-1)
Attenuation = -10 dB (the signal lost 90% of its power)
Example 2
An audio amplifier has a voltage gain of 26 dB. If the input voltage is 50 mV, what is the output voltage?
Rearrange the voltage formula: V_out = V_in × 10^(dB/20)
V_out = 50 × 10^(26/20) = 50 × 10^1.3
10^1.3 ≈ 19.95
V_out = 50 × 19.95
V_out ≈ 997.6 mV ≈ 1.0 V
When to Use It
The decibel formula is used throughout telecommunications and audio engineering.
- Calculating signal loss in cables and fiber optics
- Specifying amplifier gain in audio and radio systems
- Measuring Wi-Fi signal strength and coverage
- Acoustic measurements (sound pressure levels)