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The Decibel Scale and Signal Levels

24 min Sensors & Measurement Math

The Decibel Scale

Engineers use logarithms to express signal levels, gains, and losses across enormous ranges.

Decibel Definitions

Power: $$L = 10\log_{10}\frac{P}{P_0} \text{ dB}$$

Voltage/Amplitude: $$L = 20\log_{10}\frac{V}{V_0} \text{ dB}$$

Useful Reference Values

Example 1

An amplifier boosts a signal from 0.1 W to 10 W. Find the gain in dB.

$G = 10\log_{10}(10/0.1) = 10\log_{10}(100) = 10(2) = 20$ dB.

Example 2

A cable loses 6 dB. Input voltage is 2 V. Output voltage?

$-6 = 20\log(V_{out}/2)$ → $\log(V_{out}/2) = -0.3$ → $V_{out}/2 = 10^{-0.3} \approx 0.5$

$V_{out} \approx 1.0$ V. (−6 dB halves the voltage.)

Example 3

Signal-to-noise ratio is 40 dB. How many times larger is the signal power than the noise?

$40 = 10\log(P_s/P_n)$ → $P_s/P_n = 10^4 = 10{,}000$.

1. Amplifier: input 0.5 W, output 50 W. Gain in dB?

2. What power ratio is +30 dB?

3. −10 dB loss. What fraction of power remains?

4. Voltage doubles. How many dB?

5. SNR = 60 dB. Signal-to-noise power ratio?

6. Three stages: +10 dB, −3 dB, +20 dB. Total gain?

Show Answer Key

1. $10\log(100) = 20$ dB

2. $10^3 = 1{,}000$

3. $10^{-1} = 1/10$ (10% remains)

4. $20\log(2) \approx 6$ dB

5. $10^6 = 1{,}000{,}000$

6. $10 - 3 + 20 = 27$ dB