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The Decibel Scale and Signal Levels
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
- $+3$ dB ≈ double the power
- $+6$ dB ≈ double the voltage
- $+10$ dB = 10× power
- $+20$ dB = 100× power (10× voltage)
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?
- $40 = 10\log(P_s/P_n)$ → $P_s/P_n = 10^4 = 10{,}000$.
Practice Problems
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