Decibals

Definition

The use of decibels is an expedient mathematical method for comparing the amplitude, or power, or difference between two signals. Decibel values are often used to describe the performance of both analog and digital filters.

Power Quantities

When referring to measurements of power quantities, a ratio can be expressed as a level in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to reference value Thus, the ratio of \(P\) (measured power), to \(P_0\) (reference power) is represented by \(L_p\):

\[ L_P = 10 \log_{10} \left( \frac{P}{P_0} \right) \ \text{dB}. \]

\(P\) and \(P_0\) must measure the same type of quantity, and have the same units before calculating the ratio. If \(P = P_0\), then \(L_p = 0\). If \(P\) is greater than \(P_0\), then \(L_p\) is positive; else negative.

A difference of 10dB means a difference factor of 10. Similarly, a difference of 20dB means a difference factor of 100.

Amplitude Quantities

For amplitudes:

\[ 20 \log_{10} \left( \frac{A_1}{A_2} \right) \ \text{dB}. \]

When \(A_1\) is less than \(A_2\), the dB value is a negative number.

Amplitude Ratio (\(A_1 / A_2\))	Relative dB
\(1/1000\)	\(-60\)
\(1/100\)	\(-40\)
\(1/10\)	\(-20\)
\(1/4\)	\(-12\)
\(1/2\)	\(-6\)
\(1\)	\(0\)
\(2\)	\(6\)
\(4\)	\(12\)
\(10\)	\(20\)
\(100\)	\(40\)
\(1000\)	\(60\)