1.1 Decibel scale with signal processing applications

Introduction

The concept of decibel originates from telephone engineers who were working with power loss in a telephoneline consisting of cascaded circuits. The power loss in each circuit is the ratio of the power in to the power out, or equivivalently, the power gain isthe ratio of the power out to the power in.

Let $P_{\mathrm{in}}$ be the power input to a telephone line and $P_{\mathrm{out}}$ the power out. The power gain is then given by

Decibel

Bel is often a to large quantity, so we define a more useful measure, decibel:

If no reference level is given it is customary to use $P_{\mathrm{ref}}=1\; W$ , in which case we have:

More about decibels

Above we’ve calculated the decibel equivalent of power. Power is a quadratic variable, whereas voltageand current are linear variables. This can be seen, for example, from the formulas $P=\frac{V^2}{R}$ and $P=I^{2}R$ .

So if we want to find the decibel value of a current or voltage, or more general an amplitude we use:

Plots of the magnitude of an example filter $\left|H(if)\right|$ and its decibel equivalent are shown in .

Some basic arithmetic

The ratios 1,10,100, 1000 give dB values 0 dB, 10 dB, 20 dB and 30 dB respectively. This implies that an increaseof 10 dB corresponds to a ratio increase by a factor 10.

This can easily be shown: Given a ratio R we have R[dB]= 10 log R. Increasing the ratio by a factor of 10 we have: 10 log (10*R) = 10 log 10 + 10 log R = 10 dB + R dB.

Another important dB-value is 3dB. This comes from the fact that:

An increase by a factor 2 gives: an increase of 10 log 2≈3 dB. A“increase”by a factor 1/2 gives: an“increase”of 10 log 1/2≈-3 dB.

Decibels in linear systems

In signal processing we have the following relations for linear systems:

Other references:

Above we have used $P_{\mathrm{ref}}=1\; W$ as a reference and obtained the standard dB measure. In some applications it is moreuseful to use $P_{\mathrm{ref}}=1\; mW$ and we then have the dBm measure.

Another example is when calculating the gain of different antennas. Then it is customary to use an isotropic(equal radiation in all directions) antenna as a reference. So for a given antenna we can use the dBi measure. (i ->isotropic)

Matlab files

filter_example.m