Decibel scale with signal processing applications

Summary: Introduces the decibel scale and shows typical calculations for signal processing applications.

Introduction

The concept of decibel originates from telephone engineers who were working with power loss in a telephone line 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 is the ratio of the power out to the power in.

Let P_in be the power input to a telephone line and P_out the power out. The power gain is then given by

Gain=

P_out

P_in

(1)

Taking the logarithm of the gain formula we obtain a comparative measure called Bel.

BEL:

Gain(Bel)=log₁₀(

P_out

P_in

)

This measure is in honour of Alexander G. Bell, see Figure 1.

**Figure 1:** Alexander G. Bell

Decibel

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

Gain(dB)=10log₁₀(

P_out

P_in

)(2)

Please note from the definition that the gain in dB is relative to the input power. In general we define:

Number of decibels=10log₁₀(

P_ref

)(3)

If no reference level is given it is customary to use P_ref=1 W, in which case we have:

DECIBEL:

Number of decibels=10log₁₀(P)

EXAMPLE 1

Given the power spectrum density (psd) function of a signal x(n), S_xx(ⅈf). Express the magnitude of the psd in decibels.

We find S_xx(dB)=10log₁₀(|S_xx(ⅈf)|).

More about decibels

Above we’ve calculated the decibel equivalent of power. Power is a quadratic variable, whereas voltage and current are linear variables. This can be seen, for example, from the formulas P=

V²

and P=I²R.

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

Amplitude(dB)=20log₁₀(

Amplitude

Amplitude_ref

)(4)

This is illustrated in the following example.

EXAMPLE 2

Express the magnitude of the filter H(ⅈf) in dB scale.

The magnitude is given by |H(ⅈf)|, which gives: |H(dB)|=20log₁₀(|H(ⅈf)|).

Plots of the magnitude of an example filter |H(ⅈf)| and its decibel equivalent are shown in Figure 2.

**Figure 2:** Magnitude responses.

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 increase of 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.

EXAMPLE 3

In filter terminology the cut-off frequency is a term that often appears. The cutoff frequency (for lowpass and highpass filters), f_c, is the frequency at which the squared magnitude response in dB is ½. In decibel scale this corresponds to about -3 dB.

Decibels in linear systems

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

Y(ⅈf)=H(ⅈf)X(ⅈf)(5)

where X and H denotes the input signal and the filter respectively. Taking absolute values on both sides of Equation 5 and converting to decibels we get:

INPUT AND OUTPUT RELATIONS FOR LINEAR SYSTEMS:

The output amplitude at a given frequency is simply given by the sum of the filter gain and the input amplitude, both in dB.

Other references:

Above we have used P_ref=1 W as a reference and obtained the standard dB measure. In some applications it is more useful to use P_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)