# 0.1 Decibel scale with signal processing applications

 Page 1 / 1
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 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

$\mathrm{Gain}=\frac{{P}_{\mathrm{out}}}{{P}_{\mathrm{in}}}$
Taking the logarithm of the gain formula we obtain acomparative measure called Bel.
$\mathrm{Gain}(\text{Bel})=\lg \left(\frac{{P}_{\mathrm{out}}}{{P}_{\mathrm{in}}}\right)$
This measure is in honour of Alexander G. Bell, see .

## Decibel

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

$\mathrm{Gain}(\text{dB})=10\lg \left(\frac{{P}_{\mathrm{out}}}{{P}_{\mathrm{in}}}\right)$
Please note from the definition that the gain in dB is relative to the input power.In general we define:
$\mathrm{Number of decibels}=10\lg \left(\frac{P}{{P}_{\mathrm{ref}}}\right)$

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

$\mathrm{Number of decibels}=10\lg P()$

Given the power spectrum density (psd) function of a signal $x(n)$ , ${S}_{\mathrm{xx}}(if)$ . Express the magnitude of the psd in decibels.

We find ${S}_{\mathrm{xx}}(\text{dB})=10\lg \left|{S}_{\mathrm{xx}}(if)\right|$ .

## 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:

$\mathrm{Amplitude}(\text{dB})=20\lg \left(\frac{\mathrm{Amplitude}}{{\mathrm{Amplitude}}_{\mathrm{ref}}}\right)$
This is illustrated in the following example.

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

The magnitude is given by $\left|H(if)\right|$ ,which gives: $\left|H(\text{dB})\right|=20\lg \left|H(if)\right|$ .

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.

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 is½. In decibel scale this corresponds to about -3 dB.

## Decibels in linear systems

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

$Y(if)=H(if)X(if)$
where X and H denotes the input signal and the filter respectively. Taking absolute values on both sides of and converting to decibels we get:
The output amplitude at a given frequency is simply given by the sum of the filter gain andthe input amplitude, both in dB.

## 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

#### Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Information and signal theory' conversation and receive update notifications? By Stephen Voron By OpenStax By JavaChamp Team By Mahee Boo By OpenStax By JavaChamp Team By Dewey Compton By Mackenzie Wilcox By Rhodes By OpenStax