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
PinPin
be the power input to a telephone line and
PoutPout
the power out. The power gain is then given by
Gain=PoutPin
Gain
Pout
Pin
(1)
Taking the logarithm of the gain formula we obtain a
comparative measure called Bel.
Bel:
GainBel=log10PoutPin
Gain
Bel
10
Pout
Pin
This measure is in honour of Alexander G. Bell, see
Figure 1.
Decibel
Bel is often a to large quantity, so we
define a more useful measure, decibel:
GaindB=10log10PoutPin
Gain
dB
10
10
Pout
Pin
(2)
Please note from the definition that the gain in dB is relative to the input power.
In general we define:
Number of decibels=10log10PPref
Number of decibels
10
10
P
Pref
(3)
If no reference level is given it is customary to use
Pref=1 W
Pref
1 W
,
in which case we have:
Decibel:
Number of decibels=10log10P
Number of decibels
10
10
P
Example 1
Given the power spectrum density (psd) function of a
signal
xnxn,
Sxxⅈf
Sxx
f
. Express the magnitude of the psd in decibels.
We find
SxxdB=10log10|Sxxⅈf|
Sxx
dB
10
10
Sxx
f
.
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=V2R
P
V
2
R
and
P=I2R
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:
AmplitudedB=20log10AmplitudeAmplituderef
Amplitude
dB
20
10
Amplitude
Amplituderef
(4)
This is illustrated in the following example.
Example 2
Express the magnitude of the filter
Hⅈf
H
f
in dB scale.
The magnitude is given by
|Hⅈf|
H
f
,
which gives:
|HdB|=20log10|Hⅈf|
H
dB
20
10
H
f
.
Plots of the magnitude of an example filter
|Hⅈf|
H
f
and its decibel equivalent are shown in
Figure 2.
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),
fcfc,
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:
Yⅈf=HⅈfXⅈf
Y
f
H
f
X
f
(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
Pref=1 W
Pref
1 W
as a reference and obtained the standard dB measure. In some applications it is more
useful to use
Pref=1 mW
Pref
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)