Digital Signal Processing
Moslem Amiri, Václav Přenosil
Masaryk University
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Decibels (dB and dBm)
2
Using Logarithms to Determine Relative Signal Power
 Decibels evolution
 When comparing analog signal levels, early
specialists found it useful to define a measure of
difference in powers of two signals
 If that difference was treated as logarithm of a
ratio of powers, it could be used as a simple
additive measure to determine overall gain or
loss of cascaded electronic circuits
 Positive logarithms associated with system
components having gain could be added to
negative logarithms of those components having
loss quickly to determine overall gain or loss of
system
3
Using Logarithms to Determine Relative Signal Power
 Difference between two signal power levels
 Measured power differences smaller than one bel
were so common that it led to the use of decibel
(bel/10)
belslogdifferencePower
2
1
10 





=
P
P
dBlog10differencePower
2
1
10 





⋅=
P
P
4
Using Logarithms to Determine Relative Signal Power
5
Using Logarithms to Determine Relative Signal Power
 Fig. E-1
 Plot of logarithmic function 10·log10(P1/P2)
 Large change in function’s value when power
ratio (P1/P2) is small, and gradual change when
ratio is large
 Effect of this nonlinearity is to provide greater
resolution when ratio P1/P2 is small, giving us a
good way to recognize very small differences in
power levels of signal spectra, digital filter
responses, and window function frequency
responses
6
Using Logarithms to Determine Relative Signal Power
 For any frequency-domain sequence X(m)
 These expressions are used to convert a linear
magnitude axis to a logarithmic magnitudesquared,
or power, axis measured in dB
 Without the need for squaring operation, we
calculate XdB(m) power spectrum sequence from
X(m) sequence
dBmXmXmX
mXmX
dB |))((|log20)|)((|log10)(
|)(|)(ofspectrumpowerdiscrete
10
2
10
2
⋅=⋅=
=
7
Using Logarithms to Determine Relative Signal Power
 Normalized log magnitude spectral plots
 Division by |X(0)|2 or |X(0)| value forces the first
value in normalized log magnitude sequence
XdB(m) equal to 0 dB
 This makes it easy to compare multiple log magnitude
spectral plots
dB
X
mX
X
mX
mXdB )
|)0(|
|)(|
(log20)
|)0(|
|)(|
(log10)(normalized 102
2
10 ⋅=⋅=
8
Using Logarithms to Determine Relative Signal Power
9
Using Logarithms to Determine Relative Signal Power
 Fig. E-2(c)
 Normalization
 We can clearly see the difference in magnitudesquared
window functions in (c) as compared to
linear plots in (b)
dB
X
mX
W
mW
mW
Hanning
Hanning
Hanning
Hanning
H )
|)0(|
|)(|
(log20)
|)0(|
|)(|
(log10)( 102
2
10 ⋅=⋅=
10
Using Logarithms to Determine Relative Signal Power
11
Using Logarithms to Determine Relative Signal Power
 Fig. E-3
 We’re designing an 11-tap highpass FIR filter
whose coefficients are shown in (a)
 If center coefficient h(5) is –0.48, filter’s
frequency magnitude response |H–0.48(m)| can be
plotted as white dots on linear scale in (b)
 h(5): –0.48  –0.5, new frequency magnitude
response |H–0.5(m)| are black dots in (b)
 Difficult to see much difference on a linear scale
 Calculating two normalized log magnitude
sequences, filter sidelobe effects of changing h(5)
are now easy to see, as shown in (c)
12
Some Useful Decibel Numbers
 A few constants to memorize
 A power difference of 3 dB corresponds to a
power factor of two
 That is, if magnitude-squared ratio of two different
frequency components is 2, then
 If magnitude-squared ratio of two different
frequency components is 1/2
dB33.01(2)log10
1
2
log10differencepower 1010 ≈=⋅=





⋅=
dB33.01(0.5)log10
2
1
log10differencepower 1010 −≈−=⋅=





⋅=
13
Some Useful Decibel Numbers
Magnitude
ratio
Magnitude-squared
power (P1/P2) ratio
Relative dB
(approximate)
10-1/2 10-1 -10 P1 is one-tenth P2
2-1 2-2 = 1/4 -6 P1 is one-fourth P2
2-1/2 2-1 = 1/2 -3 P1 is one-half P2
20 20 = 1 0 P1 is equal to P2
21/2 21 = 2 3 P1 is twice P2
21 22 = 4 6 P1 is four times P2
101/2 101 = 10 10 P1 is ten times P2
101 102 = 100 20 P1 is one hundred times P2
103/2 103 = 1000 30 P1 is one thousands times P2
14
Absolute Power Using Decibels
 Another use of decibels
 To measure signal-power levels referenced to a
specific absolute power level
 In this way, we can speak of absolute power levels in
watts while taking advantage of convenience of
decibels
 The most common absolute power reference
level used is milliwatt
 dBm = dB relative to a milliwatt
dBm
milliwatt1
in watts
log10log10ofpowerabsolute 1
10
2
1
101 





⋅=





⋅=
P
P
P
P