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)
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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
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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
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Using Logarithms to Determine Relative Signal Power
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
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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


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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 
7
normalization:
we can clearly see the difference in
magnitude-squared window functions in (c)
as compared to linear plots in (b)
Using Logarithms to Determine Relative Signal Power
dB
X
mX
W
mW
mW
Hanning
Hanning
Hanning
Hanning
H
)
|)0(|
|)(|
(log20
)
|)0(|
|)(|
(log10)(
10
2
2
10


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Using Logarithms to Determine Relative Signal Power
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
h(5): –0.48  –0.5, new frequency
magnitude response |H–0.5(m)| are black
dots
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
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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 






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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
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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