Introduction to the Decibel
The decibel (dB) is a logarithmic expression of the ratio of two power levels,
P1 and P2. Derived from the Bel, a unit of sound intensity named after
Alexander Graham Bell, the decibel (one-tenth Bel) represents the smallest
perceivable change in sound level. Because of the logarithmic relationship
between powers, decibel notation allows a very large range of powers to be
described and compared. Such large ranges are frequently encountered in
communications systems; for example, power levels on a single transceiver
chassis may range from femtowatts (10–15 W) to kilowatts (103 W). Because of
the properties of logarithms, gains and losses in communications systems can
be expressed and compared in decibel units more simply than would
otherwise be possible.
Logarithms
Logarithms are exponents. That is, the logarithm (log) of a number to a given
base is the power to which the base must be raised to give the number. For
example, in the expression 102 = 100, the raised 2 is called the exponent, the
10 is called the base, and the result, 100, is the number. The exponent (2) can
also be called the logarithm of the number 100 to the base 10. Such an
expression would be written as log10 100 = 2 and would be stated in words as
“the logarithm to the base 10 of the number 100 is 2.” While any number can,
in theory, appear as a base in logarithms, those used in decibel expressions
are always base-10, or common, logarithms. (Common logarithms are distinct
from natural logarithms, which have Euler’s number, e, as the base and
which are abbreviated ln.) For common logarithms, the base is customarily
not expressed explicitly. Thus, the above expression would be written simply
as log 100 = 2 and read aloud as “the log of 100 is 2.”
Likewise, the equation 103 = 1000 is expressed as “log 1000 = 3.”
Clearly, the logarithm of any number between 100 and 1000 will fall between
2 and 3. Stated another way, the logarithm of a number between 100 and
1000 will be 2 plus some decimal fraction. The whole-number part is called
the characteristic, and the decimal fraction is the mantissa; both of these
values historically have been available in published tables of logarithms but
are now most easily determined with pocket calculators.
The inverse of a logarithm is the antilog. The antilog of a number to a
given base is simply the base raised to that number. For example (again,
assuming common logarithms), antilog 2 = 102, or 100. Both the log and
antilog can be easily determined with a pocket calculator; on some
calculators, the antilog is denoted 10x, while others may have an INV key
that provides the antilog function when paired with the log key.
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One advantage of logarithms is that very large and very small
numbers can be stated and conveniently compared. Another advantage is
that quantities expressed as logarithms can simply be added and subtracted.
Multiplication and division of large numbers is reduced to the addition or
subtraction of their logarithmic equivalents. In other words (again, assuming
the same base),
log ab = log a + log b,
log a/b = log a – log b,
(Rule 1)
(Rule 2)
log ab = b log a.
(Rule 3)
and
The Decibel as a Power Ratio
The basic expression for power ratios expressed in decibel form is
dB = 10 log
P2
.
P1
By convention, the numerator of the expression (P2) is the higher power level
so that the result is a positive number. This convention made determining
the logarithm easier when published tables were the norm but is no longer
absolutely necessary now that pocket calculators are commonplace. Note,
however, that if the smaller value appears in the numerator, the result will
be negative, thus representing a power loss (negative gain). It is important,
in any event, to keep track of negative signs and to show them in the result
when denoting a loss.
The Decibel as a Voltage or Current Ratio
Though the decibel is properly thought of as a power ratio, it is possible to
express voltages and currents as decibel relationships provided that input
and output resistances are taken into account. Since P = E2/R, one can
substitute as follows into the power relationship:
2
2
dB = 10 log
E2
R2
P2
= 10 log
2 .
P1
E1
R1
Clearing the fractions in the numerator and denominator, we obtain
2
dB = 10 log
E 2R1
2
E 1R2
.
For clarity, this expression can be rewritten as:
2
E2 R1
dB = 10 log  
,
E1 R2
which, from Rule 1 above, is equivalent to
E2
dB = 10 log  
E1
2
+ 10 log
R1
.
R2
From Rule 3, the squared term can be removed, giving
dB = 20 log
E2
R1
+ 10 log
.
E1
R2
This expression can be further simplified by recognizing that 10 log
R1
is
R2
R1
(remember that a square root is equivalent to raising a
R2
base to the one-half power). Thus, one now has
equal to 20 log
dB = 20 log
E2
+ 20 log
E1
R1
,
R2
which, by factoring, simplifies to a final result of
dB = 20 log
P=
E2 R1
.
E1 R2
Currents can be represented in decibel form by remembering that
Similarly,
I2R.
3
2
I R2
2
2
dB = 10 log
,
I R1
1
which simplifies to
dB = 20 log
I2
R2
+ 10 log
,
I1
R1
dB = 20 log
I2 R2
.
I1 R1
or
For the special case where R1 = R2 (that is, equal input and output
resistances), the relationships for voltage and current simplify to:
dB = 20 log
E2
E1
and
20 log
I2
.
I1
Reference Levels
By itself, the decibel has no absolute value; decibels represent the ratio
between two powers (or voltages, or currents). Therefore, it would not be
possible to determine, say, the output power of an amplifier without knowing
the input power, even if the decibel gain were specified. For convenience,
however, one often assigns a 0-dB reference power or voltage level and
represents quantities with respect to the reference. (Remember that 0 dB is
10 log 100. The value 100 equals 1, and the logarithm of 1 is 0.) Most
commonly encountered for both audio- and radio-frequency systems is a
power reference of 1 mW (0.001 W). Decibels referenced to 1 mW are
represented by the term dBm (lower-case m represents milliwatt). For audio
applications, 0 dBm represents 1 mW across a 600- system impedance,
whereas for radio work, the same 0 dBm represents 1 mW across 50 . Note
that these values correspond to different voltages (P = E2/R), so a meter with
a dB scale calibrated to read “0 dB” (actually 0 dBm) across 600  would not
read correctly if used in a 50- system. Though the term dBm is standard,
the impedance may not be explicitly stated and sometimes must be inferred
from the context (audio or radio).
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While dBm is the most often encountered reference in radio
communications work, other references are in wide use. Some of these include
dBV and dBmV, where 0 dB represents 1 V or 1 mV, respectively, and
which are frequently encountered in video systems and cable television; dBV
(1-V reference); dBW (1-W reference); dBk (1 kW reference, used in
broadcasting); dBu, which represents decibels referenced to a field-strength
intensity of 1 microvolt-per-meter (1 V/m); and dBc, representing levels
referenced to the system carrier. In all cases, levels may be above or below
the reference. Values above the 0-dB point should be—but are not always—
denoted with a leading + sign to minimize confusion; values below 0 dB must
be shown with a minus sign.
With a specified reference level, it is possible to use decibels to
determine exact values. For example, a microwave transmitter specified at
+27 dBm (27 dB above 0 dBm) would have a power output of 0.5 W, shown as
follows:
P2
Pout (dB) = 10 log 0.001W
27 dBm
P2
= 10 log 0.001W
2.7 dBm
P2
= log 0.001W
P2
Antilog 2.7 = 0.001W
P2
501.2 = 0.001W
(501.2)(0.001W) = P2
0.5 W = P2
Important Relationships
From inspection of the above expressions, one sees that a 10-dB power
increase is not a doubling of power, but rather an increase of 10 times. A
doubling of power corresponds to a 3-dB increase. Similarly, a halving of
power is a 3-dB decrease. A 1-dB increase in power is approximately a 25%
increase. By committing some common values to memory, one can easily
estimate power or voltage changes with decibels without resorting to pencil-
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and-paper calculations or a calculator. In the microwave transmitter example
just given, one can arrive at the same result by knowing that +30 dBm, 30 dB
above 1 mW, is 103 above 1 mW, or 1 W. The transmitter output, at +27 dBm,
is 3 dB below 1 W, which is a power reduction of half; thus, the power output
is 0.5 W. Alternatively, the power output can be arrived at by recognizing
that the output consists of multiple 10-dB, 3-dB, and 1-dB steps above 0 dBm:
the first 10-dB step, from 0 to +10 dBm, increases power from 1 mW to 10
mW; the second 10-dB step is an additional ten-times power increase, to 100
mW, or +20 dBm. Three dB above that is 200 mW; the next 3 dB brings the
power to 400 mW, or +26 dBm. The final 1-dB increase can be approximated
as a 25% increase over 400 mW. Twenty-five percent of 400 is 100, so +27
dBm is 500 mW, or 0.5 W. With a bit of practice, the communications
technician can become very proficient at determining power outputs based on
decibel readings. The most common and important relationships to remember
are as follows:
1 dB – 25% power increase (or approximately a 22% decrease)
3 dB – power doubling (100% increase) – 2× power
6 dB – 4× power; 2× voltage
10 dB – 10× power
20 dB – 100× power; 10× voltage
30 dB – 1000× power
40 dB – 10,000× power; 100× voltage
These relationships apply to decreases as well as increases: a 3-dB power
reduction is a halving of power, 10-dB decrease is a reduction to one-tenth the
original power, and so on.
Stage Gains/Losses
Decibel relationships are particularly useful for determining gains (or losses)
through pieces of equipment with multiple stages. An example involving a
superheterodyne radio receiver will illustrate how output power can be easily
calculated if the input power (or, in this case, the input voltage and
impedance) and stage gains are known.
EXAMPLE:
An 8-microvolt signal is applied to the 50-Ω input of a receiver whose stages
exhibit the following gains (all values in decibels):
RF amplifier:
Mixer:
First IF amplifier:
Second IF amp.:
Third IF amp.:
Detector:
Audio amplifier:
8
3
24
26
26
–2
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Calculate the input power in watts and dBm, and calculate the power driven
into the speaker (i.e., output of the audio amplifier.)
SOLUTION:
Because the input signal levels are given as voltage and impedance, it is first
necessary to convert to equivalent power:
E2 (8 V)2
P= R =
= 1.28 × 10–12 W.
50 
With input power known, it is easy to convert to dBm (remember that the
denominator of the decibel power formula is known—it is 1 mW).
P
dBm = 10 log 1 mW ,
= 10 log
1.28  10 12
1  10 3
= – 89 dBm .
With the input power expressed in dBm, determination of the output power is
simply a matter of adding the individual stage gains or losses:
–89 dBm + 8 dB + 3 dB + 24 dB + 26 dB + 26 dB – 2 dB + 34 dB = +30 dBm.
From the approximations given above, it should be apparent that +30 dBm is
30 dB above 1 mW, a one-thousand-times increase, so the drive power to
the speaker is 0.001 W × 1000 = 1 W.
[Example taken from Miller and Beasley, Modern Electronic Communication, 7th ed., (Upper
Saddle River, N.J.: Prentice-Hall, Inc.), pp. 147–148.]
The above example also makes a subtle but important point. Note that
the input level is referenced to a specific value (and is thus denoted by dBm)
but that individual stage gains are not. When two values expressed with
respect to a reference are compared, the resulting gain or loss is properly
expressed in dB, not in dB with respect to that reference. In other words, the
references cancel.
EXAMPLE:
The power input to an amplifier is measured as – 47 dBm, and the output is
measured as – 24 dBm. What is the gain? Is the gain properly expressed in
dB or dBm?
SOLUTION:
The gain is simply the output minus the input, or – 24 dBm – (– 47 dBm)
= 23 dB. The difference in input and output simply represents the ratio
between two power ratios, not the ratio with respect to a fixed level (1 mW in
this case), so the output is properly expressed in dB.
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