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42
Signals and Spectra Chap. 2
average power out
dB = 10 log a
average power in b = 10 log aP out
b
P in
(2–15)
DEFINITION. The decibel gain of a circuit is †
This definition gives a number that indicates the relative value of the power out with
respect to the power in and does not indicate the actual magnitude of the power levels involved.
If resistive loads are involved, Eq. (2–12) may be used to reduce Eq. (2–15) to
or
dB = 20 log a V rms out
V rms in
b + 10 log a R in
R load
b
dB = 20 log a I rms out
I rms in
b + 10 log a R load
R in
b
(2–16)
(2–17)
Note that the same value for decibels is obtained regardless of whether power, voltage, or current
[Eq. (2–15), Eq. (2–16), or Eq. (2–17)] was used to obtain that value. That is, decibels are
defined in terms of the logarithm of a power ratio, but may be evaluated from voltage or current
ratios.
If normalized powers are used,
dB = 20 log a V rms out
V rms in
b = 20 log a I rms out
I rms in
b
(2–18)
This equation does not give the true value for decibels, unless R in = R load ; however, it is
common engineering practice to use Eq. (2–18) even if R in Z R load and even if the number that
is obtained is not strictly correct. Engineers understand that if the correct value is needed, it
may be calculated from this pseudovalue if R in and R load are known.
If the dB value is known, the power ratio or the voltage ratio can be easily obtained by
inversion of the appropriate equations just given. For example, if the power ratio is desired,
Eq. (2–15) can be inverted to obtain
P out
= 10 dB>10 (2–19)
P in
The decibel measure can also be used to express a measure of the ratio of signal power
to noise power, as measured at some point in a circuit.
(S>N) dB = 10 log a P signal
b = 10 log a 8s2 (t)9
P noise 8n 2 (t)9 b
(2–20)
DEFINITION. The decibel signal-to-noise ratio is ‡
† Logarithms to the base 10 will be denoted by log( #),
and logarithms to the base e will be denoted by ln( #).
Note that both dB and the ratio P out /P in are dimensionless quantities.
‡
This definition involves the ratio of the average signal power to the average noise power. An alternative definition
that is also useful for some applications involves the ratio of the peak signal power to the average noise
power. See Sec. 6–8.