Logarithms and Decibels

1
Logarithms and Decibels
2
The Decibel

• Named for Alexander Graham Bell.
• Originally used to measure power losses in
  telephone lines.
• A Bel is the common log of the ratio of two power
  levels.
• A decibel is one-tenth of a bel.
• A Bel is not a unit of anything but simply a
  logarithmic ratio of two power levels.

3
Definition of a logarithm

• A logarithm is an exponent, but is stated
  differently.
• When we write 53125
• 5 is the base, 3 is the exponent.
• This is exponential form.
• Logarithmic form is when we say The log (to the
  base 5) of 125 is 3.

4
Notation

• log form log636 2
• exponential form 6236
• Any expression in logarithmic form may be
  indicated in exponential form.
• if any number b to the power of x N bxN, then
  the logarithm of N to the base b x. This is
  logarithmic form.

5
Base 10

• In computation the base 10 is used for
  logarithms.
• It is so convenient and common that it is not
  usually written as a subscript but is understood
  if no base is shown.
• This is similar to scientific notation where very
  large or small numbers are expressed as X 10 to
  an exponent value.
• For example 93000000 93 x106

6
Logarithms Convenience

• Comparing sounds at the threshold of hearing to
  sounds at the threshold of pain represents over a
  million fold difference in pressure levels.
• The dB as a logarithmic measure of ratios fits
  well with our perceived loudness of sound
  intensity.
• Logarithms are used to help us condense the huge
  range of SPL humans perceive into a manageable
  scale.

7
A Comparison of SPLs
A 1 db change in level is barely noticeable A
3db increase doubles MEASURED power level, but is
not perceived that way.
8
Range of Human Hearing
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Formula for dB SPL measurements

• The formula for SPL
• 20 log (p/p0)
• P0 is the reference of 20 micropascals (threshold
  of hearing).
• P the pressure level of the sound we are
  comparing to the reference level.

10
Calculating dB SPL differences

• Any sound pressure can be expressed as dB SPL by
  comparing the sound pressure to the 0 dB
  (threshold of hearing) reference point with the
  20 log formula.
• For example, how many dB SPL is a sound that is
  50 µ Pa?
• dB SPL 20 log (50µPa/20µPa)
• dB SPL 20 log (2.5 Pa)
• dB SPL 20 (.39794)
• dB SPL 7.96

11
Using the Log formulas

• First, divide the numbers in parentheses.
• Next, find the log of the result.
• Multiply that number by 20.
• We can express the difference between any 2
  pressure levels as dB using the 20 log formula.

12
Using dBs to compare two SPL levels

• Find the dB difference between 1000µPa and
  100µPa.
• dB 20 log (1000µPa/100µPa)
• dB 20 log (10Pa)
• dB 20 (1)
• dB difference 20

13
dB SPL As a Function of Distance

• SPL changes with the square of distance, meaning
  that....
• Doubling the distance results in a drop of 6 dB
  SPL.
• Halving the distance results in a 6 dB SPL
  increase.

14
dB PWL

• PWL or Lw (sound power level) is the total sound
  power emitted by a source in all directions.
• Like electrical power, PWL is measured in watts.
• Formula dB PWL 10 log (W/W0) where W0 is one
  picowatt (10-12 watt).
• Rule of thumb doubling sound pressure results in
  a 6 dB increase, whereas doubling the sound power
  level results in a 3 dB increase.

15
Calculating dB PWL

• Any sound power level can be expressed in dB PWL
  by comparing it to the 0 dB PWL reference point
  of 1pW.
• How many dB PWL is 4pW?
• dB PWL 10 log (4pW/1pW)
• dB PWL 10 log (4)
• dB PWL 10 (.60206)
• dB PWL 6.02
• We can express the difference between any two
  sound power levels (including electrical power)
  by using the 10 log formula.

16
dB PWL differences

• What is the dB difference between a 100-watt and
  350-watt amplifier?
• dB 10 log (100/350)
• dB 10 log (.2857143)
• dB 10 (-0.544068)
• dB -5.44
• The 100 watt amp is 5.44 dB less than the 350
  watt amp (or we could say the 350 watt amp is
  5.44 dB greater than the 100 watt amp).

17
The dB in Electronics

• dBs are used in audio electronics to express
  differences in power levels and voltage levels.
• In the early days of audio electronics all audio
  equipment was designed to have a 600 ohm output
  impedance.
• The dBm is a dB standard from those times and is
  not used for current audio equipment.
• Today we have the dBu, which has replaced the dBm.

18
The dB in Electronics

• Power dBm (0 dBm 1milliwatt into a 600 ohm
  load), 0dBW 1 watt into a 600 ohm load.
• dBm Power Formula 10 log (p1/.001W)
• dBW Power Formula 10 log (p1/1W)
• Voltage dBu (0 dBu .775 volts) not referenced
  to any load - chosen for historical reasons which
  is the voltage you get with 1mW in a 600 ohm
  load.
• dBu Voltage Formula 20 log (E1/.775V)
• dBV Voltage Formula 20 log (E1/1V)

19
Audio line level standards

• Today, in the United States, the professional
  line level standard is 4 dBu.
• 4 dBu audio gear generally uses balanced I/O.
• -10 dBV is the standard today for consumer audio
  gear.
• -10 dBV audio gear generally uses unbalanced I/O.

20
dB differences in power levels

• Any power level can be expressed as dBm or dBW.
• dBm and dBW both use the 10 log formula, however
  dBm uses 1mW for the 0 dB reference point dBW
  uses 1W for the 0 dB reference point.

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Calculating dBm

• How many dBm is a signal that measures 4 mW?
• dBm 10 log (4mW/1mW)
• dBm 10 log (4)
• dBm 10 (.60206)
• dBm 6.02

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Calculating dBW

• How many dBW is a signal that measures 5W?
• dBW 10 log (5W/1W)
• dBW 10 log (5)
• dBW 10 (.69897)
• dBW 6.9897

23
Calculating dB differences in power levels

• We can express the difference between any 2 power
  levels as dB (no suffix) by using the 10 log
  formula.
• We simply write the 2 voltages as a ratio.
• For example Whats the dB difference between 10
  watts and 15 watts?
• dB 10 log (10/15)
• dB 10 log (.66667)
• dB 10 (-0.1760913)
• dB -1.76, so 10W is 1.7609 dB less than 15W

24
dB differences in voltage levels

• Any voltage level can be expressed as dBu or dBV.
• dBu and dBV both use the 20 log formula, however
  dBu uses .775V for the 0 dB reference point dBV
  uses 1V for the 0 dB reference point.

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Calculating dBu

• How many dBu is a signal that measures 2 volts?
• dBu 20 log (2V/.775V)
• dBu 20 log (2.5806452)
• dBu 20 (.4117283)
• dBu 8.235

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Calculating dBV

• How many dBV is a signal that measures 2 volts?
• dBV 20 log (2V/1V)
• dBV 20 log (2)
• dBV 20 (.30103)
• dBV 6.02

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Calculating dB differences in voltage

• We can express the difference between any 2
  voltage levels as dB (no suffix) by using the 20
  log formula.
• We simply write the 2 voltages as a ratio.
• For example Whats the dB difference between 5
  volts and 10 volts?
• dB 20 log (5/10)
• dB 20 log (.5)
• dB 20 (-.30103)
• dB -6.02, so 5V is 6.02 less than 10V