Thoughts on the Decibel and the Logarithmic Scale

The acoustic dB/ The electronic dB/ The dBm

The acoustic dB 

The sound unit decibel measures acoustical and electrical power. The decibel is one tenth of a Bel; a larger unit of sound named for Alexander Graham Bell. Many telephone measurements depend on understanding the decibel. Let's look first at how the decibel measures acoustical power, then we'll turn to electrical.

The acoustical decibel measures sound intensity, not the loudness of a sound. A chainsaw and a car engine may have the same decibel level but the chainsaw seems louder because the human ear discerns higher pitched sounds more easily than low ones. An automobile wrecking yard might thus produce greater loudness than a rock quarry. The phon measures loudness.

A decibel is a single unit of sound, the first point on the decibel scale. But what makes up a unit?, a single decibel? One decibel is a relative measurement among people, it is the very limit of human hearing, the threshold point where sound can be detected by the ear and recognized by the brain. The noise produced by a single oak leaf falling to the ground is a single decibel.

A fighter jet at takeoff may produce 120 decibels. Does that mean it is 120 times more powerful than our fallen leaf? No. The difference is an unbelievable trillion times. The range of human hearing, between the softest sound you can hear, and the strongest sound you can stand, is well over a hundred million times. To measure that extraordinarily wide range we need an equally wide scale: the decibel scale.

The decibel scale, like the pH scale that measures acidity and alkalinity, and the Richter scale, which measures earthquake energy, are logarithmic, that is proportional. The decibel scale increases ten fold for every 10 decibels measured. It's base 10, that nemesis from junior high school mathematics. This scale lets us deal more easily with big numbers. The table below shows the problem and how a logarithmic scale solves it.

Let's turn away for a moment from the presentation problem, logarithmic or linear, and look at the numbers themselves. Unlike the electronic dB, the acoustic decibel has no fixed reference point. That means there is no standard distance to measure a sound from. Look at the numbers above. Leaves rustling at 10 feet obviously produce greater decibel readings than the same leaves at 20 feet. The figures above are often cited but I have never found out how far away the measuring instrument was. Could you get the same readings? Possibly. We get relatively uniform results only if the decibel measurement is tied to a specific distance from an object. Amateurs produced the table below. All measurements were from six feet.

Linear Scale 

 TIME (YEARS) But how do you present a rangeThat is far grand and greater?From when the moon first circled the earthTo when we named its craters (almost rhymes :-) For this we need a different scaleLogarithmic is its nameAnd as you can see, though this is geologyWe're not playing the linear game 

Back to electronics! 

Back to electronics!

The electronic dB

We've looked at how acoustical strength gets calculated. The decibel or dB also measures electrical strength in telecom, letting telephone personnel adjust landline and wireless systems.We'll look first at how the decibel measures gain, an increase in power. Let's compare that gain to the source putting out the signal, a hypothetical antenna somewhere in cell land. What engineers call an isotropic source. Our example below is a practical, real world problem, but difficult to understand at first reading. Go slow!

In cellular radio the carrier interference ratio or C/I helps engineers determine cell spacing and lets them adjust voice quality. The C/I is expressed in dBs. Cell sites must be spaced closely together to provide good coverage but cells with the same channels put too close together creates co-channel interference, something like cross talk on a landline telephone.

To fix cell spacing and to track down interference engineers do many tests. One test finds a base station transmitting a signal at a fixed level while an engineer measures that signal in the field, usually in a car. The car has two receivers. One receiver measures the signal from the first base station, and the other measures the signal strength coming from the cell they think the interference is coming from. Each receiver is tuned to a different channel. Signal strengths on each channel get recorded over many positions in the target cell. Engineers aim to have the first base station's signal 18dB larger than the one from the other cell site. But the versatile dB also helps with determining voice quality.

The good folks at comappls.com once put all this in context, summing up what we've just learned:

The minimum separation required between two nearby co-channel cells is based on specifying a tolerable co-channel interference, which is measured by a required carrier-to-interference ratio (C/I)s. The (C/I)s ratio also is a function of the minimum acceptable voice quality of the system. In an AMPS system, (C/I)s is equal to about 18 dB (the point at which 75% of the users call the system "good" or "excellent") and the minimum required separation, based on (C/I)s = 18dB, is about 4.6R, where R is the radius of the cell (to the point where two faces of the hexagon join).

Okay, enough engineering speak. If you want to know more about C/I, Mark van der Hoek also writes about it in the cellular radio basics article.

The dBm

The dBm stands for decibels above or below one milliwatt. We measure a signal's gain or loss by comparing it to a fixed reference point, the milliwatt, one thousandth of a watt. Things get a little complicated here. Read Mark's comments slowly. And then again for the best understanding:

"If you receive a signal at one milliwatt you have a signal strength of 0 dBm. Loss is measured in dB, and to know loss you must know the original power level. IF the original transmit power is 0 dBm and the received signal strength is 0 dBm, then you can say the loss is 0 dB.

"If you receive a signal that is 0.001 milliwatts, then you have a loss of 30 dBm. Again, same problem. You would have a signal strength of -30 dBm, and a loss of 30 dB."

"Some methods of testing analog phone lines include dialing a number that answers and provides a 1-milliwatt reference signal at 1000 Hz. The meter on the line measures the 1000-Hz signal on its end and displays a reading. Most POTs telephone lines are between -20 and -32dBm. This is true because the original signal level is known. It would be correct to say that the line loss is 20 to 32 dB. dBs measure a ratio between two levels. dBm measures a power level (not a ratio), referenced to a known power level. It is expressed as a ratio to that known power level, but it is not a ratio.

"Actually, I think I made it too complicated. In simplified form:

Really, when we say -63 dBm, we are using verbal shorthand to say, "A signal that is 63 dB lower than a 1 milliwatt signal."

Mark

Here's the same information in a linear scale and then a logarithmic scale. It reports on a transistor's instructions per second in millions.

Linear:

Logarithmic:

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