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What is a logarithm? A brief introduction. Definition and examples The decibel dB is a logarithmic unit used to measure sound level. It is also widely used in electronics, als and communication.

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What is a logarithm? A brief introduction. Definition and examples The decibel dB is a logarithmic unit used to measure sound level.

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It is also widely used in electronics, als and communication. The dB is a logarithmic way of describing a ratio.

The ratio may be power, sound pressure, voltage or intensity or several other things. Later on we relate dB to the phon and to the sone, which measures loudness. But first, to get a taste for logarithmic expressions, let's look at some s.

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4420 you have forgotten, go to What is a logarithm? For instance, suppose we have two loudspeakers, the first playing a sound with power P1, and another playing a louder version of the same sound with power P2, but everything else how far away, frequency kept the same. This example shows a feature of decibel scales that is useful in discussing sound: they can describe very big ratios using s of modest size.

But note that the decibel describes a ratio: so far we have not said what power either of the speakers radiates, only the ratio of powers. Sound pressure, sound level and dB. Sound is usually measured with microphones and they respond proportionally to the sound pressure, p.

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Now the power in a sound wave, all else equal, goes as the square of the pressure. Similarly, electrical power in a resistor goes as the square of the voltage. What happens when you halve the sound power? The log of 2 is 0. If you keep on halving the power, you have these ratios. What happens if I add two identical sounds? This frequently asked question is a little subtle, so it is discussed here on our FAQ. That is what we have done in the first graphic and sound file below. The first sample of sound is white noise a mix of a broad range of audible frequencies, analogous to white light, which is a mix of all visible frequencies.

The green line shows the voltage as a function of time. The red line shows a continuous exponential decay with time. Note, too, that a doubling of the power does not make a huge difference to the loudness. We'll discuss this further below, but it's a useful thing to remember when choosing sound reproduction equipment. Sound files and animation by John Tann and George Hatsidimitris. How big is a decibel? In the next series, successive samples are reduced by just one decibel. As you listen to these files, you will notice that the last is quieter than the first, but it is rather less clear to the ear that the second of any pair is quieter than its predecessor.

What if the difference is less than a decibel? Sound levels are rarely given with decimal places. This makes the dB a convenient size unit. You may notice that the last is quieter than the first, but it is difficult to notice the difference between successive pairs. Standard reference levels 'absolute' sound level We said above that the decibel is a ratio. So, when it is used to give the sound level for a single sound rather than a ratio, a reference level must be chosen.

This is very low: it is 2 ten billionths of an atmosphere. Nevertheless, this is about the limit of sensitivity of the human ear, in its sensitive range of frequency. Usually this sensitivity is only found in rather young people or in people who have not been exposed to loud music or other loud noises.

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Personal music systems with in-ear speakers are capable of very high sound levels in the ear, and are believed by some to be responsible for much of the hearing loss in young adults in some countries. What does 0 dB mean? This level occurs when the measured intensity is equal to the reference level. This is a small pressure, but not zero. Not all sound pressures are equally loud.

For this reason, sound meters are usually fitted with a filter whose response to frequency is a bit like that of the human ear. More about these filters below. Sound pressure level on the dBA scale is easy to measure and is therefore widely used. One reason why it is different from loudness is because the filter does not respond in the same way as the ear.

To understand the loudness of a sound, the first thing you need to do consult some curves representing the frequency response of the human ear, given below. Alternatively, you can measure your own hearing response. Another reason is that human hearing is not logarithmic.

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Logarithmic measures Why do we use decibels? The ear is capable of hearing a very large range of sounds: the ratio of the sound pressure that causes permanent damage from short exposure to the limit that undamaged ,oud can hear is more than a million. To deal with such a range, logarithmic units are useful: the log of a million is 6, so this ratio represents a difference of dB. Hearing is not inherently logarithmic in response. Logarithmic measures are also useful when a sound briefly increases or decreases exponentially over time.

This happens in many applications involving proportional gain or proportional loss. Using this filter, the sound level liud is thus less sensitive ooud very high and very low frequencies. Measurements made on this scale are expressed as dBA. The C scale varies little over several octaves and is thus suitable for subjective measurements only for moderate to high sound levels.

Measurements made on this scale are expressed as dB C. There is also a rarely used B weighting scale, intermediate between A onw C.

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The figure below shows the response of the A filter left and C Nee, with gains in dB given with respect to 1 kHz. For an introduction to filters, see RC filters, integrators and differentiators. On the music acoustics and speech acoustics sites, we plot the sound spectra in dB. The reason liud this common practice is that the range of measured sound pressures is large. It thus gives large values for sounds and infrasounds that cannot readily be heard. ISO Loudness, phons and sones, hearing response curves The phon is a unit that is related to dB by the psychophysically measured frequency response of the ear.

To convert from dB to phons, you need a graph of such. Such a graph depends on sound level: it becomes flatter at high sound levels. This graph, courtesy of Lindoslandshows the data from the International Standards Organisation for curves of equal loudness determined experimentally.

Plots of equal loudness as a function of frequency are often generically called Fletcher-Munson curves after the original work by Fletcher, H. You can make your own curves using our hearing response site. The sone is derived from psychophysical measurements which involved volunteers adjusting sounds until they judge them to be twice as loud. This allows one to relate perceived loudness to phons. Wouldn't it be great to be able to convert from Nsed which can be measured by an instrument to sones which approximate loudness as perceived by people?

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This is sometimes done using tables that you can find in acoustics handbooks. However, if you don't mind a rather crude approximation, you can say that the A weighting curve approximates the human frequency response at low to moderate sound levels, so dB A is very roughly the same as phons, over a no range of low levels. Then one can use the logarithmic relation between sones and phons described above. Recording level and decibels Meters measuring recording or output level on audio electronic gear mixing consoles etc are almost always recording the AC rms voltage see links to find out about AC and rms.

So what is the reference voltage? The obvious level to choose is one volt rms, and in this case the level is written as dBV. This is rational, and poud convenient with analog-digital cards whose maximum range is often about one volt rms. So one has to remember to the keep the level in negative dBV less than one volt to avoid clipping the peaks of the al, but not too negative so your al is still much bigger than the background noise. Sometimes you will see dBm. This used to mean decibels of electrical power, with respect to one milliwatt, and sometimes it still does.

However, it's complicated for historical reasons. When I was ,oud boy, calculators were expensive so I used dad's old slide rule, which had the factor 0. How to convert dBV or dBm into dB of sound level? There is no simple way.

It depends on how you convert the electrical power into sound power. Even if your electrical al is connected directly to a loudspeaker, the conversion will depend on the efficiency and impedance of your loudspeaker. And of course there may be a power amplifier, and various acoustic complications between where you measure the dBV on the mixing desk and where your onw are in the sound field. Intensity, radiation and dB How does sound level or radio al level, etc depend on distance from the source?

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A source that emits radiation equally in all directions is called isotropic. Consider an isolated source of sound, far from any reflecting surfaces — perhaps a bird singing high in the air. Imagine a sphere with radius r, centred on the source. The source outputs a total power P, continuously.

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