Beautiful

The beautiful (symbol B ), although apart from the international Système (IF), is of use with him. More formally, the beautiful one is a unit without dimension, expressing the order of magnitude (positive or negative) of the relationship between the absolute values of two of the same measurements dimension, one of these two measurements being a value of reference.

It is used to express the value of sizes logarithmic curves without dimension such as the level of field, the level of power, sound intensity, acoustic level of Pression or attenuation. The Logarithme S basic ten are used to obtain the numerical values of the sizes expressed in bels. For further information, to see the international standard ISO 31.

It was named in the honor of the physicist Alexandre Graham Bell.

It is a unit logarithmic curve where a difference of 1 beautiful corresponds to a ratio of 10 in power. One uses his submultiple more usually the decibel (dB). A difference in a decibel corresponds to a report/ratio of $10^\; \{1/10\}$ is about 1,259.

Formulation

One calculates the level in bels or decibels of a signal $V$ compared to a second signal $V\_0$ of the same dimension with the formules :

• $B\; \backslash \; left\; (V\; \backslash \; right)\; =\; log\_\; \{10\}\; \{\backslash \; left|\backslash \; frac\; \{V\}\; \{V\_0\}\; \backslash \; right|\}\; =\; log\_\; \{10\}\; \{\backslash \; left|V\; \backslash \; right|\}\; -\; log\_\; \{10\}\; \{\backslash \; left|V\_0\; \backslash \; right|\}$

Like beautiful equalizes 10 decibels, to convert the beautiful one into decibel, it is enough to multiply it by ten, from where:

• $dB\; \backslash \; left\; (V\; \backslash \; right)\; =\; 10\; \backslash \; cdot\; log\_\; \{10\}\; \{\backslash \; left|\backslash \; frac\; \{V\}\; \{V\_0\}\; \backslash \; right|\}\; =\; 10\; \backslash \; cdot\; log\_\; \{10\}\; \{\backslash \; left|V\; \backslash \; right|\}\; -\; 10\; \backslash \; cdot\; log\_\; \{10\}\; \{\backslash \; left|V\_0\; \backslash \; right|\}$

In the formulas above, the value of $V\_0$ measurement, is the measurement of reference. The equivalent expression using the difference of the logarithms makes it possible to simplify calculation by being unaware of the constant value of the measurement of reference, because then there exists a simple relation closely connected, which preserves the relation of order between measurements, the scale in bels (or decibels) and the simple logarithm of measurement. Consequently, the level in bels (or decibels) establishes well the Order of magnitude making it possible to compare measurements, this order also including/understanding the measurement of reference.

Independence of the measuring units

When the relationship is calculated between two values of field or tension (so that the transmitted power is proportional to the square of this value), the formula est :

• $B\; \backslash \; left\; (V^2\; \backslash \; right)\; =\; \backslash \; log\_\; \{10\}\; \{\backslash \; left|\backslash frac\{V^2\}\{V\_0^2\}\backslash right|\}\; =\; 2\; \backslash \; cdot\; \backslash \; log\_\; \{10\}\; \{\backslash \; left|\backslash \; frac\; \{V\}\; \{V\_0\}\; \backslash \; right|\}$

In other words  :

• $B\; \backslash \; left\; (V^2\; \backslash \; right)\; =\; 2\; \backslash \; cdot\; B\; (V)$, and of the same  : $dB\; \backslash \; left\; (V^2\; \backslash \; right)\; =\; 2\; \backslash \; cdot\; dB\; \backslash \; left\; (V\; \backslash \; right)$

It is seen consequently that the order of magnitude of a report/ratio expressed in bels or decibels can be calculated regardless of the fact that measurements are taken to the measure of the fields or that of the powers, except for a constant factor.

In the same way, the measuring unit can be selected arbitrarily in measured dimension. Indeed, if $\backslash \; V\; \backslash$ is the value of the measurement expressed in the unit $\backslash \; U\; \backslash$, and $\backslash \; V^\; \backslash \; premium\; \backslash$ is the value of the same measurement expressed in the of the same unit dimension $\backslash \; u^\; \backslash \; premium\; \backslash$, so that $1\; \backslash \; U\; =\; K\; \backslash \; u^\; \backslash \; prime$, where $\backslash \; K\; \backslash$ is a constant factor, alors :

• $V^\; \backslash \; premium\; =\; K\; \backslash \; cdot\; V$, and aussi : $V\_0^\; \backslash \; premium\; =\; K\; \backslash \; cdot\; V\_0$
• $B\; \backslash \; left\; (V^\; \backslash \; premium\; \backslash \; right)\; =\; \backslash \; log\_\; \{10\}\; \{\backslash \; left|\backslash \; frac\; \{K\; \backslash \; cdot\; V\}\; \{K\; \backslash \; cdot\; V\_0\}\; \backslash \; right|\}$,

and donc :

• $B\; \backslash \; left\; (V^\; \backslash \; premium\; \backslash \; right)\; =\; B\; \backslash \; left\; (V\; \backslash \; right)$, and of the same  : $dB\; \backslash \; left\; (V^\; \backslash \; premium\; \backslash \; right)\; =\; dB\; \backslash \; left\; (V\; \backslash \; right)$.

In other words, the level of any physical measurement which does not depend that of only one variable closely connected elementary can be calculated and expressed in bels or decibels, whatever the choice of the measuring units used, of their own order of magnitude (constant factor) and their effective dimension (also constant power), except for a constant factor.

Signal  report/ratio; /  noise

If S is measurement in tension or intensity of a received signal, and NR the share of noise in this signal, the signal  report/ratio; /  noise is calculated ainsi :

• $(dB)\; =\; 10\; \backslash \; cdot\; \backslash \; log\_\; \{10\}\; \backslash \; left|\backslash \; frac\; \{S\}\; \{NR\}\; \backslash \; right|$

Here, one measures the quality of reception or restitution of a signal S compared to his initial value of emission S₀, the noise corresponding to the difference NR between the two signaux :

• $N\; \backslash \; =\; \backslash \; S\; -\; S\_0$.

The signal  report/ratio; /  noise is written aussi  then;:

• $(dB)\; \backslash \; =\; \backslash \; 10\; \backslash \; cdot\; \backslash \; log\_\; \{10\}\; \backslash \; left|\backslash \; frac\; \{S\}\; \{S\; -\; S\_0\}\; \backslash \; right|\backslash \; =\; \backslash \; 10\; \backslash \; cdot\; \backslash \; log\_\; \{10\}\; \backslash \; left|\backslash \; frac\; \{S\_0\}\; \{NR\}\; +\; 1\; \backslash \; right|$

It is indicated in decibels. It is infinite ($+\; \backslash \; infin$) with the source of the original signal (of which the noise is null), and can only decrease during the transmission.

A signal  report/ratio; /  null noise translates the fact that the received signal does not make it possible any more to distinguish in a reliable way the original signal of the noise, their respective powers being equal. If the signal  report/ratio; /  noise is negative, one will perceive only the noise. The quality of a system of transmission or storage of information is thus appreciated with a signal  report/ratio; /  the positive noise (would be this only slightly) and largest possible.

The signal  report/ratio; /  noise in power measures him in decibel-Watts (dBW). It is simply the double of the signal  report/ratio; /  noise in tension or intensity, because the received power is proportional to the square of the tension or the induced intensity and mesurée :

• $(dBW)\; =\; 2\; \backslash \; cdot\; (dB)$.

Useful usual values

The profit of power of a Amplificateur is translated into décibels :

• +  0,1  dB_W corresponds to a profit slightly lower than +  2,3  %.
• +  0,5  dB_W corresponds to a profit slightly higher than +  12,2  %.
• +  1  dB_W corresponds to a profit slightly higher than +  25,9  %.
• +  2  dB_W corresponds to a profit slightly higher than +  58  %.
• +  3  dB_W corresponds to a profit slightly lower than factor 2.
• +  6  dB_W corresponds to a profit slightly lower than factor 4.
• +  7  dB_W corresponds to a profit slightly higher than factor 5.
• +  9  dB_W corresponds to a profit slightly lower than factor 8.
• +  10  dB_W (or +  1  B) correspond to a profit of factor 10 exactly.
• +  20  dB_W (or +  2  B) correspond to a profit of factor 100 exactly.
• +  30  dB_W (or +  3  B) correspond to a profit of factor 1  000 exactly.

As the power is proportional is with the square of the tension, one a:
$10\; \backslash \; cdot\; log\_\; \{10\}\; \{\backslash \; left|\backslash \; frac\; \{P\}\; \{P\_0\}\; \backslash \; right|\}\; =\; 20\; \backslash \; cdot\; log\_\; \{10\}\; \{\backslash \; left|\backslash \; frac\; \{U\}\; \{U\_0\}\; \backslash \; right|\}$

The values of profit in tension (expressed in dB (V) or decibel-volt), are double of those of the profit in power (expressed in dB_W or decibel-Watt). For example:

• +  3  dB (V) correspond to a profit of tension slightly higher than the factor 1,41 $(\backslash \; sqrt\; 2)$ ((factor 2 in power).
• +  6  dB (V) correspond to a profit of tension slightly lower than factor 2 (factor 4 in power).
• +  9  dB (V) correspond to a profit of tension slightly higher than factor 2,8 (factor 8 in power).
• +  10  dB (V) correspond to a profit of tension slightly lower than factor 3,2 (factor 10 in power).
• +  12  dB (V) correspond to a profit of tension slightly lower than factor 4 (factor 15,8 in power).
• +  20  dB (V) correspond to a profit of tension of factor 10 exactly (factor 100 in power).
• +  30  dB (V) correspond to a profit of tension slightly higher than factor 31,6 (factor 1  000 in power).

It should be noted that the level in bels or decibels of a null measurement V cannot be calculated. But one will conventionally express it by his limit $-\; \backslash \; infin$. The formula does not have on the other hand a direction if the value of measurement of reference is null. However, the formula remains used when measurement V is negative or of contrary sign to the measurement of reference, if this measurement V remains of constant sign (if not measurements cannot be only compared by their level in bels and decibels). The choice of the measurement of reference and its dimension is thus fundamental.

The same will apply to the intensity of the sound, no matter what other factors return concerned, as in any phenomenon perceptif : to increase a sound power of +  3  dB returns to practically doubling the perceived sound intensity.

Numerical applications

Correlation of the signal  report/ratio; /  noise and of the number of bits of information usable

For example in the numerical applications of data transmission to binary coding, it is necessary that the intensity or the tension of the signal received S is at least the double of that of the noise (i.e. the quadruple in power) in order to be able to decode the signal with a total reliability. It is then necessary that the signal  report/ratio; /  noise is at least equal permanently to +  6  dB in intensity or tension (or +  12  dB_W in power).

If the signal  report/ratio; /  average noise reached is higher than +  6  dB, but is not stable locally so that one cannot guarantee this minimum level of +  6  dB_W in a permanent way, it is necessary to use a system of detection and possibly of correction of error, based on the emission of an additional signal shifted in time compared to the signal to transmit, this shift making it possible to increase the probability of reliable reception of at least one of the two signals (the useful signal, and the signal of detection and correction of error). One can increase this probability (and thus the reliability of the transmission) by combining several signals of detection and correction of error shifted them also in time (but that is done at the price of an increase in the band-width necessary).

To obtain a bit of extra information, at equal power of noise and with the same quantum of time of measurement (even sampling rate), it is necessary to double the power of the received useful signal, i.e. to increase the level of this signal of +6  dB. It is not possible to amplify this received power on the side of the receiver without also doubling that of the noise.

One shows thus that for any signal of sufficient quality and of intensity or tension S, the number of separable bits of information in a way reliable and transmitted simultaneously in the same signal is directly proportional to the signal  report/ratio; /  noise in power of the received signal, each additional bit requiring 6 dB supplémentaires :

$bits\; (S)\; \backslash \; approx\; \backslash \; frac16\; \backslash \; cdot\; (dB)$, or

$bits\; (S)\; \backslash \; approx\; \backslash \; frac1\; \{12\}\; \backslash \; cdot\; (dB\_W)$.

The exact value est :

$bits\; (S)\; =\; \backslash \; frac\; \{1\}\; \{20\; \backslash \; cdot\; \backslash \; log\_\; \{10\}\; \{2\}\}\; \backslash \; cdot\; (dB)$, or

$bits\; (S)\; =\; \backslash \; frac\; \{1\}\; \{40\; \backslash \; cdot\; \backslash \; log\_\; \{10\}\; \{2\}\}\; \backslash \; cdot\; (dB\_W)$.

The exact formula above is simplified, by substitution :

$bits\; (S)\; =\; \backslash \; frac\; \{1\}\; \{20\; \backslash \; cdot\; \backslash \; log\_\; \{10\}\; \{2\}\}\; \backslash \; cdot\; 10\; \backslash \; cdot\; \backslash \; log\_\; \{10\}\; \backslash \; left|\backslash \; frac\; \{S\}\; \{S\; -\; S\_0\}\; \backslash \; right|$

c'est-with-dire :

$bits\; (S)\; =\; \backslash \; frac12\; \backslash \; cdot\; \backslash \; log\_\; \{2\}\; \backslash \; left|\backslash \; frac\; \{S\}\; \{S\; -\; S\_0\}\; \backslash \; right|$

or in an equivalent way, seen transmitter of the signal original :

$bits\; (S)\; =\; \backslash \; frac12\; \backslash \; cdot\; \backslash \; log\_\; \{2\}\; \backslash \; left|\backslash \; frac\; \{S\_0\}\; \{NR\}\; +\; 1\; \backslash \; right|$

If the formula above does not turn over an integer of bits, one can use the immediately lower entirety in the design of a system of transmission. But one can increase the useful band-width, without increasing the band-width necessary, by using a grid of coding to nonbinary numeration allowing to better approach the median number of bits characteristic of the system of transmission obtained with the formula above.

It thus appears that the bit is another unit without dimension, proportional to beautiful, and translating the same order of magnitude (with only one different base). Also the bit will be preferred with beautiful in the applications of transmission of numeric signals. It is thus essential not to confuse their symbols respectifs : “  B  ” is the symbol of beautiful, “  b  ” is that of the bit (the symbol of the byte, which is a multiple of the bit is often “  B  ”, which can lend to confusion  ; for this reason, the useful band-width of a digital system of transmission is always expressed out of bits a second rather than in bytes a second.)

Improvement of the signal  report/ratio; /  noise

The signal  report/ratio; /  noise allows to deduce bandwidth useful (out of bits a second), which is the result of the product of the number of separable simultaneous bits within the same quantum of temp (obtained by the formula above), and from frequency from quantification from signal received (expressed in Baud, unit without dimension similar to Hertz but which is characterized from it by the fact that it is quantified and results from an arbitrary choice within the sensor, nonrelated to the particular characteristics from the signal itself). These results are applied in practice in the systems design of transmission of information (data bus, Modem or another modulator or codec of broadcasting, multiplexer on fiberoptic…) and numerical systems of recording (circuit report, hard drive, numerical optical disk, magneto-optical disc…).

To reduce the perceived noise level on the side of the receiver, the only means is amplification (in dB positive) made side of the transmitter, in order to compensate for the attenuation of the signal (in dB negative) during its transmission. If one cannot avoid the attenuation of the useful signal and the amplification (in dB positive) of the noise on the totality of the transmission, one can use repeaters provided with correct filters before reamplification of the corrected signal. That supposes that the initial transmitter transmitted well a correct signal in addition to the useful signal.

But the use of correct filters induced an additional time of transmission, function directly of the temporal shift enters the principal signal and the signal of correction of error all the deus perceived by the repeater. Indeed, the repeater cannot réamplifier and to transmit the signal corrected before to have received the signal of detection and correction of error.

To avoid this temporal shift, it is possible in certain cases to use a second independent medium, if one can show that the random noise undergoes on this second medium is independent of the sudden random noise per the principal medium. A current way to limit this time is to use distinct frequency channels, multiplexed in parallel on the same physical support. This technique of transmission to self-correcting multiple channels is used in the equipment repeaters of connections, which make it possible to maintain a signal  report/ratio; /  sufficient noise at longer distances.

On the other hand, a storage system can seldom have repeater. When the density of information on the support is such as the size of elementary information becomes comparable with atomic dimensions, the unverifiable fluctuations on this scale, corresponding to the noise, will make that one will have obligatorily important deteriorations obliging the recourse to mathematical mechanisms of detection and correction of error. It is the case on all the modern Hard drives. It will be noticed that the electronic memories with chance access to dynamic cooling (DRAM) require a periodic cooling which is, literally, a repetition.

Profit and band-width of an ideal system

One will note that with an ideal system of transmission (without noise), the signal  report/ratio; /bruit and thus the number of separable bits in the same signal and the same quantum of time of measurement is infinite . This result, which can seem surprising, makes that with density quantification in time (sampling rate) equalizes, the useful band-width (out of bits a second) can be made arbitrarily large, on the condition of removing or to attenuate as much as possible sources of noise lasting the transmission of the signal, or by detecting them and eliminating them in a reliable way to the reception of the signal.

The quantitative and qualitative study of the spectrum of the effective sources of noise (as well as average techniques allowing to eliminate them or attenuate them) thus allows major technical projections, and thus an increase in the useful band-width in the systems of digital transmission or the numerical systems of recording.

However, there exist quantum limits with these evolutions, according to the principle of uncertainty of Heisenberg, by whom any measurement of a signal induces a modification of this signal, and thus the addition of a signal of noise directly related to the quantum disturbance produced by the measuring instrument of this signal.

In the case of the systems of transmission or data storage, these disturbances are generally induced by the amplifiers of signals and are all the more important as the profit (expressed in bels or decibels) of the amplifiers used is important.

Other disturbances come from the intrinsic inaccuracy and the instability of the filters band pass used, environmental factors which induce their own noise even in the event of good insulation of the medium of transmission or storage, and other variables with randomness (unforeseeable) such as  :

• the chaotic and random instability of the atomic undulatory states related on the temperature and the inherent entropy of the system,
• atomic disintegrations within the sensors or of the media of transmission or storage, and
• particles with high energy present everywhere in the universe, against which it is impossible to insulate the sensors and supports completely,

as many natural phenomena which one will have to hold account at least statistically, if one wishes to push very far the useful band-width.

See too

• Neper
• signal Report/ratio on noise
• dBa (unit): environmental measuring unit of the noise
• Comparison of the volume of current sources of noise

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