## sábado, 6 de febrero de 2010

### Significance of Octaves and Decades. The decibel unit

One decade is a factor of 10 difference between two numbers (an order of magnitude difference) measured on a logarithmic scale. It is especially useful when referring to frequencies and when describing frequency response of electronic systems, such as audio amplifiers and filters

Calculations

The factor-of-ten in a decade can be in either direction: so one decade up from 100 Hz is 1000 Hz, and one decade down is 10 Hz. The factor-of-ten is what is important, not the unit used, so 3.14 rad/s is one decade down from 31.4 rad/s.

To determine the number of decades between two frequencies, use the logarithm of the ratio of the two values:

log10 (150000 / 15) = 4 decades

-How many decades is it from 3.2 GHz to 4.7 MHz?

-How many decades is one octave?

One octave is a factor of 2, so log10(2) = 0.301 decades per octave

To find out what frequency is a certain number of decades from the original frequency, multiply by appropriate powers of 10:

-What is 3 decades down from 220 Hz?

220x10-3 =0.22Hz

-What is 1.5 decades up from 10?

10x101.5 =316.23

To find out the size of a step for a certain number of frequencies per decade, raise 10 to the power of the inverse of the number of steps:

-What is the step size for 30 steps per decade?

10 1/ 30 = 1.079775 - or each step is 7.9775% larger than the last

Graphical representation and analysis

Decades on a logarithmic scale, rather than unit steps (steps of 1) or other linear scale, are commonly used on the horizontal axis when representing the frequency response of electronic circuits in graphical form, such as in Bode plots, since depicting large frequency ranges on a linear scale is often not practical. For example, an audio amplifier will usually have a frequency band ranging from 20 Hz to 20 kHz and representing the entire band using a decade log scale is very convenient. Typically the graph for such a representation would begin at 1 Hz (100) and go up to perhaps 100 kHz (105), to comfortably include the full audio band in a standard-sized graph paper, as shown below. Where as in the same distance on a linear scale, with 10 as the major step-size, you might only get from 0 to 50.

Electronic frequency responses are often described in terms of "per decade". The example Bode plot shows a slope of -20 dB/decade in the stopband, which means that for every factor-of-ten increase in frequency (going from 10 rad/s to 100 rad/s in the figure), the gain decreases by 20 dB.

OCTAVE AND OTHER FREQUENCY INTERVALS

The octave in music

In music, the octave is the interval between two frequencies which are in the ratio of 2-to-1 (i.e. the higher frequency is exactly twice the lower frequency).  Check out the following examples.

 a one octave  rise  from 125 Hz to 250 Hz a one octave  fall  from 800 Hz to 400 Hz a two octave  rise  from 4 kHz to 16 kHz a three octave  rise  from 500 Hz to 4 kHz

The complete musical scale is generated by defining one frequency (the note A which is 440 Hz) and working out all of the other musical notes, with sharps and flats etc, from that defined frequency.

The octave in acoustics and audio

In acoustics and audio, the octave is the interval between two frequencies which are in the ratio of  10 0.3 to 1     10 0.3 = 1.995  (Calculator key strokes:  [shift]  [log]  [0]  [.]  [3]  [=] )

The standard octave intervals in acoustics are worked out starting from the Reference Frequency of 1 kHz which is 10 3.0 Hz.  The full sequence of frequencies is:

 Calculated from 10 1.5 10 1.8 10 2.1 10 2.4 10 2.7 10 3.0 10 3.3 10 3.6 10 3.9 10 4.2 Exact frequency (Hz) 31.62 63.10 125.9 251.2 501.2 1 k 1.995k 3.981k 7.943k 15.85k Nominal frequency (Hz) 31.5 63 125 250 500 1 k 2 k 4 k 8 k 16 k

On page 2 there is an image of an octave band graphic equalizer.  Look below the sliders for the nominal frequencies listed on the bottom row of the table above.

Fractions of an octave

On the Klark-Teknik graphic equalizer, on page 3, there are two sliders between each of the standard octave sliders.  For example, between the sliders at 250 Hz and 500 Hz there is one at 315 Hz and one at 400 Hz.  The sliders on this graphic equaliser are arranged at one-third-octave intervals.

 Slider: 250 Hz 315 Hz 400 Hz 500 Hz Interval: < one-third-octave > < one-third-octave > < one-third-octave > <                                           one octave                                            >

For an octave, the ratio is  10 0.3 .  The number 0.3 is the exponent (i.e. the power of ten).

We work out an interval which is a given fraction of an octave by taking the same fraction of 0.3

So, for one-third-octave, the exponent is a third of 0.3, i.e.   1/3 × 0.3  =  0.1  and  10 0.1  =  1.2589 .

This table explains how a number of fractions of an octave can be calculated.

 1-octave 1/2-octave 1/3-octave 1/6-octave 1/12-octave Value of the exponent 1 × 0.3 = 0.3 1/2 × 0.3 = 0.15 1/3 × 0.3 = 0.10 1/6 × 0.3 = 0.05 1/12 × 0.3 = 0.025 Which gives a ratio of 10 0.3 = 1.995 10 0.15 = 1.413 10 0.10 = 1.259 10 0.05 = 1.122 10 0.025 = 1.059

Octave bands

The image below is a graphics user interface (GUI) from some audio-processing computer software and it shows an octave band equaliser.  There are 10 octave bands in the audio range; hence 10 sliders.

Below the sliders, you can see the octave interval nominal frequencies *.  Each slider controls the gain for a one-octave-wide band of frequencies.  The gain, in dB, is shown in the window above each slider.

Each slider=s quoted frequency is at the centre of its respective octave band.  The quoted frequency is called the octave band centre frequency.  We identify which octave band we are controlling by quoting the octave band centre frequency.  Each of the ten octave bands reaches half an octave above the band centre frequency and half an octave below the band centre frequency.  So:

the Band Upper Limit is half an octave above (i.e. × 10 0.15) the band centre frequency; and

the Band Lower Limit is half an octave below (i.e. ÷ 10 0.15) the band centre frequency.

For the 500 Hz octave band (shown with a gain of + 2.9 dB in the image):

the exact band centre frequency is 10 2.7  (see page 1 for how this is worked out);

the band upper limit is 10 2.7 × 10 0.15  =  10 2.7 + 0.15  =  10 2.85  =  707.9 Hz

the band lower limit is 10 2.7 ÷ 10 0.15  =  10 2.7 - 0.15  =  10 2.55  =  354.8 Hz

The slider labelled 500 Hz does not adjust the gain only for the frequency of 500 Hz.  It adjusts the gain for an octave band of frequencies from 354.8 Hz up to 707.9 Hz.  500 Hz is at the centre of that octave band.  The slider adjusts the gain for the whole of the 500 Hz octave band.

The lowest frequency being processed by the software above, is not 31.5 Hz *, but is the frequency at the band lower limit of the 31.5 Hz octave band.  (See question 2(e) on page 4.)

The highest frequency being processed by the software is not 16 kHz, but is the frequency at the band upper limit of the 16 kHz octave band.  (See question 2(a) on page 4.)

The above image is of an octave band equaliser.  The following page shows an example of a one-third-octave band equaliser.  The frequency bands are narrower and the EQ control much finer.

(*  31 Hz should be labelled 31.5 Hz; and 62 Hz should be labelled 63 Hz.  Many manufacturers get these two wrong.)

One-third-octave Bands

The image below shows a one-third-octave band graphic equaliser.  There are 10 octaves in the audio range; so there are 30 one-third-octaves in the audio range.

Each one of the 30 sliders adjusts the gain of the signal for a band of frequencies which is one-third-of-an-octave wide.

These professional equalisers offer fine detailed adjustment to the equalisation (EQ) of the audio signal.

The frequency quoted under each slider is the nominal Band Centre Frequency.  So each third-octave band extends from one-sixth of an octave below the band centre frequency to one-sixth of an octave above the band centre frequency.  The two extremes are called the Band Lower Limit  and the Band Upper Limit.

Notice that the vertical position of the sliders is calibrated in dB with markings at  +12, +6, +3, 0, -3, -6, and -12 dB, showing how the power level of each band is increased or decreased relative to the flat response , 0 dB, position.

The decibel

The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity) relative to a specified or implied reference level. Since it expresses a ratio of two quantities with the same unit, it is a dimensionless unit. A decibel is one tenth of a bel, a seldom-used unit.

The decibel is widely known as a measure of sound pressure level, but is also used for a wide variety of other measurements in science and engineering (particularly acoustics, electronics, and control theory) and other disciplines. It confers a number of advantages, such as the ability to conveniently represent very large or small numbers, a logarithmic scaling that roughly corresponds to the human perception of sound and light, and the ability to carry out multiplication of ratios by simple addition and subtraction.

The decibel symbol is often qualified with a suffix, which indicates which reference quantity or frequency weighting function has been used. For example, "dBm" indicates that the reference quantity is one milliwatt, while "dBu" is referenced to 0.775 volts RMS.[1]

The definitions of the decibel and bel use base-10 logarithms. For a similar unit using natural logarithms to base e, see neper.

History

The decibel originates from methods used to quantify reductions in audio levels in telephone circuits. These losses were originally measured in units of Miles of Standard Cable (MSC), where 1 MSC corresponded to the loss of power over a 1 mile (approximately 1.6 km) length of standard telephone cable at a frequency of 5000 radians per second (795.8 Hz) and roughly matched the smallest attenuation detectable to an average listener. Standard telephone cable was defined as "a cable having uniformly distributed resistances of 88 ohms per loop mile and uniformly distributed shunt capacitance of .054 microfarad per mile" (approximately 19 gauge).[citation needed]

The transmission unit or TU was devised by engineers of the Bell Telephone Laboratories in the 1920s to replace the MSC. 1 TU was defined as ten times the base-10 logarithm of the ratio of measured power to reference power.[2] The definitions were conveniently chosen such that 1 TU approximately equalled 1 MSC (specifically, 1.056 TU = 1 MSC).[3] Eventually, international standards bodies adopted the base-10 logarithm of the power ratio as a standard unit, which was named the "bel" in honor of the Bell System's founder and telecommunications pioneer Alexander Graham Bell. The bel was a factor of ten larger than the TU, such that 1 TU equalled 1 decibel.[4] In many situations, the bel proved inconveniently large, so the decibel has become more common.

In April 2003, the International Committee for Weights and Measures (CIPM) considered a recommendation for the decibel's inclusion in the SI system, but decided not to adopt the decibel as an SI unit.[5] However, the decibel is recognized by other international bodies such as the International Electrotechnical Commission (IEC).[6] The IEC permits the use of the decibel with field quantities as well as power and this recommendation is followed by many national standards bodies, such as NIST, which justifies the use of the decibel for voltage ratios.

Alexander Graham Bell

Merits

The use of the decibel has a number of merits:

*The decibel's logarithmic nature means that a very large range of ratios can be represented by a convenient number, in a similar manner to scientific notation. This allows one to clearly visualize huge changes of some quantity. (See Bode Plot and half logarithm graph.)

*The mathematical properties of logarithms mean that the overall decibel gain of a multi-component system (such as consecutive amplifiers) can be calculated simply by summing the decibel gains of the individual components, rather than needing to multiply amplification factors. Essentially this is because log(A × B × C × ...) = log(A) + log(B) + log(C) + ...

*The human perception of, for example, sound or light, is, roughly speaking, such that a doubling of actual intensity causes perceived intensity to always increase by the same amount, irrespective of the original level. The decibel's logarithmic scale, in which a doubling of power or intensity always causes an increase of approximately 3 dB, corresponds to this perception.

Uses

Acoustics

Main article: Sound pressure

The decibel is commonly used in acoustics to quantify sound levels relative to some 0 dB reference. The reference level is typically set at the threshold of perception of an average human and there are common comparisons used to illustrate different levels of sound pressure. As with other decibel figures, normally the ratio expressed is a power ratio (rather than a pressure ratio).

A reason for using the decibel is that the ear is capable of detecting a very large range of sound pressures. The ratio of the sound pressure that causes permanent damage during short exposure to quietest sound that (undamaged) ears can hear is above a million. To deal with such a range, logarithmic units are useful: the base-10 logarithm of a trillion is 12, so a level difference of 120 dB represents a power ratio of this amount. Since the human ear is not equally sensitive to all the frequencies of sound, noise levels at maximum human sensitivity — for example, the higher harmonics of middle A (between 2 and 4 kHz) — are factored more heavily into sound descriptions using a process called frequency weighting.

Further information: Examples of sound pressure and sound pressure levels

Electronics

In electronics, the decibel is often used to express power or amplitude ratios (gains), in preference to arithmetic ratios or percentages. One advantage is that the total decibel gain of a series of components (such as amplifiers and attenuators) can be calculated simply by summing the decibel gains of the individual components. Similarly, in telecommunications, decibels are used to account for the gains and losses of a signal from a transmitter to a receiver through some medium (free space, wave guides, coax, fiber optics, etc.) using a link budget.

The decibel unit can also be combined with a suffix to create an absolute unit of electric power. For example, it can be combined with "m" for "milliwatt" to produce the "dBm". Zero dBm is the power level corresponding to a power of one milliwatt, and 1 dBm is one decibel greater (about 1.259 mW).

In professional audio, a popular unit is the dBu (see below for all the units). The "u" stands for "unloaded", and was probably chosen to be similar to lowercase "v", as dBv was the older name for the same thing. It was changed to avoid confusion with dBV. This unit (dBu) is an RMS measurement of voltage which uses as its reference 0.775 VRMS. Chosen for historical reasons, it is the voltage level which delivers 1 mW of power in a 600 ohm resistor, which used to be the standard reference impedance in telephone audio circuits.

The bel is used to represent noise power levels in hard drive specifications. It shares the same symbol (B) as the byte.

Optics

In an optical link, if a known amount of optical power, in dBm (referenced to 1 mW), is launched into a fiber, and the losses, in dB (decibels), of each electronic component (e.g., connectors, splices, and lengths of fiber) are known, the overall link loss may be quickly calculated by addition and subtraction of decibel quantities.

In spectrometry and optics, the blocking unit used to measure optical density is equivalent to −1 B. In astronomy, the apparent magnitude measures the brightness of a star logarithmically, since, just as the ear responds logarithmically to acoustic power, the eye responds logarithmically to brightness; however astronomical magnitudes reverse the sign with respect to the bel, so that the brightest stars have the lowest magnitudes, and the magnitude increases for fainter stars.

Video and digital imaging

In connection with digital and video image sensors, decibels generally represent ratios of video voltages or digitized light levels, using 20 log of the ratio, even when the represented optical power is directly proportional to the voltage or level, not to its square. Thus, a camera signal-to-noise ratio of 60 dB represents a power ratio of 1000:1 between signal power and noise power, not 1,000,000:1.[8]

Common reference levels and corresponding units

Absolute and relative decibel measurements

Although decibel measurements are always relative to a reference level, if the numerical value of that reference is explicitly and exactly stated, then the decibel measurement is called an "absolute" measurement, in the sense that the exact value of the measured quantity can be recovered using the formula given earlier. For example, since dBm indicates power measurement relative to 1 milliwatt,

*0 dBm means no change from 1 mW. Thus, 0 dBm is the power level corresponding to a power of exactly 1 mW.

*3 dBm means 3 dB greater than 0 dBm. Thus, 3 dBm is the power level corresponding to 103/10 × 1 mW, or approximately 2 mW.

*−6 dBm means 6 dB less than 0 dBm. Thus, −6 dBm is the power level corresponding to 10−6/10 × 1 mW, or approximately 250 μW (0.25 mW).

If the numerical value of the reference is not explicitly stated, as in the dB gain of an amplifier, then the decibel measurement is purely relative. The practice of attaching a suffix to the basic dB unit, forming compound units such as dBm, dBu, dBA, etc, is not permitted by SI.[9] However, outside of documents adhering to SI units, the practice is very common as illustrated by the following examples

Absolute measurements

Electric power

dBm or dBmW

dB(1 mW) — power measurement relative to 1 milliwatt. XdBm = XdBW + 30. dBW

dB(1 W) — similar to dBm, except the reference level is 1 watt. 0 dBW = +30 dBm; −30 dBW = 0 dBm; XdBW = XdBm − 30.

Voltage

Since the decibel is defined with respect to power, not amplitude, conversions of voltage ratios to decibels must square the amplitude, as discussed above.

A schematic showing the relationship between dBu (the voltage source) and dBm (the power dissipated as heat by the 600 Ω resistor)

dBV

dB(1 VRMS) — voltage relative to 1 volt, regardless of impedance.

dBu or dBv

dB(0.775 VRMS) — voltage relative to 0.775 volts.[1] Originally dBv, it was changed to dBu to avoid confusion with dBV.[10] The "v" comes from "volt", while "u" comes from "unloaded". dBu can be used regardless of impedance, but is derived from a 600 Ω load dissipating 0 dBm (1 mW). Reference voltage

dBmV

dB(1 mVRMS) — voltage relative to 1 millivolt, regardless of impedance. Widely used in cable television networks, where the nominal strength of a single TV signal at the receiver terminals is about 0 dBmV. Cable TV uses 75 Ω coaxial cable, so 0 dBmV corresponds to −78.75 dBW (-48.75 dBm) or ~13 nW.

dBμV or dBuV

dB(1 μVRMS) — voltage relative to 1 microvolt. Widely used in television and aerial amplifier specifications. 60 dBμV = 0 dBmV.

Acoustics

Probably the most common usage of "decibels" in reference to sound loudness is dB SPL, referenced to the nominal threshold of human hearing:[11]

dB(SPL)

dB (sound pressure level) — for sound in air and other gases, relative to 20 micropascals (μPa) = 2×10−5 Pa, the quietest sound a human can hear. This is roughly the sound of a mosquito flying 3 metres away. This is often abbreviated to just "dB", which gives some the erroneous notion that "dB" is an absolute unit by itself. For sound in water and other liquids, a reference pressure of 1 μPa is used.[12]

dB SIL

dB sound intensity level — relative to 10−12 W/m2, which is roughly the threshold of human hearing in air.

dB SWL

dB sound power level — relative to 10−12 W.

dB(A), dB(B), and dB(C)

These symbols are often used to denote the use of different weighting filters, used to approximate the human ear's response to sound, although the measurement is still in dB (SPL). These measurements usually refer to noise and noisome effects on humans and animals, and are in widespread use in the industry with regard to noise control issues, regulations and environmental standards. Other variations that may be seen are dBA or dBA. According to ANSI standards, the preferred usage is to write LA = x dB. Nevertheless, the units dBA and dB(A) are still commonly used as a shorthand for A-weighted measurements. Compare dBc, used in telecommunications.

dB HL or dB hearing level is used in audiograms as a measure of hearing loss. The reference level varies with frequency according to a minimum audibility curve as defined in ANSI and other standards, such that the resulting audiogram shows deviation from what is regarded as 'normal' hearing.[citation needed]

dB Q is sometimes used to denote weighted noise level, commonly using the ITU-R 468 noise weighting[citation needed]

dBZ

dB(Z) - energy of reflectivity (weather radar), or the amount of transmitted power returned to the radar receiver. Values above 15-20 dBZ usually indicate falling precipitation.[13]

dBsm

dBsm - decibel (referenced to one) square meter, measure of reflected energy from a target compared to the RCS of a smooth perfectly conducting sphere at least several wavelengths in size with a cross-sectional area of 1 square meter. "Stealth" aircraft and insects have negative values of dBsm, large flat plates or non-stealthy aircraft have positive values.[14]

Radio power, energy, and field strength

dBc

dBc — relative to carrier — in telecommunications, this indicates the relative levels of noise or sideband peak power, compared with the carrier power. Compare dBC, used in acoustics.

dBJ

dB(J) — energy relative to 1 joule. 1 joule = 1 watt per hertz, so power spectral density can be expressed in dBJ.

dBm

dB(mW) — power relative to 1 milliwatt. When used in audio work the milliwatt is referenced to a 600 ohm load, with the resultant voltage being 0.775 volts. When used in the 2-way radio field, the dB is referenced to a 50 ohm load, with the resultant voltage being 0.224 volts. There are times when spec sheets may show the voltage & power level e.g. -120 dBm = 0.224 microvolts.

dBμV/m or dBuV/m

dB(μV/m) — electric field strength relative to 1 microvolt per meter.

dBf

dB(fW) — power relative to 1 femtowatt.

dBW

dB(W) — power relative to 1 watt.

dBk

dB(kW) — power relative to 1 kilowatt.

Antenna measurements

dBi

dB(isotropic) — the forward gain of an antenna compared with the hypothetical isotropic antenna, which uniformly distributes energy in all directions. Linear polarization of the EM field is assumed unless noted otherwise.

dBd

dB(dipole) — the forward gain of an antenna compared with a half-wave dipole antenna. 0 dBd = 2.15 dBi

dBiC

dB(isotropic circular) — the forward gain of an antenna compared to a circularly polarized isotropic antenna. There is no fixed conversion rule between dBiC and dBi, as it depends on the receiving antenna and the field polarization.

dBq

dB(quarterwave) — the forward gain of an antenna compared to a quarter wavelength whip. Rarely used, except in some marketing material. 0 dBq = -0.85 dBi

Other measurements

dBFS or dBfs

dB(full scale) — the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. In digital systems, 0 dBFS (peak) would equal the highest level (number) the processor is capable of representing. Measured values are always negative or zero, since they are less than the maximum or full-scale. Full-scale is typically defined as the power level of a full-scale sinusoid, though some systems will have extra headroom for peaks above the nominal full scale.

dB-Hz

dB(hertz) — bandwidth relative to 1 Hz. E.g., 20 dB-Hz corresponds to a bandwidth of 100 Hz. Commonly used in link budget calculations. Also used in carrier-to-noise-density ratio (not to be confused with carrier-to-noise ratio, in dB).

dBov or dBO

dB(overload) — the amplitude of a signal (usually audio) compared with the maximum which a device can handle before clipping occurs. Similar to dBFS, but also applicable to analog systems.

dBr

dB(relative) — simply a relative difference from something else, which is made apparent in context. The difference of a filter's response to nominal levels, for instance.

dBrn

Lenny Z. Perez M

EES

Referencias:

http://en.wikipedia.org/wiki/Octave

www.acoustics.salford.ac.uk/.../09%20Octave%20and%20other%20intervals.doc -

http://en.wikipedia.org/wiki/Decibel

http://www.sizes.com/units/decibel.htm