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DecibelsStanHendryx,Hendryx&Associates,Sunnyvale,CA April2018

Engineersandtechniciansareoftenconfrontedwithcalculatingormeasuringcharacteristicsofsignaltransmissionsystems,includingpowerofasignal,powerlossinacircuit,gainofanamplifier,orthesensitivityofadetector.Thesignalsinvolvedmightbeelectrical,optical,radio,oracoustic.Thesequantitiesallrequiredeterminingratiosoftwonumbers—theratioofanoutputpowertoaninputpower,ortheratioofapowerleveltoastandardunitofpower.Powerismeasuredinwatts.Theoutputpower𝑃!wattsatagivenpointonatransmissionlineisseentobetheinputpower𝑃! timesalossfactorL

𝑃! = 𝐿 𝑃! Eq.1Ldependsonthelengthofthelineandtechnicalfactorsabouttheline.Inapassivetransmissionline,Lislessthan1.Ifthelineincludesanamplifierorregenerator,𝑃!couldbegreaterthen1.Listheratioof𝑃!to𝑃! .

𝐿 =𝑃!𝑃! Eq.2

Beingtheratiooftwopowers,Lisdimensionless,ofdimension1.Theseratioscantakeonverylargetoverysmallvalues,andwouldoftenrequirerepeatedmultiplicationordivisiontoobtainanoverallresultforasystem.Performingthesemultiplicationsanddivisionsisunwieldy.

Additionandsubtractionaremucheasierthanmultiplicationanddivision.In1614,theScottishmathematicianJohnNapier1inventedamethodofcalculationthatturnsmultiplicationintoadditionanddivisionintosubtraction—thelogarithm.Logarithmswerethesinglemostimportantimprovementinarithmeticcalculationbeforethemoderncomputerandhandhelddigitalcalculator.Whatmadethemsousefulistheirabilitytoreducemultiplicationtoadditionanddivisiontosubtraction.

In1924,engineersatBellTelephoneLaboratoriesadoptedthelogarithmtodefineaunitforsignallossintelephonelines,thetransmissionunit(TU).TheTUreplacedtheearlierstandardunit,milesofstandardcable(MSC),whichhadbeeninplacesincetheintroductionoftelephonecablein1896.1MSCcorrespondedtothelossofsignalpowerover1mileofstandardcable.Standardcablewasdefinedashavingaresistanceof88ohmsandcapacitanceof0.054microfaradspermile.1MSCequals1.056TU.ThelossfactorinTUwastentimesthebase-10logarithmoftheratiooftheoutputpowertotheinputpower.

In1928,BellTelephoneLaboratoriesrenamedthetransmissionunitthedecibel(dB).Theprefix‘deci’comesfromLatindecimus‘tenth’.Adecibelisonetenthofabel(B),theunitnamedinhonorofAlexanderGrahamBell,inventorofthetelephone1Napieralsoinventedtheuseofthedecimalpointtodenotefractions.

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in1879andfounder,in1885,oftheAmericanTelephoneandTelegraphCompany(AT&T).Thebelisrarelyused;thedecibelhasbecomewidelyused.Onedecibelisaboutthesmallestattenuationdetectablebyanaveragelistener,andcorrespondstoasignalpowerlossof20.6%.Interestingly,thesmallestdetectablechangeinsoundlevelbylistenersisrelativelyindependentofthelevel–about1dBatanylevel,20.6%powerreduction.Thismeansthathumanperceptionofloudnessislogarithmic.Itisthepercentagechangeinlevelthatmatters,nottheabsolutechangeinwatts.Quantitieswhosesignificanceisproportionaltoaconstantpercentchangearelogarithmic,e.g.,compoundinterest.

NotethatthelossfactorLisnotthepowerloss.Listheratiooftwovaluesofpower,adimensionlessquantity,anumber.Thepowerlossitselfis𝑃! − 𝑃! ,whichhasunitsofwatts.Byconvention,anegativedifferencerepresentsapowerloss;apositivedifferencerepresentsapowergain,aswithanamplifier.

Thepowerloss(orgain)canbeexpressedasafractionoftheinputpower𝑃! − 𝑃!𝑃!

=𝑃!𝑃!− 1 = 𝐿 − 1 Eq.3

Thelossfactorindecibels𝐿!" isdefinedtobe10timesthebase-10logarithmofL.

𝐿!" ≜ 10 log!" 𝐿 = 10 log!"𝑃!𝑃! Eq.4

ThequantityofrealinterestisthelossfactorL.DecibelisjustaconvenientunitinwhichtorepresentL.

Absolutevs.RelativePowerLevels

Whenmeasuringlossorgain,itiscustomarytoset𝑃! toanarbitraryreferencelevel,measure𝑃!anddeterminetheratioLand𝐿!" .Testinstrumentsdothemath.Thepracticalprocedureistoconnectareferencesourcetotheinstrument,noteitslevelasthereferencelevel𝑃! thenconnecttheoutput𝑃!totheinstrumentandreadthegainorlossindecibelsfromtheinstrument.Tomeasureabsolutepowerlevels,thetestinstrumentmustbecalibratedtoaninternationalstandardunitofpower,typically1milliwatt,0.001watt.Calibrationisfirstperformedwhentheinstrumentismanufacturedandperiodicallythereafter.Thesecalibrationsaretraceabletotheinternationalstandardwattusingatransferstandardmaintainedbyanationallaboratory.IntheUS,thislaboratoryisNIST,theNationalInstituteofScienceandTechnologyoftheDepartmentofCommerce.

Settingtheinstrumenttomeasureabsolutepowercausesittouseitscalibratedreferencelevelof𝑃! = 0.001watt.Theprocedureissimpler:thereisnoneedtomeasurethereferencesource,onlytheoutput,𝑃! .

Todistinguishanabsolutepowerlevelindecibelsrelativetoonemilliwatt,theunitsymboldBmisused.TheunitsymboldBisusedforrelativepowermeasurementswherethereferencepowerlevelisunspecified.

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DecibelsforFieldValues

Adecibelquantitycorrespondstoapowerratio,i.e.theratiooftwopowerlevels.Sometimesinstrumentsmeasurevoltageorcurrentinanelectricalcircuit,orelectricormagneticfieldstrengthinotherapplications,notpower.Togetadecibelvaluefromavoltage,current,orfieldlevelratiothatisthesameasifpowerweremeasured,Eq.4needstobeadjusted.Itturnsoutthatpowerisproportionaltothesquareofvoltage,current,orfieldlevels.Doublingthevoltagequadruplesthepower.Inthiscase,𝐿!"#$% = 𝑉!/𝑉! isusedandEq.4becomes

𝐿!" ≜ 20 log!" 𝐿!"#$% = 20 log!"𝑉!𝑉!= 10 log!"

𝑉!𝑉!

!

Eq.5

Logarithms

Tounderstandthedecibel,itisnecessaryfirsttounderstandlogarithms.Themathematicsusedhereistaughtinhighschool.Thepresentationisheuristic,startingwithcountingandelementarymultiplicationanddivisionwithintegers,thenbuildingtoincluderationalnumbersand,finally,allnumbers.

Thefunction𝑓 𝑥 = 𝑎!iscalledtheexponentialfunctionwithbasea,a>0.Notethatwhen𝑎 = 1, 𝑓 𝑥 = 1forallx,so𝑎 = 1isgenerallyexcluded.

Considerapositivenumberamultipliedbyitselfntimes.nisthusapositiveinteger.Letx=n.Then𝑓 𝑛 = 𝑎! = 𝑎 ∗ 𝑎 ∗ 𝑎…𝑎,i.e.arepeatedntimes.Thelogarithmof𝑎!ofbase-aisdefinedastheexponentn.

log! 𝑎! ≜ 𝑛 Eq.6

𝑎!"#! !! = 𝑎! Eq.7

Alogarithmisanexponent.Eq.7showsthatthelogarithmfunctionistheinverseoftheexponentialfunction.Supposewehaveanotherexponentialwithbaseahavingmfactors,𝑎!,wheremisalsoapositiveinteger.Ifweformtheproduct𝑎! 𝑎!,thenwehave𝑛 +𝑚repetitionsofa.However,thislongerproductisthesameas𝑎!!!.Thus,wehave

𝑎!𝑎! = 𝑎!!! Eq.8

log! 𝑎!𝑎! = log! 𝑎!!! = 𝑛 +𝑚 = log! 𝑎! + log! 𝑎! Eq.9

HerewehaveinNapier’sinventionawaytorepresenttheproductoftwonumbers𝑎!𝑎!asthesumoflogarithms𝑛 +𝑚ofthosetwonumbers.Whenthenumbersarerepresentedasexponentialswithacommonbasea,weaddtheirlogarithms.Supposenisgreaterthanmandwedivideinsteadofmultiply.Thenwehave

𝑎!

𝑎! = 𝑎!!! Eq.10

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Thisissobecausethemasinthedenominatorcancelmasinthenumerator,leaving𝑛 −𝑚asthetotalnumberofas.

log!𝑎!

𝑎! = log! 𝑎!!! = 𝑛 −𝑚 = log! 𝑎! − log! 𝑎! Eq.11

Herewelikewisehaveawaytorepresentthequotientoftwonumbers𝑎!/𝑎!asthedifferenceoflogarithms𝑛 −𝑚ofthosetwonumbers.Whenthenumbersarerepresentedasexponentialswithacommonbasea,wesubtracttheirlogarithms.

Notethat𝑎! = 1,foranya,and𝑎!!! = 𝑎! 𝑎!! = 𝑎! = 1,so𝑎!! = !!!= !!

!!.

Thenlog! 1 = 0, for all 𝑎 Eq.12

log! 𝑎!! = log!1𝑎! = log!

𝑎!

𝑎! = 0− 𝑛 = − log! 𝑎! Eq.13

Thelogarithmofthereciprocalisthenegativeofthelogarithm.Sofar,wehaveshownhowtocalculatethelogarithmofanyintegerpowerofaandthelogarithmofthereciprocalofanyintegerpowerofa,whereacanbeanypositivenumber,notrestrictedtointegers,justa>0.acannotbezero,since0! = 0anddivisionbyzeroisundefined.

Wewouldliketorepresentanynumberasachosenbaseraisedtosomepower.Itturnsoutthiscanbedoneforallpositivenumbers.However,sofar,wehaveonlyshownthatthisworkswhennandmarepositiveintegers.Wecanexpandthedomainof𝑥intheexponentialfunctionbyshowinghowtocalculate𝑓 𝑥 when𝑥isarationalnumber,i.e.theratiooftwointegers.

Consider𝑎!!,whichisdefinedtobethenumberthat,whenmultipliedbyitselfm

times,givesa,i.e.𝑎!!isthemthrootofa.Whenm=2,𝑎

!!isthesquareroot;when

m=3,𝑎!!isthecuberoot,andsoforth.Bymultiplying𝑎

!!byitselfntimes

𝑎!!

!= 𝑎

!! Eq.14

log! 𝑎!! =

𝑛𝑚

Eq.15

Wenowhaveawaytocalculatetheexponential𝑓(𝑥)for𝑥anyrationalnumber.Thisalsoworksforirrationalnumbers,numbersthatcannotbeexpressedastheratiooftwointegers,e.g.𝜋 = 3.141592…,wherethedecimalsneverrepeat.Sinceroundinganirrationalnumbertoafixednumberofdecimalplacesalwaysresultsinarationalnumber,extendingthenumberofdecimalplacesindefinitelyalsoworks.Thelogarithmfunctionwithbase𝑎 is 𝑦 = log! 𝑥.Itisdefinedastheinverseoftheexponentialfunctionwithbasea,𝑦 = 𝑎! (a > 0, a ≠ 1).

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Thedomainoflog! 𝑥 is 0,∞ ,whichistherangeof𝑎! .Therangeoflog! 𝑥 is (−∞,∞),whichisthedomainof𝑎! .Eq.16showsthisinverserelationship.

Thedomainofafunction𝑦 = 𝑓(𝑥)isthesetofvaluesof𝑥forwhichthefunctionisdefined.Therangeofthefunctionisthesetofvaluesofthefunction,𝑦.Thedomainandrangeoftheexponentialandlogarithmfunctionsareopenintervals,i.e.theydonotincludetheendpoints0or±∞.Anotherexample:𝑦 = 𝑥! and y = 𝑥areinverses.Thedomainofoneistherangeoftheother. 𝑥

!= 𝑥! = 𝑥, 𝑥 ≥ 0.

SummaryoftheRulesforExponentials

Ifa>0andb>0,thefollowingrulesholdtrueforallrealnumbers𝑥 and 𝑦.1. 𝑎! 𝑎! = 𝑎!!!2. !!

!!= 𝑎!!!

3. 𝑎! ! = 𝑎! ! = 𝑎!"4. 𝑎!𝑏! = 𝑎𝑏 !5. !!

!!= !

!

!

SummaryoftheRulesforLogarithms

Foranynumbersa>0,a≠1,b>0and𝑥 > 0,thelogarithmofbase-afunctionsatisfiesthefollowingrules:

1. ProductRule: log! 𝑏𝑥 = log! 𝑏 + log! 𝑥2. QuotientRule: log!

!!= log! 𝑏 − log! 𝑥

3. ReciprocalRule: log!!!= − log! 𝑥

4. PowerRule: log! 𝑥! = r log! 𝑥5. ConversionRule: log! 𝑥 = log! 𝑥 log! 𝑎

Applications

Threevaluesofthelogarithmbase,a,arewidelyused:10,2,ande=2.71828….Tenisusedfordecibels.Twoisusedincomputerscience.Sinceabinarynumbercomprisingnbitscantakeon2!possiblevalues,thenumberofbitsrequiredtorepresentagivenpositiveintegerNis𝑛 = log!𝑁,roundeduptothenextbit.eisEuler’sNumber,avalueofparticularimportanceincalculus.𝑒!andallofitsderivativesarethesame,𝑒! .Euler’sNumberalsoappearsinthefascinatingequation𝑒!" − 1 = 0,where𝑖! = −1.Thisequation,alsoduetoLeonardEuler(1707-1783),relatesfiveofthemostimportantconstantsinmathematics.

Logarithmsofbase-10arecalledcommonlogarithms,commonlywrittenas“log 𝑥.”Logarithmsofbase-earecallednaturallogarithms,commonlywrittenas“ln 𝑥.”Thissectionfocusesoncommonlogarithms.

Tousecommonlogarithms,atableoflogarithms(orcalculator!)isneeded.However,onlycommonlogarithmsofnumbersbetween1and10needtobe

log 𝑎!"#! !! = log! 𝑎! = 𝑥 , 𝑎 > 0,𝑎 ≠ 1, 𝑥 > 0 Eq.16

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tabulated.Each1:10intervaliscalledadecade.Thepartofalogarithmfollowingthedecimalpointiscalledthemantissa.Thewholenumberpartistheexponent.Logarithmsofnumberslessthan1orgreaterthan10areobtainedbyexpressingthenumberinscientificnotation,lookingupthesignificantpartofthenumberbinthetabletogetthemantissa,andaddingtheexponentn.

log 𝑏×10! = log 𝑏 + 𝑛 Eq.17Table1CommonLogarithms

𝑁 = 10! 𝑁 = 10!! 𝑛 = (– )log!"𝑁 𝑁 = 10! 𝑁 = 10!! 𝑛 = (– )log!"𝑁

1.0 (1.0) 0.0000 2.5119 (0.3981) 0.4000

1.1 (0.9091) 0.0414 2.75 (0.3636) 0.4393

1.1111 (0.9000) 0.0458 3.0 (0.3333) 0.4771

1.2 (0.8333) 0.0792 3.1623 (0.3162) 0.5000

1.25 (0.8000) 0.0969 3.3333 (0.3000) 0.5229

1.2589 (0.7943) 0.1000 3.5 (0.2857) 0.5441

1.3 (0.7692) 0.1139 3.9811 (0.2512) 0.6000

1.33=4/3 (0.7500) 0.1249 4.0 (0.2500) 0.6020

1.4 (0.7143) 0.1461 4.5 (0.2222) 0.6532

1.4286 (0.7000) 0.1549 5.0 (0.2000) 0.6990

1.5=3/2 (0.6667) 0.1761 5.0119 (0.1995) 0.7000

1.5849 (0.6310) 0.2000 5.5 (0.1818) 0.7404

1.6 (0.6250) 0.2041 6.0 (0.1667) 0.7782

1.67=5/3 (0.6000) 0.2218 6.3096 (0.1585) 0.8000

1.7 (0.5882) 0.2304 6.5 (0.1538) 0.8129

1.75=7/4 (0.5714) 0.2430 7.0 (0.1429) 0.8451

1.8 (0.3652) 0.2553 7.5 (0.1333) 0.8751

1.9 (0.5263) 0.2788 7.9433 (0.1259) 0.9000

1.9953 (0.5012) 0.3000 8.0 (0.1250) 0.9030

2.0 (0.5000) 0.3010 8.5 (0.1176) 0.9294

2.25=9/4 (0.4444) 0.3522 9.0 (0.1111) 0.9542

2.5 (0.4000) 0.3979 10.0 (0.1000) 1.0000

Table1warrantssomeexplanation.Thesecondcolumn,whichisthereciprocalofthenumberinthefirstcolumn,isaddedforconvenience.Thethirdcolumngivesthemantissa,thelogarithmofthenumberinthefirstcolumn.Ifthelogarithmistakentobeanegativevalue,thenitisthemantissaofthenumberinthesecondcolumn.

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Figure1isagraphoflog 𝑥 and 1/𝑥.Thelogarithmofanumberlessthan1isnegative,asshownbythegraph.Toconvertalogarithmtoadecibelvalue,multiplyby10.Toconvertadecibelvaluetoalogarithm,divideby10.SeeEq.4.Ifalevelnumberisgreaterthan10orlessthan1,expressthenumberinscientificnotation,𝑏×10!wherebisanumberbetween1and10.Enterbincolumn1andreadthemantissafromcolumn3.Addtheexponentntogetthelogarithm.SeeEq.17.Converttodecibelsbymultiplyingthelogarithmby10.

Figure1logx(red)and1/x(blue)

Example1

WhatisthelossfactorLthatcorrespondsto1dB?Entercolumn3with1/10=0.1000andreadthelossfactoras1.2589fromcolumn1.Thegainis1.2589–1=0.2589,or25.89%.SeeEq.3.Example2

WhatisthelossfactorLthatcorrespondsto–1dB?Infertheminussignandentercolumn3with1/10=0.1000.Readthelossfactoras0.7943fromcolumn2.Thelossis1–0.7943=0.2057,or20.57%.

Example3Whatdecibelvaluecorrespondsofafactorof2gainorloss?Entercolumn1with2.0andreadthelogarithmfromcolumn3as0.3010.Multiplyby10toget±3.010dB.+isa2×gain,;–isa½loss.Alternatively,express½as5.0×10!!.Entercolumn1with5.0andreadthemantissafromcolumn3as0.6990.Addtheexponent,–1,togetthelogarithm,0.6990–1=–0.3010.Multiplyby10toget–3.010dB.