8CIRCUITDESIGNFORRFTRANSCEIVERS4?VArs2++gain: A.0lHnoiseless systemSFigure1.3.The noise of S is now representedbythe input referred noise sources.The systemstillhasa voltagegain of Ay.where we have assumed that all power available at the source is delivered to theinput of thesystem.TheSNRout can be measured asA?V?SNRout(1.24)Ve+(Vn+R,In))A3Note that V? and 72 are added before squaring, to take their correlation intoaccount.The noise factor is now obtained by dividing (1.23)by (1.24),yieldingV2r +(Vn + R, In)?(Vn+ R,In)?SNRinF:(1.25)SNRoutVa.rVa.r.Thenoisefactor isusually specified fora1Hzbandwidthataparticularfrequency.Thespecificationcanbeobtainedfrom(1.25)byusingtheexpressionV2.rs=4kTRy,andiscalledthespotnoisefactortoemphasizethe1HzbandwidthDesigners are not only interested in the noise factor of a single circuit, butalso in the noise behavior of the total system (e.g. frontend), which is normallya cascade of circuits.Assuming weknow the noisefactor Fand the availablegain Gifor each separate circuit i,Friis'formula gives the expression of theoverall Frotal formcircuitsas afunction of FF2-1F3-1Fm-1Frotal=1 +(Fi-1) +Ga.1Ga.2..Ga.m-1Ga.1Ga.1Ga.2(1.26)
9RF Design:Concepts and TechnologyWe can conclude several properties from Frisformula:- The Frotul is referred to the input of thefirst circuit.- TheF of each circuit is calculated with respect tothe source impedancedriving that circuit. This stems from the use of the available gain in thedefinition.TheFofthefirststageis dominantwhenthegainofeach stageisreasonableHowever,if the first circuit exhibits a low gain or attenuation (i.e.loss),forinstance, then the noisefigure ofthe second circuitis amplified whenreferredto the input of the first circuit.AsFis dependent on the source impedance, we canminimize F for a giventwo-portsystembychoosingapropervalueforthesourceimpedance.Wetherefore simultaneouslymaximize the SNR.Assume a voltage source Vwith an impedanceZ,=R,+jX,drivinganoisysystem similar toFigure1.3The noise produced by the source impedance is modeled as V2 If we definethe input referred noise voltage as V2 = 4kT Rn-Af, and the input referrednoise current as 12 = 4kTGni - Af, the system can be treated as noiseless.Accordingto(1.25)wehaveVa +(Vn + Z,In)21Z,12Ruu(1.27)FR,R,RniV.rsforthe noisefactor, where Rni I/Gni and assuming that the two inputreferrednoise sources are not correlated. Differentiating F to R,yieldsRy = (Rm,Rmi + x3)and if we assumethat weonly have a resistive source impedance,the optimalsourceimpedancefornoisematching isfoundtobe(1.28)Ropt=RnuRniwithRru(1.29)Fmin = 1 +2Rnias the minimum noise factor.Note that the optimal source resistance for minimum noise does not coincidewith that for maximumpower transfer.The derivation above holds only when the noise sources are not correlated.If they are completely correlated,72 is completely determined byV?.For each
10CIRCUITDESIGNFORRFTRANSCEIVERSspectral component within I2,In=Y,Vn whereY,describes the correlation.Ifthecorrelation isonlypartial,wehaveI,=Y.V+Y,Vn,whereY,describesthe uncorrelation.Using these expressions we can again derive a relationshipforthe optimal sourceimpedance.For thegeneral case,theoptimal noisematchis achievedforZ,=Zopt,whiletheoptimal powertransferisobtainedfor Z,=Zin.The optimal noise impedance for a given circuit is not generallyeasy to calculate.Noisematching and power matching are in principle not obtained at the sametime.Any deviation of the source impedance from this optimal value will resultin an increase of the noise factor of the overall system. In the RF community,this deviation is normally expressed in the reflection parameters, yielding4rn-Fs.opF=Fmin +(1.30)[1+Is.opt/2 (1 -Ir.12)where rn - R./Z, is the normalized source resistance.One last design issue regarding noise is the following rule of thumb usedbyanalog designers.Assume thatthe input referred noiseof system S can berepresented by only a noise voltage source. Then (1.27) reduces toRnuF=1+(1.31)R.Thus, for equal resistance values, the noise figure is 3 dB. The circuit producesthesameamountofnoiseas the sourceresistance.Toletthe sourceresistancebedominant, the input referred noise resistance should be smaller than the sourceresistance.1.1.3Non-linearityRadiofrontend specifications are usually related to thenon-linear behavior ofthe circuits fromwhich the system isbuilt.We can think of specifications suchas gain compression, inter-modulation and blocking. We will briefly discussthese effects assuming a memoryless, time-variant system S with input x(t) andoutput y(t), with a non-linearity up to the order 35,y(t) =αx(t) +βx2(t) +yx (t)(1.32)5Note that effects of memory and higher order non-linearities may have a significant influence on the overallbehaviorof the system
RF Design: Concepts and Technology11Suppose a sinusoidal signal x(t) = A cos(ot) is applied to system (1.32),then the amplitude of thefundamental frequencyis given as3A3αA+(1.33)4from which it follows that the small signal gain α is varied with the inputampltude due to the third order non-linearity. For < 0, the gain α + isthen decreasing as a function of A.The input level at which the small signalgain has dropped by I dB is called the"1 dB compression point"(C PidB)andcan bederivedas0.145CPidB=(1.34)Obviously, the non-linearities in (1.32) will also cause higher order terms of thefundamental frequency.Infullydifferential systems,the even-orderharmonicswill be attenuated, but theywill never vanishdue tomismatches which corruptthe symmetry.Many interesting phenomena can be explained by assuming a two-tone inputsignal, i.e.x(t)=Acos(wt)+B cos(w2t),where thefirsttonerepresents thewanted signal and the second tone the interferer. Assuming the desired signalto berelated to wi,then according to (1.32), the amplitude of the fundamentaltoneisfound tobe3A3,3AB2αA+3(1.35)24Assume that the desired signal is weak compared to the interferer, A << Bthen with y <Othe small signal gain is a decreasing function of theamplitudeof the interferer.Consequently, the gain drops to zero for sufficiently large B,aneffectreferred toasblocking.Wemay eventake intoaccountamodulationofthe amplitude of the interferer, for example B(1 + m cos(omt)) cos(o2t) (withm<1)whichwill change (1.35) intom23yA33AB2m?αA+(1.36) cos(2wmt) + 2m cos(wmt)2422The formula tells us that the small signal gain of the weak desired signal ismodulated with the amplitude modulation of the strong interferer at m and2wm:ThiseffectiscalledcrossmodulationSofar wehaveonlyinvestigated theinfluence of non linearities to thefunda-mental frequency. But the non linearities give also rise to mixing terms of the6This effect is called desensitization
12CIRCUITDESIGNFORRFTRANSCEIVERSfrequencies, called inter modulation (IM).Assuming for the two-tone inputsignal both components equal or almost equal in strength then the followinginter modulation products can be obtained:- second order: βAB cos(w1 + w2)t) + βAB cos((w1-W2)t) third order:cos(21+2)t)+cos(2012)+ cos(1 + 202)1) + cos(1 - 22)Obviously,wheny w2thethird orderintermodulationterms(IM)again appear in the band of interest, that is, they cannot be removed by meansof a filter. Therefore, signals may therefore be corrupted. The metric thirdorder intercept point (I P3)has been defined as a measure of this effect.It isdefined as thepoint where theamplitudeof thefundamental (1.35)is equal tothe amplitude of the third order inter modulation (assuming A = B),91IA3_31IA3lαlA+44yielding (in (peak)voltage,not inpower),[40](1.37)AIP3=V3/ylforthe IP3, assuming α >> 2, The output IP, i then obtained as αA3.9l/42 no longer holds, and theFor practical circuits, the assumption α >>interceptpoint is oftenbeyond theallowable inputrange.It is thereforefoundby linear extrapolation of themeasured behavior for small input amplitudes.TheinputIP3(1IP3)canalsobefoundfroma singletonemeasurement.UsingtheIM3productsandagiveninputpowerPinweobtainAPII P3=(1.38)+ Pin2whereAPisthedifferencebetween themagnitudes ofthefundamentals andtheIM3productsattheoutput.Allparametersinexpression(1.38)areindecibelsThe output intercept point Ol Ps can be found to be the difference between theIIPs and the power gain G,, i.e.OIP=GpPin+P.Figure1.4 shows ageometric interpretation ofthis expression?7In general,the n-th orderintercept point isrelated tothe n-thorder intermodulation term and can befoundas follows.Define the angle of the fundamental amplitude line as ,and the angle of the amplitude of thenth-orderharmonic asSn.For the interceptpoint,weobtain tan(On)=n tan(Oy)orQ+△P=nQandAP= (P, -Pu) where Pr and Pu are thepowers at the output.Here,Q-OI Pn-P.Consequently,O1 Pn=GpPin+.IPn=Pin +