Table Of ContentRF measurements I: signal receiving techniques
F.Caspers
CERN,Geneva,Switzerland
Abstract
Forthecharacterization ofcomponents,systemsandsignalsintheRFandmi-
crowaverange, severaldedicated instruments areinuse. Inthispaperthefun-
damentals of the RF-signal sampling technique, which has found widespread
applications in‘digital’oscilloscopesandsamplingscopes,arediscussed. The
2 keyelementinthesefront-endsistheSchottkydiodewhichcanbeusedeither
1
as an RF mixer or as a single sampler. The spectrum analyser has become an
0
2 absolutely indispensable tool forRFsignal analysis. Herethefront-end isthe
RFmixerastheRFsectionofmodernspectrumanalysershasarathercomplex
n
a architecture. The reasons for this complexity and certain working principles
J
as well as limitations are discussed. In addition, an overview of the develop-
6
ment of scalar and vector signal analysers is given. For the determination of
1
the noise temperature of a one-port and the noise figure of a two-port, basic
] concepts andrelations areshown. Abrief discussion ofcommonlyusednoise
h
p measurement techniques concludes thepaper.
-
c
c
1 Introduction
a
.
s In the early days of RF engineering the available instrumentation for measurements was rather limited.
c
BesideselementsactingontheheatdevelopedbyRFpower(bimetalcontactsandresistorswithveryhigh
i
s
temperature coefficient) only point/contact diodes, and to some extent vacuum tubes, were available as
y
h signal detectors. For several decades the slotted measurement line [1] was the most used instrument
p
for measuring impedances and complex reflection coefficients. Around 1960 the tedious work with
[
such coaxial and waveguide measurement lines became considerably simplified with the availability of
1
the vector network analyser. At the same time the first sampling oscilloscopes with 1GHz bandwidth
v
7 arrived on the market. Thiswas possible due to progress in solid-state (semiconductor) technology and
4 advances in microwave elements (microstrip lines). Reliable, stable, and easily controllable microwave
2
sources are the backbone of spectrum and network analysers as well as sensitive (low noise) receivers.
3
. Thispaper willonly treatsignal receiving devices suchasspectrum analysers andoscilloscopes. Foran
1
overviewofnetworkanalysis toolsseeRFmeasurementsII:networkanalysis.
0
2
1
: 2 Basicelements andconcepts
v
i Before discussing several measurement devices, a brief overview of the most important components in
X
suchdevicesandsomebasicconceptsarepresented.
r
a
2.1 Decibel
Since the unit dB is frequently used in RF engineering a short introduction and definition of terms is
givenhere. Thedecibel istheunitusedtoexpressrelativedifferences insignalpower. Itisexpressed as
thebase10logarithm oftheratioofthepowersoftwosignals:
P [dB] = 10·log(P/P ) . (1)
0
It is also common to express the signal amplitude in dB. Since power is proportional to the square of a
signal’s amplitude, thevoltage indBisexpressedasfollows:
V [dB] = 20·log(V/V ) . (2)
0
Table1: OverviewofdBkeyvaluesandtheirconversionintopowerandvoltageratios.
Powerratio Voltageratio
−20dB 0.01 0.1
−10dB 0.1 0.32
−3dB 0.50 0.71
−1dB 0.74 0.89
0dB 1 1
1dB 1.26 1.12
3dB 2.00 1.41
10dB 10 3.16
20dB 100 10
n·10dB 10n 10n/2
In Eqs.(1) and (2) P and V are the reference power and voltage, respectively. A given value in dB is
0 0
thesameforpower ratios asforvoltage ratios. Please notethat there areno‘powerdB’or‘voltage dB’
asdBvaluesalwaysexpressaratio.
Conversely, theabsolutepowerandvoltage canbeobtained fromdBvaluesby
P[dB]
P = P0·10 10 , (3)
V [dB]
V = V0·10 20 . (4)
Logarithmsareusefulastheunitofmeasurement because
1. signalpowertendstospanseveralordersofmagnitude and
2. signalattenuation lossesandgainscanbeexpressed intermsofsubtraction andaddition.
Table1helpstoindicatetheorderofmagnitude associated withdB.
FrequentlydBvaluesareexpressedusingaspecialreferencelevelandnotSIunits. Strictlyspeak-
ing, the reference value should be included in parentheses when giving a dB value, e.g., +3 dB (1W)
indicates 3 dB at P = 1watt, thus 2W. However, it is more common to add some typical reference
0
valuesaslettersaftertheunit,forinstance, dBmdefinesdBusingareferencelevelofP =1mW.Thus,
0
0 dBm correspond to −30 dBW,where dBW indicates a reference level of P =1W. Often areference
0
impedance of50Ωisassumed. Othercommonunitsare
– dBmVforthesmallvoltages withV =1mVand
0
– dBmV/mfortheelectricfieldstrengthradiatedfromanantennawithreferencefieldstrengthE =
0
1mV/m
2.2 TheRFdiode
One of the most important elements inside all sophisticated measurement devices is the fast RF diode
orSchottkydiode. Thebasicmetal–semiconductor junction hasanintrinsically veryfastswitchingtime
of well below a picosecond, provided that the geometric size and hence the junction capacitance of the
diode is small enough. However, this unavoidable and voltage dependent junction capacity will lead to
limitations ofthemaximumoperating frequency.
The equivalent circuit of such a diode is depicted in Fig.1 and an example of a commonly used
Schottky diode canbeseeninFig.2. Oneofthemostimportant properties ofanydiode isitscharacter-
2
diodeimpedance
RFin 50 V RcaFpabcyiptoarss Videoout
Fig.1:Theequivalentcircuitofadiode
Fig.2: AcommonlyusedSchottkydiode. TheRFinputofthisdetectordiodeisontheleftandthevideooutput
ontheright(figurecourtesyAgilent)
isticwhichistherelationofcurrentasafunctionofvoltage. ThisrelationisdescribedbytheRichardson
equation [2]:
qφ qV
I = AA T2exp − B exp J −M , (5)
RC
(cid:18) kT (cid:19)(cid:20) (cid:18)kT (cid:19) (cid:21)
where A is the area in cm2, A the modified Richardson constant, k Boltzmann’s constant, T the
RC
absolutetemperature, φ thebarrierheightinvolts,V theexternalVoltageacrossthedepletionlayer,M
B J
theavalanche multiplication factorandI thediodecurrent.
This relation is depicted graphically for two diodes in Fig.3. As can be seen, the diode is not an
ideal commutator (Fig.4) for small signals. Note that it is not possible to apply big signals, since this
kindofdiodewouldburnout. However,thereexistratherlargepowerversionsofSchottkydiodeswhich
can stand more than 9kV and several 10A but they are not suitable in microwave applications due to
theirlargejunction capacity.
TheRichardson equation canberoughly approximated byasimplerequation [2]:
V
J
I = I exp −1 . (6)
s
(cid:20) (cid:18)0.028(cid:19) (cid:21)
This approximation can be used to show that the RF rectification is linked to the second derivation
(curvature) ofthediodecharacteristic.
3
I
Typical
50µA/div
LBSD
Typical
Schottky
Diode
V
50mV/div
Fig.3: Currentasafunctionofvoltagefordifferentdiodetypes(LBSD=lowbarrierSchottkydiode)
Current
Thresholdvoltage
Voltage
Fig.4: Thecurrent–voltagerelationofanidealcommutatorwiththresholdvoltage
If the DC current is held constant by acurrent regulator or alarge resistor assuming external DC
bias1,thenthetotaljunctioncurrent, including RFis
I = I = i cosωt (7)
0 0
andhencethecurrent–voltage relationcanbewrittenas
I +I +icosωt I +I icosωt
S 0 0 S
V = 0.028ln = 0.028ln +0.028ln . (8)
J
(cid:18) I (cid:19) (cid:18) I (cid:19) (cid:18)I +I (cid:19)
S S 0 S
IftheRFcurrentI issmallenough, thesecondtermcanbeapproximated byTaylorexpansion:
I +I icosωt i2cos2ωt
0 S
V ≈ 0.028ln +0.028 − +... = V +V cosωt+higherorderterms
J (cid:18) I (cid:19) (cid:20)I +I 2(I +I )2 (cid:21) DC J
S 0 S 0 S
(9)
Withtheidentitycos2 = 0.5,theDCandtheRFvoltages aregivenby
0.028 I 0.0282 V2
V = i= R i and V = 0.028ln 1+ 0 − = V − J . (10)
J I +I S DC (cid:18) I (cid:19) 4(I +I )2 0 0.112
0 S S 0 S
The region where the output voltage is proportional to the input power is called the square-law region
(Fig.5). In this region the input power is proportional to the square of the input voltage and the output
1Mostdiodesdonotneedanexternalbias,sincetheyhaveaDCreturnself-bias.
4
500
50
] LBSD
V
m
[ 5.0
r
e LBSD
w
o
p
ut 0.5
p
ut withoutload squarelawloaded
O
0.05
0.005
-50 -40 -30 -20 -10 0
Inputpower[dBm]
Fig.5: Relationbetweeninputpowerandoutputvoltage
signalisproportional totheinputpower,hencethenamesquare-law region.
The transition between the linear region and the square-law region is typically between −10 and
−20dB(Fig.5).
There are fundamental limitations when using diodes as detectors. The output signal of a diode
(essentiallyDCormodulatedDCiftheRFisamplitudemodulated)doesnotcontainaphaseinformation.
In addition, the sensitivity of a diode restricts the input level range to about −60dBm at best which is
notsufficientformanyapplications.
The minimum detectable power level of an RF diode is specified by the ‘tangential sensitivity’
whichtypically amountsto−50to−55dBmfor10MHzvideobandwidth atthedetector output[3].
Toavoidtheselimitations, anothermethodofoperating suchdiodes isneeded.
2.3 Mixer
For the detection of very small RF signals a device that has a linear response over the full range (from
0 dBm ( = 1mW) down to thermal noise = −174dBm/Hz = 4·10−21W/Hz) is preferred. An RF mixer
provides thesefeatures using 1,2, or4diodes indifferent configurations (Fig.6). Amixerisessentially
amultiplierwithaveryhighdynamicrangeimplementingthefunction
f (t)f (t) withf (t) = RFsignal andf (t) = LOsignal , (11)
1 2 1 2
ormoreexplicitly fortwosignalswithamplitude a andfrequencyf (i = 1,2):
i i
1
a cos(2πf t+ϕ)·a cos(2πf t)= a a [cos((f +f )t+ϕ)+cos((f −f )t+ϕ)] . (12)
1 1 2 2 1 2 1 2 1 2
2
Thus we obtain a response at the IF (intermediate frequency) port that is at the sum and difference
frequency oftheLO(localoscillator = f )andRF(= f )signals.
1 2
Examplesofdifferentmixerconfigurations areshowninFig.6.
5
Fig.6: Examplesofdifferentmixerconfigurations
As can be seen from Fig.6, the mixer uses diodes to multiply the two ingoing signals. These
diodesfunction asaswitch,opening differentcircuits withthefrequency oftheLOsignal(Fig.7).
Theresponse ofamixerintimedomainisdepictedinFig.8.
The output signal is always in the ‘linear range’ provided that the mixer is not in saturation with
respect to the RF input signal. Note that for the LO signal the mixer should always be in saturation to
make sure that the diodes work asa nearly ideal switch. Thephase of the RFsignal is conserved inthe
outputsignalavailable formtheRFoutput.
2.4 Amplifier
Alinear amplifier augments the input signal byafactor which is usually indicated indecibel. Theratio
between the output and the input signal is called the transfer function and its magnitude—the voltage
gainG—ismeasuredindBandgivenas
V V
RFout RFout
G[dB] = 20· or = 20·logG[lin] . (13)
V V
RFin RFin
ThecircuitsymbolofanamplifierisshowninFig.9together withitsS-matrix.
6
LO
Fig. 7: Two circuitconfigurationsinterchangingwith the frequencyof the LO where the switches representthe
diodes
LO
IF
RF
Fig.8: Timedomainresponseofamixer
0 0
S=
1 2 (cid:18) G 0 (cid:19)
Fig.9: CircuitsymbolanS-matrixofanidealamplifier
The bandwidth of an amplifier specifies the frequency range where it is usually operated. This
frequency range is defined by the −3dB points2 with respect to its maximum or nominal transmission
gain.
In an ideal amplifier the output signal would be proportional to the input signal. However, a
real amplifier is nonlinear, such that for larger signals the transfer characteristic deviates from its linear
properties valid for small signal amplification. When increasing the output power of an amplifier, a
pointisreachedwherethesmallsignalgainbecomesreducedby1dB(Fig.10). Thisoutputpowerlevel
definesthe1dBcompressionpoint,whichisanimportantmeasureofqualityforanyamplifier(lowlevel
aswellashighpower).
Thetransfercharacteristic ofanamplifiercanbedescribedintermswhicharecommonlyusedfor
RF engineering, i.e., the S-matrix (for further details see the paper on S-matrices of this School). As
implicitlycontainedintheS-matrix,theamplitudeandphaseinformationofanyspectralcomponentare
2The−3dBpointsarethepointsleftandrightofareferencevalue(e.g.,alocalmaximumofacurve)thatare3dBlower
thanthereference.
7
Fig.10:Exampleforthe1dBcompressionpoint[4]
preserved whenpassing through anideal amplifier. Forarealamplifierthe element G = S (transmis-
21
sion from port1toport2)isnotaconstant butacomplex function offrequency. Alsotheelements S
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andS arenot0inreality.
22
2.5 Interceptionpointsofnonlineardevices
Importantcharacteristicsofnonlineardevicesaretheinterceptionpoints. Hereonlyabriefoverviewwill
begiven. Forfurther information thereaderisreferredtoRef. [4].
One of the most relevant interception points is the interception point of 3rd order (IP3 point). Its
importance derives fromitsstraightforward determination, plotting theinput versus theoutput powerin
logarithmic scale (Fig.10). The IP3 point is usually not measured directly, but extrapolated from mea-
surement data at much smaller power levels in order to avoid overload and damage of the device under
test(DUT).Iftwosignals(f ,f > f )whicharecloselyspacedby∆f infrequencyaresimultaneously
1 2 1
appliedtotheDUT,theintermodulation productsappearat+∆f abovef andat−∆f belowf .
2 1
The transfer functions or weakly nonlinear devices can be approximated by Taylor expansion.
Using n higher order terms on one hand and plotting them together with an ideal linear device in log-
log
arithmic scale leads to two lines with different slopes (xn → n · logx). Their intersection point is
the intercept point of nth order. These points provide important information concerning the quality of
nonlinear devices.
In this context, the aforementioned 1dB compression point of an amplifier is the intercept point
offirstorder.
Similarcharacterization techniques canalsobeapplied withmixerswhichwithrespect totheLO
signalcannotbeconsidered aweaklynonlinear device.
2.6 Thesuperheterodyneconcept
Thewordsuperheterodyne iscomposedofthreeparts: super(Latin: over),ǫτǫρω (hetero,Greek: differ-
ent)andδυναµισ (dynamis,Greek: force)andcanbetranslatedastwoforcessuperimposed3. Different
3Thedirecttranslation(roughly)wouldbe:Anotherforcebecomessuperimposed.
8
abbrevations exist for the superheterodyne concept. In the USit is often referred to by the simple word
‘heterodyne’ and in Germany one can find the terms ‘super’ or ‘superhet’. The ‘weak’ incident signal
is subjected to nonlinear superposition (i.e., mixing or multiplication) with a ‘strong’ sine wave from a
local oscillator. At the mixer output we then get the sum and difference frequencies of the signal and
local oscillator. TheLOsignal can betuned such that the output signal isalways atthe samefrequency
or in a very narrow frequency band. Therefore a fixed frequency bandpass with excellent transfer char-
acteristics canbeusedwhichischeaper andeasierthanavariablebandpass withthesameperformance.
Awell-knownapplication ofthisprinciple isanysimpleradioreceiver(Fig.11).
RFamplifier Mixer IFamplifier Audioamplifier
BP
Bandpassfilter Demodulator
Localoscillator (oftenlockedtoaquarz)
Fig.11:Schematicdrawingofasuperheterodynereceiver
3 Oscilloscope
An oscilloscope is typically used for acquisition, display, and measurement of signals in time domain.
The bandwidth of real-time oscilloscopes is limited in most cases to 10GHz. For higher bandwidth on
repetitive signals the sampling technique has been in use since about 1960. One of the many interest-
ing features of modern oscilloscopes is that they can change the sampling rate through the sweep in a
programmed manner. This can be very helpful for detailed analysis in certain time windows. Typical
sampling rates are between a factor 2.5 and 4 of the maximum frequency (according to the Nyquist
theorem areal-timeminimumsamplingrateoftwicethemaximumfrequency f isrequired).
max
Sequential sampling (Fig.12)requires apre-trigger (required toopen thesampling gate)andper-
mitsanon-real-time bandwidth ofmorethan100GHzwithmodernscopes.
Random sampling (rarely used these days, Fig.13) was developed about 40 years ago (around
1970) forthe case whereno pre-trigger wasavailable andrelying on astrictly periodic signal topredict
apre-trigger fromthemeasuredperiodicity.
Samplingisdiscussedinmoredetailinthefollowing. Considerabandwidth-limitedtimefunction
s(t) and its Fourier transform S(f). The signal s(t) is sampled (multiplied) by a series of equidistant δ-
pulsesp(t)[5]:
+∞
p(t)= δ(t−nT ) = III(t/T ) (14)
s s
n=X−∞
wherethesymbolIII isderivedfromtheRussianletterIIIandispronounced‘sha’. Itrepresentsaseries
ofδ-pulses.
9
Fig.12:Illustrationofsequentialsampling
Fig.13: Illustrationofrandomsampling
Thesampledtimefunctions s (t)is
s
s (t) = s(t)p(t)= s(t)III(t/T )
s s
1
S (f)= S(f)∗ III(T f) (15)
s s
T
s
+∞
1 1
S (f) = S(f −mF)withF = . (16)
s
T T
s n=X−∞ s
Notethat the spectrum is repeated periodically bythe sampling process. Forproper reconstruction, one
ensuresthatoverlapping asinFig.14doesnotoccur.
If the spectra overlap as in Fig.14 we have undersampling, the sampling rate is too low. If big
gaps occur between the spectra (Fig.15) we have oversampling, the sampling rate is too high. But this
schemeapplies inmostcases. InthelimitwearriveataNyquistrateof1/T = 2f = F.
s g
Therulesmentionedaboveareofgreatimportanceforall‘digital’oscilloscopes. Theperformance
(conversion time, resolution) of the input ADC(analog–digital converter) isthe key element for single-
shot rise time. With several ADCs in time-multiplex one obtains these days 8-bit vertical resolution at
20GSa/s=10GHzbandwidth.
Another way to look at the sampling theorem (Nyquist) is to consider the sampling gate as a
harmonicmixer(Fig.16).
Thisisbasically anonlinear element(e.g.,adiode) thatgivesproduct termsoftwosignals super-
10
