Automatic Linearity (IP3) Test with Built-in Pattern Generator and Analyzer

Oct. 7, 04 ELEC5970-003/6970-003 (Guest Lecture)

1

Automatic Linearity (IP3) Test with Built-in Automatic Linearity (IP3) Test with Built-in Pattern Generator and AnalyzerPattern Generator and Analyzer

Foster Dai, Charles Stroud, Dayu YangFoster Dai, Charles Stroud, Dayu Yang

Dept. of Electrical and Computer EngineeringDept. of Electrical and Computer EngineeringAuburn UniversityAuburn University


2

PurposePurpose

• Develop Built-In Self-Test (BIST) approach Develop Built-In Self-Test (BIST) approach

using direct digital synthesizer (DDS) for using direct digital synthesizer (DDS) for

functionality testing of analog circuitry in functionality testing of analog circuitry in

mixed-signal systems mixed-signal systems

• Provides BIST-based measurement of Provides BIST-based measurement of – Amplifier linearity (IP3) Amplifier linearity (IP3) – Gain and frequency responseGain and frequency response

• Implemented in hardwareImplemented in hardware– IP3, gain, and freq. response measuredIP3, gain, and freq. response measured


3

OutlineOutline

• Overview of direct digital synthesizer (DDS)Overview of direct digital synthesizer (DDS)

• 33rdrd order inter-modulation product (IP3) order inter-modulation product (IP3)

• BIST architectureBIST architecture– Test pattern generatorTest pattern generator– Output response analyzerOutput response analyzer

• Experimental resultsExperimental results– Implementation in hardwareImplementation in hardware– IP3 MeasurementsIP3 Measurements


4

Linear vs. Nonlinear SystemsLinear vs. Nonlinear Systems

• A system is A system is linearlinear if for any inputs x if for any inputs x11(t) and x(t) and x22(t),(t),

xx11(t) (t) y y22(t), x(t), x22(t) (t) y y22(t) and for all values of (t) and for all values of

constants a and b, it satisfiesconstants a and b, it satisfies

a xa x11(t)+bx(t)+bx22(t) (t) ay ay11(t)+by(t)+by22(t)(t)

• A system is A system is nonlinear nonlinear if it does not satisfy the if it does not satisfy the

superposition law.superposition law.


5

• A system is A system is time invarianttime invariant if a time shift in input if a time shift in input results in the same time shift in output, namely,results in the same time shift in output, namely,

if if x(t) x(t) y(t), y(t), then then x(t-x(t-) ) y(t- y(t-), ), for all value of for all value of

• A system is A system is time varianttime variant if it does not satisfy the if it does not satisfy the condition. condition.

Time Invariant vs. Time Time Invariant vs. Time Variant SystemsVariant Systems


6

Memoryless SystemsMemoryless Systems

• A system is A system is memorylessmemoryless if its output does not depend on if its output does not depend on the past value of its input.the past value of its input.

• For a For a memoryless linear systemmemoryless linear system,, y(t) = y(t) = ααx(t)x(t)

wwherehereαα is a function of time if the system is time variant. is a function of time if the system is time variant.

• For a For a memorylessmemoryless nonlinear systemnonlinear system, , y(t) = y(t) = αα00 + + αα11x(t) + x(t) + αα22x²(t)+ x²(t)+ αα33x³(t) + ······x³(t) + ······ where where ααjj are in general function of time if the system is are in general function of time if the system is

time variant.time variant.


7

Dynamic SystemsDynamic Systems

• A system is A system is dynamicdynamic if its output depends on the past values of its if its output depends on the past values of its input(s) or output(s).input(s) or output(s).

• For a linear, time-invariant, dynamic system,For a linear, time-invariant, dynamic system,y(t) = h(t) * x(t)y(t) = h(t) * x(t),,

where h(t) denotes the impulse response.where h(t) denotes the impulse response.

• If a dynamic system is linear but time variant, its impulse response If a dynamic system is linear but time variant, its impulse response depends on the time origins, namely,depends on the time origins, namely,

)(),()(

).()(

),()(

txthtv

ThusthtThen

tht


8

Effects of NonlinearityEffects of Nonlinearity

• Harmonic DistortionHarmonic Distortion

• Gain CompressionGain Compression

• DesensitizationDesensitization

• IntermodulationIntermodulation

• For simplicity, we limit our analysis to memoryless, For simplicity, we limit our analysis to memoryless,

time variant system. Thus,time variant system. Thus,

...)()()()( 33

221 txtxtxty ααα (3.1)


9

Effects of Nonlinearity -- HarmonicsEffects of Nonlinearity -- Harmonics

tA

tA

tA

AA

ttA

tA

tA

tAtAtAty

α

α

α

αα

α

α

α

ααα

3cos4

2cos2

cos)4

3(

2

)3coscos3(4

)2cos1(2

cos

coscoscos)(

33

22

33

1

22

33

22

1

33

221

If a single tone signal is applied to a nonlinear system, the output generally exhibits fundamental and harmonic frequencies with respect to the input frequency. In Eq. (3.1), if x(t) = Acosωt, then

Observations:1. even order harmonics result from αj with even j and vanish if the system has odd symmetry, i.e., differential circuits. 2. For large A, the nth harmonic grows approximately in proportion to An.


10

dBA 1

Ou

tpu

t V

olt

age

(dB

V)

1dB

20logAin

• 1-dB compression point is defined as the input signal level that causes small-signal gain to drop 1 dB. It’s a measure of the maximum input range.

•1-dB compression point occurs around -20 to -25 dBm (63.2 to 35.6mVpp in a 50-Ω system) in typical frond-end RF amplifiers.

Effects of Nonlinearity – 1dB Compression Point

3

1

3

11

31311

11

3808.0145.0

1

4

3log20

αα

αα

αα

α

dB

dBdB

dB

A

dBAA

A

mW

VdBm pp

150

8/log10

2


11

Effects of Nonlinearity – IntermodulationEffects of Nonlinearity – Intermodulation

• Harmonic distortion is due to self-mixing of a single-tone signal. It can be suppressed by low-pass filtering the higher order harmonics.

• However, there is another type of nonlinearity -- intermodulation (IM) distortion, which is normally determined by a “two tone test”.

• When two signals with different frequencies applied to a nonlinear system, the output in general exhibits some components that are not harmonics of the input frequencies. This phenomenon arises from cross-mixing (multiplication) of the two signals.


12

• assume assume x(t) = Ax(t) = A11coscosωω11t+ At+ A22coscosωω22tt two tone testtwo tone test

• Expanding the right side and disregarding dc terms and harmonics, we obtain the Expanding the right side and disregarding dc terms and harmonics, we obtain the following intermodulation products:following intermodulation products:

• And these fundamental components:And these fundamental components:

tAA

tAA

tAA

tAA

tAAtAA

)2cos(4

3)2cos(

4

3:2

)2cos(4

3)2cos(

4

3:2

)cos()cos(:

121

223

121

223

12

212

213

212

213

21

212122121221

α

α

α

α

αα

Effects of Nonlinearity – Intermodulation

322113

22211222111

)coscos(

)coscos()coscos()(

tAtA

tAtAtAtAty

α

αα

tAAAAtAAAA 22

123323211

2213

3131121 cos

2

3

4

3cos

2

3

4

3:, αααααα


13

DC Term

1st Order Terms

2nd Order Terms

3rd Order terms


ttAA

ttAA

tAtA

ttAA

tAtA

tAAAA

tAAAA

AAty

AtAtx

)2cos()2cos(

)2cos()2cos(

4

3

3cos3cos4

1

)cos()cos(

2cos2cos2

1

cos)2(4

3

cos)2(4

3

)(2

1)(

coscos)(

12122

21

212122

13

23

213

13

2121212

22

212

12

222

212321

122

211311

22

212

2211

α

α

α

α

αα

αα

α


14

• OfOf particular interest are the third-order IM products at 2particular interest are the third-order IM products at 2ωω11--ωω22 and 2 and 2 ωω22--

ωω11. The key point here is that if the . The key point here is that if the differencedifference between between ωω11and and ωω22 is small, is small,

the 2 the 2 ωω11--ωω22 and 2 and 2 ωω22--ωω11 appear in the appear in the vicinityvicinity of of ωω1 1 and and ωω2.2.

ω1 ω2 ω ω1 ω2

2ω1-ω2 2ω2-ω1

ω



15

Intermodulation -- Third Order Intercept Point (IP3)Intermodulation -- Third Order Intercept Point (IP3)

• Two-tune test: Two-tune test: AA11=A=A22=A and A is sufficiently small=A and A is sufficiently small so that higher-order so that higher-order

nonlinear terms are negligible and the gain is relatively constant and equal nonlinear terms are negligible and the gain is relatively constant and equal to to αα11..

• As A increases, the fundamentals increases in proportion to A, whereas IM3 As A increases, the fundamentals increases in proportion to A, whereas IM3 products increases in proportion to A³.products increases in proportion to A³.

tt

ttAttA

ttAttA

tAAtAA

Aty

AtAtx

)2cos()2cos(

)2cos()2cos(

4

33cos3cos

4

1

)cos()cos(2cos2cos2

1

cos4

9cos

4

9

)(

coscos)(

2121

21213321

33

21212

2212

2

22

3112

31

22

21

αα

αα

αααα

α

313

3

13

231 and

3

4

4

9IIPOIPIIPA α

αααα

dBA

A

IP

dB 6.93/4

145.0

3

1


16

• Plotted on a log scale, the intersection of the two lines is defined as Plotted on a log scale, the intersection of the two lines is defined as the the third order intercept pointthird order intercept point. The horizontal coordinate of this point is . The horizontal coordinate of this point is called the called the input referred IPinput referred IP33(IIP(IIP33),), and the vertical coordinate is called and the vertical coordinate is called the the output referred IPoutput referred IP33(OIP(OIP33).).

334

3Aα

α1A

A

OIP3

IIP3 20logA

)log(20 1Aα

3

34

3log20 Aα

Intermodulation -- Third Order Intercept Point (IP3)

3

13 3

4

αα

IPA


17

Calculate IIP3 without ExtrapolationCalculate IIP3 without Extrapolation

212 1 2Freq

122

P

A1α A1α

234

3Aα 2

34

3Aα

][2

][][3 dBmP

dBPdBmIIP in

OIP3

IIP320logAin

3

34

3log20 Aα

)log(20 1Aα

P

P/2


18

Direct Digital Synthesis (DDS)Direct Digital Synthesis (DDS)

• DDS DDS generating deterministic communication generating deterministic communication

carrier/reference signals in discrete time using carrier/reference signals in discrete time using

digital hardwaredigital hardware– converted into analog signals using a DACconverted into analog signals using a DAC

• AdvantagesAdvantages– Capable of generating a variety of waveformsCapable of generating a variety of waveforms– High precision High precision sub Hz sub Hz– Digital circuitryDigital circuitry

• Small size Small size fraction of analog synthesizer size fraction of analog synthesizer size• Low costLow cost• Easy implementationEasy implementation


19

Typical DDS ArchitectureTypical DDS Architecture

1/1/ffoutout1/1/ffoutout

1/1/ffclkclk

1/1/ffoutout

1/1/ffclkclk

1/1/ffoutout

1/1/ffclkclk

ffoutout==ffclkclkFFrr

22NN

AccumAccum-ulator-ulator

NN

FrequencyFrequencyWordWord

WWSineSine

LookupLookupTableTable

RRLowLowPassPassFilterFilter

SineSineWaveWaveFFrr

clkclk

Digital CircuitsDigital Circuits

D-to-AD-to-AConv.Conv.


20

IntermodulationIntermodulation

• Two signals with different frequencies are Two signals with different frequencies are

applied to a nonlinear systemapplied to a nonlinear system– Output exhibits components that are not Output exhibits components that are not

harmonics of input fundamental frequenciesharmonics of input fundamental frequencies

• Third-order intermodulation (IM3) is criticalThird-order intermodulation (IM3) is critical– Very close to fundamental frequenciesVery close to fundamental frequencies

IM3IM3

ff11 ff22

7 87 8 freqfreq

ff11 ff22

8800 22 44 66 1010 1212 1414 1616 1818 2020 2222 2424

ff22-- ff11 ff11++ff2222ff11 22ff22

33ff11 33ff2222ff11-- ff22 22ff22-- ff11

freqfreq


21

Mathematical FoundationMathematical Foundation• Input 2-tone:Input 2-tone:

x(x(tt)=A)=A11cos cos 11tt + A + A22 cos cos 22tt

• Output of non-linear device:Output of non-linear device:y(y(tt)=α)=α00+α+α11x(x(tt)+α)+α22xx22((tt)+α)+α33xx33((tt)+)+

• Substituting x(Substituting x(tt) into y() into y(tt):):y(y(tt) = ) = ½½αα22(A(A11

22+A+A2222) )

+ [α+ [α11AA11+¾α+¾α33AA11(A(A1122+2A+2A22

22)]cos)]cos11t t + [α+ [α11AA22+¾α+¾α33AA22(2A(2A1122+A+A22

22)]cos)]cos22tt

+ + ½½αα22(A(A1122cos2cos211tt+A+A22

22cos2cos222tt ) )

+ α+ α22AA11AA22[cos([cos(11++22))tt+cos(+cos(11--22))tt]]

+ ¼α+ ¼α33[A[A1133cos3cos311tt+A+A22

22cos3cos322tt]]

+ ¾α+ ¾α33{A{A1122AA22[cos(2[cos(211++22))tt+cos(2+cos(211--22))tt]]

+A+A11AA2222[cos(2[cos(222++11))tt+cos(2+cos(222--11))tt]}]} freqfreq

11 22 2222--112211--22

¾ ¾ αα33AA22

αα11AAPP


22

3rd-order Intercept Point (IP3)3rd-order Intercept Point (IP3)

freqfreq11 22 2222--112211--22

¾ ¾ αα33AA22

αα11AAPP

Input PowerInput Power(IIP3)(IIP3)

IP3IP3 20log(20log(αα11A)A)

Out

put

Pow

erO

utpu

t P

ower

(OIP

3)(O

IP3)

P/2P/2

IM3IM3

fundamentalfundamental

PP

20log(¾20log(¾αα33AA33))

• IP3 is theoretical input power point where 3IP3 is theoretical input power point where 3 rdrd-order -order

distortion and fundamental output lines interceptdistortion and fundamental output lines intercept

• IIPIIP33[dBm]= +P[dBm]= +P inin[dBm][dBm]P[dB]P[dB]

22

Practical measurementPractical measurementwith spectrum analyzerwith spectrum analyzer


23

2-Tone Test Pattern Generator2-Tone Test Pattern Generator

LowLowPassPassFilterFilter 2-tone2-tone

WaveformWaveform

D-to-AD-to-AConv.Conv.

SineSineLookupLookupTable 1Table 1

FFrr11AccumAccum-ulator-ulator

#1#1

SineSineLookupLookupTable 2Table 2

FFrr22AccumAccum-ulator-ulator

#2#2

• Two DDS circuits generate two fundamental tonesTwo DDS circuits generate two fundamental tones

– FFrr1 & 1 & FFrr2 control frequencies tones2 control frequencies tones

• DDS outputs are superimposed using adder to DDS outputs are superimposed using adder to

generate 2-tone waveform for IP3 measurementgenerate 2-tone waveform for IP3 measurement


24

Actual 2-Tone IP3 MeasurementActual 2-Tone IP3 Measurement

DAC output x(DAC output x(tt): ):

DUT output y(DUT output y(tt):):

PP

• Outputs of DAC and DUT taken with scope from Outputs of DAC and DUT taken with scope from our experimental hardware implementationour experimental hardware implementation

• Typical Typical PP measurement requires expensive, measurement requires expensive, external spectrum analyzerexternal spectrum analyzer– For BIST we need an efficient output response analyzerFor BIST we need an efficient output response analyzer


25

Output Response AnalyzerOutput Response Analyzer

Multiplier/accumulator-based ORAMultiplier/accumulator-based ORA

• Multiply the output response by a frequencyMultiply the output response by a frequency– NN-bit multiplier, -bit multiplier, NN = number of ADC bits = number of ADC bits

• Accumulate the multiplication resultAccumulate the multiplication result– NN++MM-bit accumulator for < 2-bit accumulator for < 2MM clock cycle samples clock cycle samples

• Average by # of clock cycles of accumulationAverage by # of clock cycles of accumulation– Gives DC value proportional to power of signal at freqGives DC value proportional to power of signal at freq

• AdvantagesAdvantages– Easy to implementEasy to implement– Low area overheadLow area overhead– Exact frequency controlExact frequency control– More efficient than FFTMore efficient than FFT

XXy(y(tt))

ffxx DCDC

multipliermultiplier

accumulatoraccumulator


26

DCDC11 Accumulator Accumulator

MATLAB Simulation ResultsMATLAB Simulation Results Actual Hardware ResultsActual Hardware Results

• y(y(tt) x ) x ff22 DC DC11 ½A ½A2222αα11

• Ripple in slope due to low Ripple in slope due to low

frequency componentsfrequency components– Longer accumulation reduces Longer accumulation reduces

effect of rippleeffect of ripplefreqfreq

ff22 22ff22--ff11

¾ ¾ αα33AA22PP

αα11AA

XX

y(y(tt))

ff22

DCDC11

slope = DCslope = DC11

½A½A2222αα11


27

DCDC22 Accumulator Accumulator

slope = DCslope = DC22

33//88AA1122AA22

22αα33


• y(y(tt) x 2) x 2ff22--ff11 DC DC22 33//88AA1122AA22

22αα33

• Ripple is bigger for DCRipple is bigger for DC22

– Signal is smallerSignal is smaller

– Test controller needs to obtain Test controller needs to obtain

DCDC22 at integral multiple of 2 at integral multiple of 2ff22--ff11freqfreq

ff22 22ff22--ff11

¾ ¾ αα33AA22PP

αα11AA

XX

y(y(tt))

22ff22--ff11

DCDC22


28

BIST-based BIST-based P MeasruementP Measruement

• DCDC11 & DC & DC22 are proportional to power at are proportional to power at ff22 & 2 & 2ff22--ff11

• Only need DCOnly need DC11 & DC & DC22 from accumulators to calculate from accumulators to calculate

PP = 20 log (DC = 20 log (DC11) – 20 log (DC) – 20 log (DC22))



29

BIST ArchitectureBIST Architecture

XXTest Pattern GeneratorTest Pattern Generator

Output ResponseOutput Response AnalyzerAnalyzer

LUT2LUT2 AccumAccumff22

AccumAccum

x(x(tt)=cos()=cos(ff11)+cos()+cos(ff22))

DC1DC1

LUT1LUT1ff11

AccumAccum

LUT3LUT322ff22-f-f11

AccumAccum

DACDAC DUTDUT ADCADC

y(y(tt))

XX DC2DC2AccumAccum

DC2DC2

GainGainFreqFreqRespResp

x(x(tt)=cos()=cos(ff22))

• BIST-based IP3 measurementBIST-based IP3 measurement

– Reduce circuit by repeating test sequence for DCReduce circuit by repeating test sequence for DC22

• BIST-based Gain & Frequency Response is subsetBIST-based Gain & Frequency Response is subset


30

Experimental Implementation of BISTExperimental Implementation of BIST

• TPG, ORA, test controller, & PC interface circuitsTPG, ORA, test controller, & PC interface circuits– Three 8-bit DDSs and two 17-bit ORA accumulatorsThree 8-bit DDSs and two 17-bit ORA accumulators– Implementation in VerilogImplementation in Verilog– Synthesized into Xilinx Spartan 2S50 FPGASynthesized into Xilinx Spartan 2S50 FPGA

• Amplifier device under test implemented in FPAAAmplifier device under test implemented in FPAA• DAC-ADC PCBDAC-ADC PCB

0

200

400

600

800

1000

1200

1400

1600

Slices LUTs FFs

Total in FPGADouble ORASingle ORAPCPC FPGAFPGA

TPG/ORATPG/ORA

DAC &DAC &ADCADC

FPAAFPAADUTDUT


31

Hardware ResultsHardware Results

PP1144

BIST measures BIST measures P P 14 14 Spectrum analyzerSpectrum analyzer

P distribution for 1000P distribution for 1000

BIST BIST measurementmeasurementssmean=13.97 dB, mean=13.97 dB, =0.082 =0.082


32

More Hardware ResultsMore Hardware Results

BIST measures BIST measures P P 22 22 Spectrum analyzerSpectrum analyzer

PP2222

P distribution for 1000P distribution for 1000 BIST measurementsBIST measurementsmean=21.7 dB, mean=21.7 dB, =2.2 =2.2


33

Measurements in Noisy EnvironmentMeasurements in Noisy Environment

0

5

10

15

20

25

30

35

40

1 21 41 61 81 101 121 141 161 181 201 221 241 261

14 dB 14 dB PP BIST BISTmeasurement inmeasurement innoisy environmentnoisy environment

17 dB 17 dB PP BIST BISTmeasurement inmeasurement inless noisyless noisyenvironmentenvironment


34

BIST IP3 Measurement ResultsBIST IP3 Measurement Results

• Good agreement with actual values for Good agreement with actual values for PP < 30dB < 30dB

• For measured For measured PP > 30dB, the actual > 30dB, the actual P P is greateris greater– Good threshold since Good threshold since P P < 30dB is of most interest< 30dB is of most interest


35

ConclusionConclusion

• BIST-based approach for analog circuit BIST-based approach for analog circuit

functional testingfunctional testing– DDS-based TPGDDS-based TPG– Multiplier/accumulator-based ORAMultiplier/accumulator-based ORA

• Good for manufacturing or in-system circuit Good for manufacturing or in-system circuit

characterization and on-chip compensationcharacterization and on-chip compensation– Amplifier linearity (IP3)Amplifier linearity (IP3)– Gain and frequency responseGain and frequency response

• Measurements with hardware implementationMeasurements with hardware implementation– Accurately measures IP3 < 30dBAccurately measures IP3 < 30dB– Measurements of IP3 > 30dB imply higher valuesMeasurements of IP3 > 30dB imply higher values

Documents

Automatic Linearity (IP3) Test with Built-in Pattern Generator and Analyzer