rf module design - [chapter 3] linearity

RF Transceiver Module DesignChapter 3 Nonlinear Effects

李健榮助理教授

Department of Electronic EngineeringNational Taipei University of Technology

Outline

• Nonlinear Effects on an RF Signal

• Analysis of 1-dB-Compression Point (P1dB)

• Analysis of Second-Order Intercept Point (IP2)

• Analysis of Third-Order Intercept Point (IP3)

• Nonlinear Effect of a Cascaded System

• Nonlinear Effect on a Digitally-Modulated Signal

Department of Electronic Engineering, NTUT2/49

Nonlinear Effects

• The distortion of an RF transceiver are resulted frominternalinterferences andexternal interferences.

1) The internal interferences are generated fromthenonlineareffect of its own devices.

2) The external interference are fromoutside the transceiverand intercepted by the antenna or EM coupling.

3) Internal distortion is primarily generated frompoweramplifier.


Power Amplifier Categories

• Linear Amplifier: Class A, B, AB, and C

Classified in terms of current conduction angle

CEv

,maxCEVkneeV QV

,maxCI

Ci

QIA

AB

BC

Biased Transistor

Input Matching Output Matching


Linear Amplifier

Normalized DSi

A

C B AB

0 π 2πtω

Class Duty Cycle Theoretical Efficiency Linearity

A 100% 50% Excellent

B 50% 78.5% Moderate

AB 50~100% 50~78.5% In-Between Class-A and -B

C 0~50% 100% Poor


Nonlinear Amplifier

• Constant-envelop, nonlinear or switching-mode amplifier• Class D, E, F, S :

Transistor is driven in switching mode, theoretical efficiency 100%.

Department of Electronic Engineering, NTUT

DDV

dcL

pC

0L 0C jX

LRSt

DSiDSv

6/49

Amplifier AM/AM and AM/PM Distortion

• Modulated Input signal:

• Distorted Output signal:

( ) ( ) ( )( )cosin cv t A t t tω φ= +

( ) ( ) ( ) ( )( ), cos ,out cv t B f A t t f Aω φ θ= + +

outP 40�

0�

40− �

80− �

20

0

20−

40−

Ou

tpu

t Po

wer

(d

Bm

)

Ph

ase Sh

ift

Input Power (dBm)10− 5− 0 5 10 15 20 25

Class A

AB

C AB

A

C

AM/AM Distortion AM/PM Distortion


( )inv t ( )outv t

7/49

Nonlinear Memoryless Device (I)

• An input-output relationship of anonlinear memorylessdevice can be represented as

( ) ( ) ( ) ( ) ( )2 3 40 1 2 3 4out in in in inv t v t v t v t v tα α α α α= + + + + +⋯

( )inv t ( )outv t

inV

outV

linear

nonlinear

small signal

large signal

linear outputdistorted output

f

f

Perfect sinusoid

Harmonics


Nonlinear Memoryless Device (II)

Coefficients αi are depending on

1) DC bias, RF characteristics of the active device used in the circuit.

2) Magnitude vin of the signal.

3) When Pin < P1dB (linear region), all can be treated as constant.

• Assume the input and output impedance of the circuit are ,and ,respectively. Considering a CWinput signal with thevoltage ,the input available power is

( )inv t ( )outv t

( ) sin 2in in cv t V f tπ= ( ) ( )2 2in c in in cP f V Z f=


( )inZ f

( )outZ f


9/49

Small-signal Power Gain (Linear Gain)

• For linear operation

where Pin is the available input power andG1 is the available small-signalpower gain, which equals to

( ) ( )1 1 sin 2out in in cv t v t V tα α π= =

( )( )

2 2 2 22 211 1

1 1 1

2 2 2in cout in in in

out inout out in out out c

Z fV V V ZP P

Z Z Z Z Z f

α α α= = = =

( )( )120log 10log in c

out inout c

Z fP P

Z fα= + + ( ) ( ) ( )1 dBmout c in cP f P f G= +

( )( )1 120log 10log in c

out c

Z fG

Z fα= +

( ) sin 2in in cv t V f tπ=



( )inv t ( )outv t

Assume , we have .( ) ( )in c out cZ f Z f= 1 120logG α=

10/49

Linear Amplification

( ) ( ) dBmin cP f

1G1

1

( ) ( ) dBmout cP f

( ) ( ) dBmin cP f

1G

( ) ( ) dBmout cP f

inP

cf

f

f

1out inP P G= +


( )inv t ( )outv t

11/49

Third-order Effect

• For a single-tone input signal,

• α3 < 0 gives gain compression phenomenon

• α3 > 0 gives gain enhancement phenomenon

( ) 1cosinv t A tω=

( ) ( )3 31 1 3 1cos cosoutv t A t A tα ω α ω= +

3 31 3 1 3 1

3 1cos cos3

4 4A A t A tα α ω α ω = + +

Out-of-band Distortion (3rd Harmonic)3rd-order effect

In-band Distortion3rd-order effect

Desired Signallinear effect

( )inv t ( )outv t

( ) ( ) ( )31 3out in inv t v t v tα α= +


1 dB-Compression Point

• When the input signal becomes stronger, the output signal willnot grow proportionally but with a slower rate. It is asaturation phenomena.

1 dB

1dBOP

G

1dBIP

( )out cP f

( ) ( ) dBmin cP f1

1

• When the actual output power is 1 dB less thanthe linear extrapolated power, it reaches the 1-dB gain compression point. At this point, theinput power is called the input 1-dB-compressed power (IP1dB), the output power iscalled the output 1-dB-compressed power(OP1dB) ,and the gain is called the1-dB-compressed gain (G1dB).


( ) 3 31 3 1 3 1

3 1cos cos3

4 4outv t A A t A tα α ω α ω = + +

α3 < 0

13/49

Analysis of 1dB-Compression Point (I)

• At P1dB , the output power is compressed 1 dB, i.e.,

• The input voltage magnitude at P1dB as

311 1dB 3 1dB

20

1 1dB

34 0.891 10

A A

A

α α

α

−+ = =

( )

31 1dB 3 1dB

desired+distorted

desired 1 1dB

3410log 20log 1 dB

A AP

P A

α α

α

+= = −

11dB

3

0.145Aαα

=

( )21dB 1 1

1dB3 3

110log 30 10log 0.0725 30 18.6 10log dBm

2 in in in

AIP

R R R

α αα α

= + = + = +

( )

23

3 31 1dB 3 1dB1 1

1dB3 3

31 0.0575410log 30 10log 30 17.6 10log dBm2 out in out

A AOP

R R R

α α α αα α

+ = + = + = +


( ) ( )211 1 1

3

17.6 10log 1 dBmdBout

IP GR

α αα

= + ⋅ = + −

14/49

Analysis of 1dB-Compression Point (II)

1G

( ) dBminP

cf

cf

1out inP P G= +

( )1dB 1 1out in inP P G P G= + = + −1out inP P G= +


Measurement of P1dB

• By network analyzer in the power sweep mode:Obtain small signal gain and .

• By spectrumanalyzer :Test various input signal power level to measurement the output power spectralcontent to obtain output v.s. input power curve.

1 120logG α= 1dBG


Network Analyzer

Amplifier

Signal Generator

Amplifier

Spectrum Analyzer

16/49

Distortion Characterization (I)

• Amplifier input-output relation:

• If only one signal is present, the undesired components willbe harmonics of the fundamental, but, if there aremoresignals at input, signals will be produced with frequenciesthat are mathematical combinations of the frequencies of theinput signals, calledintermodulation products (IMPs) orintermods. It is instructive to study the results when there aretwo input signals (although we will eventually consider largenumbers of signals).



Distortion Characterization (II)

• Characterized by 1-dB gain compression, IPs , 2-toneintermodulation distortions (IMDs)

1cosinv A tω=

,1 1cosout ov G A tω=

,2 2 1cos2outv A tα ω=

,3 3 1cos3outv A tα ω=


Single-tone excitation

Nonlinear Harmonics

1ff

1ff

12 f 13 f 14 f

18/49

Distortion Characterization (III)

Designed Amplifier1f 2f

f

1f 2ff

1 22 f f− 2 12 f f−

1f 2ff

1 22 f f− 2 12 f f−

1f 2ff

1 22 f f− 2 12 f f−

IMD from AM/AM distortion

IMD from AM/PM distortion


Two-tone excitation

Nonlinear

IMProducts

• Characterized by 1-dB gain compression, IPs , 2-tone IMDs

19/49

Intercept Points

• The nonlinear properties can be described by the concept ofintercept points (IPs). The input intercept point (IIPn) is afictitious input power where the desired output signalcomponent equals in amplitude the undesired component.

( )out nP f

( )out cP f

( ) ( ) dBmin cP fIIPn1dBIP

OIPn

1dBOP

1 dB

1

1 1

n

Ou

tpu

t Po

wer

(d

Bm

)


Second-Order Nonlinear Effect (I)

• Single-tone excitation:

• For the inclusion of only the linear termand the second term,the output voltage is

( ) sin 2in cv t A tπ=

( ) ( )2

2in cin c

AP f

Z f=

( ) ( ) ( ) ( ) ( )221 2 1 2sin 2 sin 2out in in c cv t v t v t A f t A f tα α α π α π= + = +

( ) ( )2

221 2sin 2 sin 2

2 c c

AA f t A f t

α α π α π= + −

2 22 1 2

1 1sin cos2

2 2c cA A t A tα α ω α ω= + −

Out-of-band Distortion2nd-order effect

DC Offset2nd-order effect



( )in cZ f

( )inv t ( )outv t

cff

0

21/49

Second-Order Nonlinear Effect (II)

• Two-tone Excitation: ( ) 1 2sin sininv t A t B tω ω= +

( ) ( ) ( )2

1 1 2 2 1 2sin sin sin sinoutv t A t B t A t B tα ω ω α ω ω= + + +

( ) [ ]2 22 1 1 1 2

1sin sin

2A B A t B tα α ω α ω = + + +

( ) ( )2 1 2 2 1 2cos cosAB t AB tα ω ω α ω ω+ − + +

2 22 1 2 2

1 1cos2 cos2

2 2A t B tα ω α ω + − −

2 1f f−0 1f 2f 12 f 22 f1 2f f+

a b

ce

dfg

g : DC term

a, b : linear term

c : IM (down beating)d : IM (up beating)e, f : 2nd harmonic


a bg

c d

e f

22/49

Linear and 2nd-order Effects

• Linear effect:

A superscript (1) of denotes that the power content contributed from the first-order term (linear term).

• 2nd-order effect:

( ) ( ) ( ) ( )( )

1120log 10log in c

out c in cout c

Z fP f P f

Z fα= + +

( ) ( ) ( ) ( )11 dBmout c in cP f P f G= +

( )1outP


Linear Gain

( ) ( ) ( ) ( )( )( )

( )( )

22

2 2 2222 2 2 2

2 2

11 1 12

22 2 2 2 2 2 2

in c in cout c in

out c in c out c out c

AZ f Z fA

P f PZ f Z f Z f Z f

αα α

= = =

( ) ( )( )

2

220log 3 2 dBm 10log2

in c

in

out c

Z fP

Z fα= − + +

( ) ( ) ( ) ( )222 2 dBmout c in cP f G P f= +

( ) ( )( )

2

2 2 dB 20log 3 10log2

in c

out c

Z fG

Z fα= − +

Slope of 2

23/49

Second-Order Intercept Point

6 dB

6 d

B

IM2

2nd harmonic

Fundamental

Fundamental input power (dBm)

Ou

tpu

t po

wer

(d

Bm

)

6 d

B

6 dB• The 2nd-order products increase twiceas fast as the desired fundamental, thestraight lines cross. At the crossingpoint, either for the intermod or theharmonic, the fundamental and the2nd-order product have equal outputpowers.

• Since the slopes of the straight linesare known, these crossing points,called intercept points (IPs), definethe 2nd-order productsat low levels.

OIP2H

OIP2IM

IIP2IM IIP2H

6 dB

• Typically, the larger of the input oroutput intercept points is specified; soamplifiers use OIPs and mixers useIIPs. Some may even add the powerof the two fundamentals, increasingthe value of the IP by 3 dB.

6 d

B


Example

• For an amplifier with 21 dB linear gain and theOIP2H is at 17dBm, find the output 2nd harmonic power when thefundamental output signal power is−8 dBm.

( )12 2 dBmH HOIP IIP G= +OIP2H = 17 dBm

2nd harmonic

Fundamental

Fundamental input power (dBm)O

utp

ut p

ow

er (

dB

m)

IP2H

−8 dBm

25

dB

25

dB

−33 dBm

−29 dBm −4 dBm

(IIP2H )

( )17 2 21 dBmHIIP= +

( )2 4 dBmHIIP = −

( ) ( ) ( ) ( )2 2 dBmout c out c H out cP f P f OIP P f= − −

[ ] ( )8 17 8 33 dBm= − − + = −


Unequal Input Tone Power


( ) ( ) [ ] ( ) ( )2 2 2 22 1 1 1 2 2 1 2 2 1 2 2 1 2 2

1 1 1sin sin cos cos cos2 cos2

2 2 2outv t A B A t B t AB t AB t A t B tα α ω α ω α ω ω α ω ω α ω α ω = + + + + − + + + − −

( ) 1 2sin sininv t A t B tω ω= +

• If the amplitude of only one input signal changes, the harmonic of the changing signalwill change by twice as many dB as does the input, but the other harmonic will beunaffected. The IM amplitudes change by the sum of the changes in the two inputsignals; so, if only one fundamental changes, the IMs will change by the same amount.

2IIP1dBIP

2OIP

1dBOP

1f 2f

,i AP

,i BP

2 1f f−0 1f 2f 12 f 22 f1 2f f+

, , 1o A i AP P G= + , , 1o B i BP P G= +

δ

δ

2δ

δδ

26/49

Half-IF Interference (I)

• Input signal with two sinusoidal signals at f2 and f2/2

( ) 2 2

1sin sin

2inv t A t B tω ω= +

( )2

1 2 2 2 2 2

1 1sin sin sin

2 2outv t A t B t B tα ω ω α ω ω = + + +

( )2 2 22 1 2 2 2 1 2 2 2 2 2

1 1 1 1 3sin cos sin cos cos

2 2 2 2 2A B A t AB t B t A t AB tα α ω α ω α ω α ω α ω = + + + + − +

Out-of-band Distortion2nd-order effect

In-band Distortion2nd-order effect


DC Offset2nd-order effect


2 1f f−

0 21 2

ff = 2f 22 f1 2f f+

12 f

27/49

Half-IF Interference (II)

2IIP1dBIP

2OIP

1dBOP

2

1

2f 2f

,i AP,i BP

2 1f f−0 1f 2f 12 f 22 f1 2f f+

, , 1o A i AP P G= +

, , 1o B i BP P G= +

2

1

2f 2f 22 f

,o AP

,o BP


21 2

ff ≠

21 2

ff =

28/49

Half-IF Rejection

•

whereS is the sensitivity or minimum detectable power,CR is the capture ratio,which is the ratio of the desired signal and the second-order distortion when thereceiver fails to demodulate the signal.

( )1Half-IF Rejection 2

2IIP S CR= − −


2IIP1dBIP

2OIP

1dBOP

1G

CR

S

( )2out cP f

( )out cP f

( ) ( ) dBmin cP f

Half-IF rejection (IMR)

2IIP S−

2IIP S CR− −

29/49

Measurement of IP2 (I)

• Mixer: use single-tone cw test

( )2 dBmIFOIP P= ∆ +

( )12 2 dBmRFIIP OIP G P= − = ∆ +LOf RFf

RFPLOP

IFP

IFf 2 IFf

( ) dB∆


Spectrum Analyzer

30/49

Measurement of IP2 (II)

• Amplifier : use two-tone cw test

( ) ( ), ,

12 3 dBm

2 A B o A o BOIP P P= ∆ + ∆ + + +

( )12 2 dBmIIP OIP G= −

,i AP ,i BP

1f 2f

2 1f f−0 1f 2f 12 f 22 f1 2f f+

,o AP

,o BP

A∆B∆


Signal GeneratorCombiner

DUT

Spectrum Analyzer

31/49

Third-Order Nonlinear Effect (I)

• Consider only the first-order and the third-order effect of anonlinear device, i.e., .

• Single-tone excitation:The input signal contains only a sinusoidal signal , where its availablepower can be obtained as .

• In-band and out-of-band distortionsThe output voltage becomes

31 3out in inv v vα α= +

1cosiv A tω=( )2 2in inP A Z=


3 31 1 3 1cos cosoutv A t A tα ω α ω= +

3 31 3 1 3 1

3 1cos cos3

4 4A A t A tα α ω α ω = + +

( ) ( )( ) ( )1 3 31 1 1 3 1cos cos3V V t V tω ω= + +

Out-of-band Distortion3rd-order effect

In-band Distortion3rd-order effect


3rd harmonic

32/49

Third-Order Nonlinear Effect (II)

• Gain Compression or Enhancement:At f1, the amplified linear-term signal has been mixed with the third-order term

If α3 < 0 , the linear gain is compressed, otherwise, it is enhanced

( ) 31 1 3 1

3cos

4outv f A A tα α ω = +

3 0α >

( ) ( ) dBmin cP f

3 0α <

1

1


Third-Order Nonlinear Effect (III)

• Two-tone excitation:


( ) 1 2 1 2sin sin , inv t A t B tω ω ω ω= + <

i : DC term

a, b : linear term(desired signal)+inband distortion

c , d : IM3, adjacent band distortion

e, f : 3rd harmonicsg, h : out of band distortion

( ) ( ) ( )31 3out in inv t v t v tα α= +

2 2 3 33 3 1 3 1 1 3 2

3 3 9 9cos cos

2 2 4 4A B AB A A t B B tα α α α ω α α ω = + + + + +

( ) ( )2 2 3 33 1 2 3 2 1 3 1 3 2

3 3 1 1cos 2 cos 2 cos3 cos3

4 4 4 4A B t AB t A t B tα ω ω α ω ω α ω α ω+ − + − + +

( ) ( )2 23 1 2 3 1 2

3 3cos 2 cos 2

4 4A B t AB tα ω ω α ω ω+ + + +

a bi

c d fe

g h

c gfe

d

a b

h

1 22 f f−0 1f 2f 13 f 23 f

1 22 f f+2 12 f f− 1 22f f+

( ) ( )2-toneIMR 2 3 2 3in outIIP P OIP P= ∆ = − = −

∆

34/49

Third-order Intercept Point

10 dB

10

dB

IM3

3rd harmonic

Fundamental

Fundamental input power (dBm)

Ou

tpu

t po

wer

(d

Bm

)

4.7

7 d

B

4.77 dB

OIP3H

OIP3IM

IIP3IM IIP3H

4.77 dB

9.5

4 d

B

• The slopes for the 3rd-order productsare steeper than 2nd-order productssince they represent cubicnonlinearities rather than squares. IMsand harmonics change 3 dB for eachdB change in the inputs andfundamental outputs.

• Since the slopes of the straight linesare known, these crossing points,called intercept points (IPs), definethe 3rd-order productsat low levels.


( ) ( )2-toneIMR dB 2 3 inIIP P= ∆ = −

( )2 3 outOIP P= −

• Intermodulation Ratio (IMR)

∆

35/49

Example

• For an amplifier with 9 dB linear gain and theOIP3IM is at 21dBm, find the output IM3 power when the fundamental inputsignal power for each signal is−4 dBm.

( )13 3 dBmIM IMOIP IIP G= + OIP3IM = 21 dBm

IM3

Fundamental

Fundamental input power in each signal (dBm)

Ou

tpu

t po

wer

(d

Bm

)

IP3IM

5 dBm

16

dB

32

dB

−27 dBm

−4 dBm 12 dBm

(IIP3IM )

( )21 3 9 dBmIMIIP= +

( )3 12 dBmIMIIP =

( ) ( ) ( )3 2 3 dBmIM out c IM out cP P f OIP P f= − −

( ) ( )5 2 21 5 27 dBm= − − = −


Unequal Input Tone Power


( ) 1 2 1 2sin sin , inv t A t B tω ω ω ω= + <

( ) ( ) ( )3 2 2 3 31 3 3 3 1 3 1 1 3 2

3 3 9 9cos cos

2 2 4 4out in inv t v t v t A B AB A A t B B tα α α α α α ω α α ω = + = + + + + +

( ) ( )2 2 3 33 1 2 3 2 1 3 1 3 2

3 3 1 1cos 2 cos 2 cos3 cos3

4 4 4 4A B t AB t A t B tα ω ω α ω ω α ω α ω+ − + − + +

( ) ( )2 23 1 2 3 1 2

3 3cos 2 cos 2

4 4A B t AB tα ω ω α ω ω+ + + +

3IIP1dBIP

3OIP

1dBOP

1f 2f

,i AP

,i BPδ

0 1f 2f 13 f 23 f

, , 1o A i AP P G= + , , 1o B i BP P G= +δ

2δδδ 2δ3δ

37/49

Third-order Intermodulation Rejection

• From triangular A-B-C, we have

• From D-E-F, which has slope of 3, we have

• From the relations, we can obtain

third-order intermodulation rejection

1G S IMR CR x+ + = +

13 3 3 33

xIIP S IMR OIP IIP G

+ − − = = +

( )12 3 2

2IMR IIP S CR= − −

3IIP

3OIP

1dBOP

( ) ( )1 dBminP fS1G

IMR

CR

A

D

B E C F


x

38/49

Measurement of the IP3

• Amplifier : use two-tone cw test

( ), ,

13

2 i A i BOIP P P= ∆ + +

1f 2f

,i AP ,i BP

B∆

1f 2f1 22 f f− 2 12 f f−

A∆

,o AP,o BP

0


Signal GeneratorCombiner

DUT

Spectrum Analyzer

39/49

Relationship Between Products

• IMs may be predictable fromharmonics:IM2s are 6 dB higher than the 2nd-order harmonicsIM3s are 9.54 dB greater than the 3rd-order harmonicsIP3H exceeds the IP3IM by 4.77 dB

• In addition, we may be able to relate the −1-dB compressionlevel to the IP3:

( )3

1 1dB 3 1dBdesired+distorted

desired 1 1dB

3410log 20log 1 dB

A AP

P A

α α

α

+= = − 23

1dB1

30.10875

4A

αα

=

33, 1 3, 3 3,

3

4OIP IM IIP IM IIP IMA A Aα α= = 2 13,

3

4

3IIP IMAαα

=

21dB 1dB

23,

0.10875 9.64 dB3IIP IM IM

A IP

A IIP= = = −

( )1 3 1 9.64 dB 3 10.64 dBdB IM IMOP IIP G OIP= + − − = −


P1dB:

very useful result!

OIP3:

40/49

Cascaded System (I)

• We take a three-stage systemas an example of cascaded IP3and then extend to anN-stage system.

inP 1C 2C 3C

1I 2I ′ 3I ′

3I ′′2I ′′

3I ′′′

1st stage 2nd stage 3rd stage


1G 2G 3G

41/49

Cascaded System (II)

1 1inC P G=

( )3

11 2

13inP G

IIIP

=

2

1 1

1

3

in

C IIP

I P

=

inP1C

1I



1G

42/49

Cascaded System (III)

2 1 2 1 2inC C G P G G= =

( )3

1 22 1 2 2

13inP G G

I I GIIP

′ = =

( ) ( )3 33

1 21 22 2 2

2 23 3inP G GC G

IIIP IIP

′′ = =

3 3 31 2 1 2

2 2 22 13 3

in inP G G P G GI I I

IIP IIP′ ′′= + = + 2

22 2 1

2 1

1

13 3in

C

I GP

IIP IIP

=

+

inP1C

2C

1I 2I ′

2I ′′



1G 2G

43/49

Cascaded System (IV)

3 1 2 3inC P G G G=

( )3

1 23 2 3 32

13inP G G

I I G GIIP

′ ′= =

( )2

23 1 2 1

3 3 3 1 2 33 2 1

1

3 3 3in

G G GI I I I P G G G

IIP IIP IIP

′ ′′= + + = + +

( )3 3

1 23 2 3 32

23inP G G

I I G GIIP

′′ ′′= =

( ) ( )3 3 3 32 3 1 2 3

3 2 2

3 33 3inC G P G G G

IIIP IIP

′′′= =

3 1 2 32 3

1 2 33 2 1 2 1

3 2 1

1

133 3 3

tot in

intot

intot

C C G G G P

P G G GI IG G GP IIPIIP IIP IIP

= = =

+ +

1 2 1

3 2 1

1 1

3 3 3 3tot

G G G

IIP IIP IIP IIP= + +

inP 1C 2C 3C

1I 2I ′ 3I ′

3I ′′2I ′′

3I ′′′



Cascaded System (V)

• IIP3 of a N-Stage System

• The above equation shows that the IIP3 of an inter-stage isreduced by a factor of the previous stage subtotal gain. Itmeans,the back-end stage will enter saturation first.

• OIP3 of a N-Stage System

1

1 1 1 2

1 1 2 3

1 1

3 3 3 3 3

n

kNk

ntot n

GG G G

IIP IIP IIP IIP IIP

−

=

=

= = + + +∏

∑ ⋯


( ) ( )1 2 3 2 3 4 3

1 1 1 1 1 1

3 3 3 3 3 3tot T tot T N N N NOIP G IIP G IIP G G G IIP G G G IIP G IIP= = + + + +

⋅ ⋅ ⋅⋯

⋯ ⋯

( ) ( ) ( )2 3 1 3 4 2 4 5 3

1 1 1 1

3 3 3 3N N N NG G G OIP G G G OIP G G G OIP OIP= + + + +

⋅⋯

⋯ ⋯ ⋯

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Example (I)

• Calculate the cascaded OIP3 of the following stages.


21 dBm+ ∞ 25 dBm+10 dB 3 dB− 10 dB

3OIP

Gain

21 dBm+ ∞ 25 dBm+15 dB 3 dB− 10 dB

3OIP

Gain

stage 1 stage 2 stage3Gain (dB) 10 -3 10

OIP3 (dBm) 21 100 25IIP3 (dBm) 11 103 15Gain (linear) 10 0.5011872 10

OIP3(linear, mW) 125.89254 1E+10 316.22777IIP3(linear, mW) 12.589254 1.995E+10 31.622777

1/IIP3cas (linear) 0.2379221IIP3cas (linear) 4.2030556

IIP3cas (dBm) 6.2356514OIP3cas(dBm) 23.235651


OIP3 (dBm) 21 100 25IIP3 (dBm) 6 103 15Gain (linear) 31.622777 0.5011872 10




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Example (II)


21 dBm+ ∞ 25 dBm+10 dB 3 dB− 10 dB

3OIP

Gain

21 dBm+ ∞ 25 dBm+10 dB 3 dB− 15 dB

3OIP

Gain


OIP3 (dBm) 21 100 25IIP3 (dBm) 11 103 15Gain (linear) 10 0.5011872 10





OIP3 (dBm) 21 100 25IIP3 (dBm) 11 103 10Gain (linear) 10 0.5011872 31.622777

OIP3(linear, mW) 125.89254 1E+10 316.22777IIP3(linear, mW) 12.589254 1.995E+10 10



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Spectrum Regrowth

• How do we estimate ACPR of a modulated RF signal from 2-tone measurement

( )3

2-tone 6 10log dBc4

mACPR IMR

A B

= − + +

where3 2 mod

2 3 2 224 8

mm m m

A

− − = +

2 mod2

4

mm

B

− =

m denotes number of tones


Summary

• In this chapter, 2nd-order and 3rd-order nonlinear effects wereintroduced. These nonlinearities will result in harmonics andintermodulation distortions in frequency domain.

• The distortion can be easily defined using frequency-domainparameters related to signal power. It is easier to qualify thedistortion by frequency components than time-domainwaveforms. The nonlinearities can be described by P1dB andintercept points.

• The cascaded formula was also derived to showthat the IIP3of an inter-stage is reduced by a factor of the previous stagesubtotal gain. It means, the back-end stage will enter saturationfirst.
