RF Module Design - [Chapter 3] Linearity

RF Transceiver Module Design
Chapter 3 Nonlinear Effects
李健榮助理教授
Department of Electronic Engineering
National 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 from internal
interferences and external interferences.
1) The internal interferences are generated from the nonlinear
effect of its own devices.
2) The external interference are from outside the transceiver
and intercepted by the antenna or EM coupling.
3) Internal distortion is primarily generated from power
amplifier.

Power Amplifier Categories
• Linear Amplifier: Class A, B, AB, and C
Classified in terms of current conduction angle
CEv
,maxCEVkneeV QV
,maxCI
Ci
QI
A
AB
BC
Biased Transistor
Input Matching Output Matching

Linear Amplifier
Normalized DSi
A
C B AB
0 π 2π
tω
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
LRS
t
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−
OutputPower(dBm)
PhaseShift
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 a nonlinear memoryless
device can be represented as
( ) ( ) ( ) ( ) ( )2 3 4
0 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 output
distorted 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 CW input signal with the
voltage ,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
( ) ( ) ( ) ( ) ( )2 3 4
9/49

Small-signal Power Gain (Linear Gain)
• For linear operation
where Pin is the available input power and G1 is the available small-signal
power gain, which equals to
( ) ( )1 1 sin 2out in in cv t v t V tα α π= =
( )
( )
2 2 2 2
2 21
1 1
1 1 1
2 2 2
in cout in in in
out in
out out in out out c
Z fV V V Z
P P
Z Z Z Z Z f
α
α α= = = =
( )
( )120log 10log in c
out in
out c
Z f
P P
Z f
α= + + ( ) ( ) ( )1 dBmout c in cP f P f G= +
( )
( )1 120log 10log in c
out c
Z f
G
Z f
α= +
( ) sin 2in in cv t V f tπ=
( ) ( ) ( ) ( ) ( )2 3 4
( )inv t ( )outv t
Assume , we have .( ) ( )in c out cZ f Z f= 1 120logG α=
10/49

Linear Amplification
( ) ( )dBmin cP f
1G
1
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 3
1 1 3 1cos cosoutv t A t A tα ω α ω= +
3 3
1 3 1 3 1
3 1
cos cos3
4 4
A A t A tα α ω α ω
 
= + + 
 
Out-of-band Distortion (3rd Harmonic)
3rd-order effect
In-band Distortion
3rd-order effect
Desired Signal
linear effect
( )inv t ( )outv t
( ) ( ) ( )3
1 3out in inv t v t v tα α= +

1 dB-Compression Point
• When the input signal becomes stronger, the output signal will
not grow proportionally but with a slower rate. It is a
saturation phenomena.
1 dB
1dBOP
G
1dBIP
( )out cP f
( ) ( )dBmin cP f
1
1
• When the actual output power is 1 dB less than
the linear extrapolated power, it reaches the 1-
dB gain compression point. At this point, the
input power is called the input 1-dB-
compressed power (IP1dB), the output power is
called the output 1-dB-compressed power
(OP1dB) ,and the gain is called the 1-dB-
compressed gain (G1dB).
( ) 3 3
1 3 1 3 1
3 1
cos cos3
4 4
outv 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
3
11 1dB 3 1dB
20
1 1dB
3
4 0.891 10
A A
A
α α
α
−+  
= = 
 
( )
3
1 1dB 3 1dB
desired+distorted
desired 1 1dB
3
410log 20log 1 dB
A AP
P A
α α
α
+
= = −
1
1dB
3
0.145A
α
α
=
( )
2
1dB 1 1
1dB
3 3
1
10log 30 10log 0.0725 30 18.6 10log dBm
2 in in in
A
IP
R R R
α α
α α
  
= + = + = +  
   
( )
2
3
3 31 1dB 3 1dB
1 1
1dB
3 3
3
1 0.05754
10log 30 10log 30 17.6 10log dBm
2 out in out
A A
OP
R R R
α α
α α
α α
  
+      = + = + = + 
   
 
 
( ) ( )21
1 1 1
3
17.6 10log 1 dBmdB
out
IP G
R
α
α
α
 
= + ⋅ = + − 
 
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 spectrum analyzer :
Test various input signal power level to measurement the output power spectral
content 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 will
be harmonics of the fundamental, but, if there are more
signals at input, signals will be produced with frequencies
that are mathematical combinations of the frequencies of the
input signals, called intermodulation products (IMPs) or
intermods. It is instructive to study the results when there are
two input signals (although we will eventually consider large
numbers of signals).
( ) ( ) ( ) ( ) ( )2 3 4

Distortion Characterization (II)
• Characterized by 1-dB gain compression, IPs , 2-tone
intermodulation 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
1f
f
1f
f
12 f 13 f 14 f
18/49

Distortion Characterization (III)
Designed Amplifier
1f 2f
f
1f 2f
f
1 22 f f− 2 12 f f−
1f 2f
f
1 22 f f− 2 12 f f−
1f 2f
f
1 22 f f− 2 12 f f−
IMD from AM/AM distortion
IMD from AM/PM distortion
Two-tone excitation
Nonlinear
IM
Products
• Characterized by 1-dB gain compression, IPs , 2-tone IMDs
19/49

Intercept Points
• The nonlinear properties can be described by the concept of
intercept points (IPs). The input intercept point (IIPn) is a
fictitious input power where the desired output signal
component equals in amplitude the undesired component.
( )out nP f
( )out cP f
( ) ( )dBmin cP f
IIPn1dBIP
OIPn
1dBOP
1 dB
1
1 1
n
OutputPower(dBm)

Second-Order Nonlinear Effect (I)
• Single-tone excitation:
• For the inclusion of only the linear term and the second term,
the output voltage is
( ) sin 2in cv t A tπ=
( )
( )
2
2
in c
in c
A
P f
Z f
=
( ) ( ) ( ) ( ) ( )
22
1 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
22
1 2sin 2 sin 2
2
c c
A
A f t A f t
α
α π α π= + −
2 2
2 1 2
1 1
sin cos2
2 2
c cA A t A tα α ω α ω= + −
Out-of-band Distortion
2nd-order effect
DC Offset
2nd-order effect
Desired Signal
linear effect
( )in cZ f
( )inv t ( )outv t
cf
f
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 2
2 1 1 1 2
1
sin sin
2
A 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 2
2 1 2 2
1 1
cos2 cos2
2 2
A t B tα ω α ω
 
+ − −  
2 1f f−0 1f 2f 12 f 22 f1 2f f+
a b
c
e
d
fg
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:
( )
( ) ( )
( )
( )
1
120log 10log in c
out c in c
out c
Z f
P f P f
Z f
α= + +
( )
( ) ( ) ( )1
1 dBmout c in cP f P f G= +
( )1
outP
Linear Gain
( )
( )
( ) ( )
( )
( )
( )
( )
2
2
2
2 222
2 2 2 2
2 2
1
1 1 12
2
2 2 2 2 2 2 2
in c in c
out c in
out c in c out c out c
A
Z f Z fA
P f P
Z f Z f Z f Z f
α
α α
 
    = = = 
 
( )
( )
( )
2
220log 3 2 dBm 10log
2
in c
in
out c
Z f
P
Z f
α= − + +
( )
( ) ( ) ( )2
22 2 dBmout c in cP f G P f= +
( )
( )
( )
2
2 2dB 20log 3 10log
2
in c
out c
Z f
G
Z f
α= − +
Slope of 2
23/49

Second-Order Intercept Point
6 dB
6dB
IM2
2nd harmonic
Fundamental
Fundamental input power (dBm)
Outputpower(dBm)
6dB
6 dB
• The 2nd-order products increase twice
as fast as the desired fundamental, the
straight lines cross. At the crossing
point, either for the intermod or the
harmonic, the fundamental and the
2nd-order product have equal output
powers.
• Since the slopes of the straight lines
are known, these crossing points,
called intercept points (IPs), define
the 2nd-order products at low levels.
OIP2H
OIP2IM
IIP2IM IIP2H
6 dB
• Typically, the larger of the input or
output intercept points is specified; so
amplifiers use OIPs and mixers use
IIPs. Some may even add the power
of the two fundamentals, increasing
the value of the IP by 3 dB.
6dB

Example
• For an amplifier with 21 dB linear gain and the OIP2H is at 17
dBm, find the output 2nd harmonic power when the
fundamental output signal power is −8 dBm.
( )12 2 dBmH HOIP IIP G= +
OIP2H = 17 dBm
2nd harmonic
Fundamental
Outputpower(dBm)
IP2H
−8 dBm
25dB
25dB
−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 2
2 1 1 1 2 2 1 2 2 1 2 2 1 2 2
1 1 1
sin sin cos cos cos2 cos2
2 2 2
outv 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 signal
will change by twice as many dB as does the input, but the other harmonic will be
unaffected. The IM amplitudes change by the sum of the changes in the two input
signals; 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
1
sin sin
2
inv t A t B tω ω= +
( )
2
1 2 2 2 2 2
1 1
sin sin sin
2 2
outv t A t B t B tα ω ω α ω ω   
= + + +   
   
( )2 2 2
2 1 2 2 2 1 2 2 2 2 2
1 1 1 1 3
sin cos sin cos cos
2 2 2 2 2
A B A t AB t B t A t AB tα α ω α ω α ω α ω α ω
     
= + + + + − +          
2nd-order effect
In-band Distortion
2nd-order effect
Desired Signal
linear effect
DC Offset
2nd-order effect
2 1f f−
0 2
1
2
f
f = 2f
22 f1 2f f+
12 f
27/49

Half-IF Interference (II)
2IIP1dBIP
2OIP
1dBOP
2
1
2
f 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
2
f 2f 22 f
,o AP
,o BP
2
1
2
f
f ≠
2
1
2
f
f =
28/49

Half-IF Rejection
•
where S 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 the
receiver fails to demodulate the signal.
( )
1
Half-IF Rejection 2
2
IIP 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
RFP
LOP
IFP
IFf 2 IFf
( )dB∆
Spectrum Analyzer
30/49

Measurement of IP2 (II)
• Amplifier : use two-tone cw test
( ) ( ), ,
1
2 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 Generator
Combiner
DUT
Spectrum Analyzer
31/49

Third-Order Nonlinear Effect (I)
• Consider only the first-order and the third-order effect of a
nonlinear device, i.e., .
• Single-tone excitation:
The input signal contains only a sinusoidal signal , where its available
power can be obtained as .
• In-band and out-of-band distortions
The output voltage becomes
3
1 3out in inv v vα α= +
1cosiv A tω=
( )2
2in inP A Z=
3 3
1 1 3 1cos cosoutv A t A tα ω α ω= +
3 3
1 3 1 3 1
3 1
cos cos3
4 4
A A t A tα α ω α ω
 
= + + 
 
( ) ( )
( ) ( )1 3 3
1 1 1 3 1cos cos3V V t V tω ω= + +
3rd-order effect
In-band Distortion
3rd-order effect
Desired Signal
linear 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
( ) 3
1 1 3 1
3
cos
4
outv 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 harmonics
g, h : out of band distortion
( ) ( ) ( )3
1 3out in inv t v t v tα α= +
2 2 3 3
3 3 1 3 1 1 3 2
3 3 9 9
cos cos
2 2 4 4
A B AB A A t B B tα α α α ω α α ω
     
= + + + + +     
     
( ) ( )2 2 3 3
3 1 2 3 2 1 3 1 3 2
3 3 1 1
cos 2 cos 2 cos3 cos3
4 4 4 4
A B t AB t A t B tα ω ω α ω ω α ω α ω+ − + − + +
( ) ( )2 2
3 1 2 3 1 2
3 3
cos 2 cos 2
4 4
A B t AB tα ω ω α ω ω+ + + +
a bi
c d fe
g h
c g
fe
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
10dB
IM3
3rd harmonic
Fundamental
Outputpower(dBm)
4.77dB
4.77 dB
OIP3H
OIP3IM
IIP3IM IIP3H
4.77 dB
9.54dB
• The slopes for the 3rd-order products
are steeper than 2nd-order products
since they represent cubic
nonlinearities rather than squares. IMs
and harmonics change 3 dB for each
dB change in the inputs and
fundamental outputs.
• Since the slopes of the straight lines
are known, these crossing points,
called intercept points (IPs), define
the 3rd-order products at 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 the OIP3IM is at 21
dBm, find the output IM3 power when the fundamental input
signal 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)
Outputpower(dBm)
IP3IM
5 dBm
16dB
32dB
−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 3
1 3 3 3 1 3 1 1 3 2
3 3 9 9
cos cos
2 2 4 4
out 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 3
3 1 2 3 2 1 3 1 3 2
3 3 1 1
cos 2 cos 2 cos3 cos3
4 4 4 4
A B t AB t A t B tα ω ω α ω ω α ω α ω+ − + − + +
( ) ( )2 2
3 1 2 3 1 2
3 3
cos 2 cos 2
4 4
A B t AB tα ω ω α ω ω+ + + +
3IIP1dBIP
3OIP
1dBOP
1f 2f
,i AP
,i BP
δ
0 1f 2f 13 f 23 f
, , 1o A i AP P G= + , , 1o B i BP P G= +
δ
2δδδ 2δ
3δ
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 3
3
x
IIP S IMR OIP IIP G
 
+ − − = = +  
( )
1
2 3 2
2
IMR IIP S CR= − −
3IIP
3OIP
1dBOP
( ) ( )1 dBminP f
S1G
IMR
CR
A
D
B E C F
x
38/49

Measurement of the IP3
• Amplifier : use two-tone cw test
( ), ,
1
3
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 Generator
Combiner
DUT
Spectrum Analyzer
39/49

Relationship Between Products
• IMs may be predictable from harmonics:
IM2s are 6 dB higher than the 2nd-order harmonics
IM3s are 9.54 dB greater than the 3rd-order harmonics
IP3H exceeds the IP3IM by 4.77 dB
• In addition, we may be able to relate the −1-dB compression
level to the IP3:
( )
3
1 1dB 3 1dB
desired+distorted
desired 1 1dB
3
410log 20log 1 dB
A A
P
P A
α α
α
+
= = − 23
1dB
1
3
0.10875
4
A
α
α
=
3
3, 1 3, 3 3,
3
4
OIP IM IIP IM IIP IMA A Aα α= = 2 1
3,
3
4
3
IIP IMA
α
α
=
2
1dB 1dB
2
3,
0.10875 9.64 dB
3IIP 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 system as an example of cascaded IP3
and then extend to an N-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
1
1 2
13
inP G
I
IIP
=
2
1 1
1
3
in
C IIP
I P
 
=  
 
inP
1C
1I
1G
42/49

Cascaded System (III)
2 1 2 1 2inC C G P G G= =
( )
3
1 2
2 1 2 2
13
inP G G
I I G
IIP
′ = =
( ) ( )
3 33
1 21 2
2 2 2
2 23 3
inP G GC G
I
IIP IIP
′′ = =
3 3 3
1 2 1 2
2 2 2
2 13 3
in inP G G P G G
I I I
IIP IIP
′ ′′= + = + 2
2
2 2 1
2 1
1
1
3 3
in
C
I G
P
IIP IIP
=
 
+ 
 
inP 1C 2C
1I
2I′
2I′′
1G 2G
43/49

Cascaded System (IV)
3 1 2 3inC P G G G=
( )
3
1 2
3 2 3 32
13
inP G G
I I G G
IIP
′ ′= =
( )
2
2
3 1 2 1
3 3 3 1 2 3
3 2 1
1
3 3 3
in
G G G
I I I I P G G G
IIP IIP IIP
 
′ ′′= + + = + + 
 
( )
3 3
1 2
3 2 3 32
23
inP G G
I I G G
IIP
′′ ′′= =
( ) ( )
3 3 3 3
2 3 1 2 3
3 2 2
3 33 3
inC G P G G G
I
IIP IIP
′′′= =
3 1 2 3
2 3
1 2 33 2 1 2 1
3 2 1
1
1
33 3 3
tot in
intot
in
tot
C C G G G P
P G G GI IG G G
P 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 is
reduced by a factor of the previous stage subtotal gain. It
means, 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
kN
k
ntot n
G
G 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
= + + + +
⋅
⋯
⋯ ⋯ ⋯
45/49

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 stage3
Gain (dB) 10 -3 10
OIP3 (dBm) 21 100 25
IIP3 (dBm) 11 103 15
Gain (linear) 10 0.5011872 10
OIP3(linear, mW) 125.89254 1E+10 316.22777
IIP3(linear, mW) 12.589254 1.995E+10 31.622777
1/IIP3cas (linear) 0.2379221
IIP3cas (linear) 4.2030556
IIP3cas (dBm) 6.2356514
OIP3cas(dBm) 23.235651
Gain (dB) 15 -3 10
OIP3 (dBm) 21 100 25
IIP3 (dBm) 6 103 15
Gain (linear) 31.622777 0.5011872 10
OIP3(linear, mW) 125.89254 1E+10 316.22777
IIP3(linear, mW) 3.9810717 1.995E+10 31.622777
IIP3cas (dBm) 1.2356514
46/49

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
Gain (dB) 10 -3 10
OIP3 (dBm) 21 100 25
IIP3 (dBm) 11 103 15
Gain (linear) 10 0.5011872 10
OIP3(linear, mW) 125.89254 1E+10 316.22777
IIP3(linear, mW) 12.589254 1.995E+10 31.622777
IIP3cas (dBm) 6.2356514
Gain (dB) 10 -3 15
OIP3 (dBm) 21 100 25
IIP3 (dBm) 11 103 10
Gain (linear) 10 0.5011872 31.622777
OIP3(linear, mW) 125.89254 1E+10 316.22777
IIP3(linear, mW) 12.589254 1.995E+10 10
IIP3cas (dBm) 2.3610797
47/49

Spectrum Regrowth
• How do we estimate ACPR of a modulated RF signal from 2-
tone measurement
( )
3
2-tone 6 10log dBc
4
m
ACPR IMR
A B
 
= − +  
+ 
where
3 2 mod
2 3 2 2
24 8
m
m m m
A
 
 − −  = +
2
mod
2
4
m
m
B
 
−  
 =
m denotes number of tones

Summary
• In this chapter, 2nd-order and 3rd-order nonlinear effects were
introduced. These nonlinearities will result in harmonics and
intermodulation distortions in frequency domain.
• The distortion can be easily defined using frequency-domain
parameters related to signal power. It is easier to qualify the
distortion by frequency components than time-domain
waveforms. The nonlinearities can be described by P1dB and
intercept points.
• The cascaded formula was also derived to show that the IIP3
of an inter-stage is reduced by a factor of the previous stage
subtotal gain. It means, the back-end stage will enter saturation
first.

RF Module Design - [Chapter 3] Linearity

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RF Module Design - [Chapter 3] Linearity