Multiband Transceivers - [Chapter 2] Noises and Linearities

Multiband RF Transceiver System
Chapter 2 Noises and Nonlinearities
李健榮助理教授
Department of Electronic Engineering
National Taipei University of Technology

Outline
• Thermal Noise and Noise Temperature
• Noise Temperature Measurement:
 Gain Method
 Y-factor Method
• Noise Figure
• Output Noise Power of Cascaded Circuits
• Nonlinear Effects on an RF Signal
• 1-dB-Compression Point (P1dB)
• Second- and Third-order Intercept Point (IP2, and IP3)
• Nonlinear Effect of a Cascaded System
Department of Electronic Engineering, NTUT2/56

where is Boltzman’s constant
Available Thermal Noise Power
• Thermal Noise:
23
1.380 10 J/Kk 
 
NAP kTBAvailable noise power:
Thermal noise source
,n rmsvR
 KT


Noisy resistor
,n rmsv
Thevenin’s Equivalent Circuit
Noise-free resistor
R
2
, ?n rmsv 
R
R
Matched Load
2
,
2
n rms
NA
v
P kTB
R
 
 
  
,n rmsv
,
2
n rmsv


Available Noise Power
2
, 4n rmsv kTBR
Open-circuited
noise voltage?

where is Boltzman’s constant
Thermal Noise Equivalent Circuits
• Thermal Noise:
23
1.380 10 J/Kk 
 
NAP kTBAvailable noise power:
Thermal noise source
,n rmsv,n rmsvR
 KT


Thevenin’s Equivalent Circuit
Noisy resistor
Noise-free resistor
Norton’s Equivalent Circuit
Noise-free resistor
R
R
2
, 4n rmsv kTBR
,n rmsi
2
,2
,
4
4n rms
n rms
v kTB
i kTBG
R R
 
   
 
2
, 4n rmsv kTBR

Thermal Noise Power Spectrum Density
• Available noise power :
• Thermal Noise at 290 K (17 oC):
Department of Electronic Engineering, NTUT
Ideal
bandpass
filter
B
R
R
,n rmsv
NAP kTB
PSD (W/Hz, or dBm/Hz)
f (Hz)
Bandwidth B (Hz)
kT
Integrate to get noise power
0 0NAP kT BAvailable noise power:
   21
0, 0 4 10 W Hz 174 dBm HzPSDN kT 
   Power spectrum density:
5/56

Equivalent Noise Temperature (I)
• If an arbitrary source of noise (thermal or nonthermal) is
“white”, it can be modeled as an equivalent thermal noise
source, and characterized with an equivalent noise temperature.
• An arbitrary white noise source with a driving-point
impedance of R and delivers a noise power No to a load
resistor R. This noise source can be replaced by a noisy
resistor of value R, at temperature Te (equivalent temperature):
oN
R
Arbitrary
white
noise
source
R
oN
RR
eT
o
e
N
T
kB

6/56

Equivalent Noise Temperature (II)
• How to define the equivalent noise temperature for a two-port
component? Let’s take a noisy amplifier as an example.
• In order to know the amplifier inherent noise No, you may like
to measure the amplifier by using a noise source with 0 K
temperature. Is that possible?
Noisy amplifier
R
oN
aGR
0 KsT 
This means that the output noise No is
only generated from the amplifier.
Noiseless amplifier
R
o a iN G N
aGR
iN
o
i e
a
N
N kT B
G
 i o
e
a
N N
T
kB G kB
 

Gain Method
• Use a noise source with the known noise temperature Ts.
Noiseless amplifier
R
o a iN G N
aGR
i s eN kT B kT B 
sT
eT
Noisy amplifier
R
_o a i o addN G N N 
aGR
i sN kT B
sT
   o a s e a s eN G kT B kT B G kB T T   
o
s e
a
N
T T
G
 
o
e s
a
N
T T
G
 
 Need to know the amplifier power gain Ga.
 Due to the noise floor of the analyzer, the
gain method is suitable for measuring high
gain and high noise devices.

The Y-factor Method
• Use two loads at significantly different temperatures (hot and
cold ) to measure the noise temperature.
• Defined the Y-factor as
1 1a a eN G kT B G kT B 
2 2a a eN G kT B G kT B 
1 2
1
e
T YT
T
Y



11
2 2
1e
e
T TN
Y
N T T

  

R
R
1T
2T
aG
B eT 1N
2N
(hot)
(cold)
 You don’t have to know Ga.
 The Y-factor method is not suitable for measuring a very high noise device, since
it will make to cause some error. Thus, we may like a noise source with high
ENR for measuring high noise devices.
1Y 
 Sometimes, you may need a pre-amplifier to lower analyzer noise for measuring a
low noise device .
9/56

Noise Figure (NF) – (I)
• The amount of noise added to a signal that is being processed
is of critical importance in most RF systems. The addition of
noise by the system is characterized by its noise figure (NF).
• Noise Factor (or Figure) is a measure of the degradation in the
signal-to-noise ratio (SNR) between the input and output:
where Si , Ni are the input and noise powers, and So, No are the output signal
and noise powers
1i i i
o o o
SNR S N
F
SNR S N
    dB 10logNF F
Gain = 20 dB
P (dBm)
Frequency (Hz)
00
60
SNRi = 40 dB
NF = ?
P (dBm)
Frequency (Hz)
80
40
SNRo= 32 dB
72 NF = 8 dB
Noisy Amplifier

Noise Figure (NF) – (II)
• By definition, the input noise power is assumed to be the
thermal noise power resulting from a matched resistor at T0
(=290 K); that is, , and the noise figure is given as
 0
0 0
1 1ei i e
o i
kGB T TSNR S T
F
SNR kT B GS T

    
0iN kT B
  01eT F T 
Noisy
Network
G B eT
R
0T
R
i i iP S N  o o oP S N 
23
1.380 10 J/ Kk 
  
where is Boltzman’s constant0NAP kT B
   21
0 4 10 W Hz 174 dBm HzTN kT 
   
 Use the concept of SNR
 Use the concept of noise only
0 0
0 0 0
1 1o add e e
i
N kGBT N kGBT kGBT T
F
GN GkT B GkT B T
 
     
11/56

Resistive-type Passive Circuits (I)
• The circuit is with a matched source resistor, which is also at
temperature T.
• The output noise power :
• We can think of this power coming from the source resistor
(through the lossy line), and from the noise generated by the
line itself. Thus,
0P kTB
0 addedP kTB GkTB GN  
 
1
1added e
G
N kTB L kTB kT B
G

   
where is the noise generated by the line.addedN
12/56

Resistive-type Passive Circuits (II)
• The lossy line equivalent noise temperature :
• The noise figure is
where T0 denotes room temperature, T is the actual physical temperature (K). Note
that the loss L may depend on frequency.
• Output noise power :
where input thermal noise power
 
1
1e
G
T T L T
G

  
 
0
1 1
T
F L
T
    dB 10logNF F
     dBm dBm dBout inN N L NF  
 WattinN kTB
 dBminN
f
 dBmoutN
f
inN L NF 
13/56

Active Circuits
• An active circuit is with noise figure NF and available gain G.
(Note that NF and G are usually depend on frequency.)
 dBmout inS S G 
 174 10log dBminN B  
 dBmout inN N NF G  
 dBminN
f f
 dBmoutN
BW
 dBminS
f
 dBmoutS
f
BW
 dBmin inS N
f
 dBmout outS N
f
BW
 dBG
 dBNF
14/56

Multiple Stages Cascaded
• Multiple stages cascaded
where Fi is the noise factor and Gi is the available power gain of each stage.
1
1
0
1
1
N
i
i
i
j
j
F
F
G




  

2 3
1
1 1 2 1 2 1
e e eN
eT e
N
T T T
T T
G G G G G G 
    

1eT
1G 2G
2eT eNT
NG
g T addkT G NgkT
1ekT 2ekT eNkT
gkT  T g eTkG T T
eTkT
1 2T NG G G G 
1 1 1g ekT G kT G
 1 1 1 2 2 2g e ekT G kT G G kT G 
 1 2 1 1 2 2g N e N e N eN NkT G G G kT G G kT G G kT G      
1
1 2
0
i
T N j
j
G G G G G


  
  01eT F T 
Cascade System
Equivalent System
  32
1
1 1 2 1 2 1
1 11
1 1 N
N
F FF
F F
G G G G G G 
 
      

1st stage dominate less significant
15/56

Output Noise Power of Cascaded Circuits (II)
• When the noise temperature and gain of each stage are determined,
the overall noise temperature and gain of the whole system can be
obtained.
• Use the following methods to calculate the output noise ,
(1) Cascade Formula
(2) Walk-Through method
(3) Summation method
1 1 dBL 
1 300 KT  
1 300 KT  
3 4 dBL 
2 150 KeT  
2 25 dBG 
4 700 KeT  
4 30 dBG 
50 KsT  
oN
stage1 stage2 stage3 stage4
oN
16/56

Cascade Formula Method
   1 1 11 1.259 1 300 77.7 KeT L T     
   3 3 31 2.512 1 300 453.6 KeT L T     
150 453.6 700
77.7 275.42 K
0.794 0.794 316.23 0.794 316.23 0.398
eTT     
  

   23 21
1.38 10 50 275.42 =4.5 10 Watts Hz= 173.5 dBm Hzs eTk T T  
      
0 173.5 1 25 4 30 dBm HzN      
1 1 dBL 
1 300 KT  
1 300 KT  
3 4 dBL 
2 150 KeT  
2 25 dBG 
4 700 KeT  
4 30 dBG 
50 KsT  
oN
Stage 1 Teff :
Stage 3 Teff :
System equivalent noise temperature and output noise :
17/56

Walk-Through Method – Stage 1
• Calculate the noise signal from stage to stage. At first,
calculate the noise density stage by stage:
 Antenna noise:
 Cable 1 noise:
23 19
1.38 10 50=6.9 10 mW Hz 181.6 dBm HzskT  
     
   1 1 11 1.259 1 300 77.7 KeT L T     
23 19
1 1.38 10 77.7=10.72 10 mW Hz 179.7 dBm HzekT  
     
1 1 dBL 
1 300 KT  
1 300 KT  
3 4 dBL 
2 150 KeT  
2 25 dBG 
4 700 KeT  
4 30 dBG 
50 KsT  
oN
Stage Input A Input B Sum Output Noise Density (dBm/Hz)
1 181.6 179.7 177.5 178.5
2 178.5
18/56

1 1 dBL 
1 300 KT  
1 300 KT  
3 4 dBL 
2 150 KeT  
2 25 dBG 
4 700 KeT  
4 30 dBG 
50 KsT  
oN
 LNA Noise:
23 19
2 1.38 10 150=2.07 10 mW Hz 176.8 dBm HzekT  
     
1 181.6 179.7 177.5 178.5
2 178.5 176.8 174.6 149.6
3 149.6

1 181.6 179.7 177.5 178.5
2 178.5 176.8 174.6 149.6
3 149.6 172.0 149.6 153.6
4 153.6
1 1 dBL 
1 300 KT  
1 300 KT  
3 4 dBL 
2 150 KeT  
2 25 dBG 
4 700 KeT  
4 30 dBG 
50 KsT  
oN
 Cable 2 Noise:
   3 3 31 2.512 1 300 453.6 KeT L T     
23 19
2 1.38 10 453.6=6.26 10 mW Hz 172 dBm HzekT  
     

1 1 dBL 
1 300 KT  
1 300 KT  
3 4 dBL 
2 150 KeT  
2 25 dBG 
4 700 KeT  
4 30 dBG 
50 KsT  
oN
 Gain amplifier noise:
23 19
4 1.38 10 700=9.66 10 mW Hz 170.2 dBm HzekT  
     
1 181.6 179.7 177.5 178.5
2 178.5 176.8 174.6 149.6
3 149.6 172.0 149.6 153.6
4 153.6 170.2 153.5 123.5

Summation Method
• Each noise source is individually taken through the various
gains and loses to the output, and the sum of all output noises
is just the total output noise (Superposition).
 For stage1:
 For stage2:
 For stage3:
 For stage4:
181.6 1 25 4 30 131.6 dBm Hz      
179.7 1 25 4 30 129.7 dBm Hz      
176.8 25 4 30 125.8 dBm Hz     
172 4 30 146 dBm Hz    
170.2 30 140.2 dBm Hz   
1 1 dBL 
1 300 KT  
1 300 KT  
3 4 dBL 
2 150 KeT  
2 25 dBG 
4 700 KeT  
4 30 dBG 
50 KsT  
oN
oN
Noise Contributor Output Noise Density (dBm/Hz)
Environment 131.6
Stage 1 129.7
Stage 2 125.8
Stage 3 146.0
Stage 4 140.2
Total 123.5
22/56

Noise Figure Method
1 1 dBL 
1 300 KT  
1 300 KT  
3 4 dBL 
2 150 KeT  
2 25 dBG 
4 700 KeT  
4 30 dBG 
50 KsT  
oN
Atten1 Amp2 Atten3 Amp4
Gain (dB) -1 25 -4 30
Gain 0.79432823 316.227766 0.39810717 1000
T 300 150 300 700
F 1.26785387 1.51724138 2.56402045 3.4137931
NF (dB) 1.03069202 1.81054679 4.08921484 5.33237197
Cumumlatvie Gain 0.79432823 251.188643 100 100000
Fcas 1.26785387 1.91902219 1.92524867 1.9493866
NFcas (dB) 2.89897976
Gcas (dB) 50
Ni (Ts=50 K) (dBm) -181.611509
No=Ni+Gcas+NFcas -128.7125 Wrong!Since NF is defined@290 K
Fcas=1+(Te/T0)
Te 275.322114
No=Gcas(kTsB+kTeB) 4.4894E-16 -123.47807 Correct!

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.

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 t         
 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
  sin 2in in cv t V f t    2
2in c in in cP f V Z f
 inZ f
 outZ f
         2 3 4
26/56

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 t   
 
 
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

    
 
 120log 10log in c
out in
out c
Z f
P P
Z f
        1 dBmout c in cP f P f G 
 
 1 120log 10log in c
out c
Z f
G
Z f
 
  sin 2in in cv t V f t
         2 3 4
Assume , we have .   in c out cZ f Z f 1 120logG 
27/56

Linear Amplification
   dBmin cP f
1G
1
1
   dBmout cP f
1G
   dBmout cP f
inP
cf
f
f
1out inP P G 
28/56

Third-order Effect
• For a single-tone input signal,
• α3 < 0 gives gain compression phenomenon
• α3 > 0 gives gain enhancement phenomenon
  1cosinv t A t
   3 3
1 1 3 1cos cosoutv t A t A t    
3 3
1 3 1 3 1
3 1
cos cos3
4 4
A A t A t    
 
   
 
Out-of-band Distortion (3rd Harmonic)
3rd-order effect
In-band Distortion
3rd-order effect
Desired Signal
linear effect
     3
1 3out in inv t v t v t  

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
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 t    
 
   
 
α3 < 0
30/56

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
 

  
  
 
 
3
1 1dB 3 1dB
desired+distorted
desired 1 1dB
3
410log 20log 1 dB
A AP
P A
 


  
1
1dB
3
0.145A



 
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
 
 
  
       
   
 
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
 
 
 
  
            
   
 
 
   21
1 1 1
3
17.6 10log 1 dBmdB
out
IP G
R



 
      
 
31/56

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  1dBG
Network Analyzer
Amplifier
Signal Generator
Amplifier
Spectrum Analyzer
33/56

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 t
,1 1cosout ov G A t
,2 2 1cos2outv A t 
,3 3 1cos3outv A t 
Single-tone excitation
Nonlinear Harmonics
1f
f
1f
f
12 f 13 f 14 f
35/56

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
36/56

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
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 t
 
 
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 t        
   
2
22
1 2sin 2 sin 2
2
c c
A
A f t A f t

     
2 2
2 1 2
1 1
sin cos2
2 2
c cA A t A t      
Out-of-band Distortion
2nd-order effect
DC Offset
2nd-order effect
Desired Signal
linear effect
 in cZ f
cf
f
0
38/56

Second-Order Nonlinear Effect (II)
• Two-tone Excitation:   1 2sin sininv t A t B t  
     
2
1 1 2 2 1 2sin sin sin sinoutv t A t B t A t B t        
   2 2
2 1 1 1 2
1
sin sin
2
A B A t B t    
 
     
   2 1 2 2 1 2cos cosAB t AB t              
2 2
2 1 2 2
1 1
cos2 cos2
2 2
A t B t   
 
    
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
39/56

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
  
 
     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

 
 
       
 
 
 
 
2
220log 3 2 dBm 10log
2
in c
in
out c
Z f
P
Z f
   
 
     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
  
Slope of 2
40/56

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     

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 v  
1cosiv A t
 2
2in inP A Z
3 3
1 1 3 1cos cosoutv A t A t    
3 3
1 3 1 3 1
3 1
cos cos3
4 4
A A t A t    
 
   
 
   
   1 3 3
1 1 1 3 1cos cos3V V t V t   
Out-of-band Distortion
3rd-order effect
In-band Distortion
3rd-order effect
Desired Signal
linear effect
3rd harmonic
43/56

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 t  
 
  
 
3 0 
3 0 
1
1

Third-Order Nonlinear Effect (III)
• Two-tone excitation:
  1 2 1 2sin sin ,inv t A t B t     
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 t  
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 t       
     
          
     
   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 t              
   2 2
3 1 2 3 1 2
3 3
cos 2 cos 2
4 4
A B t AB t        
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     

45/56

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)

46/56

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
dBm
16dB
32dB
27 dBm
4 dBm 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    

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 AP
P A
 


   23
1dB
1
3
0.10875
4
A



3
3, 1 3, 3 3,
3
4
OIP IM IIP IM IIP IMA A A   2 1
3,
3
4
3
IIP IMA



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:
48/56

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
49/56

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
50/56

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
51/56

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
    


  
53/56

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
54/56

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
55/56

Summary
• The measuring methods of the equivalent noise temperature (and
thus the NF) are the practical procedure corresponding to the noise
theory. Each method has its own pros and cons.
• The calculation of a cascade system output noise was also
introduced by using cascade formula, walk-through, and output
summation methods.
• Besides, 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.

Multiband Transceivers - [Chapter 2] Noises and Linearities

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