Multiband Transceivers - [Chapter 2] Noises and Linearities
56
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
Multiband Transceivers - [Chapter 2] Noises and Linearities
1. Multiband RF Transceiver System Chapter 2 Noises and
Nonlinearities Department of Electronic Engineering National Taipei
University of Technology
2. 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
3. where is Boltzmans constant Available Thermal Noise Power
Thermal Noise: 23 1.380 10 J/Kk NAP kTBAvailable noise power:
Thermal noise source ,n rmsvR KT Noisy resistor ,n rmsv Thevenins
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? Department of
Electronic Engineering, NTUT3/56
4. where is Boltzmans constant Thermal Noise Equivalent
Circuits Thermal Noise: 23 1.380 10 J/Kk NAP kTBAvailable noise
power: Thermal noise source ,n rmsv,n rmsvR KT Thevenins Equivalent
Circuit Noisy resistor Noise-free resistor Nortons 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 Department of
Electronic Engineering, NTUT4/56
5. 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 BAvailable noise power: 21 0, 0 4 10 W Hz 174 dBm
HzPSDN kT Power spectrum density: 5/56
6. 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):
Department of Electronic Engineering, NTUT oN R Arbitrary white
noise source R oN RR eT o e N T kB 6/56
7. Equivalent Noise Temperature (II) How to define the
equivalent noise temperature for a two-port component? Lets 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 Department of
Electronic Engineering, NTUT7/56
8. 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. Department of Electronic Engineering,
NTUT8/56
9. The Y-factor Method Use two loads at significantly different
temperatures (hot and cold ) to measure the noise temperature.
Defined the Y-factor as Department of Electronic Engineering, NTUT
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 dont
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
10. 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 Department of Electronic Engineering,
NTUT10/56
11. 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 Department of Electronic Engineering, NTUT 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
Boltzmans 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
12. 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, Department of Electronic Engineering, NTUT
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
13. 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 Department of Electronic Engineering,
NTUT 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
14. Active Circuits An active circuit is with noise figure NF
and available gain G. (Note that NF and G are usually depend on
frequency.) Department of Electronic Engineering, NTUT 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
15. Multiple Stages Cascaded Multiple stages cascaded where Fi
is the noise factor and Gi is the available power gain of each
stage. Department of Electronic Engineering, NTUT 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 NgkT 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
16. 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 Department of
Electronic Engineering, NTUT 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
17. Cascade Formula Method Department of Electronic
Engineering, NTUT 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
stage1 stage2 stage3 stage4 Stage 1 Teff : Stage 3 Teff : System
equivalent noise temperature and output noise : 17/56
18. 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: Department of Electronic
Engineering, NTUT 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 stage1 stage2 stage3 stage4 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
19. Walk-Through Method Stage 2 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 LNA Noise: 23 19 2 1.38 10 150=2.07 10 mW Hz 176.8
dBm HzekT Stage Input A Input B Sum Output Noise Density (dBm/Hz) 1
181.6 179.7 177.5 178.5 2 178.5 176.8 174.6 149.6 3 149.6
Department of Electronic Engineering, NTUT19/56
20. Walk-Through Method Stage 3 Stage Input A Input B Sum
Output Noise Density (dBm/Hz) 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
stage1 stage2 stage3 stage4 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
Department of Electronic Engineering, NTUT20/56
21. Walk-Through Method Stage 4 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 Gain amplifier noise: 23 19 4 1.38 10 700=9.66 10 mW
Hz 170.2 dBm HzekT Stage Input A Input B Sum Output Noise Density
(dBm/Hz) 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 Department of
Electronic Engineering, NTUT21/56
22. 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: Department of
Electronic Engineering, NTUT 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 stage1 stage2 stage3 stage4 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
23. 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 stage1 stage2 stage3
stage4 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! Department of Electronic
Engineering, NTUT23/56
24. 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.
Department of Electronic Engineering, NTUT24/56
25. 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 Department of
Electronic Engineering, NTUT25/56
26. Nonlinear Memoryless Device (II) Coefficients i 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 t 2 2in c
in in cP f V Z f Department of Electronic Engineering, NTUT inZ f
outZ f 2 3 4 0 1 2 3 4out in in in inv t v t v t v t v t 26/56
27. 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 Department of Electronic
Engineering, NTUT 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 Assume , we have . in c out cZ f Z f 1 120logG
27/56
28. 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 Department of Electronic
Engineering, NTUT inv t outv t 28/56
29. Third-order Effect For a single-tone input signal, 3 < 0
gives gain compression phenomenon 3 > 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 inv t outv t 3 1 3out in inv t v t v t
Department of Electronic Engineering, NTUT29/56
30. 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). Department of Electronic
Engineering, NTUT 3 3 1 3 1 3 1 3 1 cos cos3 4 4 outv t A A t A t 3
< 0 30/56
31. 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
Department of Electronic Engineering, NTUT 21 1 1 1 3 17.6 10log 1
dBmdB out IP G R 31/56
32. 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 Department of
Electronic Engineering, NTUT32/56
33. 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 Department of Electronic Engineering, NTUT Network Analyzer
Amplifier Signal Generator Amplifier Spectrum Analyzer 33/56
34. 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 0 1 2 3 4out in in in inv t v t v t v t v t Department of
Electronic Engineering, NTUT34/56
35. 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 Department of Electronic Engineering, NTUT Single-tone excitation
Nonlinear Harmonics 1f f 1f f 12 f 13 f 14 f 35/56
36. 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
Department of Electronic Engineering, NTUT Two-tone excitation
Nonlinear IM Products Characterized by 1-dB gain compression, IPs ,
2-tone IMDs 36/56
37. 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)
Department of Electronic Engineering, NTUT37/56
38. 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 Department of Electronic Engineering,
NTUT in cZ f inv t outv t cf f 0 38/56
39. 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 f0 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 Department of
Electronic Engineering, NTUT a bg c d e f 39/56
40. 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
Department of Electronic Engineering, NTUT 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
41. 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
Department of Electronic Engineering, NTUT41/56
42. 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 Fundamental input power
(dBm) 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 Department of Electronic Engineering,
NTUT42/56
43. 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 Department of Electronic
Engineering, NTUT 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
44. 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 3 < 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 dBmin cP f 3 0 1 1 Department of Electronic Engineering,
NTUT44/56
45. Third-Order Nonlinear Effect (III) Two-tone excitation:
Department of Electronic Engineering, NTUT 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 f2 12 f
f 1 22f f 2-toneIMR 2 3 2 3in outIIP P OIP P 45/56
46. Third-order Intercept Point 10 dB 10dB IM3 3rd harmonic
Fundamental Fundamental input power (dBm) 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. Department of Electronic
Engineering, NTUT 2-toneIMR dB 2 3 inIIP P 2 3 outOIP P
Intermodulation Ratio (IMR) 46/56
47. 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 Department of Electronic Engineering,
NTUT47/56
48. 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 Department of Electronic Engineering,
NTUT P1dB: very useful result! OIP3: 48/56
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 3I2I 3I 1st stage 2nd stage 3rd stage Department
of Electronic Engineering, NTUT 1G 2G 3G 49/56
50. 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 1st stage 2nd stage 3rd stage
Department of Electronic Engineering, NTUT 1G 50/56
51. 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 1st stage 2nd
stage 3rd stage Department of Electronic Engineering, NTUT 1G 2G
51/56
52. 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 3I2I 3I 1st stage 2nd stage 3rd
stage Department of Electronic Engineering, NTUT52/56
53. 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
Department of Electronic Engineering, NTUT 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
54. Example (I) Calculate the cascaded OIP3 of the following
stages. Department of Electronic Engineering, NTUT 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 stage 1 stage 2
stage3 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
1/IIP3cas (linear) 0.7523759 IIP3cas (linear) 1.3291229 IIP3cas
(dBm) 1.2356514 OIP3cas(dBm) 23.235651 54/56
55. Example (II) Department of Electronic Engineering, NTUT 21
dBm 25 dBm 10 dB 3 dB 10 dB 3OIP Gain 21 dBm 25 dBm 10 dB 3 dB 15
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
stage 1 stage 2 stage3 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 1/IIP3cas (linear) 0.5806201 IIP3cas (linear) 1.7222967 IIP3cas
(dBm) 2.3610797 OIP3cas(dBm) 24.36108 55/56
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. Department of
Electronic Engineering, NTUT56/56