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# Multiband Transceivers - [Chapter 2] Noises and Linearities

Noises and Linearities

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### Multiband Transceivers - [Chapter 2] Noises and Linearities

1. 1. 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
2. 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. 3. 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? Department of Electronic Engineering, NTUT3/56
4. 4. 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 Department of Electronic Engineering, NTUT4/56
5. 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 BAvailable noise power:    21 0, 0 4 10 W Hz 174 dBm HzPSDN kT     Power spectrum density: 5/56
6. 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. 7. 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   Department of Electronic Engineering, NTUT7/56
8. 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. 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 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
10. 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. 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 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
12. 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. 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. 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. 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 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
16. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 26. 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 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. 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. 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. 29. 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  inv t  outv t      3 1 3out in inv t v t v t   Department of Electronic Engineering, NTUT29/56
30. 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             α3 < 0 30/56
31. 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. 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. 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. 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. 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. 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. 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. 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. 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 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 Department of Electronic Engineering, NTUT a bg c d e f 39/56
40. 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. 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. 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. 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. 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 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     dBmin cP f 3 0  1 1 Department of Electronic Engineering, NTUT44/56
45. 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 f2 12 f f 1 22f f    2-toneIMR 2 3 2 3in outIIP P OIP P       45/56
46. 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. 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. 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. 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 Department of Electronic Engineering, NTUT 1G 2G 3G 49/56
50. 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. 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. 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 3I2I 3I 1st stage 2nd stage 3rd stage Department of Electronic Engineering, NTUT52/56
53. 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. 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. 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. 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