Phase-Noise and Amplitude-Noise Measurement of DACs and DDSs Claudio - - PowerPoint PPT Presentation

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Phase-Noise and Amplitude-Noise Measurement of DACs and DDSs Claudio - - PowerPoint PPT Presentation

Dec 17, 2023 22 likes •273 views

Phase-Noise and Amplitude-Noise Measurement of DACs and DDSs Claudio E. Calosso (1) , A. Carolina Crdenas Olaya (1) , Enrico Rubiola (2),(1) (1) INRiM, Torino, Italy (2) FEMTO-ST Institute, Besancon, France Extended version IFCS 2019 Outline

slide-1
SLIDE 1

Phase-Noise and Amplitude-Noise Measurement of DACs and DDSs

Outline

  • Introduction
  • Method
  • Experimental details
  • Results

1

Claudio E. Calosso(1), A. Carolina Cárdenas Olaya(1), Enrico Rubiola(2),(1)

(1) INRiM, Torino, Italy (2) FEMTO-ST Institute, Besancon, France

home page http://rubiola.org

IFCS 2019 Extended version

slide-2
SLIDE 2

Basic DDS Scheme

2

The contents n of the m-bit register is interpreted as a complex number

<{z} ={z} <{z} ={z}

z plane zk zk−1

θk−1 θk

k k

time t = k/νc

quantity digital analog state variable n θ = 2π n D

  • assoc. complex

z = ejθ modulo D = 2m 2π increment N η = 2π N D time k, 0, 1, 2, . . . t = k/νs clock freq. νs

  • utput freq. ν0 = N

D νs

LUT

DAC

D-type register adder

DAC

sin cos carry

  • utput

control word clock

integer: nk = (nk−1 + N) mod D phase: θk = (θk−1 + η) mod 2π

N νs ν0 = N D νs

complex: zk = zk−1 exp(jη)

High resolution D = 248 νs = 1 GHz ∆ν = 3.55 µHz

slide-3
SLIDE 3

High-Speed DACs Have DDSs Inside

3

NCO

11675-002

SDIO SCLK CS IRQ RESET SYNCOUT0– SYNCOUT0+ PROTECT_OUT1 PROTECT_OUT0

DAC PLL SERDES PLL POWER-ON RESET SERIAL I/O PORT CONFIG REGISTERS CLK_SEL PLL_CTRL DACCLK PLL_LOCK SYNCHRONIZATION LOGIC DAC ALIGN DETECT HB1

TXEN0 TXEN1 SERDIN7± VTT SERDIN0±

CLOCK DATA RECOVERY AND CLOCK FORMATTER

SYNCOUT1+ SYNCOUT1–

REF AND BIAS

I120 SYSREF+ SYSREF– SDO

HB3 HB2 DACCLK

OUT3+ OUT3–

INV SINC

fDAC

÷4, ÷8 NCO COMPLEX MODULATION PHASE ADJUST Q-GAIN I-GAIN SYSREF Rx

CLK+ CLK–

MODE CONTROL DACCLK CLK Rx HB3 HB2 HB1 Q-OFFSET I-OFFSET HB1 HB3 HB2 MODE CONTROL HB3 HB2 HB1 FSC FSC

OUT2+ OUT2–

DACCLK

OUT1+ OUT1–

INV SINC

fDAC

÷4, ÷8 NCO COMPLEX MODULATION PHASE ADJUST Q-GAIN I-GAIN Q-OFFSET I-OFFSET FSC FSC

OUT0+ OUT0–

CLOCK DISTRIBUTION AND CONTROL LOGIC PDP1 PDP0

quadrature gain DC offset AD9144 JESD 204 B I Q I Q If no internal NCO

  • Implement the NCO

in FPGA

  • The brute force of the

JESD204B suffices

  • IP NCOs available
  • Minimal NCO not

difficult to implement

A n a l

  • g

D e v i c e s , A D 9 1 4 4 d a t a s h e e t

slide-4
SLIDE 4

The Beast to Kill

4 L(ƒ), dBc (two DDSs)

Y.Gruson

Not a real challenge, but low enough to spend attention Aka, the lowest noise we have seen in a DDS AD9915

slide-5
SLIDE 5

Traditional Measurement Methods

5

NCO NCO

τ1 τ2 τc

FFT analyzer DAC DAC in ref 2 ref 1

PM-noise analyzer

ADC NCO NCO

τ1 τ2 τc

DAC DAC ADC ADC ADC CORDIC CORDIC FFT analyzer in ref

PM-noise analyzer

  • Low output power, ≈0 dBm, the mixer takes

≈10 dBm —> amplifier

  • Some commercial instruments have only one

input (FSWP , E5052A), ref 1 and ref 2 come from internal synthesizers

  • DACs and DDSs have higher resolution and

clock frequency than ADCs

  • At least 100 averages for 10 dB noise rejection,

1000 for 15 dB

  • How long does it take going down to ≤ 1 mHz?

Cross-spectrum and averaging —> known problems Alternative? Saturated mixer Fully digital

slide-6
SLIDE 6

Bridge (Interferometric) AN-PN Measurement

6

Also: Ivanov, IEEE T UFFC 45(6) 1526-1536, Nov 1998 Rubiola, RSI 73(6) 2445-2457, Jun 2002 Rubiola & al, RSI 70(1) 220-225, Jan 1999

v(t) = V0 cos(ωt) + x sin(ωt) − y sin(ωt) φ = x/V0 α = y/V0 Noisy signal Synchronous detection [vΔ 2 cos(ωT)] * hlp = x vΔ(t) = x sin(ωt) − y sin(ωt) Suppressed carrier [vΔ 2 sin(ωT)] * hlp = y Use the detected x and y to estimate AM and PM noise alternate LO path

  • Suppress the carrier
  • Amplify and detect the noise sidebands
  • PN results from (detected noise) / carrier

R u b i

  • l

a & a l , R S I 7 ( 1 ) 2 2

  • 2

2 5 , J a n 1 9 9 9

slide-7
SLIDE 7

Bridge (Interferometric) AN-PN Measurement

Benefits

  • Lowest background noise
  • Amplifier NF —> white noise
  • No up-conversion of near-dc 1/ƒ

with high carrier rejection

  • Low 50-60 Hz pickup due to

Microwave gain before detection

  • The noise in the LO arm is rejected

(amplification allowed)

  • No AM noise pickup, as in the

saturated mixer 7 Annoying

  • Difficult alignment
  • Narrow band setup

Also: Ivanov, IEEE T UFFC 45(6) 1526-1536, Nov 1998 Rubiola, RSI 73(6) 2445-2457, Jun 2002 Rubiola & al, RSI 70(1) 220-225, Jan 1999

alternate LO path

  • Suppress the carrier
  • Amplify and detect the noise sidebands
  • PN results from (detected noise) / carrier

R u b i

  • l

a & a l , R S I 7 ( 1 ) 2 2

  • 2

2 5 , J a n 1 9 9 9

slide-8
SLIDE 8

How Amplifier’s Flicker Works

8

PM AM noise-free amplifier

u(t) v(t) y(t) x(t) input

  • utput

near-dc noise

stopband stopband passband

ν0 Sv(f)

(output)

Su(f)

(input)

PSD

up-conversion

ν

internal input noise

  • No carrier
  • White noise only in the RF region
  • Flicker in the near DC region
  • No RF flicker
  • RF noise sidebands result from up-

conversion

  • AM and PM noise ≈ independent of

carrier power

slide-9
SLIDE 9

Bridge with a Commercial Instrument?

9

Mixer instrument Saturated-mixer instruments

  • Only some give full access to the

mixer

  • Modern instruments use two channels

and correlation

  • One channel cannot be used alone

Digital instruments

  • Nope, extracting the phase requires a

carrier

slide-10
SLIDE 10

Controlled Amount of Residual Carrier

10

AN

PN

PN

not to scale

AN

not to scale

PN Vθ = −j ⇣ V2ejθ/2 − V1e−jθ/2⌘ = Ceα+jψ

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Vη = V2(1 + η/2) − V1(1 − η/2) = Ceα+jψ

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(A) AM and PM amplification (B) AM and PM cross-amplification

V1 = V0eα1+jϕ1 V2 = V0eα2+jϕ2

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θ

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PN AN

V2(1 + η/2)

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V1(1 − η/2)

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V2ejθ

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V1e−jθ

<latexit sha1_base64="VyMVm7eKFNCI1usCwjCMsP/KQgc=">ACLXicdVC7SgNBFJ31GeMjiZY2g0GwMewqvrqAjaWCiUISw+zkrhmdnV1m7gphWez9BVutLfwWC0HEzsLCX3CSKjogYHDOfcy5x4/lsKg6z46I6Nj4xOTuan89MzsXKFYmq+bKNEcajySkT72mQEpFNRQoITjWAMLfQlH/vlu3z+6AG1EpA6xF0MrZKdKBIztFK7WKi3PThJV8+a2AVkWbtYdivuAPQb2XC9nU2Pep9KuVp6u3y5er/fb5eciWYn4kICrlkxjQ8N8ZWyjQKLiHLNxMDMePn7BQalioWgmlg+QZXbZKhwaRtk8hHajfN1IWGtMLfTsZMuya315f/NPrK9oEJlumzYHLdZpRpjo0socOvzOU/kyHwXYrFSpOEBQfhgsSTGi/epoR2jgKHuWMK6FvY/yLtOMoy04b5v7qof+T+prFW+9snZgK3TJEDmySJbICvHIFqmSPbJPaoSThFyTG3Lr3DkPzpPzPBwdcT53FsgPOK8f9Xqrfg=</latexit>

ηV0

<latexit sha1_base64="4X0T0nCkpWMyDGKfcaol4akd7c=">ACJHicdVC7SgNBFJ2NGmN8P7CxGRTBKuxGfHUBG0sF8wATwuzkbjJkdnaZuSuEZX/CVms/wD8QWzuxsPFXdJIoRNEDA4dz7uWeOX4shUHXfXNyU9Mz+dnCXHF+YXFpeWV1rWaiRHOo8khGuEzA1IoqKJACY1YAwt9CXW/fzr069egjYjUJQ5iaIWsq0QgOEMrNZqAjNbabntlxy25I9AJcuB6J4ce9b6UncrG0Pt43HzvL3q5JudiCchKOSGXPluTG2UqZRcAlZsZkYiBnvsy5cWapYCKaVjgJndNcqHRpE2j6FdKRObqQsNGYQ+nYyZNgzv72h+Kc3VLQJTLZLmyOX6zSjTHVohD0YnzOU/kyHwXErFSpOEBQfhwsSTGiw8ZoR2jgKAeWMK6F/R/lPaYZR9tr0Tb3XQ/9n9TKJW+/VL6wFbpkjALZItkj3jkiFTIGTknVcKJDfkltw5986z8+K8jkdztfOvkB5/0TzuqnJg=</latexit>

AN AN PN PN

AN

ηV0

<latexit sha1_base64="4X0T0nCkpWMyDGKfcaol4akd7c=">ACJHicdVC7SgNBFJ2NGmN8P7CxGRTBKuxGfHUBG0sF8wATwuzkbjJkdnaZuSuEZX/CVms/wD8QWzuxsPFXdJIoRNEDA4dz7uWeOX4shUHXfXNyU9Mz+dnCXHF+YXFpeWV1rWaiRHOo8khGuEzA1IoqKJACY1YAwt9CXW/fzr069egjYjUJQ5iaIWsq0QgOEMrNZqAjNbabntlxy25I9AJcuB6J4ce9b6UncrG0Pt43HzvL3q5JudiCchKOSGXPluTG2UqZRcAlZsZkYiBnvsy5cWapYCKaVjgJndNcqHRpE2j6FdKRObqQsNGYQ+nYyZNgzv72h+Kc3VLQJTLZLmyOX6zSjTHVohD0YnzOU/kyHwXErFSpOEBQfhwsSTGiw8ZoR2jgKAeWMK6F/R/lPaYZR9tr0Tb3XQ/9n9TKJW+/VL6wFbpkjALZItkj3jkiFTIGTknVcKJDfkltw5986z8+K8jkdztfOvkB5/0TzuqnJg=</latexit>

η ⌧ 1

<latexit sha1_base64="SZhto5WhuMGk8c2hlUeAoXAX750=">ACJHicdVC7SgNBFJ2NGjW+4gMbm0ERrMJuxFcXsLGMYGIgG8Ls5G4yZHZ2mbkrhCU/Yau1H+AfiK2dWNj4KzpJFT0wMDhnHu5Z06QSGHQdV+d3NT0TH52br6wsLi0vFJcXaubONUcajyWsW4EzIAUCmoUEIj0cCiQMJl0D8d+ZdXoI2I1QUOEmhFrKtEKDhDKzV8QOZL6bWLO27JHYN+Iweud3LoUe9T2alsPN7X3x82q+1VJ+93Yp5GoJBLZkzTcxNsZUyj4BKGBT81kDeZ1oWqpYBKaVjQMP6a5VOjSMtX0K6Vj9vpGxyJhBFNjJiGHP/PZG4p/eSNEmNMNd6o9drMhZapDY+zB5Jyh9Gc6DI9bmVBJiqD4JFyYSoxHTVGO0IDRzmwhHEt7P8o7zHNONpeC7a5r3ro/6ReLn7pfK5rdAlE8yRLbJN9ohHjkiFnJEqRFOJLkmN+TWuXOenGfnZTKacz531skPOG8fdUanhg=</latexit>

θ ⌧ 1

<latexit sha1_base64="J4IHuBVyFZ1KYf5mwCtRj0SfmAg=">ACJnicdVC7SgNBFJ31GeMr0dJmMAhWYVfx1QVsLCOYB2aDzE7uJkNmZ5eZu0JY8gPWtlpZ+DV2ImLjX1gJThIFT0wcDjnXubcEyRSGHTdZ2dqemZ2bj63kF9cWl5ZLRTX6iZONYcaj2WsmwEzIWCGgqU0Ew0sCiQ0Aj6xyO/cQnaiFid4SCBdsS6SoSCM7TSuY89QOZL6V0USm7ZHYN+I3ud7TvUe9TKVm797g/eqlelF05vxOzNMIFHLJjGl5boLtjGkUXMIw76cGEsb7rAstSxWLwLSzceQh3bJKh4axtk8hHavfNzIWGTOIAjsZMeyZ395I/NMbKdqEZrhF/bHLdTakTHVobG+dfGco/ZkOw8N2JlSIig+CRemkmJMR53RjtDAUQ4sYVwLex/lPaYZR9ts3jb3VQ/9n9R3yt5uefUVuiSCXJkg2ySbeKRA1IhJ6RKaoQTRa7JDbl17p0H59F5moxOZ876+QHnNcPU/SpKA=</latexit>

ψ = (ϕ2 − ϕ1)/η

<latexit sha1_base64="+prn9WgaT/AaSBIjs69H4sSoGC0=">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</latexit>

ψ = (α2 − α1)/θ

<latexit sha1_base64="dM1WH+tPtl1ebdGmqQxcm8sK25M=">ACPXicdVBaxNBGJ2tbRamIvRQ8dLIX24HZmNTaXQqEXDx5aNJAN4RvJ7PJkNnZeZbISw5evKneNWzh/4Kf4C3It68OklasKU+GHi893M915SaOWQsR/B0oPlhyuPVtdqj5+sP31WbzvuLy0QrZFrnPbTcBJrYxso0Itu4WVkCVanifj45l/lFap3JzhpNC9jIYGpUqAeilfn0rLpw63I1BFyPoR6+vCd/bj3EkEfr1bRayd823vEVZ2GS8xZueRE3OWER5yObYPto8/X354cXnk34jWIkHuSgzaVBocO6CswJ7FVhUQstpLS6dLECMYSgvPDWQSder5kmdMcrA5rm1j+DdK7+u1FB5twkS/xkBjhyd72ZeK83U6xL3XSHxnNX2GpKwQxo7nMuvnOU3r4O01avUqYoURqxOC4tNcWczqkA2WlQD3xBIRVPh8VI7Ag0Bde83d1EP/TzpRyN+E0amvkJEFVslL8orsEk4OyBF5T05ImwjyiXwhX8m34HvwM7gKfi1Gl4LrnQ1yC8Gfv8J/sAM=</latexit>

✏ = (↵2 − ↵1)/⌘

<latexit sha1_base64="LNwhWIPkjRPhFoKZoreytQAowLY=">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</latexit>

✏ = ('2 − '1)/✓

<latexit sha1_base64="6dG+ZVh21eVPYIauLZBm48Q1RZ4=">ACQ3icdVDBShxBFOwx0eiqyRovIbk0yoIesulZSaKHgOAlhxwUsio4y/Km943b2NM9dL8RlmG/wK/JNTmGfES+wYuI10B6ZxWixIKGouo9+lWlhVaehPgdzTx5Ojv3bH6hsbi0/PxFc+Xlobelk9iVlt3nIJHrQx2SZHG48Ih5KnGo/Rsb+IfnaPzypqvNCqwl8OpUZmSQEHqN1sJFl5paz5tJOfgiqHqd97esXjzXUJDJOg310X7vYh3Pgu2qJGTbjrZjHt8r67quDq59fXl/s91eiuWRgZmjIanB+5NYFNSrwJGSGseNpPRYgDyDUzwJ1ECOvlfVeca8FZQBz6wLzxCv1X83Ksi9H+VpmMyBhv6hNxH/60U5zM/bvGkdqWrxhzMgNuQc/qd5/z+dZRt9yplipLQyOlxWak5WT4plA+UQ0l6FAhIp0I+LofgQFKovRGau6uHP04O+14q905CBUKNsU8e8PW2AaL2Ue2yz6zfdZlkl2wb+w7+xH9i6j6+hmOjoT3e6snuI/vwFg8ey4Q=</latexit>

jθV0

<latexit sha1_base64="ZUJe6VscVYfgydsbZeyBa0mkuZM=">ACJ3icdVDLSgMxFM34tr7qAzdugqXgqsxUfO0KblxWsGOhLSWTZtpoJhmSO0IZ+hlude3WPxA/wJ3o0i/RdKrQih4IHM65l5x7glhwA67kxNz8zOzS8s5paWV1bX8usbvlGJpqxGlVC6HhDBJesBhwEq8eakSgQ7DK4Ph36lzdMG67kBfRj1opIV/KQUwJWalw1oceAYL/tvMFt+RmwGPkwPVODj3sfSuFytbzo/5tF1trztzY6iScQkUEGMaXhuDK2UaOBUsEGumRgWE3pNuqxhqSQRM60yzARat0cKi0fRJwpo5vpCQyph8FdjIi0DO/vaH4pzdUtAnNoIibmUt1OsBEdrCyt46+MxhPpoPwuJVyGSfAJB2FCxOBQeFhabjDNaMg+pYQqrm9D9Me0YSCrTZnm/upB/9P/HLJ2y+Vz2FLhphAe2gXbSHPHSEKugMVENUaTQLbpD986D8+K8Om+j0Sne2cTcD5+AJ7fKiK</latexit>

jθV0

<latexit sha1_base64="ZUJe6VscVYfgydsbZeyBa0mkuZM=">ACJ3icdVDLSgMxFM34tr7qAzdugqXgqsxUfO0KblxWsGOhLSWTZtpoJhmSO0IZ+hlude3WPxA/wJ3o0i/RdKrQih4IHM65l5x7glhwA67kxNz8zOzS8s5paWV1bX8usbvlGJpqxGlVC6HhDBJesBhwEq8eakSgQ7DK4Ph36lzdMG67kBfRj1opIV/KQUwJWalw1oceAYL/tvMFt+RmwGPkwPVODj3sfSuFytbzo/5tF1trztzY6iScQkUEGMaXhuDK2UaOBUsEGumRgWE3pNuqxhqSQRM60yzARat0cKi0fRJwpo5vpCQyph8FdjIi0DO/vaH4pzdUtAnNoIibmUt1OsBEdrCyt46+MxhPpoPwuJVyGSfAJB2FCxOBQeFhabjDNaMg+pYQqrm9D9Me0YSCrTZnm/upB/9P/HLJ2y+Vz2FLhphAe2gXbSHPHSEKugMVENUaTQLbpD986D8+K8Om+j0Sne2cTcD5+AJ7fKiK</latexit>

Vθ = j ⇣ V2ejθ/2 − V1e−jθ/2⌘ = Ceα+jψ

<latexit sha1_base64="JZnCGCq3QMyAnzNsKf8/axf/I8g=">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</latexit>
slide-11
SLIDE 11

Generalization, for Small η

11

η/2

< l a t e x i t s h a 1 _ b a s e 6 4 = " 2 G A Q O 9 F / C W + e l R m x w U c D d / 4 U N F g = " > A A A B 6 X i c Z Z D N S g M x E M d n / a z 1 q + r R S 7 A V P J S 6 u x 7 Z s G L x w r 2 A 9 q l Z N N s G 5 p k l y Q r l K U P
  • S
d R b 7 6 O L + B b + A h m 2 x 6 H Q j 5 8 Z 8 Z Z v 4 T J p x p 4 7 r f z t r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 T h V h D Z J z G P V C b G m n E n a N M x w 2 k k U x S L k t B 2 O 7 / J 8 + 4 k q z W L 5 a C Y J D Q Q e S h Y x g
  • 2
V O p U e N f j S r / R L Z b f m z g K t g r e A 8 u P z K L R L 3 3 1 B j F J B Z W G c K x 1 1 3 M T E 2 R Y G U Y 4 n R Z 7 q a Y J J m M 8 p F 2 L E g u q g 2 y 2 7 x S d W 2 W A
  • l
j Z J w 2 a q X 8 7 M i y w G Z U t X 9 e
  • X
P Q E x F W Q 1 H N E p H e m m G i W 6 C j M k k N V S S + Y g
  • 5
c j E K L e N B k x R Y v j E A i a K 2 S R G W G F i b H H K V r 7 3 r L Z V W j 5 N e + q 5 j / 4 5 b
  • 7
v w M U 4 B T O 4 A I 8 u I Y 6 3 E M D m k C A w w u 8 w 4 c z d p 6 d V + d t X r r m L H p O 4 F 8 4 n 7 9 q e
  • 4
b < / l a t e x i t >

η/2

< l a t e x i t s h a 1 _ b a s e 6 4 = " 2 G A Q O 9 F / C W + e l R m x w U c D d / 4 U N F g = " > A A A B 6 X i c Z Z D N S g M x E M d n / a z 1 q + r R S 7 A V P J S 6 u x 7 Z s G L x w r 2 A 9 q l Z N N s G 5 p k l y Q r l K U P
  • S
d R b 7 6 O L + B b + A h m 2 x 6 H Q j 5 8 Z 8 Z Z v 4 T J p x p 4 7 r f z t r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 T h V h D Z J z G P V C b G m n E n a N M x w 2 k k U x S L k t B 2 O 7 / J 8 + 4 k q z W L 5 a C Y J D Q Q e S h Y x g
  • 2
V O p U e N f j S r / R L Z b f m z g K t g r e A 8 u P z K L R L 3 3 1 B j F J B Z W G c K x 1 1 3 M T E 2 R Y G U Y 4 n R Z 7 q a Y J J m M 8 p F 2 L E g u q g 2 y 2 7 x S d W 2 W A
  • l
j Z J w 2 a q X 8 7 M i y w G Z U t X 9 e
  • X
P Q E x F W Q 1 H N E p H e m m G i W 6 C j M k k N V S S + Y g
  • 5
c j E K L e N B k x R Y v j E A i a K 2 S R G W G F i b H H K V r 7 3 r L Z V W j 5 N e + q 5 j / 4 5 b
  • 7
v w M U 4 B T O 4 A I 8 u I Y 6 3 E M D m k C A w w u 8 w 4 c z d p 6 d V + d t X r r m L H p O 4 F 8 4 n 7 9 q e
  • 4
b < / l a t e x i t >

B C A

γ1

<latexit sha1_base64="w0SV1uasOZla5S7ghfwo/wJK7g=">AB63icZDNSgMxEMdn/az1q+rRS7AVPJSyWw96s+DFYwX7ge1Ssm2DU2yS5IVytKn0JOoN9/GF/AtfASz2x60HQj58Z8Zv4TxJxp47rfztr6xubWdmGnuLu3f3BYOjpu6yhRhLZIxCPVDbCmnEnaMsxw2o0VxSLgtBNMbrN854kqzSL5YKYx9QUeSRYygo2VHiv9ERYCD7zKoFR2a24eaBW8BZRvfiCP5qD01R9GJBFUGsKx1j3PjY2fYmUY4XRW7CeaxphM8Ij2LEosqPbTfOMZOrfKEIWRsk8alKt/O1IstMBmXLV/VqEz0FMRVANRzRJKh3phgmv/ZTJODFUkvmIMOHIRCgzjoZMUWL41AImitktERljhYmx5yla+96y2Vo12veZa1+Xy83PkdoACncAYX4MEVNOAOmtACAhJe4B0+HOE8O6/O27x0zVn0nMC/cD5/AS0Ejyc=</latexit>

γ2

<latexit sha1_base64="IBkCgELn+RqUpuFZgENYKGAkyA8=">AB63icZDNSgMxEMdn/az1q+rRS7AVPJSyux70ZsGLxwr2A9tSsm2DU2yS5IVytKn0JOoN9/GF/AtfASz2x60HQj58Z8Zv4TxJxp47rfztr6xubWdmGnuLu3f3BYOjpu6ShRhDZJxCPVCbCmnEnaNMxw2okVxSLgtB1MbrN8+4kqzSL5YKYx7Qs8kixkBsrPVZ6IywEHviVQans1tw80Cp4Cyjf/EAejUHpqzeMSCKoNIRjrbueG5t+ipVhNZsZdoGmMywSPatSixoLqf5hvP0LlVhiMlH3SoFz925FioQU246r9swqdgZ6KoBqIapZQOtRLM0x43U+ZjBNDJZmPCBOTIQy42jIFCWGTy1gopjdEpExVpgYe56ite8tm12Fl/zLmv+vV+u/M7QAFO4QwuwIMrqMdNKAJBCS8wDt8OMJ5dl6dt3npmrPoOYF/4Xz+Ai6Djyg=</latexit>

O

(1 − β1)e−jγ1

<latexit sha1_base64="0W3BYJzOcDNtqG8td6M9owmPZxM=">AB/nicZVDLSsNAFL3xWeur6rKbYBUaknqQncW3LisYFvBxDKZ3tSxM0mYmQglFPRndCXqzo/wB/wLP8FJ24WPA8OcOfde7pwTJwp7Tif1szs3PzCYmGpuLyurZe2thsqziVFs05rG8DIhCziJsaY5XiYSiQg4doLBaV7v3KFULI4u9DBX5B+xEJGiTZSt1Te2XMPvA16br7eJ0d3Hp9IoR5jXa6pYpTc8aw/xN3SionXzBGs1v68HoxTQVGmnKi1JXrJNrPiNSMchwVvVRhQuiA9PHK0IgIVH42NjGyd43Ss8NYmhNpe6z+nMiIUILom6q58w6VEzUQTUQ1bwgVaj+7NDhsZ+xKEk1RnSyIky5rWM7z8LuMYlU86EhEpmfmnTGyIJ1SaxorHv/jX7n7TrNfewVj+vVxrVSQ5QgDJswx64cAQNOIMmtIDCAzBK7xZ9aj9Wy9TFpnrOnMFvyC9f4Nt9qVhw=</latexit>

( 1 + β

2

) e

j γ

2

<latexit sha1_base64="bkf7cmQXzb2CjU0+8Lv1Lrom/o=">AB/XicZVDLTsMwENzwprwK3OASUZBAVFVSDnADiQtHkChFakrluJvW1E4i20Gqogp+Bk4IuPET/AB/wSfgpD1AO5Ll8eyu1jN+zJnSjvNtTU3PzM7NLywWlpZXVteK6xs3KkokxRqNeCRvfaKQsxBrmOt7FEInyOdb93ntXrDygVi8Jr3Y+xKUgnZAGjRBupVdza3XcPR81aVUP8C69zpECPMY7LaKJafi5LAniTsipdMfyHZKn57YgmAkNOVGq4TqxbqZEakY5DgpeojAmtEc62DA0JAJVM809DOw9o7TtIJLmhNrO1b8TKRFKEN0tmzvrUBlRfeGXfVHOClIFamyHDk6aKQvjRGNIhyuChNs6srMo7DaTSDXvG0KoZOaXNu0Sag2gRWMfXfc7CS5qVbco0r1qlo6Kw9zgAXYh3YBxeO4Qwu4BJqQOEJXuAdPqxH69l6td6GrVPWaGYT/sH6/AVI4pVQ</latexit>

θ − γ2

<latexit sha1_base64="taPK0W9HTI4Me7eil+P4WKpLGg=">AB9HicZDNTsJAEMen+IX4BXr0ogmHpC0eNCbJF48YiJoIoRMly1s2G2b3amGEN9ET0a9+S+gG/hI7gFDgqTbPaX/8xkZv5BIoUhz/t2ckvLK6tr+fXCxubW9k6xtNsycaoZb7JYxvouQMOliHiTBEl+l2iOKpD8NheZvnbB6NiKMbGiW8o7AfiVAwJCt1i6XDNg04Um7j0pht3bYLZa9qjcJdxH8GZQvfmASjW7xq92LWap4REyiMfe+l1BnjJoEk/yp0E4NT5ANsc/vLUaouOmMJ6s/uUdW6blhrO2LyJ2ofzvGqIxCGlTsn1WYDMxIBZVAVbKENqGZm0HheWcsoiQlHrHpiDCVLsVu5oDbE5ozkiMLyLSwW7psgBoZWZ8K9nx/thFaNWq/m1dl0r172pD5CHfTiAY/DhDOpwBQ1oAoNHeIF3+HAenGfn1XmbluacWc8e/Avn8xezGpI+</latexit>

θ + γ1

<latexit sha1_base64="zjVr89CXfG7gcg2mxmiILc1wB3M=">AB9HicZBLSgNBEIZrfBtfoy7dDCaCYAgzcaE7BTcuFcwDkhBqOj1JY/fM0F0TCSE30ZWoO0/iBbyFR7AnycJHQdMf1VRVX+YSmHI9z+dhcWl5ZXVtfXCxubW9o67u1c3SaYZr7FEJroZouFSxLxGgiRvpqjCiVvhPdXeb4x5NqIJL6jUco7CvuxiARDslLX3S21acAJT9p9VAq7QanrFv2KPw3vPwRzKF58wTRu5Hu5ewTPGYmERjWoGfUmeMmgSTfFJoZ4anyO6xz1sWY1TcdMbT1SfekV6XpRo+2LypurPjEqo5AGZfvnFSYHM1JhOVTlPKFNZP7MoOi8MxZxmhGP2WxElEmPEi93wOsJzRnJkQVkWtgtPTZAjYysTwV7fvD32P9Qr1aC0r1tlq89Gc+wBocwCEcQwBncAnXcAM1YPAT/AKb87QeXSenZdZ6YIz79mHX+G8fwOujZI7</latexit>

<

<latexit sha1_base64="sduLxP7BnzwNeE/boEyew0Qbyj4=">AB5nicZDNTsJAEMen+IX4hXr0gmHghp0URvknjxiMYCTRku2xhw+62d2akIZX0JNRbz6QL+Bb+AhuCweFSTb7y39mMvOfIGZUacf5tgpr6xubW8Xt0s7u3v5B+fCoraJEYuLhiEWyGyBFGBXE01Qz0o0lQTxgpBNMbrN854lIRSPxqKcx8TkaCRpSjLSRvGr/gVQH5YpTd/KwV8FdQOXmB/JoDcpf/WGE06Exgwp1XOdWPspkpiRmalfqJIjPAEjUjPoECcKD/Nl53Z0YZ2mEkzRPaztW/HSniM9rpk/q1AZqCkPagGvZQmpQrU0Q4fXfkpFnGgi8HxEmDBbR3bm2R5SbBmUwMIS2q2tPEYSYS1uUzJ2HeXza5Cu1F3L+qN+0aleTm/AxThBE7hHFy4gibcQs8wEDhBd7hwxpbz9ar9TYvLViLnmP4F9bnL6KtjR0=</latexit>

=

<latexit sha1_base64="QRCNIRDHfsl8u2wa06r2grSEq1c=">AB5nicZDNSsNAEMcn9avWr6pHL8FW8FBKUgW9WfCitwqmLbShbLabdunuJuxuhBL6CnoS9eYD+QK+hY/gJu1B24Flf/xnhpn/BDGjSjvOt1VYW9/Y3Cpul3Z29/YPyodHbRUlEhMPRyS3QApwqgnqakW4sCeIBI51gcpvlO09EKhqJRz2Nic/RSNCQYqSN5FX797w6KFecupOHvQruAio3P5BHa1D+6g8jnHAiNGZIqZ7rxNpPkdQUMzIr9RNFYoQnaER6BgXiRPlpvuzMPjPK0A4jaZ7Qdq7+7UgRVxzpc38WYXKQE15UAt4LUtIFaqlGTq89lMq4kQTgecjwoTZOrIz/aQSoI1mxpAWFKzpY3HSCKszWVKxr67bHYV2o26e1FvPDQqzcv5HaAIJ3AK5+DCFThDlrgAQYKL/AOH9bYerZerbd5acFa9BzDv7A+fwGhJY0c</latexit>
  • 1
<latexit sha1_base64="VrGMJ5Gdxw2ZamhlCHQCNtZ7yMs=">AB53icZBLTsMwEIYn5VXKq8CSjUWLxKJUSVjAjkpsWBaJPqQmqhzXaU3tJLIdpKrqGWCFgB34QLcgiPgpF1AO5LlT/MaOafIOFMadv+tgpr6xubW8Xt0s7u3v5B+fCoreJUEtoiMY9lN8CKchbRlma024iKRYBp51gfJvlO09UKhZHD3qSUF/gYcRCRrA2UvC86pOtV+u2HU7D7QKzgIqNz+QR7Nf/vIGMUkFjThWKmeYyfan2KpGeF0VvJSRNMxnhIewYjLKjyp/m2M3RmlAEKY2lepFGu/u2YqE1qOa+bMKlYGaiKAWiFqWkCpUSzN0eO1PWZSkmkZkPiJMOdIxykyjAZOUaD4xgIlkZktERlhios1pSsa+s2x2Fdpu3bmsu/dupeHO7wBFOIFTOAcHrqABd9CEFhB4hBd4hw+LWc/Wq/U2Ly1Yi5j+BfW5y/MpI0o</latexit>

η/2

<latexit sha1_base64="2GAQO9F/CW+elRmxwUcDd/4UNFg=">AB6XicZDNSgMxEMdn/az1q+rRS7AVPJS6ux70ZsGLxwr2A9qlZNsG5pklyQrlKUPoSdRb76OL+Bb+Ahm2x60HQj58Z8Zv4TJpxp47rfztr6xubWdmGnuLu3f3BYOjpu6ThVhDZJzGPVCbGmnEnaNMxw2kUxSLktB2O7/J8+4kqzWL5aCYJDQeShYxgo2VOpUeNfjSr/RLZbfmzgKtgreA8u0PzKLRL31BjFJBZWGcKx13MTE2RYGUY4nRZ7qaYJmM8pF2LEguqg2y27xSdW2WAoljZJw2aqX87Miy0wGZUtX9eoXPQExFWQ1HNE0pHemGiW6CjMkNVS+Ygo5cjEKLeNBkxRYvjEAiaK2S0RGWGFibHKVr73rLZVWj5Ne+q5j/45bo7vwMU4BTO4AI8uIY63EMDmkCAwu8w4czdp6dV+dtXrmLHpO4F84n79qeo4b</latexit>

V∆ = ηejθ

<latexit sha1_base64="Xnir8WFRJal/fExzbBHoGf/Z5ls=">AB/nicZDJSgNBEIZr3I1b1KOXxih4CGEmHvQiCnrwqGCi4MTQ06kxbXoWumuEMAT0ZfQk6s2H8AV8Cx/BnsSDyw9Nf9RC1V9BqQh1/1wxsYnJqemZ2ZLc/MLi0vl5ZWmSTItsCESleiLgBtUMsYGSVJ4kWrkUaDwPOgdFvnzW9RGJvEZ9VNsRfw6lqEUnGyoXV7baLb9I1TE93wkzvAqv/Gpa3Gw0S5X3Jo7FPsP3jdU9j9hqJN2+d3vJCKLMCahuDGXnptSK+eapFA4KPmZwZSLHr/GS4sxj9C08qGJAdu0kQ4LE21fTGwY/dmR8hEnLpV+xcVpgDTj4JqEFWLhDah+TODwt1WLuM0I4zFaESYKUYJK27BOlKjINW3wIWdksmulxzQfZiJWvf+2v2PzTrNW+7Vj+tVw7qozvADKzBOmyBztwAMdwAg0QcA+P8AKvzp3z4Dw5z6PSMe7ZxV+yXn7AltRlpo=</latexit>

PM1 AM1

θ

<latexit sha1_base64="yO1WSrJ1TBwulQ/1aSPJB6jp4TU=">AB6XicZBLSgNBEIZr4ivGV9Slm8ZEcBHCzLjQnQE3LiOYByRD6On0JE16HnTXCHkELoSded1vIC38Aj2TLQpKDpj7+qPrLT6TQaNvfVmFjc2t7p7hb2ts/ODwqH5+0dZwqxlslrHq+lRzKSLeQoGSdxPFaehL3vEnd1m+8SVFnH0iNOEeyEdRSIQjKRutU+jnS6qBcset2HmQdnCVUbn8gj+ag/NUfxiwNeYRMUq17jp2gN6MKBZN8XuqnmieUTeiI9wxGNOTam+X7zsmFUYkiJV5EZJc/dsxo6EOKY5r5s8qdAZ6Gvo1P6xlCaUDvTIDgxtvJqIkR6xYglQRjktkmQ6E4Qzk1QJkSZkvCxlRhuY4JWPfWTW7Dm237lzV3Qe30nAXd4AinME5XID19CAe2hCxhIeIF3+LAm1rP1ar0tSgvWsucU/oX1+QsjY6Y</latexit>

PM2 AM2

DrK rLG = 1 I( Dcos((M) sin((M) sin((M) cos((M)G Dr>( + r>( rA( + rA(G u ) ) u

<latexit sha1_base64="2GAQO9F/CW+elRmxwUcDd/4UNFg=">AB6XicZDNSgMxEMdn/az1q+rRS7AVPJS6ux70ZsGLxwr2A9qlZNsG5pklyQrlKUPoSdRb76OL+Bb+Ahm2x60HQj58Z8Zv4TJpxp47rfztr6xubWdmGnuLu3f3BYOjpu6ThVhDZJzGPVCbGmnEnaNMxw2kUxSLktB2O7/J8+4kqzWL5aCYJDQeShYxgo2VOpUeNfjSr/RLZbfmzgKtgreA8u0PzKLRL31BjFJBZWGcKx13MTE2RYGUY4nRZ7qaYJmM8pF2LEguqg2y27xSdW2WAoljZJw2aqX87Miy0wGZUtX9eoXPQExFWQ1HNE0pHemGiW6CjMkNVS+Ygo5cjEKLeNBkxRYvjEAiaK2S0RGWGFibHKVr73rLZVWj5Ne+q5j/45bo7vwMU4BTO4AI8uIY63EMDmkCAwu8w4czdp6dV+dtXrmLHpO4F84n79qeo4b</latexit> <latexit sha1_base64="2GAQO9F/CW+elRmxwUcDd/4UNFg=">AB6XicZDNSgMxEMdn/az1q+rRS7AVPJS6ux70ZsGLxwr2A9qlZNsG5pklyQrlKUPoSdRb76OL+Bb+Ahm2x60HQj58Z8Zv4TJpxp47rfztr6xubWdmGnuLu3f3BYOjpu6ThVhDZJzGPVCbGmnEnaNMxw2kUxSLktB2O7/J8+4kqzWL5aCYJDQeShYxgo2VOpUeNfjSr/RLZbfmzgKtgreA8u0PzKLRL31BjFJBZWGcKx13MTE2RYGUY4nRZ7qaYJmM8pF2LEguqg2y27xSdW2WAoljZJw2aqX87Miy0wGZUtX9eoXPQExFWQ1HNE0pHemGiW6CjMkNVS+Ygo5cjEKLeNBkxRYvjEAiaK2S0RGWGFibHKVr73rLZVWj5Ne+q5j/45bo7vwMU4BTO4AI8uIY63EMDmkCAwu8w4czdp6dV+dtXrmLHpO4F84n79qeo4b</latexit> <latexit sha1_base64="w0SV1uasOZla5S7ghfwo/wJK7g=">AB63icZDNSgMxEMdn/az1q+rRS7AVPJSyWw96s+DFYwX7ge1Ssm2DU2yS5IVytKn0JOoN9/GF/AtfASz2x60HQj58Z8Zv4TxJxp47rfztr6xubWdmGnuLu3f3BYOjpu6yhRhLZIxCPVDbCmnEnaMsxw2o0VxSLgtBNMbrN854kqzSL5YKYx9QUeSRYygo2VHiv9ERYCD7zKoFR2a24eaBW8BZRvfiCP5qD01R9GJBFUGsKx1j3PjY2fYmUY4XRW7CeaxphM8Ij2LEosqPbTfOMZOrfKEIWRsk8alKt/O1IstMBmXLV/VqEz0FMRVANRzRJKh3phgmv/ZTJODFUkvmIMOHIRCgzjoZMUWL41AImitktERljhYmx5yla+96y2Vo12veZa1+Xy83PkdoACncAYX4MEVNOAOmtACAhJe4B0+HOE8O6/O27x0zVn0nMC/cD5/AS0Ejyc=</latexit> <latexit sha1_base64="IBkCgELn+RqUpuFZgENYKGAkyA8=">AB63icZDNSgMxEMdn/az1q+rRS7AVPJSyux70ZsGLxwr2A9tSsm2DU2yS5IVytKn0JOoN9/GF/AtfASz2x60HQj58Z8Zv4TxJxp47rfztr6xubWdmGnuLu3f3BYOjpu6ShRhDZJxCPVCbCmnEnaNMxw2okVxSLgtB1MbrN8+4kqzSL5YKYx7Qs8kixkBsrPVZ6IywEHviVQans1tw80Cp4Cyjf/EAejUHpqzeMSCKoNIRjrbueG5t+ipVhNZsZdoGmMywSPatSixoLqf5hvP0LlVhiMlH3SoFz925FioQU246r9swqdgZ6KoBqIapZQOtRLM0x43U+ZjBNDJZmPCBOTIQy42jIFCWGTy1gopjdEpExVpgYe56ite8tm12Fl/zLmv+vV+u/M7QAFO4QwuwIMrqMdNKAJBCS8wDt8OMJ5dl6dt3npmrPoOYF/4Xz+Ai6Djyg=</latexit> < l a t e x i t s h a 1 _ b a s e 6 4 = " W 3 B Y J z O c D N t q G 8 t d 6 M 9
  • w
m P Z x M = " > A A A B / n i c Z V D L S s N A F L 3 x W e u r 6 r K b Y B U U a k n q Q n c W 3 L i s Y F v B x D K Z 3 t S x M m Y m Q g l F P R n d C X q z
  • /
w B / w L P 8 F J 2 4 W P A 8 O c O f d e 7 p w T J J w p 7 T i f 1 s z s 3 P z C Y m G p u L y y u r Z e 2 t h s q z i V F F s 5 r G 8 D I h C z i J s a a Y 5 X i Y S i Q g 4 d
  • L
B a V 7 v 3 K F U L I 4 u 9 D B B X 5 B + x E J G i T Z S t 1 T e 2 X M P v A A 1 6 b r 7 e J d 3 H p 9 I
  • R
5 j X a 6 p Y p T c 8 a w / x N 3 S i
  • n
X z B G s 1 v 6 8 H
  • x
T Q V G m n K i 1 J X r J N r P i N S M c h w V v V R h Q u i A 9 P H K I g I V H 4 2 N j G y d 4 3 S s 8 N Y m h N p e 6 z + n M i I U I L
  • m
6 q 5 8 w 6 V E z U U Q T U Q 1 b w g V a j + 7 N D h s Z + x K E k 1 R n S y I k y 5 r W M 7 z 8 L u M Y l U 8 6 E h h E p m f m n T G y I J 1 S a x
  • r
H v / j X 7 n 7 T r N f e w V j + v V x r V S Q 5 Q g D J s w x 6 4 c A Q N O I M m t I D C A z z B K 7 x Z 9 9 a j 9 W y 9 T F p n r O n M F v y C 9 f 4 N t 9 q V h w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " b k f 7 c m Q X z b 2 C j j U + 8 L v 1 L r
  • m
/
  • =
" > A A A B / X i c Z V D L T s M w E N z w p r w K 3 O A S U Z B A V F V S D n A D i Q t H k C h F a k r l u J v W 1 E 4 i 2 G q
  • g
p + B k 4 I u P E T / A B / w S f g p D 1 A O 5 L l 8 e y u 1 j N + z J n S j v N t T U 3 P z M 7 N L y w W l p Z X V t e K 6 x s 3 K k
  • k
x R q N e C R v f a K Q s x B r m m m O t 7 F E I n y O d b 9 3 n t X r D y g V i 8 J r 3 Y + x K U g n Z A G j R B u p V d z a 3 X c P P R 8 1 a V U P 8 C 6 9 9 z p E C P M Y 7 L a K J a f i 5 L A n i T s i p d M f y H H Z K n 5 5 7 Y g m A k N N O V G q 4 T q x b q Z E a k Y 5 D g p e
  • j
A m t E c 6 2 D A J A J V M 8 9 D O w 9
  • 7
T t I J L m h N r O 1 b 8 T K R F K E N t m z v r U B l R f e G X f V H O C l I F a m y H D k 6 a K Q v j R G N I h y u C h N s 6 s r M
  • 7
D a T S D X v G K
  • Z
O a X N u S S a g 2 g R W M f X f c 7 C S 5 q V b c
  • r
1 q l
  • 6
K w 9 z g A X Y h h 3 Y B x e O 4 Q w u 4 B J q Q O E J X u A d P q x H 6 9 l 6 t d 6 G r V P W a G Y T / s H 6 / A V I 4 p V Q < / l a t e x i t > <latexit sha1_base64="taPK0W9HTI4Me7eil+P4WKpLGg=">AB9HicZDNTsJAEMen+IX4BXr0ogmHpC0eNCbJF48YiJoIoRMly1s2G2b3amGEN9ET0a9+S+gG/hI7gFDgqTbPaX/8xkZv5BIoUhz/t2ckvLK6tr+fXCxubW9k6xtNsycaoZb7JYxvouQMOliHiTBEl+l2iOKpD8NheZvnbB6NiKMbGiW8o7AfiVAwJCt1i6XDNg04Um7j0pht3bYLZa9qjcJdxH8GZQvfmASjW7xq92LWap4REyiMfe+l1BnjJoEk/yp0E4NT5ANsc/vLUaouOmMJ6s/uUdW6blhrO2LyJ2ofzvGqIxCGlTsn1WYDMxIBZVAVbKENqGZm0HheWcsoiQlHrHpiDCVLsVu5oDbE5ozkiMLyLSwW7psgBoZWZ8K9nx/thFaNWq/m1dl0r172pD5CHfTiAY/DhDOpwBQ1oAoNHeIF3+HAenGfn1XmbluacWc8e/Avn8xezGpI+</latexit> < l a t e x i t s h a 1 _ b a s e 6 4 = " z j V r 8 9 C X f G 7 g c g 2 m x m i I L c 1 w B 3 M = " > A A A B 9 H i c Z Z B L S g N B E I Z r f B t f
  • y
7 d D C a C Y A g z c a E 7 B T c u F c w D k h B q O j 1 J Y / f M F T C S E 3 Z W
  • O
/ i B b y F R 7 A n y c J H Q d M f f 1 V R V X + Y S m H I 9 z + d h c W l 5 Z X V t f X C x u b W 9
  • 6
7 u 1 c 3 S a Y Z r 7 F E J r
  • Z
  • u
F S x L x G g i R v p p q j C i V v h P d X e b 4 x 5 N q I J L 6 j U c
  • 7
C v u x i A R D s l L X 3 S 2 1 a c A J T 9 p 9 V A q 7 Q a n r F v 2 K P w 3 v P w R z K F 5 8 w T R u u u 5 H u 5 e w T P G Y m E R j W
  • G
f U m e M m g S T f F J
  • Z
4 a n y O 6 x z 1 s W Y 1 T c d M b T 1 S f e k V V 6 X p R
  • +
2 L y p u r P j j E q
  • 5
A G Z f v n F S Y H M 1 J h O V T l P K F N Z P 7 M
  • O
i 8 M x Z x m h G P 2 W x E l E m P E i 9 3 w O s J z R n J k Q V k W t g t P T Z A j Y y s T w V 7 f v D 3 2 P 9 Q r 1 a C r 1 t l q 8 9 G c + w B
  • c
w C E c Q w B n c A n X c A M 1 Y P A A T / A K b 8 7 Q e X S e n Z d Z 6 Y I z 7 9 m H X + G 8 f w O u j Z I 7 < / l a t e x i t > <latexit sha1_base64="sduLxP7BnzwNeE/boEyew0Qbyj4=">AB5nicZDNTsJAEMen+IX4hXr0gmHghp0URvknjxiMYCTRku2xhw+62d2akIZX0JNRbz6QL+Bb+AhuCweFSTb7y39mMvOfIGZUacf5tgpr6xubW8Xt0s7u3v5B+fCoraJEYuLhiEWyGyBFGBXE01Qz0o0lQTxgpBNMbrN854lIRSPxqKcx8TkaCRpSjLSRvGr/gVQH5YpTd/KwV8FdQOXmB/JoDcpf/WGE06Exgwp1XOdWPspkpiRmalfqJIjPAEjUjPoECcKD/Nl53Z0YZ2mEkzRPaztW/HSniM9rpk/q1AZqCkPagGvZQmpQrU0Q4fXfkpFnGgi8HxEmDBbR3bm2R5SbBmUwMIS2q2tPEYSYS1uUzJ2HeXza5Cu1F3L+qN+0aleTm/AxThBE7hHFy4gibcQs8wEDhBd7hwxpbz9ar9TYvLViLnmP4F9bnL6KtjR0=</latexit> <latexit sha1_base64="QRCNIRDHfsl8u2wa06r2grSEq1c=">AB5nicZDNSsNAEMcn9avWr6pHL8FW8FBKUgW9WfCitwqmLbShbLabdunuJuxuhBL6CnoS9eYD+QK+hY/gJu1B24Flf/xnhpn/BDGjSjvOt1VYW9/Y3Cpul3Z29/YPyodHbRUlEhMPRyS3QApwqgnqakW4sCeIBI51gcpvlO09EKhqJRz2Nic/RSNCQYqSN5FX797w6KFecupOHvQruAio3P5BHa1D+6g8jnHAiNGZIqZ7rxNpPkdQUMzIr9RNFYoQnaER6BgXiRPlpvuzMPjPK0A4jaZ7Qdq7+7UgRVxzpc38WYXKQE15UAt4LUtIFaqlGTq89lMq4kQTgecjwoTZOrIz/aQSoI1mxpAWFKzpY3HSCKszWVKxr67bHYV2o26e1FvPDQqzcv5HaAIJ3AK5+DCFThDlrgAQYKL/AOH9bYerZerbd5acFa9BzDv7A+fwGhJY0c</latexit> <latexit sha1_base64="VrGMJ5Gdxw2ZamhlCHQCNtZ7yMs=">AB53icZBLTsMwEIYn5VXKq8CSjUWLxKJUSVjAjkpsWBaJPqQmqhzXaU3tJLIdpKrqGWCFgB34QLcgiPgpF1AO5LlT/MaOafIOFMadv+tgpr6xubW8Xt0s7u3v5B+fCoreJUEtoiMY9lN8CKchbRlma024iKRYBp51gfJvlO09UKhZHD3qSUF/gYcRCRrA2UvC86pOtV+u2HU7D7QKzgIqNz+QR7Nf/vIGMUkFjThWKmeYyfan2KpGeF0VvJSRNMxnhIewYjLKjyp/m2M3RmlAEKY2lepFGu/u2YqE1qOa+bMKlYGaiKAWiFqWkCpUSzN0eO1PWZSkmkZkPiJMOdIxykyjAZOUaD4xgIlkZktERlhios1pSsa+s2x2Fdpu3bmsu/dupeHO7wBFOIFTOAcHrqABd9CEFhB4hBd4hw+LWc/Wq/U2Ly1Yi5j+BfW5y/MpI0o</latexit> < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 G A Q O 9 F / C W + e l R m x w U c D d / 4 U N F g = " > A A A B 6 X i c Z Z D N S g M x E M d n / a z 1 q + r R S 7 A V P J S 6 u x 7 Z s G L x w r 2 A 9 q l Z N N s G 5 p k l y Q r l K U P
  • S
d R b 7 6 O L + B b + A h m 2 x 6 H Q j 5 8 Z 8 Z Z v 4 T J p x p 4 7 r f z t r 6 x u b W d m G n u L u 3 f 3 B Y O j p u 6 T h V h D Z J z G P V C b G m n E n a N M x w 2 k k U x S L k t B 2 O 7 / J 8 + 4 k q z W L 5 a C Y J D Q Q e S h Y x g
  • 2
V O p U e N f j S r / R L Z b f m z g K t g r e A 8 u P z K L R L 3 3 1 B j F J B Z W G c K x 1 1 3 M T E 2 R Y G U Y 4 n R Z 7 q a Y J J m M 8 p F 2 L E g u q g 2 y 2 7 x S d W 2 W A
  • l
j Z J w 2 a q X 8 7 M i y w G Z U t X 9 e
  • X
P Q E x F W Q 1 H N E p H e m m G i W 6 C j M k k N V S S + Y g
  • 5
c j E K L e N B k x R Y v j E A i a K 2 S R G W G F i b H H K V r 7 3 r L Z V W j 5 N e + q 5 j / 4 5 b
  • 7
v w M U 4 B T O 4 A I 8 u I Y 6 3 E M D m k C A w w u 8 w 4 c z d p 6 d V + d t X r r m L H p O 4 F 8 4 n 7 9 q e
  • 4
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rN Os = 1 I D cos(M) sin(M) − sin(M) cos(M)G r-( − -* /( − /*s w ) ) w

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  • m
/
  • =
" > A A A B / X i c Z V D L T s M w E N z w p r w K 3 O A S U Z B A V F V S D n A D i Q t H k C h F a k r l u J v W 1 E 4 i 2 G q
  • g
p + B k 4 I u P E T / A B / w S f g p D 1 A O 5 L l 8 e y u 1 j N + z J n S j v N t T U 3 P z M 7 N L y w W l p Z X V t e K 6 x s 3 K k
  • k
x R q N e C R v f a K Q s x B r m m m O t 7 F E I n y O d b 9 3 n t X r D y g V i 8 J r 3 Y + x K U g n Z A G j R B u p V d z a 3 X c P P R 8 1 a V U P 8 C 6 9 9 z p E C P M Y 7 L a K J a f i 5 L A n i T s i p d M f y H H Z K n 5 5 7 Y g m A k N N O V G q 4 T q x b q Z E a k Y 5 D g p e
  • j
A m t E c 6 2 D A J A J V M 8 9 D O w 9
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  • 7
D a T S D X v G K
  • Z
O a X N u S S a g 2 g R W M f X f c 7 C S 5 q V b c
  • r
1 q l
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slide-12
SLIDE 12

Bridge vs Modulation-Index Amplification

12

ref

coupler

V1 V2 VΣ V∆ DAC DAC DAC

select AM- PM detect

× N

FFT analyzer same ƒ

noise rejected

high ƒs phase digital PN analyzer in ref

coupler

V1 V2 VΣ V∆ DAC DAC

× N

high ƒs FFT analyzer same ƒ arbitrary ƒ

20–50 dB carrier rejection noise rejected

AM detect

lowest residual carrier

synch detect

  • Reference has the same frequency
  • Reference phase selects AM/PM

detection

  • Arbitrary reference frequency
  • Residual carrier selects AM/PM detection
  • Either AM or PM detector
  • Still low W & 1/ƒ noise

Adjust amplitude and phase with the DDS control word Bridge (Interferometer) Modulation-Index Amplification wideband wideband flexible s i m p l e

slide-13
SLIDE 13

Detectors

  • Direct digitization of the RF signal
  • High background noise
  • Heavy correlation and averaging
  • AVG only partially trusted
  • Flexible, ƒin ≠ ƒref
  • Laboratory (FEMTO, Holmes, Miles…)
  • Brand (Jackson Lab, Symmetricom)

13

  • Power detector & spectrum analyzer
  • Simple lab implementation
  • Fair background noise
  • No AN —> PN pollution
  • Optional correlation and averaging
  • No known commercial instrument

All-digital phase meters Simple amplitude detectors

  • E. Rubiola, arXiv:physics/0512082v1, Dec 2005
  • .TXT, .CSV) used

<10 min at +25°C 90-264 VAC, < 20W +35°C operating, -20°C to +50°C storag

Park Run Dr., Suite 100, Las Vegas, x: (702) 233-1073, www.jackson-labs.com sales@jackson-labs.com

10−200 Ω k 100 50 Ω to external

video out rf in

Ω ~60 pF

slide-14
SLIDE 14

Alignment

14

(1) Full carrier suppression (2) Add a small carrier

PN AN

ηV0

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AN PN

jθV0

<latexit sha1_base64="ZUJe6VscVYfgydsbZeyBa0mkuZM=">ACJ3icdVDLSgMxFM34tr7qAzdugqXgqsxUfO0KblxWsGOhLSWTZtpoJhmSO0IZ+hlude3WPxA/wJ3o0i/RdKrQih4IHM65l5x7glhwA67kxNz8zOzS8s5paWV1bX8usbvlGJpqxGlVC6HhDBJesBhwEq8eakSgQ7DK4Ph36lzdMG67kBfRj1opIV/KQUwJWalw1oceAYL/tvMFt+RmwGPkwPVODj3sfSuFytbzo/5tF1trztzY6iScQkUEGMaXhuDK2UaOBUsEGumRgWE3pNuqxhqSQRM60yzARat0cKi0fRJwpo5vpCQyph8FdjIi0DO/vaH4pzdUtAnNoIibmUt1OsBEdrCyt46+MxhPpoPwuJVyGSfAJB2FCxOBQeFhabjDNaMg+pYQqrm9D9Me0YSCrTZnm/upB/9P/HLJ2y+Vz2FLhphAe2gXbSHPHSEKugMVENUaTQLbpD986D8+K8Om+j0Sne2cTcD5+AJ7fKiK</latexit>

initial state (null) AM PM amplification AM PM cross- amplification

  • Sub-binary search by inspection

with a spectrum analyzer

  • Start from phase
  • Alternate phase and amplitude
  • Real: modulation index amplification
  • Imaginary —> AM-PM interchange

and model-index ampli

  • Best results with –20 to –40 dB

DAC 1 D A C 2

reference

≈FSR – 0.5 dB

initial state (random)

DAC 2

null the carrier

slide-15
SLIDE 15

Accuracy

  • The upper bits are spent for carrier

rejection

  • Lower bits determine the accuracy of

the residual carrier

15

20 dB rejection 0.1 —> 3.33 bits Accuracy 12–1–3.33 = 7.67 bits 0.5% —> 0.04 dB Probably not needed for AN / PN

Example, ENOB = 12

40 dB rejection 0.01 —> 6.67 bits Accuracy 12–1–6.67 = 4.33 bits 5% —> 0.4 dB Still OK for AN / PN

sign carrier rejection gain accuracy

MSB LSB

DAC register

Trade accuracy vs carrier (and 1/ƒ) rejection

slide-16
SLIDE 16

The Full Scheme

16

DAC DAC

JESD204B

phase-noise analyzer in ref

A1 B A2 C

V1 V2 VΣ V∆

τ1 τ2

Computer & FPGA FFT analyzer

power detector Agilent 35670A Symmetricom 5125A

Herotek DHM124AA 40dB 1kΩ

monitor DC AD9144 board

150 MHz 150 MHz LT1028

τc

synth

1 GHz

+2 dBm 0 dBm or +10 dBm

–3 dBm 86 mV 8.6 V

att 3 or 13 dB 3 dB coupler 3 dB coupler –56…–26 dBm –52…–22 dBm balun 40 dB

  • r 60 dB

+3 dBm or +13 dBm

VM VD DAC NCO

I Q +2 dBm balun

DAC

I Q balun balun 150 MHz unused

τ'

1

τ'

2

125 MHz

  • utput

D

NCO

τc

Mini-Circuits parts Amplifier ZKL-1R5+, 40 dB gain Amplifier ZFL-500LN, 24 dB gain 3-dB couplers SFSC-2-4-S+ Filter BLP-150+

No amplification Low noise contribution AD9144-FMC-EBZ card Z-Board ZC706 (Zynq -7000 & peripherals)

slide-17
SLIDE 17

Background Noise

  • DAC output +2 dBm
  • Amplifier input –5 dBm 


(7 dB loss DAC—> amplifier)

  • Thermal energy –174 dBm/Hz
  • Ampli noise figure F = 2 dB
  • Sφ = Sα = –167 dBrad2/Hz


(–174+5+2) 17

DAC

A1 B

V1 VΣ V∆

τ1

AD9144 board

150 MHz

τc

+2 dBm 3 dB coupler 3 dB coupler –56…–26 –52…–22 dBm balun 40 dB

  • r 60 dB

+3 +1

τ'

125 MHz

  • utput

D

White PM Noise Flicker PM Noise

∆

3 dB coupler –52…–22 dBm 40 dB

  • r 60 dB
  • Ampli flicker –130 dB at 1 Hz


(conservative)

  • Carrier rejection 30 dB
  • Expected –160 dBrad2/Hz at 1 Hz
  • More severe limitation from the noise

analyzer

slide-18
SLIDE 18

PN Measured with the Phase Detector

18

Phase noise measured with the phase meter (same information) Raw data Accounting for gain

slide-19
SLIDE 19

AN Measured with the Phase Detector

19

Amplitude noise measured with the phase meter (same information) Raw data Accounting for gain

slide-20
SLIDE 20

AN & PN Measured with Power Detector

20

The dispersion of results will be fixed AM Noise PM Noise

slide-21
SLIDE 21

Comparison

21

PN AN

  • Best results with 30 dB gain
  • The results with the two

methods match perfectly

  • Clock jitter contributes to the

difference, PN > AN Gain 30 dB bump

DAC DAC

τ1 τ2

slide-22
SLIDE 22

Exact 1/ƒ PN Over 7.5 Decades

22

  • PN between two outputs

(Microsemi 5125A)

  • Unfortunately, the axes were

misaligned (θ≈60º)

  • Exact 1/ƒ slope, ±1 dB over

7.5 decades

  • Insufficient averaging below

0.7 mHz

AD9144, two channels

7.5 decades 125 MHz out 1 GHz sampling 50 dB gain exact 1/ƒ slope

Fractional noise PSD, dB/Hz

–170 –150 –130 –110 –90 –70 –50

c–1 = 1.5x10–11

slide-23
SLIDE 23

How We Solved the Angle

  • Use trusted 1/ƒ levels for calibration
  • Gain 50 dB
  • Reasonably trusted
  • Consistent lowest-noise point

23

DrK rLG = 1 I( Dcos((M) sin((M) sin((M) cos((M)G Dr>( + r>( rA( + rA(G u ) ) u

<latexit sha1_base64="2GAQO9F/CW+elRmxwUcDd/4UNFg=">AB6XicZDNSgMxEMdn/az1q+rRS7AVPJS6ux70ZsGLxwr2A9qlZNsG5pklyQrlKUPoSdRb76OL+Bb+Ahm2x60HQj58Z8Zv4TJpxp47rfztr6xubWdmGnuLu3f3BYOjpu6ThVhDZJzGPVCbGmnEnaNMxw2kUxSLktB2O7/J8+4kqzWL5aCYJDQeShYxgo2VOpUeNfjSr/RLZbfmzgKtgreA8u0PzKLRL31BjFJBZWGcKx13MTE2RYGUY4nRZ7qaYJmM8pF2LEguqg2y27xSdW2WAoljZJw2aqX87Miy0wGZUtX9eoXPQExFWQ1HNE0pHemGiW6CjMkNVS+Ygo5cjEKLeNBkxRYvjEAiaK2S0RGWGFibHKVr73rLZVWj5Ne+q5j/45bo7vwMU4BTO4AI8uIY63EMDmkCAwu8w4czdp6dV+dtXrmLHpO4F84n79qeo4b</latexit> <latexit sha1_base64="2GAQO9F/CW+elRmxwUcDd/4UNFg=">AB6XicZDNSgMxEMdn/az1q+rRS7AVPJS6ux70ZsGLxwr2A9qlZNsG5pklyQrlKUPoSdRb76OL+Bb+Ahm2x60HQj58Z8Zv4TJpxp47rfztr6xubWdmGnuLu3f3BYOjpu6ThVhDZJzGPVCbGmnEnaNMxw2kUxSLktB2O7/J8+4kqzWL5aCYJDQeShYxgo2VOpUeNfjSr/RLZbfmzgKtgreA8u0PzKLRL31BjFJBZWGcKx13MTE2RYGUY4nRZ7qaYJmM8pF2LEguqg2y27xSdW2WAoljZJw2aqX87Miy0wGZUtX9eoXPQExFWQ1HNE0pHemGiW6CjMkNVS+Ygo5cjEKLeNBkxRYvjEAiaK2S0RGWGFibHKVr73rLZVWj5Ne+q5j/45bo7vwMU4BTO4AI8uIY63EMDmkCAwu8w4czdp6dV+dtXrmLHpO4F84n79qeo4b</latexit> <latexit sha1_base64="w0SV1uasOZla5S7ghfwo/wJK7g=">AB63icZDNSgMxEMdn/az1q+rRS7AVPJSyWw96s+DFYwX7ge1Ssm2DU2yS5IVytKn0JOoN9/GF/AtfASz2x60HQj58Z8Zv4TxJxp47rfztr6xubWdmGnuLu3f3BYOjpu6yhRhLZIxCPVDbCmnEnaMsxw2o0VxSLgtBNMbrN854kqzSL5YKYx9QUeSRYygo2VHiv9ERYCD7zKoFR2a24eaBW8BZRvfiCP5qD01R9GJBFUGsKx1j3PjY2fYmUY4XRW7CeaxphM8Ij2LEosqPbTfOMZOrfKEIWRsk8alKt/O1IstMBmXLV/VqEz0FMRVANRzRJKh3phgmv/ZTJODFUkvmIMOHIRCgzjoZMUWL41AImitktERljhYmx5yla+96y2Vo12veZa1+Xy83PkdoACncAYX4MEVNOAOmtACAhJe4B0+HOE8O6/O27x0zVn0nMC/cD5/AS0Ejyc=</latexit> <latexit sha1_base64="IBkCgELn+RqUpuFZgENYKGAkyA8=">AB63icZDNSgMxEMdn/az1q+rRS7AVPJSyux70ZsGLxwr2A9tSsm2DU2yS5IVytKn0JOoN9/GF/AtfASz2x60HQj58Z8Zv4TxJxp47rfztr6xubWdmGnuLu3f3BYOjpu6ShRhDZJxCPVCbCmnEnaNMxw2okVxSLgtB1MbrN8+4kqzSL5YKYx7Qs8kixkBsrPVZ6IywEHviVQans1tw80Cp4Cyjf/EAejUHpqzeMSCKoNIRjrbueG5t+ipVhNZsZdoGmMywSPatSixoLqf5hvP0LlVhiMlH3SoFz925FioQU246r9swqdgZ6KoBqIapZQOtRLM0x43U+ZjBNDJZmPCBOTIQy42jIFCWGTy1gopjdEpExVpgYe56ite8tm12Fl/zLmv+vV+u/M7QAFO4QwuwIMrqMdNKAJBCS8wDt8OMJ5dl6dt3npmrPoOYF/4Xz+Ai6Djyg=</latexit> < l a t e x i t s h a 1 _ b a s e 6 4 = " W 3 B Y J z O c D N t q G 8 t d 6 M 9
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  • 4
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For small η

ty is U6Fi = n12/# with n12 = 1.48 × 10122 (F ℎ12 = 8 × 10126 %12 = 3.4 × 10122 U6Fi = n12/# n12 = 1.48 × 10122 s are ℎ12 = 8 × 10126 and %12 = 3.4 × 10122 (a n = ℎ (1 − cos h) + % cos h

n2 = ℎ12(1 − cos6 h) + %12 cos6 h h = 1.03 59∘ U F = j0.74 F + 0.26 F m

h = 1.03 59∘ U6Fi = j0.74 FG + 0.26 FIm

n2 = ℎ12(1 − cos6 h) nally h = 1.03 rad (59∘). U6Fi = j0.74 FG

Solve Result Trusted, from the “comparison”

slide-24
SLIDE 24

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slide-25
SLIDE 25

Conclusions

  • Original method
  • Simple & highly suitable to DACs
  • Can be automated
  • RF gain —> little/no need of correlation
  • Results are consistent
  • Modulation-index amplification
  • AM <—> PM conversion
  • Flicker observed over a wide span
  • Exact 1/ƒ, <1 dB discrepancy / 7.5 decades

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