nonlinear analysis of direct sampling mixer using f-matrix

Nonlinear Analysis of Direct

Sampling Mixer using F-matrix

Tokyo Institute of Technology

Araki・Sakaguchi Lab

Kousuke Aio2010/7/16

Content

Background

Analysis with Fundamental Matrix

Nonlinear System

Periodic Time-Variant System

Analysis of Charge Sampling Circuit

Simulation

Conclusion and Future Work

2010/7/16 1

Content

Background


Nonlinear System



Simulation


2010/7/16 2

Background

Software-defined Radio Software-defined Radio

LNA TA ADC DSPDSM SCF IFA

LNTA

Analog

Discrete-Time

Discrete Time Receiver

System with various modulation method

and frequency on a RF circuit.

⇒ Software-defined Radio

TA : Voltage ⇒ Current

DSM : charge sampling

decimation

SCF : RF Filtering

•Power consumption

•Small size

2010/7/16

Digital

Discrete-TimeAnalog

Continuous-Time

3

Down conversion and decimation

⇒ Demand Moderation of ADC

Discrete time processing

⇒ Reconfigurable Technology

Background

Direct Sampling Mixers (DSM)

Charge Sampling

⇒Frequency Conversion

Charge Shared Filtering

⇒RF Filtering

Decimation

⇒ Less power loss due to low speed ADC

MOS Switches and capacitors

⇒ One chipping operation

Reconfigurability

⇒ Filter response change easily.

•Basic of DSM

•Clock Timing Chart

Features

Noise-reduction

Improvement of filter property

Nonlinear distortion

Problem

Research Propose : Design of distortion compensation method

Nonlinear distortion

sampling decimation

filtering

output

input

2010/7/16 4

Background

Complexity of Nonlinear DSM

2010/7/16

Nonlinear TA

Nonlinear MOS Switch


Periodic time-variant system has frequency conversion.

Memory Effect

DSM has frequency characteristic (memory effect) ,

so nonlinearity is expressed as a function of not only

amplitude, also operation frequency.

2 tone signal

IMD3

5

Content

Background


Nonlinear System



Simulation


2010/7/16 6

Advantage


2010/7/16

New Analysis Method

The frequency conversion and Volterra series function are

introduced into Fundamental matrix.

Complex circuit can be divided into some F-matrix. ⇒ flexibility

Calculation of cascade connection is matrix calculation. ⇒ easy

out

out

21212121

21212121

out

out

22

22

11

11

in

in

I

V

DDBCCDAC

DBBACBAA

I

V

DC

BA

DC

BA

I

V

•Cascade Connection with F-matrix

A1 B1

C1 D1

Vin

Iin

A2 B2

C2 D2

Vout

Iout

Inverse of (1, 1) component ⇒ Frequency Characteristic : Vout/Vin

7


2010/7/16 8

Approach

Nonlinear Periodic Time-Variant :

NLPTV system (ex. Nonlinear Mixer, DSM)

Linear Periodic Time-Variant :

LPTV System (ex. Linear Mixer)

Nonlinear Time-Invariant

System (ex. PA)

Linear Time-Invariant :

LTI System (ex. LC Filter)

⇒ Conventional F-matrix target

Nonlinear System

Volterra Series Function

9

21212121213

111112

1

)()()(),,(

)()(),(

)()()(

ddXXXH

dXXH

XHY

Overall property of DSM can’t be expressed as power series function

because DSM have frequency characteristic . ⇒ Volterra Series

Volterra Series Function

Expression of F-matrixMultidimensional Transfer Function

ωI

ωV

ωI

ωV

Y

Y

X

X

DC

BA入力 : X 出力 : Y

DC

BA

This A has each dimensional functionsA1(ω1), A2(ω1, ω2), A3(ω1,ω2,ω3)

2010/7/16

Nonlinear System

Cascade Connection of Volterra Series

2010/7/16

X Y Z

GG : X ⇒ Y

H : Y ⇒ Z

F : X ⇒ Z

)()()()()()(

)()()(

2112212212211212

111111

ωHωH,ωωG,ωωHωωG,ωωF

ωHωGωF

)(()()(

3/)()()(P2

)()()(

3121113213

113223212

321332113213

ωHωHωH,ω,ωωG

ωH,ωωHω,ωωG

,ω,ωωHωωωG,ω,ωωF

)(

)(

10

0HG

)(

)(

10

0H

10

0G

)(

)(

Z

Z

Z

Z

X

X

I

V

I

V

I

V

10

FH

Eq. 1

321 ,,operation n permitatio : P where


Linear Periodic Time-Variant (LPTV) System

2010/7/16 11

),(),(

)(),()(

TtHtH

txtHty

n

n nXHY )()()( LO

Frequency Conversion

period time:

output : )(

input : )(

T

ty

tx

PTV transfer function : H(ω, t)

Fourier series expansion (Eq. 2)

Ttjn

n

n

tjn

n

dtetHT

H

eHtH

0

LO

LO

),(1

)(

)(),(

)()()( LO txeHtytjn

n

n

Frequency Conversion

Fourier Transform

frequencyclock : /2LO T

Content

Background


Nonlinear System



Simulation


2010/7/16 12


13

2121222123112122212 )()()()()()()(A ddIIIadIIaIaI

)(

)(

A/0

A 0

)(

)(

2

2

TA1

1

I

V

RI

V5

1

2

2

4

133

3

122

1

11

2

mmmm

mm

m

gggga

gga

ga

3

3

2

2

1

mA/V 1.86

mA/V 3.10

mA/V 3.11

m

m

m

g

g

g

Nonlinearity of TAPower series expansion is approximated from measurement of the

prototype TA.

F-matrix expression of nonlinear TA

IP3 → 3rd coefficient

I1 I2

V2V1

RTA

3

in3

2

in2in1 )()()()( tVgtVgtVgtI mmmTA

2010/7/16

IP2 → 2nd coefficient

(Eq. 3)

V2

I2

V1

I1

V1 V2

I1 I2


Nonlinearity of MOS Switch

14

MOS transistor is dealt as ideal switch and nonlinear resistor.

2

2

1

1

10

)(R1

)(

)(

I

Vt

I

V

3

23on

2

22on21on21 )()()()()( : state ON tIrtIrtIrtVtV

3

23off

2

22off21off21 )()()()()( : state OFF tIrtIrtIrtVtV

2010/7/16

2121222123

112122

212

)()()()(

)()()(

)()()()R(

ddIIItr

dIItr

ItrIt

ron1 50 Ω roff1 1 MΩ

ron2 -10 kV/A² roff2 0

ron3 -200 MV/A³ roff3 0

Nonlinear Resistor R(t) (Table.1)

F-matrix expression of nonlinear MOS switch


Analysis Procedure

2010/7/16 15

1. Expression as F-matrix

2. Matrix and Volterra series operation

for cascade connection. (Eq.1)

3. Derivation of Inverse of (1.1)

component at F-matrix. (Appendix.1)

4. Fourier series expansion. (Eq.2)

out

out

CMOSPTA

in

inFFFF

)(

)(

I

V

I

V

),,,( ),,,( ),,(F 3213212111 tGtGtG

),,,( ),,,( ),,( 32132121 tHtHtH

),,( ),,( ),( 3213,212,1, nnn HHH Multidimensional & PTV

Transfer Function

Charge Sampling Circuit

1

01F ,

10

)R(1F

1/1

01F ,

A/0

A0F

CMOS

0

P

TA

TA

HCj

t

RR

Content

Background


Nonlinear System



Simulation


2010/7/16 16

Simulation

Simulation

17

Circuit Simulation (ADS)Simulation Parameter

Pin -15 dBm CH 40 pF

fLO 500 MHz C0 0.3 pF

R0 4 kΩ VG 0 ~ 1.8 V

A (Eq. 3) R(t) (Table.1)

•Input Frequency : fin1 + fLO and fin2 + fLO (2 tone signal)

⇒ Output Frequency are fin1, fin2, and 3rd harmonic signal are down

converted near 3fin1 and 3fin2 due to nonlinearity.

•Differential operation

⇒2nd distortion is canceled

fin1 fin2 3fin1 3fin2

Frequency Spectrum

Po

we

r

Fundamental

IMD33rd harmonic

＊IMD3 → 3rd Inter Modulation Distortion

＊VG →gate voltage of switch

2010/7/16

Freq

Simulation

2010/7/16 18

LPTV System : Frequency Response

•Frequency Response •Frequency Response near fLO

If input frequency is near nfLO, MATLAB and ADS results are almost same.

First, we assume that charge sampling circuit is linear (a2=a3=0, ron2=ron3=0)

Simulation

2010/7/16 19

NLPTV System : Frequency Spectrum

From frequency spectrum, small difference is shown between MATLAB and

ADS, but these are almost matched at fundamental and IMD3 components.

• fin1 = 0.2MHz, fin2 = 0.3MHz • fin1 = 0.3MHz, fin2 = 0.5MHz

Simulation

2010/7/16 20

NLPTV System : AM-AM Characteristic

Difference between MATLAB and ADS is less than 0.5dB.

IIP3 (3rd Input Intercept Point) is about -4 ~ -3dBm.

• fin1 = 0.2MHz, fin2 = 0.3MHz(fIMD31=2fin1-fin2=0.1MHz, fIMD32=2fin2-fin1=0.4MHz)

• fin1 = 0.3MHz, fin2 = 0.5MHz(fIMD31=2fin1-fin2=0.1MHz, fIMD32=2fin2-fin1=0.7MHz)

Content

Background


Nonlinear System



Simulation


2010/7/16 21


Conclusion

Future Work

The simple nonlinear periodic time-variant system can be

analyzed with F-matrix, Volterra series function and Fourier

series expansion.

When input frequency is near the clock frequency, this

analysis has same result with circuit simulation result.

Improvement of this analysis model (more exact, more

complex circuit architecture)

Invention of efficient distortion compensation method.

2010/7/16 22

Appendix.1

2010/7/16

Volterra Inverse

X Y ZG H

)()(G)(G

),(G),(H

)(G)(H

2111211

212212

1

1

111

G

)(G)(G)(G)(G/)(G

),(G),(G2),,(G),,(H 3121113211

321

322321232133213

0),,(F

0),(F

1)(F

3213

212

11

If G is inverse function of H, G can be expressed as this.

23

nonlinear analysis of direct sampling mixer using f-matrix

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