chapter 1 - noise

RF IC Technology - Noise1


IntroductionDefinition of NoiseSources of High Frequency Noise

Thermal NoiseShot NoiseFlicker Noise

Propagation of Noise Through a SystemNoise Through a Linear FilterSystems With Several Noise SourcesFrequency Transformation of Noise

Noise RepresentationsNoise FigureNoise Temperature

Equivalent Input NoiseCascade of Noisy NetworksMeasurement of Noise

Contents


References for Noise

• Selected Papers, Provided by Lecturer

• Design of Analog CMOS Integrated Circuits, Behzad Razavi,Chapter 7, pp.201 -245

• Analysis and Design of Analog Integrated Circuits, Gray andMeyer, 4th Ed., Chapter 11, pp. 748 – 807

• Design with Operational Amplifiers and Analog Integrated Circuits, Franco, Third Ed., Chapter 7, pp. 311 - 346

• Analog Integrated Circuit Design, David Johns, Chapter 4, pp. 181 – 220, Ed. 1997


Types of Noise

• External Noise– May be of random or regular nature from outside

sources– Interaction between the circuit and the outside

world– Interaction between different parts of circuit

• Internal Noise– Random signals due to the natural phenomenon


Intrinsic Noise

• Thermal Noise • Shot Noise• Flicker Noise (1/f)• Burst or Popcorn Noise


RF Signal Along With Noise

RF f

RF f

RX

Good

RX

Bad

RF f

SSnn(f(f))


Definition of Noise

• Noise is a RANDOM PROCESS

• The value of noise can not be predicted at any time

• The average power for most types of noise is predictable by observing noise over a long time


Quantitative Definition of NoiseStatistical Models

Probability Density Function (PDF): p(n)

1)( =∫+∞

∞−

dnnp

p(n) dn = probability of n1 < n < n1 + dn

∫2

1

)(n

n

dnnp = P( n1 < n < n2 )


Distribution of PDFGaussian Model

Distribution of Probability Density Function (PDF): p(n)

−= 2

2

21exp

21)(

nn

nnpσσπ

)(np

n

2σn

+ σn- σn

68.0)( =∫+

−

dnnpn

n

σ

σ

P( - σn < n < + σn ) = 0.68


Average Noise Power

dtRtv

TP

T

T Lav ∫

+

−

=2/

2/

2 )(1

dtRtx

TP

T

T LTav ∫

+

−∞→=

2/

2/

2 )(1 lim

dttxT

PT

TTav ∫

+

−∞→=

2/

2/

2 )(1 lim

dttnT

tnT

TT ∫

+

−∞→=

2/

2/

22 )(1 lim)(

For a periodic (T) voltage signal v(t) across a load RL :


Noise in Time Domain

Noise is expressed as a Fourier Series over a period T :

∑+∞=

−∞=

=

K

KK t

TKjXtn 2 exp)( π

0 2 exp)( XtTKjXtn

K

KK =

= ∑

+∞=

−∞=

π

∑+∞=

=

∗+=K

KKK XXXtn

1

20

2 2)(


Noise in the Frequency Domain

)(1 tnf)(tn

)(21 tnfBPF

f1

1 Hz

( )2

)( 1fSnNoise average power in a 1 Hz bandwidth around a frequency f1 :


Power Spectral Density (PSD) of Noise

∫=2

1

)()(2f

fn dffStn

The Amount of mean-squared noise over a finite bandwidth ∆ f = f1 - f2

ftnfS

ffStn

n

n

∆=

∆=

)()(

)()(2

2

fVfS n

n ∆=

2

)(


Sources of Noise

• Thermal Noise, Johnson’s Noise• Shot Noise• Flicker Noise (1/f)• Burst or Popcorn Noise


Thermal Noise• A thermally (thermal energy) generated noise due to

random motion of the charge carriers; electrons• The average noise power is proportional to:

– Temperature– Frequency bandwidth (spectrum) of the thermal noise

fKTBKTPn ∆==

KTfPS n

n =∆

=Power Spectrum Density : (Watt/Hz)

K = 1.38 x 10 -23 J/oK

(Watt)


Thermal Noise

Power Spectrum Density (W/Hz)

Frequency (Hz)

KT

0

Time

Instantaneous Noise Voltage


Resistor Thermal Noise

KTBRVP n

n ==4

2

RKTBVn 42 =

GKTBIn 42 =

R

Noisy Resistor

R

)(tvn

R

2nV

Noiseless Resistor

≡ ≡


Root-Mean-Square Value of Noise

RKTBVn 4=

GKTBIn 4=

THz 2.6=<<hKTfValid up to very high frequencies of 10 GHz,

R

nV

Noiseless Resistor

Planck’s constant, h = 6.67 x 10 -34 J.S


Thermal Current Noise

GKTBIn 4=

R ni

NoiselessResistor

GKTBIn 42 =


PSD of Thermal Noise

SSVV(f)(f) (V2/Hz)

f (Hz)

4RKT

0

RKTBVP nav 42 ==

fVfS n

n ∆=

2

)(

KTRfSn 4)( =

(V2)

(V2/Hz)


Thermal Noise of Series Resistors

KTBRV Sn 42 =

KTBRRVn )(4 212 +=

22

21

2nnn VVV +=

22

21 nnn VVV +=

KTBRVi

in 42 ∑=

21 nnnT XXX +=

221

2 )( nnnT XXX +=

2122

21

2 2 nnnnnT XXXXX ++=


Thermal Noise of Parallel Resistors

KTBGI TnT 42 =

KTBGGInT )(4 212 +=

22

21

2nnnT III +=

22

21 nnnT III +=

KTBGIi

inT 42 ∑=


Effective Bandwidth

2)(11|)(|RC

Hω

ω+

=

2|)(|)()( ωHfSfS nino =

2)(11)(RC

KTfSno ω+=

effnono KTBRCKTdffSP === ∫

∞

4 )(

0

Parallel Resistor and Capacitor:

R

2niV

C noV

+

-

)()(

)(sVsVsH

ni

no=


Effective Bandwidth

2)(11|)(|RC

Hω

ω+

=

2)(11)(RC

KTfSno ω+=

effno KTBRCKTP ==

4

R

2niV

C noV

+

-

Power Spectrum Density (W/Hz)

f (Hz)

KT

0RC

Beff 41

=

noS~


Noise Voltage at the Output

RCKT

RVP n

no 44

2

==

R

2niV

C noV

+

-

)()(

)(sVsVsH

ni

no=

CKTVn =2

CKTV rmsn =,


MOSFET’s Channel Thermal Noise

The most significant source of the thermal noise generatedin the channel is called Drain NoiseDrain Noise or Channel NoiseChannel Noise;

For long channel devices operating in the saturation region

Long channel: γ = 2/3 Submicron : γ = 2.5 for 0.25 micron technologyγ : is bias-dependent parameter

fgKTI mn ∆= 42 γ

Ref.: Y. TsividisfgKTI dn ∆= 0

2 4 γ


Induced Gate Noise

+_

DS

GVGS VD

ndingi

fCg

fgKTigsm

dong ∆

=

2

2

))(/5(24

απδ

dogKTγ4

f

fing∆

2

Slope=20 dB/decade

αtf5

Fluctuating channel potential couples capacitively into the gate terminal, causing a noise gate current

- δ is gate noise coefficient Typically assumed to be 2γ- Correlated to drain

+_


Induced Gate Noise

fCKTi gsng ∆= 222

1516 ω

WLCC oxgs 32

=

“Analysis and Design of Analog Integrated Circuits,” Gray andMeyer, 4th Ed., Chapter 11, pp. 748 – 807


Noise Parameter As a Function of Vd

Drain Voltage VDrain Voltage VDSDS


Correlation factor between the induced gate noise and the channel drain noise

22

*

dg

dg

ii

iijCB =+


Thermal Noise in MOSFETs (Cont.)

The ohmic sections also contribute thermal noise. In a relatively wide transistor:

G

D

S

RG1 RG2RGn

RG1+ RG2+ … + RGn= RG



In which R1 =RG /3

We will hereafter neglect the thermal noise due to the

ohmic sections of MOS devices

_+

_+

_+

RD

RS

R1

2, 1Rn

V

2,RSnV

2,RDnV


Noise Sources in a CMOS Amplifier



In which R1 =RG /3

While the thermal noise caused in channel is controlled by gm, the thermal noise caused by RGcan be reduced in a folded device, and the thermal noise caused by RS and RD are negligibledue to small resistor values.

We will hereafter neglect the thermal noise due to the

ohmic sections of MOS devices⇒

_+

_+

_+

RD

RS

R1

2, 1Rn

V

2,RSnV

2,RDnV


Shot Noise

• Associated with discrete packets of charge emission, or the charge carriers crossing a boundary; potential barrier region,

• Depends directly on the direct component of the current

p n# of Holes # of Electrons

j

t

iJ ID


Shot Noise

fqIi DCn ∆= 22

DCn

i qIfifS 2)(

2

=∆

=Power Spectrum Density :

∫ −=−= ∞→

T

DTDn dtIiT

Iii0

222 )(1lim)(

It is valid for the fmax (less than or) comparable to 1/τ , where τ is the junction transit time

(A2)

(A2 /Hz)


Shot Noise

−= 2

2

21exp

21)(

σσπiip

fqIDC∆= 22σ

fqIDC∆= 2σ

)( nIp

In

2 σI

+σI-σI IDCurrent amplitude lies betweenID ± σI for 68 percent of the time


Flicker Noise

• Flicker noise or Contact noise occur due to the imperfect contact“Contamination” between two conducting materials that causes the conductivity to fluctuate in the presence of a dc current.

• mean-square flicker noise current in 1 Hz frequency band: where Kf is the flicker noise coefficient, I is the dc current, m is the flicker noise exponent, and n~1.

• Flicker noise is modeled by a noise current source in parallel with the device.

One-over-f-noise/ low frequency noise/ pink noise

n

mf

f fIK

i =2

(A2 /Hz)


Flicker Noise, (1/f) Noise• Associated with the fluctuation in carrier density due to trapped

electrons “Crystal defects”, for example, the dangling bonds existing in the MOS oxide-substrate interface.

ffIKI fn ∆= β

α2

1/f GenerationDevicermsnI ,

Kf : constant for a particular deviceα : constant in the range 0.5 to 2β : constant of about unity

ffIKI frmsn ∆= β

α

,

(A2)


Flicker Noise PSD

fIK

fIfS fn

i

α

=∆

=2

)(

Si(f)

fLog scale

Log

scal

e

• In devices exhibiting high flicker noise levels, this noise source may dominate the device noise at frequencies well into mega hertz

• In general, Kf is an unknown constant• May varies by orders of magnitude from one device type to the next• Vary widely for different transistors or integrated circuits from the

same process wafer

(A2 /Hz)


Flicker Noise in MOSFETsThe flicker noise in MOSFETs can be easily modeled as a voltagesource in series with the gate and roughly given by:

K is a process-dependent constantK = 10-25 (V2 F), ∆f = 1Hz(Behzad Razavi)

Flicker noise spectrum:

• Devices with areas of several thousands square microns in low noise applications

• PMOS devices exhibit less 1/f noise than NMOS transistors.

fWLCKVox

n12 = (V2)


Flicker Noise in MOSFETs (Cont.)

SO WHAT HAPPENS TO TOTAL FLICKER NOISE IF THE LOWEREND OF THE BAND APPROACHES ZERO?

1. Signals in most applications do not contain significant low frequency components.

2. The logarithmic dependence of the flicker noise power upon fLallows some margin for error in selecting fL


Flicker Noise in MOSFETs (Cont.)

KTg

WLCKfg

fWLCKgKT m

oxm

coxm c 8

31)32(4

'2

'

=⇒=

Weak dependence for a given L and thus relatively constant “corner frequency”, leads to fc falling in the vicinity of 500KHz to 1MHz for submicron transistors.


Propagation of Noise Througha System

For a linear and noiseless system:

2|)2(|)()( fHfSfS io π=

)(ωH)( fSi )( fSo

R

C

|)(| ωHRC

1 ω


Propagation of Noise Througha System

R

C

vni(t) vno(t)

SSnini (f)(f) (V2/Hz)

f (Hz)

4RKT

0 f (Hz)

SSnionio(f(f)) (V2/Hz)

4RKT

0RC41


= ∑

∞

−∞=

tTkj

kki Xn π2exp

= ∑

∞

−∞=

tTkj

TkH

kko Xn ππ 2exp2

2

2*

)]2()[(

)]2(][ lim2[)(

fHfSfHTXXfS

i

KKTo

π

π

=

= ∞→


2H1H

22

21 |)2(||)2(|)()( fHfHfSfS io ππ=

221 ||)()( HHfSfS io =

)( fSi )( fSo


Example

+ Ainv

v n1 v n 2

( )Avvvv nninout 21 ++=+

++

= VVAVAV nn

mout

2

2

2

1

22

22

2

Signal Noise

) cos( tVv min ω=


Example

Vout

v n1 v n 2

( )fA π2vin + +

NoiseSignal

22

221

20

22 |)2(| |)2(|)(

2 nnm

out VfAVfAffVV ++−= ππδ


Example

+ +

2nv1nv

inv outv)2( fA π

)2( fB π


( )fAvvV n

neq π22

1+=

( ) ( ) ( )( ){ }vv

fBfAvfffAVV

nn

no

out

fAB 2

2

2

2

222

10

22

2

2Re2

2 222π

δ πππ++

++

−=

( ) ( )

+=fA

fBvV neq ππ

2122


Systems with Several Noise Sources

~

n 1

n 2

n j... +

-

~

~

~

outout nV +LZinV


~

n 1

n 2

n j... +

-

~

~

~

LZ

01

=n

02

=n

0=nj... +

-

Effective Input Noise

Noiseless Network

~

~

~~

~ LZ

Noisy Network

inV

inV

inn

outout nV +

outout nV +


Procedure for Output Noise• Turn off all noise sources, then determine the

output• Turn off the input voltage (current) source and all

noise sources except one. – If the noise source set on is correlated to other noise

sources in the system, turn these sources on as well– Analyze the system for mean-squared noise output

• Repeat the above step until all the noise sources exhausted

∑=

=N

iioutTotalout VV

1

22 )(


Example

)(21 2

321

222nno VVAAv ++=

A

++

+

A

)(1 tv vn2)(2 tvvn3

vn1

dv ov+-))()(()( 12 tvtvAtvd −=

)(21 2

22

122 VVAvds +=

( ) )(21 2

22

122

32

222 VVAvvAv nnd +++=

( )23

22

22nndn vvAv +=


Frequency Transformation of Noise

f f

Power Power

⇔Function ofRF System

RF


×( )tvV n+1

( )( )tVVVV nout +=12

V 2

Mixer

( )tVVVVV nout 212+=

Wanted Noise

( ) [ ]eeV tjtjcc

tA ωω −+=21

[ ] [ ]{ }∑∞

=

−++=1

02 expexp2k

kkk tjtjaaV ωω( ){ }

( ) ( ) ( ){ }( ) ( ) ( ){ }∑

∑∞

=

+−−

∞

=

−−+

−

++

+++

++=

1

1

021

4

4

2

k

tjtjk

k

tjtjk

tjtj

eeaeeaeeaVV

ckck

ckck

cc

tA

tA

tA

ωωωω

ωωωω

ωω


[ ]∑∞

−∞=

=l

lln tjXV ωexp

Tl

l

πω 2=

( )( )

∑+

∞

−∞=+

∞

−∞==

∞

−∞=

∞

=−

∑

∑

∑

l ll

k bytranslated

bytranslated

kk

XXa

XXXXaatvV l

lll

lln

kk

*2

0

1

*

2**

42ωω

( )

+

++

−= ∑

∞

=fSaffSffSaaS nknkn

k

kkout f 2

01

*

4


• Noise Figure; Noise Factor• Noise Temperature

Noise Measures


Noise Figure

oo

ii

NSNSF

//

=

o

i

NSNSF

)/()/(

=NoSo

NiSi

PPPPF

//

=

i

o

o

i

NN

SSF =

Noise Factor:

FF log10(dB) =Noise Figure:


Noise Figure

oo

ii

NSNSF

//

=

sR

sV

NoisyNetwork

LR

nPAG

NiPNoP

+

-

NoiselessNetwork

neP AG

NiP

nNiA

No

PPGP

+=


Noise Figure

i

o

o

i

oo

ii

NN

SS

NSNSF ==

//

eAa NGN =

iA

o

iA

iAa

NGN

NGNGNF =

+=

i

e

NNF += 1

Ni

Ne

PPF += 1


Noise Temperature

BKTR

VP s

rmsnN ==

4

2,

Maximum available noise power from a noisy resistor at temperature Ts:

NoisyNetwork aP

KBPT a

e =Noise Temperature:

LR BKTP ea =


Noise Temperature

Ni

Ne

NoSo

NiSi

PP

PPPPF +== 1

//

BKTP eNe =

BKTP sNi =

s

e

TTF += 1

A

a

A

nNe G

NGPP ==Where:


Characterization of Two-Port Noisy Network

Using z-parameters

NoisyNetwork1v 2v

1i 2i

NoiselessNetwork1v 2v

2i1i 1nv 2nv

12121111 nvizizv ++=

22221212 nvizizv ++=

01121 ==

=iin vv

02221 ==

=iin vv


Example

2v

1i 1nv 2nv1R

3R

2R 2i

1v

Noiseless

NoisyNetwork

1R

3R

2R 1R 2R1i 1ne 2ne

3ne

3R

2i

1v 2v

Q:Verify correlation

between vn1 and vn2


Characterization of Tow-Port Noisy Network

Using y-parameters

NoisyNetwork1v 2v

1i 2i


1i 2i

1ni 2ni

12121111 nivyvyi ++=

22221212 nivyvyi ++=

01121 ==

=vvn ii

02221 ==

=vvn ii

Q:Derive in1 and in2

in terms of vn1 and vn2



211

121

2

212

122

1

nnn

nnn

vzvzi

vzvzi

∆−

+∆

=

∆+

∆−

=

Derive in1 and in2in terms of vn1 and vn2

nn

nn

n

n

YVIVZIVVZI

VZIV

−=−=

−=

+=

−

−

1

1 )(

2221212

2121111

nnn

nnn

vyvyivyvyi

−−=−−=



Derive vn1 and vn2 in terms of in1 and in2

nn

nn

n

n

ZIVIYVIIYV

IYVI

−=−=

−=

+=

−

−

1

1 )(

211

121

2

212

122

1

nnn

nnn

izizv

iyiyv

∆−

+∆

=

∆+

∆−

=


Characterization of Two-Port Noisy Network

Using (ABCD)-parameters

NoisyNetwork1v 2v

1i 2i


2i

ni

1i nv

nviBAvv +−+= )( 221

niiDCvi +−+= )( 221

Q:Derive in and vn

in terms of vn1 and vn2


Derive in and vnin terms of vn1 and vn2


2i

ni

1i nv

221

111 nnn v

zzvv −=

221

1nn v

zi −

=

By characterizing the network in terms of z parameters:


Derive in and vnin terms of in1 and in2


2i

ni

1i nv

221

111 nnn i

yyii −=

221

1nn i

yv −

=

By characterizing the network in terms of y parameters:


Minimum Noise Figure

• Find Noise Figure: F = f(is,Ys,vn,in)• if in=inu+inc derive Fmin

Hint: assume

• Express optimum value of Ys, called Yopt ,for Fmin

NoiselessNetworksi 2v

2i

ni

nv1i

sY

sss jBGY +=ncnc vYi = ccc jBGY +=


Input Noise in Terms of Noise Figure “F”

NoiselessNetwork

neP AG

NiP

nNiA

No

PPGP

+=

Ni

ne

PPF += 1

Ni

totalNi

PP

F ,=

)(, BKTFFPP oNitotalNi ==

))(1(, BKTFPPP oNitotalNine −=−=

))(1( BKTFGPGP oAneAn −==


Networks in Cascade

sR

sVLR

NiP NoisyNetwork

1nP1AG

1NoP+

-

NoisyNetwork

2nP2AG

2NoP

1F 2F

))(1()( 211,2 BKTFBKTFGP ooAtotalNi −+=

112

112

2

GFF

PGGPF

NiAA

No −+==

NiAA

No

PGGPF

12

2=

totalNiANo PGP ,222 =

21,2 neNototalNi PPP +=

)( 011,111 BKTFGPGP AtotalNiANo ==


Networks in Cascade

sR

sVLR

NiP NoisyNetwork

1nP1AG

1NoP+

-

NoisyNetwork

2nP2AG

2NoPNoisy

Network

3nP3AG

NoP

1F 2F 3F

11

11

2

321 GG

FGFFF −

+−

+=

iAAA

o

NGGGNF

321

=



• Find Noise Figure: F = f(is,Ys,vn,in)• if in=inu+inc derive Fmin

Hint: assume

• Express optimum value of Ys, called Yopt ,for Fmin

NoiselessNetworksi 2v

2i

ni

nv1i

sY

sss jBGY +=ncnc vYi = ccc jBGY +=

+

_



2

2

s

sc

iiF =

snnssc YViii ++−=

)(2)()( 2222snnssnnssnnssc YViiYViiYViii +−++=++−=

0)( =+ snns YVii

222 )( snnssc YViii ++= 2

2)(1s

snn

iYViF +

+=


ncnun iii +=

ncnc VYi =

ncnun VYii +=

2*ncnn VYVi =

2

*

n

nnc

ViVY =

2

2))((1s

nscnu

iVYYiF ++

+=

BGkTi ss 02 4=

BRkTV nn 02 4=

BGkTi unu 02 4=

BGkTBRkTjBGjBGBGkT

Fs

nccssu

0

02

0

444

1++++

+=

[ ]22 )()(1 cscss

n

s

u BBGGGR

GG

+++++=


cs BB −=

2)(1 css

n

s

uBB GG

GR

GGF

cs+++=−=

0=−=

s

BB

dGdF

cs

0)()(22

2

2 =

+−++−=−=

s

cscssn

s

u

s

BB

GGGGGGR

GG

dGdF

cs

)( 22coptnu GGRG −=

n

ucopts RGGGG +== 2


cn

ucoptoptopt jBRGGjBGY −+=+= 2

2min )(1 copt

opt

n

opt

uYY GG

GR

GGFF

opts+++== =

)2()(1 222

min ccoptoptopt

n

opt

coptn GGGG

GR

GGGRF +++−+=

)(21 coptn GGR ++=


[ ]22min )()()(2 optscs

s

n

s

uoptcn BBGG

GR

GGGGRFF −+++++−=

[ ]22min )()( optsopts

s

n BBGGGRFF −+−+=

2

min optss

n yygrFF −+=

sssss

s jbgYjBG

YYy +=

+==

00

optoptoptoptopt

opt jbgYjBG

YY

y +=+

==00

2

min optss

n YYGRFF −+=



• Find Noise Figure: F = f(vs,Zs,vn,in)• if vn=vnu+vnc derive Fmin

Hint: assume

• Express optimum value of Zs, called Zopt ,for Fmin

NoiselessNetworksv

2v

2i

ni

nv1isZ

sss jXRZ +=ncnc iZv = ccc jXRZ +=

+

_

+-

chapter 1 - noise

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