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1Shuenn-Yuh Lee EE/CCU Chapter 1Communication and Biologic IC Lab.

Radio Frequency Integrated Circuits

Instructor : Shuenn-Yuh Lee

National Chung Cheng UniversityDepartment of Electrical Engineering

Office : 431 TEL : (05)2720411-33223, BP : 0921565137E-mail : [email protected]

Table of Content : 1. Basic Concepts in RF Design2. Modulation and Detection3. Multiple Access Techniques and Wireless Standards4. Transceiver Architectures5. Transmission line and match networks6. LNA and Mixer7. Oscillators8. Phase-Locked Loops9. Frequency Synthesizer10. Power Amplifier


數位邏輯設計與實習

數位系統設計與實習電子學

VLSI設計導論與實習

混合訊號佈局設計類比積體電路

數位系統快速雛型設計混合訊號積體電路

VLSI測試理論

VLSI設計流程整合實務

MOS元件分析與模擬

SOC導論

訊號處理架構設計VLSI系統設計與高階合成

類比與混合積體電路測試通訊積體電路設計

射頻積體電路設計

射頻通訊概論

課程規劃流程


Reference Books• Texbook : RF Microelectronics, Behzad Razavi, 1998

• Design of Analog CMOS Integrated Circuits, Behzad Razavi, 2001

• The Design of CMOS Radio-Frequency Integrated Circuits, 1998

• Analog Integrated Circuit Design, David A. Johns, and Ken Martin, 1997

• CMOS Analog Circuit Design, Second Edition, Phillip E. Allen ,and Douglas R.

Holberg, 2002

• Circuit Design for RF Transceivers, Domine Leenaerts, Hohan van der Tang, and

Cicero Vaucher, 2001

• Architectures for RF Frequency Synthesizers, Cicero S. Vaucher, 2002

• Phase-Locked Loops for Wireless Communications : Digital, Analog and Optical

Implementations, Second Edition, 2002

Grade Factors1. Homework 30%, Project 70%


Introduction to Wireless Systems

• Complexity Comparison

• Design Bottleneck

• Applications

• Analog and Digital Systems

• Choice of Technology

• Effect of Nonlinearity


Complexity Comparison• A simple FM transceiver

– FM transmitter

Q1operates as both oscillator and a

frequency modulator

– FM receiver

Q1operates as both oscillator and a

frequency demodulator


• RF section of a cellphone– This circuit is orders of magnitude more complex than analog FM circuit

Complexity Comparison (Cont.)


Design Bottleneck• RF and baseband processing in a transceiver

– The baseband section is more complex – The RF section is still the design bottleneck of

the entire system

• Three reasons– Disciplines required in RF design– RF design Hexagon

– Design toolsComputer-aided analysis and synthesis tools for RF ICs are still in theirinfancy


• PCS (person communication system)

– Pagers and Cellular phones

• WLANs (wireless local area network)

• GPS (global positioning system)

• RF IDs (RF identification systems)

• Home Satellite Network

Applications


Analog and Digital Systems• Block diagram of a generic analog RF system

– (a) transmitter (b) receiver


• Block diagram of a generic digital RF system– (a) transmitter, (b) receiver– voice compression : reduce bit rate and required bandwidth– coding and interleaving : reduce the bit error rate

– pulse shaping : shape the rectangular pulse

Analog and Digital Systems (Cont.)


Choice of Technology• Performance,cost,and time to market are three critical

factors influencing the choice of technologies in the

competitive RF industry

• Level of integration,form factor, and prior(successful)

experience play an important role in the decisions made by

the designers

• Must resolve a number of practical issues

– Substrate coupling of signals that differ in amplitude by 100 dB

– Parameter variation with temperature and process.

– Device modeling for RF operation (been resolved)


Choice of Technology (Cont.)• GaAs

– High frequency, high power, and high cost

– Widely used in power amplifier and front-end switched

• Silicon Bipolar

– Higher integration, high power, and lower cost

– Widely used in transceiver

• CMOS

– Highest integration, lowest cost, and low power

– Low operation frequency


Basic Concepts in RF Design

Outline• Nonlinearity and Time Variance

– Effects of Nonlinearity

– Cascaded Nonlinear Stages

• Intersymbol Interference

• Random Processes and Noise

– Random Processes

– Noise

• Sensitivity and Dynamic Range

• Passive Impedance Transformation


Nonlinearity and Time Variance• Linear system

for all values of the constants a and b

• Nonlinear system– Not satisfy above condition– With nonzero initial conditions or

finite “offsets”

• Time invariant

),()()()()()(,)()(

2121

2211

tbytaytbxtaxtytxtytx

+→+→→

)()(,)()( ττ −→−→ tytxtytxwe assume the switch is on if vin1>0 and off otherwise.

tAtvtAtv

in

in

222

111

cos)(cos)(

ωω

==

•Simple switching circuit •Nonlinear time-variant system

The path of interest is from vin1 to vout. The system is nonlinear because the control is only sensitive to the polarity of vin1, and time variant because vout also depends on vin2


( )

( )

−=

−∗=

∑

∑∞+

−∞=

+∞

−∞=

12

12

2/sin

2/sin)()(

Tnfv

nn

Tnf

nnfvfv

inn

ninout

ππ

δππ

Nonlinearity and Time Variance (Cont.)• Linear time-variant system

– The path of interest is from vin2 to vout, then the system is linear but time variant

– Vout can also be considered as the product of vin2 and a square wave toggling between 0 and 1


Memoryless• Its output does not depend on the past value of its input

• For a memoryless linear system

• For a memoryless nonlinear system

)()( txty α=

L++++= )()()()( 33

2210 txtxtxty αααα

Equivalent

RtsVvItvT

inSout ⋅

= )(exp)( 1


Odd symmetry and dynamic• Odd symmetry (called differential or balanced)

– For x(t)→-x(t) :– Example : bipolar differential pair

• Dynamic : if its output depends on the past values of its input(s) or output(s)– Linear time-invariant, dynamic system

– Linear time-variant, dynamic system

h(t) denotes the impulse response

L+−+−= )()()()( 33

2210 txtxtxty αααα

)(*)()( txthty =

)(*),()( txthty τ=

T

inEEout V

vRIv2

tanh=

( )( )

TinTin

TinTin

TinTin

TinTin

TinTin

TinTin

VVVV

VVVV

EE

VVVV

VVVV

EEccout

VVEE

cVVC

VVEE

cVVC

eeeeRI

eeeeRIVVV

eRIV

eIi

eRIV

eIi

2/2/

2/2/

//

//

21

/2/2

/1/1

1111

11

11

−

−

−

−

−−

+−

=

++−−+

=−=

+≈⇒

+=

+≈⇒

+=

α

α


Effects of Nonlinearity• For memoryless, time-variant, nonlinear system

)()()()( 33

221 txtxtxty ααα ++≈

• Effects of nonlinearity

– Harmonics

– Gain compression

– Desensitization and blocking

– Cross modulation

– Intermodulation


Harmonics• If x(t)=Acosωt, and )()()()( 3

32

21 txtxtxty ααα ++≈

• Two observations– Odd symmetry : Even-order harmonics result from αj with enen j are

vanish– The amplitude of the n th harmonic consists of a term proportional to An

Harmonics

tAtAtAAA

tAtAtAty

ωαωαωααα

ωαωαωα

3cos4

2cos2

cos4

32

coscoscos)(3

32

23

31

22

333

2221

++

++=

++=


• For small-signal gain of a circuit , harmonics are negligible– For example, α1A is much greater than all the other factors that contain

A, then the small-signal gain is equal to α1

– For bipolar differential pair

• In most circuit of interest, the output is a “compressive”or ”saturating” function of the input – Gain = α1+3α3A2/4, where α3 < 0, and A ↑ ⇒ gain ↓

• 1-dB compression point

RgVRI

vv

mT

EE

in

out ===21α

Gain Compression

3

11

12

131

145.0

1log2043log20log20

αα

ααα

=

−=+=

−

−

dB

dBin

out

Athen

dBAAA


Desensitization and Blocking• Desensitization

– A weak,desired signal along with a strong interferer.

– Since a large signal tends to reduce the ”average” gain of the circuit,the weak signal may experience a vanishingly small gain. Called ”desensitization”

• Blocking:– Gain= (α1+3α3A2

2/2) , A2 ↑ ⇒ gain ↓ ⇒ gain = 0 ⇒called “blocked”

– In RF design,the term”blocking signal” usually refers to interferers that desensitize a circuit even if the gain does not fall to zero

– Many RF receivers must be able to withstand blocking signals 60 to 70 dB greater than the wanted signal

L

L

+

+=

+

++=

<<+=

tAA

tAAAAty

AAfortAtAtx

112231

12213

31311

21

2211

cos23

cos23

43)(

coscos)(

ωαα

ωααα

ωω


Cross Modulation

• A weak signal and a strong interferer pass through a

nonlinear system

– For example : If the amplitude of the interferer is modulated by a

sinusoid A2(1+mcosωmt)cos ω2t

( )

L+

++++=

++=

tAtmtmmAty

ttmAtAtx

mm

m

11

222231

2211

coscos22cos22

123)(

coscos1cos)(

ωωωαα

ωωω

– Desired signal at the output contains amplitude modulation at ωm and 2 ωm

– Cross modulation arises in amplifiers that must simultaneously process many independent signal channels


• Harmonics distortion in a low-pass filter : fall out of the passband

Intermodulation (IM)

• Intermodulation distortion (two tone test)

– When two signals with different frequencies are applied to a

nonlinear system,the output in general exhibits some

components that are not harmonics of the input frequencies

– This phenomenon arises from “mixing”(multiplication) of the

two signals


Third-order IM product• Assume x(t)=A1cosω1t+ A2cosω2t, thus

• Intermodulation product

• Fundamental components

( ) ( )

( ) ( )

( ) ( )tAAtAA

tAAtAA

tAAtAA

121

223

121

223

12

212

213

212

213

21

212122121221

2cos4

32cos4

3:2

2cos4

32cos4

3:2

coscos:

ωωαωωαωω

ωωαωωαωω

ωωαωωαωωω

−++±=

−++±=

−++±=

tAAAA

tAAAA

22

12332321

12213

3131121

cos23

43

cos23

43:,

ωααα

ωαααωωω

+++

++=

322113

22211222111 )coscos()coscos()coscos()( tAtAtAtAtAtAty ωωαωωαωωα +++++=


Third-order IM product effect• Intermodulation in a nonlinear system

– Assume A1=A2=A

• Corruption of a signal due to Intermodulation– For example : if α1A=1Vpp, and 3 α3A2/4=10mVpp

⇒ the IM components are at -40dBc


Third intercept point (IP3)• Growth of output components in an intermodulation test

– ω1, ω2 ∝ A, 2 ω1 ±ω2 ∝ A3

– Input IP3(IIP3)

– Output IP3 (OIP3)

– IP3 is a unique quantity that by itself can serve as a means of comparing the linearity of different circuits


IP3 calculation• Let x(t)=Acosω1t+Acos ω2t

– If α1>>9 α3A2/4 – Input IP3=AIP3 and output IP3=α1AIP3

( ) ( ) L+−+−+

++

+=

tAtA

tAAtAAty

123

3213

3

22

3112

31

2cos432cos

43

cos49cos

49)(

ωωαωωα

ωααωαα

3

13

33331

34

43

αα

αα

=⇒

=

IP

IPIP

A

AA


Quick IP3 measurement• Denote : Ain : the input level at each frequency

Aω1, ω2 : the amplitude of the output at ω1, ω2

AIM3 : the amplitude of the IM3 product• Quick method

( ) inIMIP

inIPIM

in

IP

inin

in

IM

AAAA

AAAA

AA

AAA

AA

log20log20log2021log20

log20log20log20log20

134

4/3

32,13

22332,1

2

23

23

13

3

1

3

2,1

+−=⇒

−=−⇒

==≈

ωω

ωω

ωω

αα

αα


Example• Asig,in=1µVrms, AIP3=70mVrms (≅-10dBm), Aint,in=1mVrms

– Asig,in : signal amplitude, Aint,in : interferer amplitude

• Relation between A1-dB and AIP3

( )( ) dB

AAA

AA

AA

AA

AAA

AAA

AA

in

IPinsig

outIM

out

in

insig

outIM

outsig

outin

insigoutsig

in

out

insig

outsig

8.139.4101

10701033

236

3int,

23,

,3

int,

int,

,

,3

,

int,int,

,,

int,

int,

,

,

≈=×

××=

⋅==⇒

=⇒≈

−

−−

dBAA

IP

dB 6.9

34

145.0

3

1

3

1

3

1 −≈=−

αα

αα


Cascaded nonlinear stage

[ ][ ][ ]

( ) L++++=

+++

+++

++=⇒

++=

++=

)(2)(

)()()(

)()()(

)()()()(

)()()()(

)()()()(

33

312211311

333

2213

233

2212

33

22112

313

212112

33

2211

txtx

txtxtx

txtxtx

txtxtxty

tytytyty

txtxtxty

βαβααβαβα

αααβ

αααβ

αααβ

βββ

ααα

33122113

113 23

4βαβααβα

βα++

=IPA

• Proper choice of the values and signs of the terms in the denominator can yield an high IP3

• As a worst-case estimate, relations among AIP3, AIP3,1 and AIP3,2

– α1↑, the overall IP3 ↓, because with higher gain in the first stage, the second stage sense large input level, thereby producing much greater IM3 products

22,3

21

1

222

1,311

33122113

23 2

312431

IPIPIP AAAα

ββα

βαβαβααβα

++=++

=


Intermodulation in cascade of two stage

L+++= 23,3

21

21

22,3

21

21,3

23

11

IPIPIPIP AAAAβαα

• General expression for three or more stages


Intersymbol interference• Linear time-invariant systems can also “distort” a signal if they

do not have sufficient bandwidth– Example : low-pass filter

• Intersymbol interference (ISI)– Each bit level is corrupted by decaying tails created by

previous bits– Leads to higher error rate in the detection of random

waveforms– Reduce ISI methods : pulse shaping in the transmitter

equalization in the receiver


Pulse shape• It is less susceptible to interference with its shifted replicas

– All other pulse go through zero at the point when the present pulse reaches its peak

– If the bit stream is sampled at t=KTs, no ISI exists

• For a pulse shape, p(t)

• Nyquist’s condition for the spectrum of a pulse shape that gives no ISI– Using a train of impulses to sample this pulse

0001)(≠=

==kifkifkTp s

11

11)(

)()()(

=

−⇒

=

−∗⇒

=−⋅

∑

∑

∑

ss

ss

transformFourier

s

TkfP

T

Tkf

TfP

tkTttp

δ

δδ


Raised cosine filter• Pulse shape design problem

– The filter required to produce the rectangular spectrum becomes quite complex in both the transmit and receive paths

– The sinc waveform decays slowly with time, introducing considerable ISI in the presence of timing errors in the sampling command

• A pulse shape often employed in Nyquist signaling is related to a “raised consine” spectrum– Raised-consine pulse

( )( )

( )

s

sss

ss

ss

s

s

s

s

Tf

Tf

TTfTT

TfTfP

TtTt

TtTttp

210

21

21

21cos1

2

210)(

/41/cos

//sin)( 222

α

ααααπ

ααπα

ππ

+>=

+<<

−

−−+=

−<<=

−=

– 0 < α< 1 is the “rolloff” factor


Raised cosine filter (cont.)• p(t) decays faster than a sinc function

• For α = 0, p(t) reduces to a sinc function• P(f) is similar to box spectrum but with smooth edges

• Trade-off in the choice of α– Decay rate in the time domain and the excess bandwidth in the

frequency domain

– Typical values of α are between 0.3 and 0.5

• Raise-cosine filtering


Random process• Random processes are an integral part of communications,

used to represent both signals and noise

• A random (actually a “stochastic”) process can be defined as “a family of time functions”

• Questions– How is a random process characterized ?

– What aspects of its statistics are important ?

– How are these aspects incorporated in system analysis ?

– Ordinary signal generator : output waveform

which is predictable and well-defined

– Random signal : e.g., voice going through a

phone line, statistics output obtained from

multiple measurements


Review of Random Processes• Probability : P{x=5} or P{x≦5}• Random variable

– Discrete processes : tossing coins– Continuous processes : temperature, noise voltage, and received signal

amplitude or phaseContinuous random variable X, representing a random process with real continuous samples x, where -∞<x< ∞0≦P{X ≦x0} ≦1

• Cumulative distribution function (CDF), FX(x)

)()(}{)5()()()4(

0)()3(1)()2(

0)()1(}{)(

1221

2121

xFxFxxxPxxifxFxF

FFxF

xXPxF

XX

XX

X

X

X

X

−=≤<≤≤

=−∞=∞≥

≤=


Review of Random Processes (Cont.)• Probability density function (PDF), fx(x)

xallfordxxdFxf X

x ,0)()( ≥=

– Properties

1)()3(

)()()2(

)()()(}{)1( 2

11221

=

=

=−=≤<

∫∫

∫

∞

∞−

∞−

duuf

duufxF

dxxfxFxFxxxP

X

x

XX

x

x XXX

• Joint CDF associated with random variables X and YFXY(x,y)=P{X≤x and Y≤y}– Properties

∫∫

∫ ∫∞

∞−

∞

∞−==

=≤<≤<

∂∂∂

=

dyyxfxfdxyxfyf

dxdyyxfyYyandxXxP

yxFyx

yxf

xyxxyy

x

x

y

y xy

xyxy

),()(),()(

),(}{

),(),(

2

1

2

12121

2

– X and Y are statistically independent fxy(x,y)=fx(x)fy(y)


Review of Random Processes (Cont.)• Some important probability density functions

– Uniform distribution PDF

– Gaussian PDF

where m is the mean of the distribution, and σ2 is the variance

– Rayleigh PDF

bxaforab

xfx ≤≤−

=1)(

∞<<∞−= −− xforexf mxx

22 2/)(

221)( σ

πσ

∞<≤= − rforerrf rr 0)(

22 2/2

σ

σ

• Expected value– Discrete random variables

– Continuous random variables

– Satisfy linear operation

}{}{1∑=

===N

iii xXPxXEx

∫∞

∞−== dxxxfXEx x )(}{

}{}{}{)2(}{}{)1(

YEXEYXEXcEcXE

+=+=


Review of Random Processes (Cont.)• Expected value of a function of a random variables

– If we have a random variables x, and a function y=g(x) that maps values from x to a new random variables y

– Discrete random variable

– Continuous random variable

• Nth moment of the random variable, X

}{)()}({}{1∑=

====N

iii xxPxgxgEyEy

∫∞

∞−=== dxxfxgxgEyEy x )()()}({}{

∫∞

∞−== dxxfxxEx x

nnn )(}{

• Variance, σ2 : second moment of X after subtracting the mean

of X ( ){ } ( )2222

222

}2{

)(

xxxxxxE

dxxfxxxxE x

−=+−=

−=−= ∫∞

∞−σ

– The root-mean-square (rms) value of the distribution is σ– If a zero-mean random voltage is represented by the random variable x,

the power delivered to a 1Ω load by this voltage source will be equal to the variance of x


Review of Random Processes (Cont.)• Expexted value of the joint PDF

• Expexted value of two independent random variables

• Autocorrelation : how rapidly the sample values vary with time

∫∞

∞−+= dttxtxR )()()( * ττ

– R(0)≥R(τ), and R(τ)= R(-τ)

– R(0) is the normalized energy of the signal

– For stationary random processes, such as noise processes

R(τ)=E{x*(t)x(t+τ)}

}{}{)()(}{ yExEdyyyfdxxxfxyExy yx === ∫ ∫∞

∞−

∞

∞−

∫∞

∞−== dxdyyxfyxgyxgEyxg xy ),(),()},({),(


Review of Random Processes (Cont.)• For stationary random processes

– Power spectral density (PSD) (frequency domain)

∫∞

∞−

−= ττω ωτdeRS j)()(

– Find the autocorrelation from a known PSD

∫∞

∞−= ωω

πτ ωτdeSR j)(

21)(

– For a noise voltage, v(t), the PSD, Sv(ω), represents the noise power density in the spectral domain, assuming a 1Ω load resistor

WdffSdSRtvEtvP vvL ∫∫∞

∞−

∞

∞−===== )2()(

21)0()}({)( 22 πωωπ


Thermal Noise• Thermal noise, also known as Nyquist, or Johnson, noise, is

caused by the random motion of charge carriers– Generated in any passive circuit element that contains resistors, lossy

transmission lines, and other lossy components

• Noise Voltage– (a) A resistor at temperature T produces the noise voltage vn(t)

– (b) The random noise voltage generated by a resistor at temperature T

kTBRVn 4=Where k=1.380x10-23 J/K is Boltzmann’s constant

T is the temperature, in degree Kelvin (K)B is the bandwidth, in Hz, R is the resistance, in Ω


Thermal Noise (Cont.)• Noise power : defined as the maximum power that can be

delivered from the source to a load resistor– The load is conjugately matched to the source

kTBR

VP nn =

=

12

2• B ↓ ⇒ PL↓

• T ↓ ⇒ PL↓

222)(

)(

0nkTBPS

dfSP

nn

B

B nn

===

= ∫−ω

ω• PSD

• Autocorrelation )(222

1)( 00 τδωπ

τ ωτ ndenR j == ∫∞

∞−

• PDF (white gaussian noise) 22 2/

221)( σ

πσn

n exf −=


Statistical Ensembles• Characterization : e.g., the noise voltage across a resistor,

requires a “doubly infinite” set : an infinite number of measurements, each for an infinite length of time– The large of set of resistor noise voltage is called an “ensemble,” and

each of the waveforms is called a “sample function”

• How do we measurement the average value of the noise voltage of a resistor ?

∫+

−∞→=

2/

2/)(1lim)(

T

TTdttn

Ttn

– This notion of dc component of a random signal is called the “time average”

– Another definition : averaging over sample function

Called “ensemble average”P(n) : probability density function of the process

∫+

−∞→=

2/

2/

22 )(1lim)(T

TTdttn

Ttn

∫+∞

∞−= dnnPtntn n )()()(

∫+∞

∞−= dnnPtntn n )()()( 22 Called “mean square” power (with respect to

a 1-Ω resistor)


• Assume binary signals are transmitted in the presence of bandlimited white guassian noise

Basic Threshold Detection

)()()( tntstr +=– r(t) : received signal, s(t) : transmitted signal voltage, – n(t) : noise voltage, zero mean and variance σ2

• Input signal and noise voltagefor a basic threshold detection system

• Possible outcomesof threshold detection


• For gaussian PDF noiseProbability of Error

{ }

∫∫ ∞−

−−

∞−==

<+==

2/

2

2/)(2/

00)1(

0

220

0

2)(

2/)()(

v vrv

r

e

dredrrf

vtnvtrPP

πσ

σ

– Using the change of variable 20 2/)( σrvx −=

20

0)1(

221

0

2

σπvxwheredreP

x

xe == ∫

∞ −

∫∞ −==x

ue duexerfcwherexerfcP

22)()(21

0)1(

π– By a similar analysis we can find– The probability of error dependent on the ratio v0/σ (signal-to-noise

ratio, SNR)– Since erfc(x) decreases monotonically with x, large SNR results in lower

probability of error

)1()0(ee PP =


Probability of Error (Cont.)• Graphical interpretation of

the probability of error forthreshold detection

• Probability of error versusSNR for threshold detection


Measurement of Spectrum• Apply the signal to a bandpass filter with a 1-Hz bandwidth

centered at f and measure the average output power over a sufficiently long time (1 second)

∫ −=

=∞→

T

T

T

Tx

dtftjtxfXwhere

TfX

fS

0

2

)2exp()()(

)()( lim

π

• Algorithm for PSD estimation

Many sample functions Spectrum calculation

Spectrum average


Measurement of Spectrum (Cont.)• Two-sided spectrum

– Sx(f) is an even function off for real x(t)

– The total power carried by x(t) in the frequency range [f1 f2]

dffSdffSdffSf

f x

f

f x

f

f x ∫∫∫ =+−

−

2

1

2

1

1

2

)(2)()(

• One-sided spectrum


• Equivalent noise temperature of an arbitrary white noise source

Equivalent Noise Temperature

kBNTe 0=

• Equivalent noise temperature of a noisy amplifier– Assume Ni=0, and N0 will be due only to the noise generated by the

amplifier itself

GkBNTe 0=– Ideal noiseless amplifier with a resistor at a temperature

– Input equivalent noise source Ni=kTeB


Noise in Linear System• In wireless radio receiver, both desired signals and undesired

noise pass through various stages, such as RF amplifiers, filters, and mixers– Study the general case of transmission of noise through a linear system

• Autocorrelation and power spectral density in linear systemX(t)

Rx(τ), Sx(ω)

y(t)

Ry(τ), Sy(ω)

h(t)H(ω)

( )

( ) ( ) ( )

( ) ( ) ( )ωωω

ττττ

τττ

ττ

xy

xx

y

SHS

RhhdudvvuRvhuh

dudvutxutxEvhuhtytyER

duutxuhty

duutxuhty

2

)()()(

)}()({)()()}()({

)()()(

)()()(

=

⊗−⊗=−+=

−+−=+=

−+=+

−=

∫ ∫∫ ∫

∫∫

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

∞

∞−


Noise in Linear System (Cont.)• Gaussian white noise through an ideal low-pass filter

– ni(t) and no(t) : noise and signal voltages in the time domain– Ni and No : for average powers of noise and signals

– Since the input noise is white, the two-sided PSD of the input noise is constant )(

2)( 0 fallnfSni =

– Output PSD

– Output noise power

∆>

∆<==fffor

fffornfSfHfS

io nn||0

||2)()()(02

( ) 00 )(20

fnfSfN n ∆=∆= proportional to the filter bandwidth

– Noise shaping in a linear system


Noise in Linear System (Cont.)• Gaussian white noise through an ideal integrator

– ni(t) and no(t) : noise and signal voltages in the time domain– Ni and No : for average powers of noise and signals

ni(t)

Ni

n0(t)

N0

( )∫T

dt0

' ( ) ( )Tjej

H ω

ωω −−= 11

( ) ( ) ( ) ( )( )

( )

2sin

2

sin2

)(2

sinsin2/sin4

cos2211

022

0

22020

0

22

2

2

2

2

22*2

TndxxxTn

dffTfTTndffHnN

fTfTT

ffTT

TeeHHHTjTj

=

=

==

===

−=

−−==

∫

∫∫∞

∞−

∞

∞−

∞

∞−

−

ππ

π

ππ

ππ

ππ

ωω

ωω

ωωωω

ωω


Noise in Device• Thermal in device

= mn gkTI

3242

– The factor 2/3 may need to be replaced with higher values for L < 1 μm

– The distributed gate resistance of MOSFETs also contributes thermal noise

• Shot noise in deviceqIIn 22 =

– q is the charge of an electron and I the average current

• Flicker noise in device– Arise from random trapping of charge at the oxide-silicon interface of

MOSFETs

– Note that : nonlinearity or time variance in circuits such as mixers or oscillators can translate the 1/f –shaped spectrum to the RF range

fWLCKVOX

n12 =


Input-Referred Noise• Representation of noise by input noise generators

• Example : assume one dominant source of thermal noise– MOS amplifier – Equivalent input noise generators

2222222nDinnmnDnm IZIgandIVg ==

)3/(8),3/(8

324

222

2

inmnmn

mnD

ZgkTIgkTV

gkTI

==⇒

=

If |Zin| → ∞, In2 → 0, and Vn2 is

sufficient to represent the noise


Noise Figure (NF)• Define : NF=SNRin/SNRout

– SNRin and SNRout are the signal-to-noise ratios measured at the input and output

– If a system has no noise, then SNRout=SNRin, regardless of the gain

( )[ ]( )[ ]( )

( )

( )s

snn

RS

snn

RS

snnRS

snnRS

in

vsnnRS

invout

RS

inin

kTRRIV

VRIV

VRIVVNF

RIVV

V

ARIVV

VASNR

VVSNR

41

1

2

2

2

2

22

22

2

2222

222

22

22

++=

++=

++=

++=

++=

=

α

α

αα


Noise Figure (Cont.)• For simulation purposes

( )

( )[ ]

s

outn

s

snns

s

snns

kTRAV

kTRARIVkTRA

kTRRIVkTRNF

41

414

44

2

2,

2

22

2

=

++=

++=

– Where A=αAv and V2n,out represents the total noise at the output

• Calculation of noise figure of resistor RP( )

( ) ( )P

S

SP

PSPS

PS

Pv

PSoutn

RR

kTRRRRRRkTNF

RRRA

RRkTV

+=+

=

+=

=

14

1||4

||4

2

2

2,

– RP ↑⇒ NF ↓ ⇒ not coincide with that for maximum power transfer (RS=RP)


Noise Figure (Cont.)• Another example

– Feedback amplifier with input match

– Noise equivalent circuitNeglecting body effect, channel length modulation, parasitic capacitance

DmmSin RggRR

12 111

+==

The noise current of M2 flows through RS/2,

Thus the output noise voltage

( ) DmSnn RgRIV 12 2/=

By miller effect

( )SmD

out RgRR 212

+=

The noise current of RD and M1 is multiplied by the output resistance of the circuit


Noise Figure (Cont.)

• The noise figure

( )Sm

Smm

DSm

Sv

outn

RgRgg

RRg

kTRAV

NF

21

22

12

2

2,

1321

321

41

+

+++=

=

( ) ( )

( )22

2

1

221

22

221

22

221

2

221

222

221

2,

143

8432

14

41

414

SmD

mD

DmSmDmS

SmD

nRD

DmSnDmSoutn

RgRkTgRkT

RgRkTgRgkTR

RgRII

RgRIRgkTRV

+

++

+=

+++

+

=

• The total output noise power

– Subject to the condition gm2RS=(1+gm1RD)-1


• The total noise power at the input of the first stage

Noise figure of cascaded stages

( )( )21

212

2

1

1111

21, ||

Sin

inRS

Sin

inninSninn RR

RVRR

RVRRIV+

+

+

+=

• The total noise power at the input of the second stage

( )2

12

22212

2

21

221

21,

22, ||

+

++

+

=outin

inninoutn

inout

invinninn RR

RVRRIRR

RAVV

• The total output noise power of the cascade2

2

22

22,

2

+

=outL

Lvinntotal RR

RAVV


Noise figure of cascaded stages (Cont.)• The total voltage gain from Vin to Vout

Lout

Lv

inout

inv

inS

intotalv RR

RARR

RARR

RA+++

=2

221

21

1

1,

• The overall noise figure

( ) ( )S

inS

inv

noutn

S

nSnS

SoutL

Lvinn

totvtot

kTR

RRRA

VRIkTR

VRIkTR

kTRRRRAV

ANF

411

44

411

2

1

1

21

2212

211

2

2

22

22,2

,

+

++

++=

+

=

NF of the first stage

– For special case where RS=Rin1=Rout1=Rin2

( )

21

21

21

222

1

14

1

v

Sv

nSntot

ANFNF

kTRAVRINFNF

−+=

++=


Noise figure of cascaded stages (Cont.)• Another derivation : using “available power gain”

– The available output power of stage 1

1

21

2

1

12, 4

1

outv

inS

ininavout R

ARR

RVP

+

=

– The available source power

S

inavsource R

VP4

2

, =

– The noise figure of stage 2 with respect to a source impedance Rout1

1

21

2

1

1

out

Sv

inS

inP R

RARR

RA

+

=

( )

( ) ( )P

RR

S

inS

inv

noutn

S

nSnStot

out

noutnR

ANF

NFkTR

RRRA

VRIkTR

VRIkTRNF

kTRVRINF

out

S

out

14

114

4

41

1

1

,2,12

1

1

21

2212

211

1

2212

,2

−+=

+

++

++=

++=

• For m stage)1(11

21

11)1(1−

−++

−+−+=

mpp

m

ptot AA

NFA

NFNFNFL

LFriis equation :


Noise figure of Lossy Circuits• Example : (a) LC attenuator, (b) lossy circuit matched at input

and output

(a) (b)

Antenna LNA– Circuit for noise figure

calculation– equivalent circuit

S

out

TH

inoutin

outTHoutSinin

RR

VVPPLlosspower

RVPandRVP

2

2

22

/

)4/()4/(

==

==


Noise figure of Lossy Circuits (Cont.)• Output noise voltage

( )22

2, 4

outL

Loutoutn RR

RkTRV+

=

• Voltage gain from Vin to Vout

outL

L

in

THv RR

RVVA

+=

• Noise figure

LkTRV

VkTRNFSTH

inout ==

414 2

2

• Cascade of lossy filter and LNA

( )LNA

LNA

LNAfilttot

NFLLNFL

LNFNFNF

⋅=−+=

−+= −

1

11


Sensitivity

minmin, log10/174 SNRBNFHzdBmPin +++−=

out

RSsig

out

in

SNRPP

SNRSNRNF

/==

outRStotsig SNRNFPP ⋅⋅=,

BSNRNFPP

BSNRNFPP

dBdBHzdBmRSdBmin

outRStotsig

log10|||| min/min,

,

+++=

⋅⋅⋅=

HzdBmkTRkTRPin

SRS /1744

41

−===

• The minimum signal level that the system can detect with acceptable signal-to-noise ratio

– Psig denotes the input signal power and PRS the source resistance noise power

– The overall signal power is distributed across the channel bandwidth, B,

– Assume conjugate matching at the input, we obtain PRS as the noise power that Rs delivers to the receiver


• Upper end of the DR : maximum input level in a two-tone test for which the third-order IM products do not exceed the noise floor– Calculation of IP3

– Maximum input power

Dynamic range (DR)

)( ,, GPPandGPP inIMoutIMinout +=+=

32 ,3 inIMIIP

in

PPP

+=

32 3

max,FPP IIP

in+

=

2

2,

,3

inIMinin

outIMoutinIIP

PPP

PPPP

−+=

−+=

• Spurious-free dynamic range (SFDR)

( ) min3

min3

min,max, 3)(2

32 SNRFPSNRFFPPPSFDR IIPIIP

inin −−

=+−+

=−=

• For example, if a receiver with NF=9dB, PIIP3=-15dBm, and B=200KHz requires an SNRmin=12dB, then SFDR=53dB

Input IM3 = noise floor

F=-174 dBm+NF+10logB


Passive impedance transformation • At RF, we often resort to passive circuits to transform

impedances- from high to low and vice versa or from complex to real and vice versa

• Equivalent series and parallel RC circuits

( )ωSSS CRQ /1= ωPPP CRQ =

1

11

2 +++=⇒

+=

+

sCRsCRsCCRRsCR

sCsCR

sCRR

SSPPSPPSSP

S

SS

PP

P

( ) SPSSS

P CCCwhereRQCR

R ≈=≈≈ 22

1ω

– For s=jω, RPCP=1/(RSCSω2) and RPCP+ RSCS-RPCS=0, Assuming RP>>RS, we have CP≅CS and

If Q is relatviely high (>5) and the band of interest relatively narrow, then one network can be converted to the other


Passive impedance transformation (Cont.) • For high-frequency transformers : exhibit loss, capacitive

coupling between the primary and the secondary– Complicating the design and requiring careful modeling

• Other approaches to impedance transformation– Impedance transformation by means of a capacitive divider

Boost the value of RP by a factor (1+CP/C1)2

( )[ ] ( )[ ]( )

PP

tot

PPeqtot

eqStotPPS

RCCR

CCCCCC

CRRandCRR

2

1

11

22

1

/

/1/1

+≈

+≈≈

≈≈ ωω


Passive impedance transformation (Cont.) – Impedance transformation by means of an inductive divider

Boost the value of RP by a factor (1+L1/LP)2

LS

RS

Leq

RS

PSPSSPSPPP

SSPP

PP

RRsRLsRLsLLRsL

RsCsLRsLRsLsL

+++=⇒

++=+

+

2

11

– For s=jω, RSRP=LPLSω2 and LPRP= LPRS+LSRP, Assuming RP>>RS, we have LP≅LS and

( )SP

P

P

PS LLLwhere

QR

RLR ≈=≈≈ 2

2ω ( ) ( )

( )

PP

tot

Peqtot

S

eqtot

P

PS

RLLR

LLLLRL

RandRLR

2

1

1

22

1

+≈

+≈≈

≈≈ωω


Passive impedance transformation (Cont.) – Transformation of a resistance to a lower value

( ) PSPP

S CCwhereCR

R ≈≈ 21ω

– In the vicinity of resonance, L1 and CS resonate and the network is

approximately equivalent to a resistor equal to 1/(C2Pω

2RP)

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