ECE 590Microwave Transmission for

Telecommunications

Noise and Distortion in Microwave Systems

March 18, 25, 2004

Random Processes

1dx)x(f:unity is

functiondensity y probabilit aunder area totalthen the

1)F( and du)u(f)x(F then 0,)F(- since

dx)x(f)x(F)x(For 0dx

)x(Fd)x(f

(PDF),Function Density y Probabilit

)x(F)x(F}xXP{xsuch that

}xX{P)x(F Function,on Distributi Cumulative

1}xP{X0such that x, values,sample

continuous real with process random repesents X

X

x

XX

x

x

X1X2XX

X

1X2X21

X

0

2

1

Random Processes

PDF][Rayleigh r0for er

)r(f

variance; mean,m where

PDF][gaussian x-for e2

1)x(f

;on]distributi [uniform )ab/(1)x(f sPDF' Basic

)y(f)x(f)y,x(f :Y and X of PDFs theofproduct theis

PDFjoint t then theindependenlly statistica are Y and X If

dxdy)y,x(f}yYy and xXx{P

or )yx,(Fyx

)y,x(f PDF,Joint

}yY andx X{P)y,x(F : variablesrandom Two

22

22

2

1

2

1

/r2r

2

2/)mx(

2X

X

YXXY

x

x

y

y

XY2121

XY

2

XY

XY

Expected Values

(x)dxf)x(g)}x(g{EE{y} y

becomes this variablerandom continuous thefor

}xx{P)g(x)}x(g{EE{y} y then y, variablerandom new a

x tofrom valuesmaps that g(x)yfunction a and variablerandom xIf

(x)dx xfE{X} x : variablesrandom continuousfor and

}xX{PxE{X} x : variablesrandom discreteFor

E{X}or x as X variablerandom of average)or mean (or valueExpected

deviation. standard and variance,mean, theassuch averages

lstatisticaon rely must tic,determinisnon are variablesRandom

x

i

N

1ii

x

i

N

1ii

Expected Values

.xx}xx2xE{x

(x)dx )(}){( variance.the

ofroot square the, ison distributi theof (rms) square-mean -root The

X. ofmean thegsubtractinafter X ofmoment second

thegcalculatinby found is X variablerandom theof , variance,The

(x)dx}{ x: variablerandom theofmoment n theassuch

averages, lstatisticaorder higher find to x g(x)Let

2222

222

2

nth

n

x

xnn

fxxxxE

fxxE

Expected Values

E{x}E{y}(y)dy(x)dxy}E{x, xy

:y and x , variablesrandomt independen For two

y)dxdy(x,),(y)}E{g(x, y)g(x, :PDFjoint theinvolves

variablesrandom twooffunction a of valueexpected The

yx

xy

yfxf

fyxg

Autocorrelation and Power Spectral Density

.de)(R)S(function

ationautocorrel theof ansformFourier tr as defined is ),S((PSD),

density spectralpower theprocesses, random stationaryFor

domainfrequency in thefunction ation autocorrel theof Spectra

)}t(x)t(x{E)R( as defined isfunction ation autocorrel

theprocesses, noise assuch processes, random stationaryFor

)dt(tx)t(x)R( is )(t version xshifted

- timea and x(t)of conjugate theofproduct theof average timethe

as x(t)signal ticdeterminiscomplex aFor ation.autocorrelby Done

. with timesignals random and ticdeterminisboth oftion Representa

tj

*

*

Autocorrelation and Power Spectral Density

v(t).of PSD theis )(S where

df)f2(Sd)(S2

1)0(R)}t(v{E)t(vP

as found becan

load 1 a assuming domain, )(frequency spectral in thedensity power

noise therepresentsdensity spectralpower the voltage,noise For

de)(S2

1)R( :PSDknown

a fromation autocorrel thefind toused becan transforminverse The

v

vv22

L

j

Noise in Microwave Circuits• Result of random motions of charges or

charge carriers in devices and materials

• Thermal noise (most basic type)– thermal vibration of bound charges (also called

Johnson or Nyquist noise)

• Shot noise – random fluctuations of charge carriers

• Flicker noise– occurs in solid-state components and varies

inversely with frequency (1/f -noise)

Noise in Microwave Circuits

• Plasma noise– random motion of charges in ionized gas such

as a plasma, the ionosphere, or sparking electrical contacts

• Quantum noise– results from the quantized nature of charge

carriers and photons; often insignificant relative to other noise sources

Noise power and Equivalent Noise Temperature

band. microwave he through tup sfrequenciefor validis This

ion)approximat Jeans-(Rayleigh kTBR4V toreduces This

.in is resistance R, and Hz,in bandwidth B K,in is T constant,

sBoltzmann'k constant, sPlanck'h where1e

hfBR4V

law,radiation body black sPlanck'by

given valuerms nonzero abut value,average zero a has voltageThis

. terminalsitsat n fluctuatio voltagesmall a produce motions random

These T. re, temperatu the toalproportionenergy kinetic awith

motion randomin are electrons hein which tresistor a Consider

n

kT/hfn

Noise power and Equivalent Noise Temperature

22 2/n

2n

0nn

2n

2n

e2

1)n(f

:mean zeroith gaussian w is noise whiteof PDF

B) toB-from is range (frequency noise (white) thermalofdensity

spectralpower sided- two theis which 2/n2/kTB2/P)(S

:frequency oft independen be alsomust density

spectralpower thesofrequency oft independen isPower density.

spectralpower theof in terms drepresente be alsocan power noise The

GHz 1000f0 from

freq.) all of makeup noise, (whitefrequency of ceindependen theNote

power noise availmax thegives which kTB)R4/(VRIP

power maximumfor resistance equal of load a togconnnectinBy

:circuit equivalentvenin eTh itsby replaced becan resistor Noisy

Noise in Linear Systems

)()()()(R

:responseinput theofation autocorrel theof

nconvolutio double by thegiven isoutput theofation autocorrel

that theshowsresult This .v)dudv-(th(u)h(v)R

v)dudv-u)x(t-x(th(u)h(v)E{)}()({)(R

is y(t) ofation autocorrel theso v)dv-h(v)x(t)y(t

is y(t) of version shifted timeand u)du-h(u)x(ty(t)

is responseoutput theh(t), is system theof response

input the wheresysteminvariant -elinear tim aConsider

y

-

x

-

y

-

-

xRhh

u

tytyE

Noise in Linear Systems

)(S)(H)(S

tosimplifies above then the,d)eth()(H

h(t) of ansformFourier tr theis )H( Since

.dudvde)(Reh(v)e)u(hd)e(R

find ,dd that so v-u variable,of change awith

dudvdv)e-u(Rh(u)h(v) d)e(R

:)S( of in view R of ansformfourier tr by the

obtained isdensity spectralpower theof in termsresult Equivalent

x

2

y

-

tj-

-

jx

vjuj

-

j-y

- -

j-x

-

j-y

y

Gaussian white noise through an ideal low-pass filter

bandwidth.filter the toalproportion

ispower noiseoutput or the n f)f(S)f2(N

ispower noiseoutput Then the

fffor 0

fffor 2

n

)f(S)f(H)f(S is PSDoutput then

f. allfor ,2

n)f(S :constant is

noiseinput theof PSD theso white,is noiseinput The

PSD. sided- two theaccomodate tofrequency negative

and positivefor (defined f frequency, cutoff awith

H(f) function, transfer ah filter wit pass-low aGiven

0n0

0

n

2

n

0n

0

i0

i

Gaussian white noise through an ideal integrator

2

Tndx

x

)xsin(

2

Tndf

fT

)fTsin(

2

Tn df)fH(

2

n

N ispower noiseoutput theand f, allfor ,2

n)f(S : noise

input with whitefT

)fTsin(T

)f(

)fT(sin2/Tsin4

Tcos22e1e1)()HH()H(

time.intervaln integratio theis T where,e1j

1 )H(

function, transfer a with integratoran Given

0

-

2

0

-

22020

00

n

22

2

2

2

2

22

TjTj*2

Tj

i

Mixing of noise: frequency conversion

half.

inpower noise thereduces mixing theso 2

d)t(cos2

)}t({cosE)}t(n{E)}t(cos)t(n{E

)}t(v{EN : v(t)of variance thefrom found becan power output

average The ).t( cos n(t) v(t)ismixer idealized theof

output The n(t). oft independen is and 20 interval on the

ddistributeuniformly is , phase, the where), t cos( is signal

oscillator local The (t)}.E{n power, average of variance

with signal noisegaussian whitedbandlimite a as n(t)Given

22

0

02

2

022

022

20

0

0

22

Mixing of noise: frequency conversion

over time. of variation,

phase ensemble over the is averaging ewhether th

obtained ispower output average same or the

2

dt tcos2

dt)}t(v{ET

1N

tcos}tcos)t(n{E)}t(v{E

:phase the

withoutconsidered is signal oscillator local theIf

2/2

0

0220

T

0

20

022

0222

0

Basic Threshold Detection

used.been hasfunction density y probabilitgaussian a where

dr2

edr)r(f}2/v)t(nv)t(r{PP

mitted.been trans has "0" a when Pfor Similarly

mitted.been trans has "1" awhen detection in error an of

y probabilitP :error ofy Probabilit /2;vr(t) :threshold

;0s(t) "0" Binary 1;s(t) "1" Binary

. of varianceandmean zero has n(t) wheren(t)s(t)r(t)

noise.gaussian whitedbandlimite of presence in the

ed transmittsignalsbinary with systemion Communicat

2/v

2

2/)vr(2/v

r00(1)e

(0)e

(1)e0

2

0

220

0

Basic Threshold Detection

voltage.noise theof valuerms theis and

voltagesignal maximum vsince SNR ratio,-to-noise tosignal the

considered becan /v ratio The .2/ vof thresholda from resulting

symmetry thefrom P)22/v(erfc 5.0)erfc(x 5.0P

.due

2erfc(x) asritten function werror ary complement theto

related is intergral this.22/v x where,dxe

P Then

2/)vr( let x form standard a toP reduce To d.transmitte

being is "1"binary a when vof mean value abut with gaussian,

also is r(t) signal receive themean, zeroith gaussian w is n(t) Since

0

00

(0)e

200

(1)e

x

u

200x

x(1)e

20

(1)e

0

2

0

2

Graphical Representation of Probability of Error for Basic Threshold Detection

Noise Temperature and Noise Figure

.GkB/NT with thesourcenoisy aby driven being

amplifier noiseless a toequivalent is source noiselessan by

drivenamplifier noisy A .Kelvin degreesin expressed T

re, temperatunoise equivalentby their zedcharacteri becan

systemsreceiver and components wirelessso : kB/NT

load. a to Npower noise a delivering R resistance a

toequivalent is source theSo frequency. offunction anot

white,is noise when valid-re temperatunoise tEquivalen

0e

e

0e

0

Noise FigureNoisy Rf and microwave components can be characterized by anequivalent noise temperature.

An alternative is the noise figure which is the degradation of the signalto noise ratio between the input and the output of the component, or F = (Si/Ni)/ (S0/N0) 1. The input noise power, Ni = k T0 B; Pi= Si+ Ni ; P0= S0+ N0; S0= G Si; N0= kGB(T0+ Te) ;

Noise FigureSo F = [(Si/ k T0 B)]/ [(G Si / k G B (T0+ Te)] =(T0+ Te)/ T0 = 1 + Te/ T0 1.

Or the temperature of the noisy network Te = (F - 1) T0 .

Let Nadded = noise power added by the network, the output noisepower, N0= G (Ni+ Nadded)

So F = [(Si/ Ni)]/ [(G Si / G (Ni+ Nadded)] = 1 + Nadded/ Ni

Noise Figure of a Lossy LineLossy transmission line (attenuator) held at a physical temperature, T.Power Gain, G<1 so power loss factor = L =1/G>1

If the line input is terminated with a matched load at temperature T, then the output will appear as a resistor of value R and temperature T.

Output Noise power is the sum of the input noise power attenuated through the lossy line plus the noise power added by the lossy line itself .

Noise Figure of a Lossy LineSo the output Noise power, No = kTB = G(kTB + Nadded), whereNadded is the noise generated by the line. Therefore,

Nadded = {(1/G) - 1 }kTB = (L-1) kTB

The equivalent noise temperature Te of the lossy line becomes:

Te = Nadded / KB = (L - 1) T; and the noise figure is

F = 1 + Te / T0 = 1 + (L - 1) T / T0

Noise Figure of Cascaded ComponentsConsider a cascade of two components having power gains G1 and G2, noise figures F1 and F2 and noise temperatures T1 and T2. Find overall noise figure, T and noise temperature T of the cascadeas if it were the single component with Ni = k T0 B.

Using noise temperatures, the noise power at the output of the firststage is N1= G1 k B T0+ G1 k B Te1;

and the output at the second isN0= G2 N1+ G2 k B Te2 = G1 G2 k B (T0 + Te1 + Te2 / G1)

Noise Figure of Cascaded Components

For the equivalent single system: N0= G1 G2 k B (T0 + Te)

So the noise of the cascade system is Te = Te1 + Te2 / G1

Recall F = 1 + Te/ T0 so the cascade system

F = 1+ Te1/ T0 + Te2 / (G1 T0) = F1 + ( F2 - 1) / G1;

more generallyTe = Te1 + Te2 / G1 + Te3 / (G1G1)F = F1 + ( F2 - 1) / G1+ ( F3 - 1) / G1 G2

Noise Figure of a Passive Two-Port NetworkImpedance mismatches may bedefined at each port in terms ofthe reflection coefficients, as shown in the diagram.

Assume the network is attemperature, T and the inputnoise power is N1 = k T B is applied to the input of the network.

The available output noise at port 2 is N2 = G21 k T B + G21 Nadded

the noise generated internally by the network (referenced at port 1). G21 is the available gain of the network from port 1 to port 2.

Noise Figure of a Passive Two-Port NetworkThe available gain can be expressedin terms of the S-parameters of thenetwork and the port mismatches asG21 = power available from networkdivided by power available fromsource = { |S21|2 (1- | s | 2)}/| 1+S11s | 2(1- | out | 2) and the output mismatch is out = S22+ S12S21s /(1- S11s )From N2=k T B, findNadded = (1/G21-1)k T B, and the equivalent noise temperature isTe = Nadded /kB = T(1- G21)/ G21, and F = (1/G21-1)T/T0

Can apply to examples mismatched lossy line and Wilkinsonpower divider.

Gain Compression

General non-linear network with an input voltage vi and and outputvoltage v0 can be expressed in a Taylor series expansion:

v0 = a0 + a1vi + a2vi2 + a3vi

3 + … where the Taylor coefficients are given by:

a0 = v0 (0) (DC output); {rectifier converting ac to dc}

a1 = dv0 / dvi| vi =0 (linear output) ; {linear attenuator or amplifier}

a2 = d2v0 / dvi2| vi =0 (squared output) ; {mixing and other frequency

conversion functions}

Gain Compression

Let vi = V0 cos 0t then evaluate v0 = a0 + a1vi + a2vi2 + a3vi

3 + …

v0 = a0 + a1 V0 cos 0t + a2 V0 2 cos 2 0t + a3 V0

3 cos 3 0t + …

=( a0 + ½ a2 V0 2 ) + (a1 V0 + ¾ a3 V0

3 ) cos 0t +½ a2 V0

2 cos 20t + ¼ a3 V0 3 cos 30t + …

This result leads to the voltage gain of the signal component atfrequency 0

Gv = v0 (0 )

/ vi (0 ) = (a1 V0 + ¾ a3 V0

3 ) / V0 = a1 + ¾ a3 V0 2

(retaining only terms through the third order)

Gain Compression Gv = v0 (0) / vi (0) = (a1 V0 + ¾ a3 V0

3 ) / V0 = a1 + ¾ a3 V0 2

here we see the a1 term plus a term proportional to the square of themagnitude of the amplitude of the input voltage. The coefficient a3 is typically negative; so the gain of the amplifier tens to decreasefor large values of V0. This is gain compression or saturation.

Intermodulation DistortionFor a single input frequency, or tone, 0, the output will consist of harmonics of the input signal of the form, n 0, for n = 0, 1, 2, ….Usually these harmonics are out of the passband of the amplifier,but that is not true when the input consists of two closely spacedfrequencies. Let vi = V0(cos 1t + cos 2t ); where 1 ~ 2. Recallv0 = a0 + a1vi + a2vi

2 + a3vi3 + … ; hence

Intermodulation DistortionThe output spectrum consists of harmonics of the form, m1+n2

with m, n = 0, 1, 2, 3, … These combinations of the two inputfrequencies are call intermodulation products, with order |m| + |n|.Generally, they are undesirable; however, in cases, for example amixer, the the sum or difference frequencies form the desiredoutputs. Note that they are both far from 1 and 2. But theterms 21 - 2 and 22 - 1 are close to 1 and 2. Which causesthird-order intermodulation distortion.

Third-Order Intercept PointPlot of first and third-order products of the output versus input poweron a log-log plot hence the slopes represent the powers.

Dynamic Range