2009/08/24-28 2009/08/24-28. 2009/08/24-28 fundamentals of antenna theory electromagnetic potentials...

射电天文基础射电天文基础姜碧沩姜碧沩

北京师范大学天文系北京师范大学天文系

2009/08/24-282009/08/24-28 日，贵州大学日，贵州大学

2009/08/24-282009/08/24-28日日射电天文暑期学校射电天文暑期学校

Fundamentals of Antenna Theory Fundamentals of Antenna Theory

• Electromagnetic potentials• Green’s function for the wave equation• The Hertz Dipole• The reciprocity theorem• Descriptive antenna paramters

– The power pattern– The definition of the main beam– The effective aperture– The concept of antenna temperature


Electromagnetic PotentialsElectromagnetic Potentials

DJH

BE

B

D

cc

c14

1

0

4

AB

Maxwell’s equations

AEAEAE ccc

110

1


The Lorentz GaugeThe Lorentz Gauge

c

1ˆ

ˆ AA

0

0

22

2

c

c

A

Neither A nor Φ are completely determined by the definitions. An arbitrary vector can be added to A without changing the resulting B. While E will be affected, unless Φ is also changed. The induced parameter Λ is free.

4

4

22

22

c

ccJAA


Green’s Function for the Wave Green’s Function for the Wave EquationEquation

The form of wave equation

),(1

22 tf

vx

cv

Helmholtz wave equation

),(),()( 22 xx Fk cvk //

Green’s function

),(),()( 22 x'xx'x Gk


SolutionsSolutionsSolution of the wave equation

''

)'

,'(

4

1),( 3xd

tv

ft

xx

xxx

x

Solution of the Maxwell equation

''

)'

,'(),( 3xdv

t

ctA

xx

xxxJ

x

''

)'

,'(1

),( 3xdvt

t

xx

xxx

x


The Hertz DipoleThe Hertz Dipole

• An infinitesimal dipole with a length Δl and a cross-section q

rti

r

lI

cAz

2exp


Rotor or curl of vector in cylindrical Rotor or curl of vector in cylindrical coordinatescoordinates

1ˆ

ˆ

ˆ

z

z

z

AAe

z

A Ae

z

Ae A

A


Rotor or curl of vector in spherical Rotor or curl of vector in spherical coordinatescoordinates

(sin )1ˆ

sin

( )1ˆ

sin

1ˆ

r

r

r

A Aer

rAAe

r r r

Ae rAr r

A


Electric and Magnetic FieldsElectric and Magnetic Fields

rti

ririr

lIiE

2exp

2

121

1sin

2 2

rti

ririr

lIiEr

2exp

2

121cos2

2 2

rtirir

lIiH

2exp

21

1sin

2


Radiation FieldRadiation Field

rti

r

lIiE

2

expsin

2

rti

r

lIiH

2

expsin

2

2

22* sin

24)Re(

4 r

lIcc

HES

2

23

2

lIc

P


The Reciprocity TheoremThe Reciprocity Theorem

• The antenna parameters for receiving and transmitting are the same

2112 IUIU


Descriptive Antenna ParametersDescriptive Antenna Parameters--The Power Pattern--The Power Pattern

),( PS

),(1

),(max

n PP

P

The power pattern

Normalized power pattern

Directivity

dP

PG

),(

),(4),(

The Hertz dipole 2sin),( nP 2sin2

3),( G


The Definition of the Main BeamThe Definition of the Main Beam

4

2

0 0

nnA sin),(),( ddPdP

lobemain

nMB ),( dP

A

MBB

Amax

4

GD

Main beam efficiency


BeamwidthBeamwidth

MB

12EWMB


The Effective ApertureThe Effective Aperture

S/ee PA gAe AA

2e

max

4

A

GD

2Ae A


The Concept of Antenna TemperatureThe Concept of Antenna Temperature

dPBAW ),(),(2

1ne

AkTW

dP

ddPTT

),(

sin),(),(),(

n

00nb

00A


ExamplesExamples• The full width half power (FWHP) angular size,θi

n radians, of the main beam of a diffraction pattern from an aperture of diameter D isθ≈1.02λ/D . (a) Determine the value of θ, in arc min, for the human eye where D=0.3cm, at λ=5×10-5cm. (b) Repeat for a filled aperture radio telescope, with D=100m, atλ=2cm, and for the very large array interferometer (VLA), D=27km andλ=2cm.


Cont’dCont’d

• Hertz usedλ≈26cm for the shortest wavelength in his experiments. (a) If Hertz employed a parabolic reflector of diameter D ≈2m, what was the FWHP beam size? (b) If the Δl ≈0.3cm, what was the radiation resistance from equation (5.42)? (c) Hertz’s transmitter was a spark gap. Suppose the current in the spark was 0.5A, what was the average radiated power from equation (5.41)?


Cont’d Cont’d

• For the Hertz dipole, P(θ)=P0sin2 θ. Use equation (5.51), (5.53) and (5.59) to obtain ΩA, ΩMB, ηB and Ae.

• (a) Use the equations in previous problem, together with equations (5.51), (5.60), and (5.63) to show that for a source with an angular size <<the telescope beam, TA=SνAe/2k. Use the relations above and equation (5.64) to show that TA=ηBTB where TB is the observed brightness temperature.


• (b) Suppose that a Gaussian-shaped source has an actual angular size θs and actual peak temperature T0. This source is measured with a Gaussian-shaped telescope beam size θB. The resulting peak temperature is TB. The flux density Sν, integrated over the entire source, must be a fixed quantity, no matter what the size of the telescope beam. Use this argument to obtain a relation between temperature integrated over the telescope beam TB

Show that when the source is small compared to the be

am, the main beam brightness temperature and further the antenna temperature

.

2s

2B

2s

0B

TT

2BS0B )/( TT

2BS0BA )/( TT


• Suppose that a source has T0=600K, θ0

=40”, θB=8’, and ηB=0.6, what is TA?


Signal Processing and Receivers Signal Processing and Receivers

• Signal processing and stationary stochastic processes

• Limiting receiver sensitivity• Incoherent radiometers• Coherent receivers• Low-noise front ends and IF amplifiers• Summary of presently used front ends• Back ends: correlation receivers, polarimeters

and spectrometers


Signal Processing and Signal Processing and Stationary Stochastic ProcessesStationary Stochastic Processes

• Stationary stochastic processes x(t)

• Probability density, expectation values and ergodicy

• Autocorrelation and power spectrum

• Linear systems

• Gaussian random variables

• Square-law detectors


Probability Density Function pProbability Density Function p(x)(x)

• Definition– The probability that at any arbitrary momen

t of time the value of the process x(t) falls within an interval (x-½dx, x+ ½ dx)


Expectation ValuesExpectation Values• Of the random variable x

• Of a function f(x)

• Of the transformation y=f(x)

dxxxpxE )(}{

dxxpxfxfE )()()}({

)('

)()()(}{xf

dxxpxfdyyypyE xy


Mean Value and Time AverageMean Value and Time Average

• Mean value

• Dispersion

• Time average

}{xE

}{}{ 222 xExE

T

TT

dttxfT

xf ))((2

1)( lim


Autocorrelation and Power Autocorrelation and Power SpectrumSpectrum

• Fourier transform

• Mean-squared expected value

T

T

ti

T

dtetxX2/1

2/1

2)()( lim

T

T

tiT dtetxX

2/1

2/1

2)()(

T

T

T

T

stiT dsdtetxsxEXE

2/1

2/1

2/1

2/1

)(22)()(})({


Dirichlet’s TheoremDirichlet’s Theorem

b

a

b

a

b

y

x

a

dxyxfdydyyxfdx ),(),(


Cont’dCont’d• ACF: Auto Correlation Function

• PSD: Power Spectral Density

• Wiener-Khinchin theorem– ACF and SPD of an ergodic random proce

ss are Fourier transform pairs

)}()({)}()({)( txtxEsxsxER TTT

2)(

1lim)( XE

TS T

T

deRS i2)()(

deSR i2)()(


Linear SystemsLinear Systems

dthxdhtxty )()()()()(

dtethH ti 2)()(

dhtxEtyE )()()(

xy H )0(


ACF and PSDACF and PSD

)()()()( hhRR xxyy

dtdhthtRR xxyy

)()()()(

2)()()( HSS xy


Gaussian Random VariablesGaussian Random Variables

• Normally distributed random variables or Gaussian noise– Probability density distribution is a Gaussia

n function with the mean μ=0

• FT of a Gaussian is also a Gaussian

22 2/

2

1)(

xexp

0xE 22 xE

1 t


Square-law DetectorSquare-law Detector

)()( 2 taxty


Probability Density PProbability Density Pyy(y) (y)

)(xy

)('

)(

)('

)(

)('

)()(

2

2

1

1

n

nxxxy x

xp

x

xp

x

xpyp

)()()( 21 nxxxy


For a Gaussian Variable x(t)For a Gaussian Variable x(t)

0

0 02

exp2

1)( 2

yfor

yfora

y

ayypxxy

)0(33)(

)0()()(22422

22

xx

xx

RaatyE

aRataxEtyE

)(2)0()( 222 xxy RRaR

')'()'(2)()0()( 222 dSSaRaS xxxy


Limiting Receiver SensitivityLimiting Receiver Sensitivity

• Radio receivers are devices that measure the spectral power density

• Basic units– The reception filter with the power gain transfer fu

nction G(ν) defining the spectral range to which the receiver responds

– The square-law detector used to produce an output signal that is proportional to the average power in the reception band

– The smoothing filter with the power gain transfer function W(ν) that determines the time response of the receiver


The Minimum Noise Possible The Minimum Noise Possible with a Coherent Systemwith a Coherent System

• Heisenberg uncertainty principle

• Coherent system– Phase reserved

• Minimum noise

– For incoherent detectors, this limit does not exist since phase is not reserved

– In the cm and even mm wavelength regions, this noise temperature limit is quite small, e.g. at 2.6mm, it is 5.5K, while at optical wavelengths, it is about 104K

4/htE 2/1 n

k

hT

)minimum(rx


The Fundamental Noise LimitThe Fundamental Noise Limit• Sources of system noise

– Receiver– atmosphere – spillover of the antenna consisting of ground

radiation that enters into the system through far side lobes

– Noise produced by unavoidable attenuation in wave guides and cables

• System temperature

– Addition of all the noise temperature iTTsys


SensitivitySensitivity

• Nyquist sampling theorem– The highest frequency which can be accur

ately represented is less than one-half of the sampling rate. If we want a full 20 kHz bandwidth, we must sample at least twice that fast, i.e. over 40 kHz.

sysT

T


Bandwidth and TimeBandwidth and Time

dG

dG

2

2

)(

)(

2

1

dW

W

)(

)0(

1

sysT

T


Some FiltersSome Filters


Receiver StabilityReceiver Stability

• Large gains of the receiver system of the order of 1014

• Variations of the power gain enter directly into the determination of limiting sensitivity

G

G

T

T

sys


The Dicke SwitchThe Dicke Switch

sys

RA

sys T

TT

G

G

T

T


Generalized System SensitivityGeneralized System Sensitivity2

1

G

GK

T

T


Best Integration TimeBest Integration Time

10

2

G

G

1

1

m


Receiver CalibrationReceiver Calibration

GTTz

GTTz

)(

)(

rxHH

rxLL

1LH

rx

y

yTTT

LH / zzy


Incoherent RadiometersIncoherent Radiometers• Coherent radiometers

– The phase of the received wave field is preserved– A full reconstruction of the wave field is allowed– In the sub-mm and far-infrared region, coherent ra

diometers with high sensitivities and large bandwidths become difficult to build and operate

• Incoherent radiometers– Phase is not preserved– Bolometers

• Offer a wider bandwidth and sufficient sensitivity• Sensitive to all polarizations• A wide range of effects that depend on the intensity of th

e radiation is used for such detector• For astronomical purposes, thermal detectors made of s

emiconductor dominate the field of continuum measurements in the mm and sub-mm regions


Bolometer RadiometersBolometer Radiometers

• Theory– The resistance R of a material varies with

the temperature. When radiation is absorbed by the bolometer material, the temperature varies and this temperature change is a measure of the intensity of the incident radiation

• Characteristics– Intrinsically broadband devices


Device Device

The power from an astronomical source, P0, raises the temperature of the bolometer elements by ΔT, which is much smaller than the temperature T0 of the heat sink. Heat capacity, C, is analogous to capacitance, and the quantity conductance, g, is analogous to electrical conductance, G, which is 1/R. The noise performance of bolometers depends critically on the thermodynamic temperature T0 and on the conductance G. The temperature change causes a change in the voltage drop across the bolometer.


What is a bolometer and how does it work


Requirements for a Bolometer in astronomyRequirements for a Bolometer in astronomy

• Energy balance equation

• Thermal time constant

• Amplitude of the temperature variation

• Requirements– Respond with maximum temperature step to a given power

input– Have a short thermal time constant – Produce a detector noise as close to the theoretical minimum

as possible

PTGdt

TdC

G

PT /te

G

PT GC /

2

0

)2(1 tiG

PT


The Noise Equivalent Power of a BolometerThe Noise Equivalent Power of a Bolometer

• Noise sources– Johnson noise in the bolometer– Thermal fluctuations, or phonon noise– Background photon noise– Noise from the amplifier and load resistor

• Cooling will reduce all of these noise contributions• For ground bolometers the background photon no

ise will determine the noise of the system

BGph 2NEP kT


Currently Used Bolometer SystemsCurrently Used Bolometer Systems

• Large bolometer arrays– MAMBO2

• 117 element array, the IRAM 30m telescope, 1.3mm, angular resolution of 11″, FOV of 240″

– SCUBA• 37 pixel array at 0.87mm and 91 element array at 0.45m

m, the JCMT 15m telescope

• Types of bolometers– Superconducting bolometers– Polarization measurements– Spectral line measurements


Coherent ReceiversCoherent Receivers

• Basic components

• Semiconductor junctions

• Front end

• Back end


Basic ComponentsBasic Components

• Thermal noise of an attenuator– Noise temperature of an attenuator TN

• Cascading of amplifiers

nn

TGGG

TGG

TG

TT S121

3S21

2S1

1SS

111


MixersMixers

• The unit that is performing the actual frequency conversion– In principle any device with a nonlinear

relation between input voltage and output current can be used as a mixer

• Local Oscillator (LO)

• Intermediate Frequency


）（

）（

）（

）（

）（

frequency sum 2)(2sin

frequency difference 2)(2sin

LO of harmonic 2nd 224sin21

signal of harmonic 2nd 224sin21

component DC )(21

LSLS

LSLS

LL2

SS2

22

tVE

tVE

tV

tE

VEI

The thick arrows are the given values. There are two methods to specify the system: (a) when two input frequenciesνL and νS are given, (b) νL and νIF are specified. In this case, signals from both the upper sideband (νL+νIF) and lower sideband (νL–νIF) contribute to the IF signal


Semiconductor JunctionsSemiconductor Junctions


Low-Noise Front Ends and IF amplifiersLow-Noise Front Ends and IF amplifiers• The mixer converts the RF frequency to the fixed IF freque

ncy where the signal is amplified by the IF amplifier. The main part of the amplification is done in the IF. The IF should only contribute a negligible part to the system noise temperature. But because usually some losses are associated with frequency conversion, the first mixer may be a major source for the system noise. Two ways exist to decrease this contribution– Choosing a low loss nonlinear element to mix the signal to a lower

frequency– Placing a low-noise amplifier before the mixer

• Receiver frontends in historical order– Uncooled mixers– Maser amplifiers– Parametric amplifiers– HEMT: High Electron Mobility Transistors– SIS: Superconducting mixers– HEB: Hot Electron Bolometers


Summary of Presently Used Front EndsSummary of Presently Used Front Ends

• Single pixel receiver systems– λ>3mm, HEMT– mm and sub-mm, SIS– λ <0.3mm, superconducting HEB

• Multibeam systems– HEMTs front ends are rather simple systems, there has

been a trend to build many receivers in the focal plane• Effelsberg, Parkes, Lovell, FCRAO

– SIS• IRAM, CSO, Onsala

– Bolometers: cooled GeGa bolometers, TES (Transition Edge Sensors)


Correlation Receivers and PolarimetersCorrelation Receivers and Polarimeters

T

TT

xy

dssysxT

RsysxsysxE

)()(2

1lim

)()()()()(

)cos(2

1

)4cos(2

1

2

1lim)cos(

2

1

)2sin()2sin(2

1lim)()(

yxyx

T

T

yxT

yxyx

T

T

yxT

yx

EE

dttT

EE

dtttT

EEtytx

Cross-correlation function for two ergodic processes


Cont’d Cont’d

• Mixing two partially coherent signals with an LO preserves the correlation between the two signals at the intermediate frequencies

)2/()22(sin)(

)2/()22(sin)(

LLIF

LLIF

yy

xx

tVEty

tVEtx

)cos(2

1)()( 22

IFIF yxyxEEVtytx


Block Diagram of Correlation Block Diagram of Correlation Receiver Receiver


Correlation ReceiverCorrelation Receiver• Components

– The signals from the antenna and from the reference are input to a 3dB hybrid

– The two outputs of the hybrid are amplified by two independent radiometer receivers which share a local oscillator

– The IF signals are correlated

• Advantage – The stability is the same as that of a Dicke receiver

• Limiting sensitivity

2ref

2A212

1UUGGU

2

sysT

T


PolarimeterPolarimeter


4

3

21

21

const 2

const 2

)(const

)(const

zV

zU

zzQ

zzI

4

3

21

21

const 2

const 2

)(const

)(const

zU

zQ

zzV

zzI

Circular polarized wave

Linearly polarized wave


SpectrometersSpectrometers• Spectral information contained in the radiation

field– The receivers must be single sideband– The frequency resolution is usually small– In the kHz range, the time stability must be high– Local oscillator frequency should be stable

• Types – Multichannel Filter Spectrometer– Fourier and Autocorrelation Spectrometer– Acousto-Optical Spectrometer– Chirp Transfer Spectrometer


Multichannel Filter Multichannel Filter SpectrometerSpectrometer

• Multi separate channels simultaneously measure different parts of the spectrum

• Design aims– The shape of the bandpass for the individual chann

els must be identical– The square-law detectors must have identical chara

cteristics– Thermal drifts should be as identical as possible

• Problems – Stability requirement is very high– The flexibility in varying the spectral resolution is lo

w


Fourier and Autocorrelation Fourier and Autocorrelation SpectrometerSpectrometer


Autocorrelation SpectrometerAutocorrelation Spectrometer

• Wiener-Khinchin theorem– Autocorrelation function and SPD are FT p

airs

• Digital autocorrelator• Advantages

– Flexibility• Various spectral resolution or bandwidth

– Stability • long-time integration possible

• Sampling and quantization


One-bit QuantizationOne-bit Quantization

)0)(( 1

)0)(( 1)(

tx

txty


Autocorrelation FunctionAutocorrelation Function

0)()(0)()(

)()()(

txtxPtxtxP

tytyERy

)0(

)(arcsin

2)(

x

xy R

RR

)(2/sin)0()( yxx RRR

arcsin law

van Vleck clipping correction


Basic HardwareBasic Hardware• A filter limits the IF input frequency band from 0 to B• A clipper transforms the signal x(t) into clipped signa

l y(t)• y is then sampled at equidistant time intervals by the

sampler, which multiplies y with a very short pulse at the clock frequency 2B

• The samples are shift at the clock frequency into a shift register

• The shift register content with time delay is compared with the un-delayed sample

• The autocorrelation value Ry is calculated • Ry is converted to Rx


FT of the measured ACFFT of the measured ACF

• Lag window

• Measure PSD

• Frequency resolution

m

mw

0

1)( )2(sinc2)( mmW

)()(~ WSS

m 605.0

0

605.0

N


Hanning WindowHanning Window

• Lag window

• Frequency resolution

otherwise 0

for 2

cos)(

2

Hm

mw

)2sin()2(1

2)2(sin

2H mm

mmm cW

00

121

NNm


SensitivitySensitivity

intsys

2/

tT

T


ImprovementsImprovements

• Multi-bit digitization– For a 2bit or 4 level autocorrelator, the sen

sitivity factor will be replaced by 1.14

• Large bandwidth


Acousto-Optical SpectrometerAcousto-Optical Spectrometerss

• Theory – A sound wave causes periodic density variat

ion in the medium which it passes. These density variations in turn cause variations in the bulk constantsεand n of the medium, so that a plane EM wave passing through this medium will be affected. Such a medium will cause a plane monochromatic EM wave to be dispersed.

– Acousto-Optical Spectrometer


CharacteristicsCharacteristics

• Dispersion angle is proportional to the frequency of the acoustic wave

• Intensity of the diffracted light is proportional to the acoustic power

• Resolution

• Bandwidth

sc

cos v

l

coscos

c

c0 v

LN

cc / vL


Chirp Transform SpectrometerChirp Transform Spectrometer

• In contrast to AOS, the CTS is a device which makes exclusively use of radio technology, instead of both radio and optical technologies. For not too wide bandwidth, CTS is an alternative to AOS.


Pulsar BackendsPulsar Backends

• Pulsars– The pulsar signals change rapidly with time

• Measurements– Determination of average pulse shapes

• Short time constant due to the short pulse period• Narrow bandwidth due to frequency dispersion • Many individual pulses are added together to

obtain a pulse profile

– Searches for periodic pulses with unknown periods


Pulse Dispersion and Dispersion Pulse Dispersion and Dispersion RemovalRemoval

• Pulse dispersion can be described by a transfer function in the time or frequency domain. A filter can be constructed to remove this dispersion for a limited frequency range either by hardware or by software techniques


Pulsar SearchesPulsar Searches

• Rail filtering– Convolution of the received signal with a

matching filter whose impulse response is given by rectangular functions spaced at the assumed period


ExercisesExercises

• The equivalent noise temperature of a coherent receiver Tn which corresponds to the NEP of a bolometer is determined by using the relation NEP=2k Tn(Δν)1/2. For Δν=50GHz, determine Tn for NEP=10-16W Hz-1/2. A bolometer receiver system can detect a 1mK source in 60s at the 3σlevel with the bandwidth of 100GHz. How long must one integrate to reach this RMS noise level with a coherent receiver with a noise temperature of 50K and bandwidth of 2GHz?


ExercisesExercises

• The definition of a decibel, db, is

If a 30db (i.e. gain 1000) amplifier with a noise temperature of 4K is followed by a mixer with a noise temperature of 1000K, what is the percentage contribution of the mixer to the noise temperature of the total if Tsys=Tstage1+Tstage2/Gainstage1 ?

input

outputlog10dbP

P


ExerciseExercise• (a) When observing with a double-sideband

coherent receiver system, an astronomical spectral line might enter from either upper or lower sideband. The upper sideband frequency is 115GHz and the lower sideband frequency is 107GHz. What is the intermediate frequency? What is the local oscillator frequency?

• (b) to decide whether the line is actually in the upper or lower sideband, the observer increases the local frequency by 100kHz. The signal moves to lower frequency. Is the spectral line from the upper or lower sideband?


ExerciseExercise

• Given below is the rms noise as a function of time for an acousto-optical spectrometer used in Chile on the 1.2m telescope of the National University of Chile at the Cerro Tololo Interamerican Observatory. There were 172 calibration measurements between July 1993 and August 1994. All were 10min scans. After baseline subtraction these spectra had an rms noise of 0.151±0.026K, where the uncertainty is the rms scattering about the mean. Averaging the spectra in groups of four, there were 43 spectra with 40min integration. The rms noise was 0.086±0.013K. Next, averaging groups of 16 scans, then groups of 64 scans, and lastly all 172 scans (28.5 hrs of integration), one has the following rms(160min)= 0.056±0.007K, rms(640min)= 0.039±0.004K, rms(1720min)= 0.031K. Over what period of time did the noise in this system follow a t-1/2 relation?


HomeworkHomework

• On two days, labelled as 1 and 2, you have taken data which are represented by Gaussian statistics. The mean values are x1 and x2, with σ1

andσ2. Assume the average is given by x=fx1+(1-f)x2 and the corresponding . Determine the value of f which gives the smallest σ by differentiating the relation for σand setting the result equal to zero. Show that

22

221

22 )1( ff

2222

221

412

1222

21

422

222

21

21

122

21

22

xxx


HomeworkHomework

• For an input voltage signal calculate the Fourier transform, autocorrelation function and power spectrum. Note that this function extends (formally) to negative times. This frequently used generalization allows a simplification of the mathematics.

• Repeat previous calculation for

tAtv 2sin)(

tAtv 2cos)(


HomeworkHomework

• Calculate the power spectrum Sν for the function v(t)=A for –τ/2<t<τ/2, otherwise v(t)=0, by taking the FT to obtain V(ν) and then squaring this. Next, graphically form the autocorrelation function. This is done by (1) reflecting the function about the y axis, (2) shifting the function along the x axis, (3) summing the common area, (4) repeating the process for different values of the shift and (5) plotting the common area as a function of τ. Finally transform this result to obtain Sν. Show that these two methods give the same result.


102)/( GG

HomeworkHomework• What is the minimum noise possible with a coherent receive

r operating at 115GHz? At 1000GHz, at 1014Hz?• If the bandwidth of a receiver is 100MHz, how long must on

e integrate to reach an RMS noise which is 0.1% of the system noise with a total power system? Repeat for a Dicke switched system, and for a correlation system. Now assume that the receiver system has an instability described by . For a time dependence we take , and K=1. On what time scale will the gain instabilities dominate uncertainties caused by receiver noise? If one wants to have the noise decrease as what is the lowest frequency at which one must switch the input signal against a comparison?

2)/()/1(/ GGKTT sysRMS 00 6

1 10

t/1

2009/08/24-28 2009/08/24-28. 2009/08/24-28 fundamentals of antenna theory electromagnetic potentials...

Documents

beamwidth slide

hertz dipole slide

maxwell equation slide

effective aperture slide

radiation field slide

crosssection q slide

wave equation solution

power pattern directivity