a6525 lecture 8 = radio lecture 3 - cornell...

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A6525 Lecture 8 = Radio Lecture 3 �•  Radiometry, brightness temperatures and their

physical meaning •  Manipulating the electric field as a voltage in

receiver systems and relating it to radiation microphysics

•  Practical telescope sensitivity issues: – Radio source confusion

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Mixers �•  Heterodyned systems use mixers •  A mixer is simply a multiplier •  Refers specifically to the product of an RF signal

selected by the antenna and a monochromatic “local oscillator” (LO) signal

•  Purpose: •  Shift the RF signal in frequency for easier processing •  Can shift different RFs to a common “intermediate

frequency” (IF) to simplify electronics –  AM radio

•  Shifting an an IF < RF is easier to digitize •  Analog: a nonlinear device: diodes •  Digital: exact to within number of bits of precision +

accuracy of ADC = analog-to-digital converter 2

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Mixers �

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•  Based on identities

•  Easier:

•  Shift operator in frequency space •  Let A = Fourier representation of RF signal •  Let B = monochromatic signal (local oscillator) •  Multiplication à lower and upper sidebands •  Often one of the sidebands is removed by filtering •  Implemented with analog devices or digitally

2 cosA cosB = cosA+B + cosA�B

2 cosA sinB = sinA+B � sinA�BeiAeiB = ei(A+B)

[eiA + e�iA]⇥ [eiB + e�iB ] = ei(A+B) + e�i(A�B) + c.c.

2 cosA cosB = cosA+B + cosA�B

2 cosA sinB = sinA+B � sinA�B

=)

A simple heterodyned system�

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LNA Bandpass Filter

LO

RFIF

IF bandwidth ΔνIF Centered on upper or

lower sideband

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Spectrum�

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0 ⌫IF ⌫RF�⌫RF �⌫IF

RF spectrum selected by feed antenna + LNA

ΔνRF ΔνRF

0 ⌫IF ⌫RF�⌫RF �⌫IF

IF spectrum after filter ΔνIF ΔνIF

0 ⌫RF

Spectrum after mixer ΔνRF ΔνRF ΔνRF ΔνRF

⌫RF � ⌫LO ⌫RF + ⌫LO�(⌫RF + ⌫LO) �(⌫RF � ⌫LO)

Lower sideband

See NRAO WEB site www.nrao.edu

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I(t) / |v(t)|2

LNA Bandpass Filter

Square-Law Detector Integrator

E(x, t) Single polarization system, one pixel

I(t)

voltage / ˆ

E · pp = unit vector for antenna polarization (x, y, RHCP, LHCP)

Stochastic signal �I

The time-integrated output has fluctuations given by the radiometer equation

�I

hIi=

1p�⌫T

The radiometer equation is a consequence of summing an ensemble of sinusoids

with random phases and the central limit theorem.

hIi

1.  Radiometry �•  Radio telescopes are used as radiometers to measure

power incident on an antenna •  A single reflector antenna + single “feed” antenna at the focal point = a

single pixel •  Significant (Nobel prize) single-pixel science:

–  Discovery of the Cosmic Microwave Background –  Discovery of Pulsars –  Indirect detection of gravitational waves from a pulsar binary

•  Spectroscopy = frequency resolved radiometry •  Sensitivity of a radiometer is characterized in terms of the

system temperature: •  Tsys = Tsky + Treceiver + Tspillover + ...

•  m wavelengths: Galactic noise from synchrotron radiation dominates

•  cm wavelengths: receiver + thermal Galactic noise •  mm wavelengths: receiver + atmospheric noise

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Background Sky Noise

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Radiometry Fundamentals

In contrast to optical and infrared astronomy where it is natural to think of photon detection, radioastronomy generally involves signals formed from large numbers of photons that are, hence, well withinthe classical regime. Radio signals from natural sources are inherently random, and this randomnessdetermines (i.e. limits), in part, the ability of a radio telescope to discriminate individual radio sources.

A single-antenna radio telescope with standard receiver is a radiometer that measures the power levelof the radiation at the frequency ⌫, in the polarization, and in the solid angle to which the telescope issensitive. The ability of the telescope to identify an individual radio source depends on:

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1. Radiometer noise: noise produced by the sky background and by the telescope itself. The skybackground consists of an isotropic 3 Kelvin ‘cosmic’ background (the blackbody radiation leftover from the big bang); a broadly distributed synchrotron background from galactic cosmic rayelectrons; and a thermal background from plasma in the galactic disk. The relative contributions ofthese backgrounds and instrumental noise depend strongly on radio frequency.

2. Radio source ‘confusion:’ radiometer variations caused by the ensemble of sources that are in thetelescope beam at any instant. For a fixed antenna, as the Earth rotates, individual sources passthrough the beam and the number of sources in the beam is a Poisson random variable; thus theradiometer output varies with time. To single out any given source, its flux density must be greaterthan the root-mean-square variation due to confusion.

3. Gain variations: All real world amplifiers and detectors have output levels that vary with time dueto various macroscopic (e.g. temperature) and microscopic (1/f noise) causes.

4. Radio frequency interference (RFI): Sources of RFI range from the ridiculous to the sublime,being caused by car ignitions; arcing in motors and relays (e.g. the Space Science Bldg. eleva-tor); lightning; aircraft and police transmissions; satellites. RFI is noticeably less at night and onweekends, as well as at remote locations and, generally speaking, at higher frequencies (f � 30

GHz).

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Blackbody Radiation. Radiation from a blackbody source is described by a brightness distribution(a.k.a. specific intensity; units = energy/time/area/ frequency/solid angle)

B

⌫

=2h⌫3

c

2

1

e

h⌫/kT � 1(1)

where h = Planck’s constant, ⌫ = frequency, c = speed of light, k = Boltzmann’s constant, and T =temperature. At radio frequencies, h⌫ ⌧ kT , so expanding the exponential into a Taylor series of twoterms, we get the Rayleigh-Jeans approximation

B

⌫

=2kT

�

2, (2)

where � ⌘ c/⌫. It is therefore conventional to refer to radiation intensities in terms of an effectiveradiation temperature. This is done even in cases where the radiation mechanism is known to differfrom blackbody radiation.

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Tb =�2B⌫

2k⇡ �2F⌫

2k⌦s

⌦s ⇡ ⇡✓2s/4

✓s = angular diameter

Brightness temperature

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1.  What is the meaning of Tb?�•  Thermal sources:

–  In some conditions Tb is closely related to the physical temperature in the source (gas clouds, etc.)

•  Nonthermal sources: – Tb is related to the mean particle energy in the source

(equivalent to kT) •  These statements are true for incoherent sources

(particles radiating independently) •  Coherent sources:

–  Jupiter, solar bursts, brown dwarfs, interstellar and circumstellar masers, pulsars

– Tb is much larger than any possible physical temperature or mean particle energy (1042 K for pulsars)

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Radio Transient Phase Space�

Filling phase space with hypothetical new discoveries:

Maximal giant pulse emission from pulsars

Prompt Gamma-ray emission

Evaporating black holes

ETI’s asteroid radar

What else?

Type II

Type III

Jup DAM BD LP944-20

ADLeo

UVCeti

IDV ISS

GRBISS

Tb =S

2k�⇥

Tb =S

2k

✓D

�W

◆2

= 1020.5 KSmJy

✓Dkpc

�GHzWms

◆2

Equivalent brightness temperature (RJ regime):

FRBs

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Radio Telescopes as Radiometers: A radio telescope used as a single antenna with a radiometermeasures power, not brightness or specific intensity. The power level for a radio telescope may beexpressed as an integral

P

⌫

=1

2A

e

Z Z

�⌫

d⌦d⌫ B

⌫

(⌦)Pn

(⌦), (3)

where

1. The factor of 1/2 accounts for the acceptance by a single antenna of only a single polarizationwhereas the radiation field from the source is assumed to be unpolarized (equal power in twoopposite or orthogonal polarizations).

2. Ae

= effective area of antenna ⌘ ⌘⇥ geometric area ⇡ 0.6⇡R2A

, where ⌘ is the aperture efficiency

of the telescope that measures the uniformity by which the main reflector is illuminated by the feedantenna at the focus.

3. B⌫

= brightness distribution of the sky as a function of frequency ⌫ and direction solid angle ⌦.4. P

n

(⌦) = the antenna power pattern. This is the response of the antenna as a function of direction,normalized so that the maximum response is 1 in the boresight direction and < 1 in other directions.The width of the ‘main lobe’ of the power pattern is ✓

A

= FWHM (full width at half maximum)⇡ �/D

A

, DA

= antenna diameter. The solid angle of the main beam is ⌦A

=Rmain beam d⌦P

n

(⌦) ⇡⇡(✓

A

/2)2.5. �⌫ = receiver bandwidth = range of frequencies centered on some frequency ⌫. We always have

�⌫ ⌧ ⌫.6. Equation 3 can be viewed as a convolution integral of the antenna power pattern with the sky. In a

drift scan of the sky, the measured width of the response will be a combination of the beam widthand the source size.

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Extended Sources: For a source with extent ⌦s

� ⌦A

(such as the 3K cosmic background), thebrightness distribution is constant over the telescope beam and over the receiver bandpass, so

P

⌫

⇡ 1

2�⌫A

e

⌦A

B

⌫

. (4)

Recall that Ae

/ D

2A

while ⌦A

/ (�/DA

)2. Their product is therefore independent of the telescopediameter; moreover, it can be shown that A

e

⌦A

= �

2. Using this fact and assuming that the extendedsource radiates as a blackbody, we get

P

⌫

⇡ kT�⌫. (5)

Thus (1) measurement of the power level of an extended source determines the radiation temperatureof the source (the radiation temperature may or may not be related to the physical temperature in thesource); (2) the ability to measure the temperature is just as good for a small antenna as a large one, tothe extent that the source is extended for both. For example, extended 21 cm emission from the Galaxyis as easily detected with a 4 meter antenna as with a 300 meter antenna (Arecibo).

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Point Sources: If ⌦s

⌧ ⌦A

, the source is said to be a point source, since it is unresolvable with thetelescope beam. Now, assuming the antenna power response P

n

to be constant over the source (and thatthe beam center points directly at the source �! P

n

= 1), we have

P

⌫

⇡ 1

2�⌫A

e

⌦s

B

⌫

(max). (6)

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Flux density: Another useful quantity is flux density F

⌫

; (flux = rate at which energy passes through asurface; ‘density’ refers to flux per unit frequency). Flux density has units of energy/time/area/frequency.A useful unit of flux density is the Jansky (named after Karl Janksy who, for Bell Labs in the early1930’s, discovered radio emission from the Galactic center while investigating sources of noise in trans-Atlantic radio communications):

1 Jy ⌘ 10�26 watts

m2 Hz⌘ 10�23 erg

s cm2 Hz. (7)

The flux densities of thousands of radio sources are tabulated, having been determined in sky surveysmade at frequencies from about 25 MHz up to 100 GHz (a range of 4000:1).

The relationship between flux density and brightness distribution is

F

⌫

=

Zd⌦ B

⌫

(⌦). (8)

We can rewrite this expression as

F

⌫

⌘ ⌦s

B

⌫

(max), (9)

where we effectively define the solid angle of the source ⌦s

.

Returning to the telescope response for a point source and using the flux density, we have

P

⌫

=1

2�⌫A

e

F

⌫

. (10)

Note that, unlike an extended source, the power measured from a point source scales linearly with thetelescope area; bigger antennas measure more power.

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I(t) / |v(t)|2

LNA Bandpass Filter

Square-Law Detector Integrator


I(t)

voltage / ˆ


Stochastic signal �I

The time-integrated output has fluctuations given by the radiometer equation

�I

hIi=

1p�⌫T

The radiometer equation is a consequence of summing an ensemble of sinusoids

with random phases and the central limit theorem.

hIi

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Telescope Response in Terms of Effective Temperature: Suppose we assume that the power level wemeasure is due to an extended source at temperature T whereas it actually is due to a point source offlux density F

⌫

. Equating equations (1) and (2) and solving for temperature gives

T =

✓A

e

2k

◆F

⌫

. (11)

The quantity A

e

/2k serves as a conversion factor from flux density to temperature:

G ⌘ A

e

2k= 2.7⇥ 10�3 K/Jy DA = 4m

= 1.5⇥ 10+1 K/Jy DA = 300m (Arecibo) (12)

As an example, a 1 Jy source produces a 15K deflection at Arecibo compared with a system temperaturethat is 40K at 1.4 GHz or 100K at 0.4 GHz. This is a very large deflection.

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System Temperature and the Radiometer Equation: It is convenient to describe the properties ofthe receiver system as well as the radiation field in terms of temperature. The system temperaturemeasures the power level when the telescope is pointed at ‘blank’ sky and includes contributions fromthe sky background and noise in antenna, cable, and receiver components. The system temperaturequantifies the mean power level. All contributions to the radiometer temperature are random in nature,so the receiver output is actually noiselike. This is easily demonstrated by viewing the signal on anoscilloscope or listening to the signal through a loudspeaker. (You can demonstrate this with your FMstereo by listening to the hiss when it is tuned in between two stations; some of this hiss is due tosynchrotron radiation from cosmic ray electrons in our galaxy’s magnetic field).

Radiometer noise in a receiver behaves according to the familiar 1/pN law from statistics. That is, if

we have a receiver with bandwidth �⌫ and if we average the signal output over a time ⌧ , the numberof independent fluctuations that is summed is N = �⌫⌧ . The root mean square fluctuation �

T

of theradiometer divided by the mean temperature T

sys

is the radiometer equation:

�

T

T

sys

=1p�⌫⌧

. (13)

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Demonstration of Radio Spectral Analysis�•  Radio spectrum computed from real-time FFT •  Bandpass defined by several elements in the

receiver chain •  Noise = random ~ “white” noise = radiometer

noise •  RFI = radio frequency interference •  Broad spectral line = Galactic atomic hydrogen •  Effective temperature of the spectral line ~ spin

temperature of H ~ ambient temperature •  Needed for any telescope usage: how do we

calculate noise and signal levels? 24

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3.8 m Space Sciences Radio Telescope 3.8m diameter f/0.4 prime focus optics Single polarization channel RF = 1.4 GHz Three mixing stages Can detect the 21 cm atomic

hydrogen line as well as with the Green Bank Telescope (100m) and Arecibo (305m)

Why?

25 Rooftop Telescope Upgrade

Laura Spitler

January 25, 2011

1 Intro

2 Hardware

Figure 1: Diagram of three panels

2.1 RF panel

2.2 IF1

2.3 IF2

2.4 Gain through system

Table 4 shows the gain of each component, the accumulative gain after component, and the maximuminput power each component can tolerate if such a limit exists.

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Block diagram of “front-end” receiver �

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Figure 3: Overview of RF/IF path and iBOB block diagram

Software for a410

Several scripts and executables are used to interface the iBOB spectrometer with the two standard a410data collection scripts /scripts/chart recorder rspec and /scripts/spectrum analyzer6. Details of their useare provided in DOC2BWRITTEN.

Added to �/bin

start rspec - Two arguments: dwell time num points. Kills any existing gulp process and starts new gulpprocess. Must be ’sudo’ed so a410 is given permission to run it as root.stop rspec - Kills any running gulp process. Must be ’sudo’ed so a410 is given permission to run it as root.load mod - Checks to see if appropriate parallel port modules are loaded correctly and loads them if not.Needs to be ’sudo’ed. Run by iBOB configuration scripts.ch pcapfile - Changes permission on pcap files generated by gulp so non-root users can access them. Needsto be ’sudo’ed.gulpfilterbank rspec - Converts .pcap file to filterbank (.fil) formatted file.bandpass - sigproc program that averages a filterbank file over time. The default is to average the entire fileinto one spectrum. Note the scripts may reference the same program in /../lspitler/mysvn/sigproc-4.3/.

Added to �/script

chart recorder rspec - Chart recorder script modified to use ibob roof specchart recorder.sm - Modified (not added) to include cr3 macro for roof specspectrum analyzer6 - Spectrum analyzer version that uses roof specspectrum analyzer6.sm - SM script modified for roof specrspec get spec - Primary script for getting spectra/power from iBOBstarting HI spec - Programs the iBOB’s personality and configures the software registers. Writes status to”starting ibob.log” and make use of ”load ibob.imp”.

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Digital“back end” receiver �

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Detectability of sources: The level of radiometer noise determines how well we can measure thesystem temperature. More importantly, it determines how well we can determine that, rather thanlooking at just blank sky, the telescope is pointed towards an actual source. In order to be detectable,the source must increase the system temperature by an amount that is distinguishable from the noise.A useful rule of thumb is that a source must increase the temperature by an amount that is 5 times thestatistical variation; i.e.

T

source

⌘ GF

⌫

� 5�T

=5T

sys

(�⌫⌧ )1/2. (14)

Equivalently, the flux density must satisfy:

F

⌫

� 5Tsys

G(�⌫⌧ )1/2. (15)

Clearly, if we maximize the product of bandwidth and time constant, we can measure weaker sources.

For Tsys

= 100K, �⌫ = 40 MHz, and ⌧ = 1 sec, and using the above values for G, we find detectionlevels

F

⌫

� 0.079 Jy

G= 30 Jy 4 m

= 0.0053 Jy 300 m. (16)

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The way sources are detected is by performing a differential measurement on and off the source (e.g.measure the radiometer level when the telescope beam is pointed toward the source; subtract from thisthe level when it is pointed at one or more adjacent positions) and testing whether the difference islarger than the noise fluctuations. Alternatively, sometimes drift scans are performed by pointing thetelescope to a position through which the source will drift due to Earth’s rotation. A point source willproduce a response that is proportional to a one-dimensional cut through the beam P

n

(⌦). Drift scansare useful for 1) discovering new sources; 2) mapping out the beam; and 3) testing the pointing of thetelescope.

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Other factors: There is a limit to how well one can increase the time-bandwidth product as opposed toincreasing G / A

e

. This is because other factors come into play, especially source confusion, whichresults from there being multiple sources within the telescope beam at a given sky position. Movingfrom one sky position to another, these sources change, thereby producing a radiometer fluctuation thatcan exceed that produced by radiometer noise alone. If so, measurements are said to be ‘confusionlimited’ rather than radiometer noise limited. Telescopes with larger beams (i.e. small antennas at largewavelengths) have larger confusion variations. The rms confusion fluctuation is roughly

�

c

⇡ 3700⇣

⌫

1 GHz

⌘�0.7⌦A

Jy. (17)

For the Arecibo telescope at 1.4 GHz, �c

⇡ 2mJy. A source must be stronger than 5�c

⇡ 10 mJy to beconvincingly detected.

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The Astrophysical Journal, 758:23 (14pp), 2012 October 10 Condon et al.

Figure 4. Noise levels in the final image. The rms noise σn in the ring of radiusρ in the sky image (dashed curve) and in the weighted P (D) distribution ofpoints inside the circle of radius ρ. Abscissa: offset from the pointing center(arcmin). Ordinate: root-mean-square noise (µJy beam−1).

Figure 6 is a profile plot of the central portion of ourS/N-optimized wideband sky image. The intensity scale canbe inferred from the highest peaks, which are truncated atSp = 100 µJy beam−1. This image is confusion limited in thesense that the rms fluctuations are everywhere larger than thenoise levels plotted in Figure 4.

3. THE P (D) DISTRIBUTION

Historically, confusion was measured from the probabilitydistribution P (D) of pen deflections D on a chart-recorderplot of fringe amplitudes from the single baseline of a two-element radio interferometer (Scheuer 1957). On a single-dish oraperture-synthesis array image, the corresponding “deflection”D at any pixel is the intensity in units of flux density perbeam solid angle. The observed D at any point is the sum ofthe contribution from the noise-free source confusion and thecontribution from image noise. The source confusion and noisecontributions are independent of each other, so the observedP (D) distribution is the convolution of the source confusion andnoise distributions, and the variance σ 2

o of the observed P (D)distribution is the sum of variances of the noise-free sourceconfusion (σ 2

c ) and the noise (σ 2n ) distributions:

σ 2o = σ 2

c + σ 2n . (10)

The goal is to extract the source confusion distribution and itswidth

σc =!σ 2

o − σ 2n

"1/2 (11)

from the observed deflections and the measured noise. Ifσc ≪ σn, then ∂σc/∂σo ≈ (σo/σc) ≈ (σn/σc) ≫ 1 and smallerrors in the measured σo (or σn) cause large errors in theextracted σc. This is the reason that only fairly low-resolution(σc > σn) images can be used to measure confusion. Also, tothe extent that other sources of error (e.g., uncleaned sidelobes)are present but not accounted for, Equation (11) will tend tooverestimate the source confusion.

The noise in our weighted sky image is low at the center butincreases with radial distance ρ from the pointing center, and thenoise eventually overwhelms the confusion at large ρ. On theother hand, increasing the radius ρ of the circular region inside

Figure 5. Effective frequencies in the final image. The effective frequency ⟨ν⟩ isthe frequency at which the flux density of a source with spectral index α = −0.7equals the local flux density in the ring of radius ρ in the sky image (dashedcurve) or the average flux density of the weighted P (D) distribution of pointsinside the circle of radius ρ. Abscissa: offset from the pointing center (arcmin).Ordinate: effective frequency (GHz).

Figure 6. Confusion profile. This profile plot shows the 3 GHz confusionamplitude in an 8 arcsec FWHM beam, truncated at µmJy beam−1.

which the P (D) distribution is measured increases the numberN of beam solid angles sampled and thus acts to decrease thestatistical uncertainty ∆σo in the width of the observed P (D)distribution. For each thin ring of radius ρ covering N beamsolid angles, we use Equation (10) to calculate

(∆σo)2 =#

∂σo

∂σc∆σc

$2

+#

∂σo

∂σn∆σn

$2

, (12)

where∂σo

∂σc= σc

σoand

∂σo

∂σn= σn

σo. (13)

Then#

∆σo

σo

$2

= 1σ 4

o

%σ 4

c

#∆σc

σc

$2

+ σ 4n

#∆σn

σn

$2&, (14)

where (∆σc/σc) ∼ N−1/2 and (∆σn/σn) = (2N )−1/2

(Equation (7)). In the limit σn ≫ σc,

#∆σo

σo

$2

≈#

12Nσ 4

o

$σ 4

n . (15)

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LNA Bandpass Filter

Baseband Mixer


voltage / ˆ


Real and imaginary voltage (in phase and quadrature components)

Quadrature baseband mixer:

cos 2⇡⌫LOt

sin 2⇡⌫LOt

⇡/2

LO

Lowpass Filter

Lowpass Filter

ADC

ADC

It

Qt

Bandpass Filter

In phase component

Quadrature component

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Why are complex signals useful in the analysis of receiver systems?

1. Complex quantities like ei!t are easier to manipulate than cos!t, etc.

2. Many receiver systems are narrowband i.e. �⌫ ⌧ ⌫, for which the signal may be written as complex compo-

nents.

e.g.

v(t) / E�(t) = R�"(t)e�i!0t

where

v(t) = real voltage

E�(t) = narrowband electric field (scalar here)

"(t) = complex baseband field (scalar), slowly varying

e�i!0t= fastly varying “carrier” frequency

3. The output of a quadrature baseband mixer yields I,Q components that may be considered the real and imagi-

nary parts of ✏(t).

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Utility of Baseband Signal �

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•  Complex baseband voltage is related to the Fourier transform of the selected electric field

•  Can show that

•  Significance? – Access to fundamental units of radiation in relativistic

sources – Have access to phase as well as amplitude of the

radiation field •  Important for pulsar applications (coherent dedispersion) •  Necessary for interferometry

E�(t) =�"(t)e2⇡i⌫RFt

E�(t) = selected (scalar) E(t) field in RF bandwidth"(t) = complex baseband field

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Radiation Field and Receiver Voltages

A basis vector for EM waves is a plane waveei(k·x�!t) ! ⌘ 2⇡⌫

so a general radiation field has a Fourier representation

E(x, t) =

ZZdx d! e

E(k,!) ei(k·x�!t).

An observer at xjobs

using a telescope that is selective in direction, po-larization, and frequency via the antenna response and receiver filters,etc. can measure a voltage at RF frequencies

VRF

(t) /ZZ

antenna beam and bandwidth

dk d! [

eE(k,!) · p] ei(k·x�!t)

p = antenna polarization vector, assumed independent of directionand frequency (not generally true)

By inspection, the RF voltage contains the information in the selectedFourier components. Through appropriate manipulation, these Fouriercomponents can be used.

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For a narrowband system with center frequency ⌫0

and bandwidth �⌫,the RF voltage fluctuates on time scales ⇠ ⌫�1

0

. At GHz frequenciesthis is ⇠ 1 ns.

While digitizers (AGCs) exist that can sample this fast, much higherfrequencies cannot be sampled. For this and other reasons, it is usefulto mix the signal to lower frequencies. The ultimate shift is all the wayto baseband, where ⌫

0

is mapped to zero frequency and the bandwidthis �⌫/2.

Puzzle: does this mean that the bandwidth has been halved and that wecan satisfy the sampling theorem differently than before?

A: No, the same number of samples is needed because the samplingmust be of a complex signal (real and imaginary parts).

What is the response of the system to a point source that emits an im-pulse (delta functions in direction and time)?

A: From the student!

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Two examples of the power of phase�•  Imaging with aperture synthesis = manipulation

of the correlation function between fields measured between multiple pairs of antennas

•  Dedispersion of a pulsar signal to remove the effects of propagation through the ionized interstellar medium (deconvolution of digital data)

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Cygnus A�

Radio image Very Large Array Aperture Synthesis

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Sub-nanosec shot pulse from the Crab Pulsar �

39

Requires digital filtering of the electric field components selected over a 2.5 GHz bandwidth to remove phase wrapping imposed by dispersive propagation through the interstellar medium

Interstellar Dispersion�•  Ionized gas in the ISM is dispersive: refractive index

is frequency dependent (chromatic) à  The group velocity of a pulse is therefore frequency dependent à  The arrival time of a pulse is chromatic à  The phases of different Fourier components are altered

systematically

•  The spread in arrival times depends on the receiver bandwidth

40

�t =8.3µs DM�⌫

⌫3

⌫ = frequency in GHz

�⌫ = bandwidth in MHz

DM =

Z D

0ds ne(s) = dispersion measure pc cm�3

ne = electron density

Crab pulsar:DM = 56.791 pc cm�3

⌫ = 9 GHz�⌫ = 2.5 GHz�t = 1620 µs

9/17/15

21

Points �•  Beamwidths: θ ≈ λ / D

– Applies to all detectors of wavelike phenomena – Optical/IR telescopes, radio telescopes, our

eyes) •  Effective temperatures (pretend radiation is

blackbody in the Rayleigh-Jeans regime) •  Unit for flux density: the Jansky, used to

characterize radio sources •  Radiometer noise: analogous quantity to

readout noise + photon counting noise in CCDs (with different statistics)

41

a6525 lecture 8 = radio lecture 3 - cornell...

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