Lecture Notes

NRAO Lecture Program - Summer 1964

RADIO INTERFEROMETRY

by

N. J. KeenNational Radio Astronomy Observatory

Green Bank, West Virginia

RADIO INTERFEROMETRY

N. J. KeenNational Radio Astronomy- Observatory

Green Bank, West Virginia

Introduction

The structural limitations on the size of large radio telescopes limit

the angular resolution of these instruments. Hence, we must either look for'

other methods of obtaining high ..resolution observations (lunar occultations)

or we must artificially build an antenna of the large aperture required ("ap-

erture synthesis"). The fundamental telescope in aperture synthesis is the

interferometer. The method of obtaining the equivalent very large aperture

beam is to obtain the phase and amplitude of each interferometer record, and

add all interferometer records together-as a Fourier series ( 1 ). For conven-

ience we will .,consider a one-dimensional antenna.

If we consider a 300-foot one-dimensional. radio antennaoperating at

S= 10 cm, the "fan-beam" width (to .half-power) is radians 4 arc minutesd

If we wish to obtain narrower beams at a given wavelength, we have to increase

the diameter of the antenna. For a 10 arc.second fan-beam at " = 10 cm, a

7200-foot ..long antenna is required;

The Fourier Transformation

It is necessary to consider the properties of Fourier transformations

in order to obtain a clearer picture of high resolution techniques in radio

astronomy. A simple account of Fourier transformations is given by Jennison( 2 )

(3) (4)More rigorous treatments are given by Arsac and Sneddon .From the law

of superposition of electromagnetic wayes, the far-field pattern of an antenna

on the sky is the Fourier transformation of the aperture (illumination) func-

tion of the antenna. Similarly, the Fourier transformation of brightness dis-

tribution of a source (at infinity) is the field distribution (due to the source)

on the observing ,plane. In the same way that the conjugate Fourier quantities

in electrical theory are time and frequency, so are the conjugate Fourier quan-

tities in interferometry angle and spatial frequency.

Let us consider some general properties of the Fourier transformation.

For any antenna of one-dimensional angular beam .pattern G(9), observing a

source with one-dimensional brightness distribution T(e), the recorded output

power is given by

.R(e ) =. G(e.-O ) T(.O) dO (1)ooall 8

where 8 represents the direction. in which the power is G(0-0 ). This is knowno o

as a "convolution", and is sometimes abbreviated to

R(0 ) = G(e o ) * T(),o o

The Fourier transformation of R(60) is

r(u).= g(u). t(u), "(2)

where u is the spatial frequency and g(u) and t(u.) are the .Fourier transfor-

mations of G and T, respectively. For the Fourier pair r(u) and R(O), we '

have the important general relationship..

-3-

This may be compared with the resultant two linear electrical filters.

In this case we have

1 2

F(t ) = F1(to) * F2(t)

F(t ) is known as-the (unnormalized)cross-correlation coefficient, and

is the Fourier transformation of the resultant filter function f(m). Because

of the close resemblance between f(mo) and r(u), and due to the limited spatial

frequency band observed at one time by an interferometer, we may consider the

interferometer to be a spatial frequency filter.

Interferome ter Geometry

Let us consider a two-antenna interferometer whose individual antennas

are on an East-West baseline. Interferometer fringe maxima will be observed

when path differences (via each antenna).,from source to central point are

an integral number of wavelengths (fig. I). The instrumental meridian of the

two antenna interferometer is the direction in which the paths (via each an-

tenna) are equal. The meridian is sometimes referred to as the position of

the 'tentral fringe". For East-West interferometer baselines, sources of all

declinations pass through the local meridian and. interferometer meridian si-

multaneously (see fig. II); for non-East-West baselines, the coincidence of

r(u) = g(u) . t(u)

R(8) = G(9) * T(O)

-4-

meridians only occurs for sources which pass through the local zenith (i.e.,

whose declination is equal to geographic latitude).

Away from the instrumental meridian two. important effects occur.

(a) The apparent baseline, as seen by the source, is reduced. This

is known as "baseline forshortening", and is proportional to cos 9

(see fig. I).

(b) For sources at declinations other than zero, non-meridian obser-

vations result in, an apparent rotation of the projected interferometer

fringes -.and baseline (on the source). We call this angle p:i4,ir

the angle between celestial North-South and the baseline projection

on the sources RO° - 3 ..

otes o c fringes drawn on the celestial sphere comprisea series of semi-

circular,arcs, whose planes are parallel (see fig. II). Since sources at all

declinations move at the same hour-angle rate, and fringes at higher decli-

nations have larger hour-angle separations, the rate at which a source crosses

the fringes is lower at higher declinations. If 6 is' the source declination,

the rate of crossing the fringes is proportional to cos 6 sin p.

For an East-West interferometer, the rotation of the projected base-

line is given by

tan p = cos 6 cot H '

where H is the hour angle of the source.

Example. An interferometer comprises two antennas on'an East-West base-

line. Determine.the observed fringe frequency when viewing a source of 6 = 300

-5-

two hours before meridian transit (H = 0), The maximum fringe frequency (at

8 = 0, H = 0) is 0.2 cycle per second.

H = 2 hours, i.e., G = 300

Hence fringe rate = 0.2 cos 5 sin p cos e

= 0.2 cos2 (300 ) sin ,p

tan p = cosec 30° cot 30°

p = 730 54'

fringe rate = 0.144 cycles per second

The Treatment of Interferometer' Data

The interferometer output is fringes, whose phase and amplitude must

be retained in order that all spatial frequency contributions may be combined

to give the synthetic antenna beam. Again, we consider a one=-dimensional

antenna.

Since the interferometer is a spatial frequency filter, its output

represents one component .of the Fourier transform of the brightness distri-

bution. In fig. III several one=dimensional source distributions are shown,

together with their Fourier transformations: the transformation amplitudes

are known as "fringe visibility curves'". For zero spatial frequency the

fringe visibility is unity (V = 1).

The Interferometer System

The N.R.A.O. interferometer system is shown in block schematic form

in fig. IV. Analog multiplication followed by integration is equivalent to

(5).correlation, and Blum has shown that two advantages accrue from correlation

(rather than addition and detection). They are

(a) a slight improvement in system signal-to-noise ratio, and

(b) large reduction in system gain fluctuations.

In order to fix interferometer fringes relative to the baseline, sig-

nal and image bands are both accepted prior to superheterodyning the R.F.

It has been shown theoretically (by Read ( 6 ) ) and experimentally that, for

equal response to signal and image bands, sources may be tracked by insert-

ing delays in the I.F. circuit, whilst fringes remain fixed on the sky.

Interferometer Signal-to-Noise Ratio

The output of the interferometer is a sine wave of amplitude propor-

tional to the unresolved source flux: i.e., proportional to'fringe visibility

and total source flux density. Two typical records are shown in fig. V.

The system noise (assumed gaussian) has an r.m.s. value proportional to T-i/2

for a given I.F; bandwidth, whose t is the integration time. For a sine wave

plus gaussian noise, the maximum signal-to-noise ratio for a given source

\and system is when

sine wave period

.-- .. f!2n R

assuming integration is in a low pass RC filter. For the present N.R.A.O.

interferometer (July 1964), the signal-to-noise ratio is

TA _ 3.2 VS

R"MS.-

-7-

where T is the amplitude of the interferometer fringes, ATRMS is the.. r.m.s.ARS

noise, V. is the fringe visibility, S is the total source flux density (in

flux units) and r is the integration time constant in seconds.

Final Note

No attempt has been made here to consider interferometer baselines

oblique to E.ast-We.st, as at N.R.A.O., due to the considerable complexity of

the geometry. Further information on the N,R.A.O. interferometer may be

found in reference (7).

References

(1) Ryle, M., Nature, 180, 110 (1957)

(2) Fourier Transforms and Convolutions for the Expermentalist, R. C. Jennison;

Pergamon (1961)

(3) Transformation .de Fourier .et .Thorie des Distributions, J. Arsac; Dunod(1961)

(4) Fourier Transforms, I. N. Sneddon, McGraw-Hill (1951)

(5) Blum, E. J., Annales d'Astrophysique, 22, 139-162 (1959); English trans-

lation at N.R.A.O.

(6) Read, R. B. , Astrophysical Journal, 138, 1 (1963)

(7) Keen, N. J., N.R.A.O. Electronics Division Internal Report, No. 14 (1963)

Examples

1. An interferometer uses two 85-foot antennas (as at NRAO) separated by

900 meters on an East-West baseline. The operating wavelength is 1420 MHz.

Determine the fringe rate for a radio-source at 5 = 00

(a) at instrumental meridian transit angle (zero hour angle);

(b)' 3 hours.away from instrumental meridian transit.

2. Determine the fringe rate for the above interferometer observing a source

of 6 = 40 ° at 4 hours from the instrumental meridian transit. Why may

we not use this point on an East-West visibility curve?

3. For the interferometer in Example 1, calculate

(a) the time constant to give maximum signal-to-noise ratio

in 1(a) and 2;

(b) the signal-to-noise ratio.for a point source of 8 flux units

in 1(a) and 2, using these time: constants.

4. Determine the time constant and signal-to-noise ratio in case 1(b) where

the source has 80 flux units, and is in fact two equal point .sources

separated by one minute of arc in an East-West direction. Repeat the

calculation with a source of equal flux, which is 'in fact a double gaus-

sian in an,East-West direction with half-power widths and separation

equal to one minute of arc.

[Hint: Use fig. IV in both cases, remembering that the scale on the brightness

distribution varies inversely as the scale on the visibility curve.]

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