Tetsuo Sasao and André B. Fletcher

Introduction to VLBI Systems
Chapter 3
Lecture Notes for KVN Students
Partly Based on Ajou University Lecture Notes
Issued July 14, 2005
revised September 25, 2005
revised October 28, 2005
revised November 11, 2005
(to be further edited)Radio Interferometry
Contents
1 Fundamentals of Radio Interferometry
1.1 Two Explanations of VLBI . . . . . . . . . . . . . . . . . . . .
1.1.1 VLBI System . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 “Geodetic” Explanation — Noise Approach . . . . . .
1.1.3 “Astrophysical” Explanation — Monochromatic Wave
Approach . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Superposition of Monochromatic Waves . . . . . . . . .
1.1.5 Fringe Pattern Appears within an Envelope . . . . . .
1.1.6 Peak of the Envelope . . . . . . . . . . . . . . . . . . .
1.1.7 Fringe Pattern Enclosed by the Envelope . . . . . . . .
1.2 Elements of Stationary Random Processes . . . . . . . . . . .
1.2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Random Processes in Linear Systems . . . . . . . . . .
1.2.3 Stationary Random Processes . . . . . . . . . . . . . .
1.2.4 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . .
1.2.5 Stationary Random Processes in Linear Systems . . . .
1.2.6 Spectra of Stationary Random Processes . . . . . . . .
1.2.7 Spectra of Outputs of Linear Systems . . . . . . . . . .
1.2.8 Two Designs of Spectrometers . . . . . . . . . . . . . .
1.2.9 Fourier Transforms of Stationary Random Processes . .
1

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5
6
7
9
9
10
12
13
26
28
30
33
35
41
43
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1.3 The White Fringe . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 A Simple Interferometer . . . . . . . . . . . . . . . . .
1.3.2 Received Voltages as Stationary Random Processes . .
1.3.3 Cross–Correlation of Received Voltages . . . . . . . . .
1.3.4 Fringe Pattern Enclosed by Bandwidth Pattern . . . .
1.3.5 Amplitude and Phase Spectra of the Correlated Signals
1.3.6 Coherence Interval in the Sky . . . . . . . . . . . . . .
1.3.7 Fringe Spacing in the Sky . . . . . . . . . . . . . . . .

49
49
50
51
55
58
59
60

2 A Realistic Radio Interferometer61
2.1 Source Structure, Visibility and Intensity . . . . . . . . . . . . 63
2.1.1 Source Coherence Function . . . . . . . . . . . . . . . . 63
2.1.2 Power of Electric Field Incident from a Certain Direction 65
2.1.3 Poynting Flux . . . . . . . . . . . . . . . . . . . . . . . 66
2.1.4 Electric Field and Radio Source Intensity — Electromagnetics and Astronomy . . . . . . . . . . . . . . . . 69
2.1.5 Field of View of a Radio Telescope . . . . . . . . . . . 70
2.1.6 Power Pattern of a Receiving Antenna . . . . . . . . . 72
2.1.7 New Dimensions of Voltage and Electric Field . . . . . 74
2.1.8 How Does A Radio Interferometer View the Universe? 75
2.1.9 Complex Visibility . . . . . . . . . . . . . . . . . . . . 79
2.2 Frequency Conversion in Radio Interferometers . . . . . . . . 80
2.2.1 Response of RF Filters . . . . . . . . . . . . . . . . . . 81
2.2.2 Fourier Transform of IF Voltage Signal . . . . . . . . . 83
2.2.3 Cross–Correlation of Fourier Transforms of IF Signals . 84
2.2.4 Roles of Low–Pass Filters . . . . . . . . . . . . . . . . 86
2.2.5 Relationship between RF Spectrum and IF Spectrum . 88
2.3 Delay Tracking and Fringe Stopping . . . . . . . . . . . . . . . 91
2.3.1 General Form of Cross–Power Spectrum of IF Signals . 91
2.3.2 “Expected Correlation” of IF Signals . . . . . . . . . . 95
2.3.3 Radio Source Tracking by an Interferometer . . . . . . 98
2.3.4 Requirements to the Delay Tracking and Fringe Stopping100
2.3.5 Insertion of Instrumental Delay at IF Band . . . . . . . 104
2.3.6 Separation of Delay Tracking and Fringe Stopping Due
to Frequency Conversion . . . . . . . . . . . . . . . . . 106
2.3.7 Actual Implementations of Delay Tracking . . . . . . . 108
2.3.8 Actual Implementations of Fringe Stopping . . . . . . . 109
2.4 Correlator Outputs . . . . . . . . . . . . . . . . . . . . . . . . 113
2.4.1 Multiplier and Integrator . . . . . . . . . . . . . . . . . 113
2.4.2 Single Sideband (USB or LSB) Reception . . . . . . . . 115
2.4.3 Correlator Outputs in Single Sideband Reception . . . 117
2

2.4.4
2.4.5
2.4.6
2.4.7

Fringe Amplitude and Fringe Phase . . . . . . . . . . . 119
Group Delay and Fringe Frequency . . . . . . . . . . . 120
Complex Correlator . . . . . . . . . . . . . . . . . . . . 122
Projected Baseline . . . . . . . . . . . . . . . . . . . . 124

3 Source Structure and Correlated Flux Density
125
3.1 Basic Equations of Image Synthesis . . . . . . . . . . . . . . . 126
3.1.1 Complex Visibility as an Observable . . . . . . . . . . 126
3.1.2 Visibility and Intensity in EW–NS Coordinate System 126
3.1.3 Approximation of a Celestial Sphere by a Tangent Plane129
3.1.4 Interferometer is a Fourier Transformer . . . . . . . . . 133
3.2 uv–Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.2.1 uv–Trajectories . . . . . . . . . . . . . . . . . . . . . . 135
3.2.2 Synthesized Beams . . . . . . . . . . . . . . . . . . . . 139
3.3 Correlated Flux Density of a Source with Gaussian Intensity(Brightness) Distribution . . . . . . . . . . . . . . . . . . . . . 142
3.4 What Can VLBI Observe? . . . . . . . . . . . . . . . . . . . . 146
4 Signal–to–Noise Ratio in Radio Interferometry147
4.1 Statistical Model of a Correlator Output . . . . . . . . . . . . 148
4.1.1 Signal + Noise Inputs to a Correlator . . . . . . . . . . 148
4.1.2 Integration of the Multiplier Output . . . . . . . . . . 148
4.1.3 Statistical Expectation of the Correlator Output . . . . 149
4.1.4 Dispersion of the Correlator Output . . . . . . . . . . . 150
4.2 Useful Formulae Related to the Correlator Output . . . . . . . 150
4.2.1 Fourth Order Moment Relation . . . . . . . . . . . . . 151
4.2.2 Multiplier Output as a Stationary Random Process . . 155
4.2.3 Time Averaging Operator . . . . . . . . . . . . . . . . 156
4.2.4 Power Spectrum of Multiplier Output . . . . . . . . . . 157
4.2.5 Power and Cross–Power Spectra of Input Voltages . . . 158
4.2.6 Antenna Temperature and System Noise Temperature . 159
4.3 Sensitibity of a Radio Interferometer . . . . . . . . . . . . . . 160
4.3.1 Standard Deviation Due to the Noise . . . . . . . . . . 160
4.3.2 Signal in the Correlator Output . . . . . . . . . . . . . 162
4.3.3 Signal–to–Noise Ratio of the Correlator Output . . . . 163
4.3.4 Additional Remarks on the SN√
R Formula . . . . . . . 164
4.3.5 A Simple Interpretation of the B τa Factor . . . . . . 165
4.3.6 Formulae for SN R and Minimum Detectable Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

3

1

Fundamentals of Radio Interferometry

1.1

Two Explanations of VLBI

Two quite different explanations on the principles of Very Long Baseline
Interferometry (VLBI) are given in the literature. The two alternative explanations are illustrated in Figure 1.

VLBI

VLBI

Figure 1: Receptions of noise (left) and monochromatic wave (right) with
VLBI. This picture is based on a drawing originally provided by Dr. Katsuhisa Sato.

1.1.1

VLBI System

The VLBI system itself is described in almost the same way in these two
explanations:Two or more antennas are located at distant stations. They observe the same radio source at the same time. The observed data4

are recorded on magnetic media, such as magnetic tapes, withaccurate time marks generated by independent, but highly stable and well synchronized, clocks (or, better to say, frequency
standards). The recorded media are sent to a correlation center,
where they are played back and mutually multiplied and averaged(integrated) for some duration of time. This “multiplication andintegration” procedure is called “correlation processing”.
The geometry of the observation is also presented in a similar way:
The radio wave from the same radio source must travel slightly
further to reach antenna ② (right hand one in Figure 1) thanantenna ① (left hand one), with a small time delay τg . The delay
τg is determined by the geometric configuration of the antennas
and the radio source, and hence it is called the “geometric delay”.
The difference begins with the treatment of the signal from the radiosource.
1.1.2“Geodetic” Explanation — Noise Approach

One explanation, shown in the left panel of Figure 1, which is favored ingeodetic VLBI, regards the signal as a random noise time series.
If we simply multiply and average the two played–back data
streams recorded at the same time, we must get a nearly zero
result in most cases, since we are in effect averaging products of
two random noise time series, which is also random noise, with all
possible positive and negative values. But if, and only if, we shift
the playback timing of the record from the antenna ② exactly
by the geometric delay τg , while keeping the playback timing of
the record from the antenna ① unchanged, then the noise patterns from the same source in the two records ① and ② coincide.
Therefore, the product of the two time series always gives positive values (since plus times plus is plus, and minus times minus
is also plus) and the integration yields some finite positive value.
Thus, we “get the correlation” of the two records. By carefully
adjusting the time shift value so that the maximum correlation
is obtained, we precisely determine the geometric delay with anaccuracy of 0.1 nsec (10−10 sec) or better, which is sufficient todetermine the plate movements of the continents with typical
speeds of a few cm / year.
51.1.3“Astrophysical” Explanation — Monochromatic Wave Approach

Another explanation, shown in the right panel of Figure 1, which is favored in
astronomical VLBI for very high angular resolution imaging of radio sources,
regards the signal as a monochromatic sine wave.
Two waves from the same source with an angular sky frequency ωarrive at two antennas, giving rise to sinusoidal oscillations witha small time offset τg due to the geometric delay. Therefore, theplayed–back data records from antennas ① and ② are proportional to sin(ωt) and sin(ω(t − τg )), respectively. Their productis then proportional to1sin(ωt) sin(ω(t − τg )) = {cos(ωτg ) − cos(2ωt − ωτg )}.
2(1)

It is clear that the contribution of the rapidly oscillating second
term in the right hand side of equation (1), at a frequency twiceas large as the sky frequency, is almost nullified after time averaging (integration) over some duration. Therefore, only the firstterm, which is proportinal to cos(ωτg ), is left after the correlation
processing.
This term expresses a sinusoidal interferometric fringe pattern on
the sky, because the argument ωτg varies with the source direction in the sky. In particular, since ωτg = 2πcτg /λ, where c is thelight velocity, and λ = 2πc/ω is the wavelength, the fringe patternreverses its sign when the path length difference cτg changes bya half wavelength λ/2, as expected from the standard theory ofinterferometry. The angular distance corresponding to the separation between two successive peaks of the fringe pattern is called
the “fringe spacing”. If the radio source is more extended than
the fringe spacing, contributions from various elements of thesource are mutually compensated in the correlation processing,
due to the different signs of the fringe pattern over the extended
source. Therefore, the strength of an extended radio source issignificantly diminished in the VLBI output. If the source is sufficiently compact compared with the fringe spacing, on the other
hand, the amplitude of the source strength is almost the sameas what is measured by a single dish radio telescope. Thus, the
VLBI output contains information on the source structure. By
analysing the VLBI data obtained with various fringe patterns,
6

we can obtain a detailed image of the source structure, with surprisingly high angular resolutions of 1 milliarcsecond, or better.
Each of these two explanations, if examined separately, seems clear andinternally consistent. But it looks as if they are explaining completely diferent observational technologies, having no common feature at all. Nevertheless, they are the explanations of the same VLBI instrument, observing thesame radio source, with the same antennas, receivers, frequency standards,
magnetic tapes, and correlators. Then, how can we understand the two explanations from a unified point of view?
1.1.4

Superposition of Monochromatic Waves

Both of the above two explanations deviate from reality on the same point,
but in opposite directions. This point is the spectrum of the received signal.
The noise approach implicitly assumes that the spectrum of the signal
is white, i.e. the amplitude of the spectrum is finite, and more or less constant, in a very wide range of frequency. While this assumption may not
be too bad for the radio wave propagating in space, it is certainly not valid
for the received signal, which must be band–limited due to the frequency
characteristics of the optical and receiving systems of element antennas.
The monochromatic–wave approach, on the other hand, assumes an infinitely narrow bandwidth, when it talks about a wave having a certain frequency. But this is, of course, far from the reality (Figure 2).

ω

ω

.

White spectrum in ’geodetic’ explanation. Line spectrum in ’astrophysical’ explanation.

ωBut the reality is a ’band-limited white spectrum’.

Figure 2: Different source signal spectra assumed in the two explanations.
7So, what will come out, if we take a more realistic picture, by summing up
the monochromatic waves with different frequencies, spread within a certain
bandwidth? Figure 3 shows an answer.
Combined Cosine Fringe Patterns with Slightly Different Frequencies
1

fringe patterns of monochromatic waves

Fringe Patterns with Different Frequencies within 10 % of Center Frequency-2

combined fringe patterns

0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.5

-1

-0.5
0
0.5
1
bandwidth times geometric delay

1.5

-1

2

-2

-1.5

-1
-0.5
0
0.5
1
bandwidth times geometric delay

1.5

2

Figure 3: Fringe patterns of 11 monochromatic waves with slightly different
frequencies within a bandwidth which is 10 % of the central frequency (left),
and their superposition (right).
Here, we summed up 11 fringe patterns of monochromatic waves (leftpanel of Figure 3), which have slightly different frequencies, distributed ateven intervals within a bandwidth B, centered at ν0 = ω0 /2π, to generate
the superposed pattern shown in the right panel of the same figure. The lowermost curve, in the left panel of Figure 3, shows the fringe pattern cos(ωl τg )with the lowest angular frequency ωl = 2π(ν0 − B/2), while the uppermost
one shows the fringe pattern cos(ωu τg ) with the highest angular frequency
ωu = 2π(ν0 + B/2).
The horizontal axes of both panels in Figure 3 show the geometric delay
τg , multiplied by the bandwidth B, within a range of −2 ≤ Bτg ≤ 2. We took
the center of the horizontal axis at τg = 0, since the noise approach predicts
that the finite correlation is obtained only when the playback timing of one
record is shifted by the geometric delay. This shift is made to align the tworecords, as if the same wave front is received at the same time by the twoantennas. This is obviously equivalent to effectively reducing the geometricdelay to zero. Therefore, we assume the simplest case, where the source
direction is nearly perpendicular to the baseline, so that τg ' 0 from the
beginning.
We assumed here that the bandwidth is equal to 10 % of the central
frequency (ν0 = 10B).

8

1.1.5Fringe Pattern Appears within an Envelope

The right panel of Figure 3 shows a rapid oscillation, enclosed by a more
slowly varying envelope. The rapid oscillation has 10 peaks and valleys,
within an interval of ∆(Bτg ) = 1. The number 10 here is nothing but
the ratio ν0 /B. So, this corresponds to the fringe pattern cos(ω0 τg ) at thecentral angular frequency ω0 = 2πν0 , as expected in the monochromatic–
wave approach.
But the fringe pattern here does not have a constant amplitude. Instead,
it is enclosed by an envelope which takes a maximum value at τg = 0, when
the two time series, obtained from the same source with two antennas, aremost coincident with each other. This reminds us of the explanation of thecorrelation result in the noise approach.
1.1.6

Peak of the Envelope
Combined 100 Cosine Fringe Patterns with Different Frequencies
1

combined fringe patterns

0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-20

-15

-10
-5
0
5
10
bandwidth times geometric delay

15

20

Figure 4: Superposed fringe patterns of 100 monochromatic waves, withslightly different frequencies, contained within a bandwidth equal to 10 % of
the central frequency.
In order to see the point more clearly, we make our model still closer toan actual continuum spectrum, by increasing the number of monochromatic9

waves to 100, but keeping the same bandwidth (B = ν0 /10), and show
the superposed fringe patterns over a wider range of the horizontal axis:−20 ≤ Bτg ≤ 20. The result is given in Figure 4.
Now it is clear that the correlation result of the superposed monochromatic waves has sufficiently large amplitude only within a small range of thegeometric delay around τg = 0, which is roughly given by −1/B ≤ τg ≤ 1/B.
Although there are a number of sidelobes due to the finite bandwidth, theamplitude of these sidelobes rapidly decreases with increasing |τg |. Therefore,
Figure 4 is actually quite close to what is expected in the noise approach.
As a matter of fact, in a standard procedure for geodetic VLBI, the peak
position of the envelope of the fringe pattern, such as the one shown in Figure
4, is searched by effectively shifting the playback timing of one record against
the another. The peak is obtained at the time shift value which makes the tworecords most coincident, as if the same wave front was received at the sametime by the two antennas. The best time shift value thus obtained yields anestimate of a quantity called the “group delay”, which will be explained laterin more detail. After some corrections for systematic effects, the group delay
serves as a good estimate of the geometric delay, which is further analysed
to obtain scientific results in geodesy, geophysics, and astronomy.
Since the horizontal axis of the Figure 4 stands for Bτg , the larger the
bandwidth, the narrower the envelope is, in terms of the geometric delay
τg . Therefore, the accuracy (or statistical error) of determination of thegeometric delay in geodetic VLBI will be proportional to 1/B. Also, theaccuracy must be inversely proportional to the signal–to–noise ratio S/N of
the observation, since the higher the S/N , the finer we can determine thepeak position of the envelope. Although we do not know the exact number ofthe proportinality coefficient yet, we just assume that the coefficient is close
to 1, for the purpose of a rough estimation.
So, if the bandwidth B is 500 MHz and the S/N is 20, then the expected
accuracy is around1
1== 10−10 = 0.1 nsec.
8(S/N )B
20 × 5 × 10
Therefore, we can already understand, at least qualitatively, how a 0.1 nsec
accuracy is achieved in geodetic VLBI.
1.1.7

Fringe Pattern Enclosed by the Envelope

Now, as an opposite extreme, let us adopt a narrower bandwidth B = ν0 /40,
compared with the central frequency ν0 , and look at a narrower range:−0.2 ≤ Bτg ≤ 0.2. The result is given in Figure 5, which clearly shows
10

Combined Fringe Patterns within a Band 2.5% of Center Frequency
1

combined fringe patterns

0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

bandwidth times geometric delay

Figure 5: Superposed fringe patterns of monochromatic waves, with slightly
different frequencies, contained within a bandwidth equal to 2.5 % of thecentral frequency.
the fringe pattern cos(ω0 τg ), at the central angular frequency ω0 = 2πν0 of
the band, which is quite similar to the one expected in the monochromatic–
wave approach.
Therefore, we can conclude that the two explanations are talking about
two extreme cases, corresponding to the very wide and very narrow bandwidths, of a common signal, which is composed of the fringe pattern at thecentral frequency enclosed by the envelope pattern, whose sharpness is determined by the bandwidth. Geodetic VLBI uses the envelope pattern todetermine the peak position, where the signals obtained at two antennas aremost coincident with each other, to get a good estimate of the geometricdelay τg . VLBI source imaging uses the fringe pattern, which appears within
a limited central range of the envelope pattern, to derive the high angularresolution structures of astronomical radio sources. Both tasks can be donewith the same VLBI telescope.
Although we could obtain, at least qualitatively, a unified view on theapparently quite different two approaches, the above discussion assumed theensemble of a finite number of monochromatic waves, which are still notvery realistic. More rigorous treatment of the signals with band–limited

11

continuum spectra can be obtained in the so–called “white fringe theory”(e.g., Thompson, Moran and Swenson, 2001), which is based on the statisticaltheory of the stationary random processes.

1.2

Elements of Stationary Random Processes

The radio waves coming from astronomical sources are mostly generated bychaotic processes occuring in the source regions. For example, the thermalradiation is caused by the thermal random motions of the atoms, molecules
and free electrons, while the synchrotron radiation emerges from the random
explosive processes, which accelerate relativistic electrons in magnetic fields.
Hence, the electromagnetic fields, or the voltages in the receiving systems,
associated with the cosmic radio waves, mostly show characteristics of theGaussian random noise time sereis, as the “geodetic explanation” assumed.
A mathematical tool, which well describes such a random noise time sereis,is the statistical theory of the stationary random processes. Therefore, we
briefly introduce here basic elements of the theory, to the extent which will beneeded in following discussions. For deeper understanding, one can consult
with standard textbooks, for example, “Probability, Random Variables, andStochastic Processes, 2nd Edition” by Athanasios Papoulis (1984) .

Theory of
turbulence

Information
theoryReduction ofexperiments

Economic
forecastingStatisticalphysics

Trends inpopulation
Qualitycontrol(Stationary) Random Process
Spectroscopy
Correlation
processing

Data analysis
Polarization

Receiversensitivity

Digitalsampling

Figure 6: Statistical theory of random processes is a powerful tool for avariety of scientific disciplines.
The statistical theory of the random (or Stochastic) processes has wide
12

applications to many disciplines of radio astronomy, as well as other naturaland human sciences, as illustrated in Figure 6.
In the antenna theory, the basic framework was the electromagnetics,
and the vector algebra was used as the main mathematical tool. In thetheory of radio interferometry, however, we will no longer newly deal with theelectromagnetics. Instead, we will intensively use the theory of the stationary
random process as the fundamental tool for the mathematical developmentof the theory.

1.2.1

Basic Concepts

Random (or Stochastic) Process
A process x(t) is called “random (or Stochastic) process”, if it is a functionof time t, and, if its value x(t) at any time t is a random variable, i.e., mayvary from trial to trial (see Figure 7).
If we characterize each trial of an experiment by an outcome ζ of theexperiment, the random process can be represented as a function of both tand ζ, i.e., as x(t, ζ).
x(t)trial 1
trial 2
trial 3
trial 4
trial 50ttFigure 7: A random process is a function of time whose value at any time t
is a random variable.
The random process is a mathematical model of any time–varying and,in general, deterministically unpredictable process. The properties of therandom process are usually described in terms of statistical quantities, suchas probability distribution, probability density, expectation, correlation, covariance, etc.

13

Probability Distribution and Probability Density
Let us first consider a real random process x(t).
Let us denote a probability for x(t) at a specific time t not to exceed
a certain number x, as P {x(t) ≤ x}. Also, let us denote a probability foroccurence of multiple events, x(t1 ) not to exceed x1 , x(t2 ) not to exceed x2 ,· · ·, and x(tn ) not to exceed xn , as P {x(t1 ) ≤ x1 , x(t2 ) ≤ x2 , · · · , x(tn ) ≤xn }.
Now, the first–order probability distribution F (x; t) of the random process x(t) is defined as:
F (x; t) = P {x(t) ≤ x}.

(2)

Likewise, the second–order probability distribution F (x1 , x2 ; t1 , t2 ) is defined as:
F (x1 , x2 ; t1 , t2 ) = P {x(t1 ) ≤ x1 , x(t2 ) ≤ x2 },(3)and the n–th–order probability distribution F (x1 , · · · , xn ; t1 , · · · , tn ) is defined as:
F (x1 , · · · , xn ; t1 , · · · , tn ) = P {x(t1 ) ≤ x1 , · · · , x(tn ) ≤ xn }.

(4)

On the other hand, the first–order probability density f (x; t) of the random process x(t) is defined as a derivative of the distribution F (x; t) withrespect to x:∂F (x; t)f (x; t) =.
(5)∂x
Since, by definition,
F (x + ∆x; t) − F (x; t)∂F (x; t)
= lim
,
∆x→0
∂x
∆x
the probability density has a meaning:
P {x(t) ≤ x + ∆x} − P {x(t) ≤ x}
∆x→0∆x
P {x < x(t) ≤ x + ∆x}
.
= lim
∆→0
∆x

f (x; t) = lim(6)

Likewise, the second–order probability density f (x1 , x2 ; t1 , t2 ) is definedas:f (x1 , x2 ; t1 , t2 ) =∂ 2 F (x1 , x2 ; t1 , t2 ),∂x1 ∂x2

14(7)

and the n–th–order probability density f (x1 , · · · , xn ; t1 , · · · , tn ) is defined as:∂ n F (x1 , · · · , xn ; t1 , · · · , tn ).
(8)∂x1 · · · ∂xn
Generally speaking, if t is continuous, we need infinite number of various
orders of probability distributions, in order to properly describe a random
process. In many practical cases, especially in cases of stationary random
processes, however, it is sufficient to take into account first– and second–order distributions only, as we will see later.
f (x1 , · · · , xn ; t1 , · · · , tn ) =Following general properties are satisfied for probability distributions anddensities, as evident from their definitions:• F (∞; t) = 1,
• F (x1 ; t1 ) = F (x1 , ∞; t1 , t2 ),• f (x; t) ≥ 0 (i.e., F (x; t) is a monotonically increasing function of x),
•

Zx2

x1

f (x; t) dx = P {x1 < x(t) ≤ x2 },• f (x1 ; t1 ) =
•

Z∞

Z∞

f (x1 , x2 ; t1 , t2 ) dx2 ,−∞f (x; t) dx = 1.

−∞

Now, let us consider a case, where a random process z(t) is a complex
process:z(t) = x(t) + i y(t),where a real part x(t) and an imaginary part y(t) are real random processes,
and i is the imaginary unit.
The probability distribution of the complex random process z(t) is definedby the joint probability distribution of x(t) and y(t). Thus, the n–th–orderprobability distribution is defined as:
F (z1 , · · · , zn ; t1 , · · · , tn ) = F (x1 , · · · , xn ; y1 , · · · , yn ; t1 , · · · , tn )= P {x(t1 ) ≤ x1 , · · · , x(tn ) ≤ xn , y(t1 ) ≤ y1 , · · · , y(tn ) ≤ yn },(9)and the n–th order probability density is defined as:
f (z1 , · · · , zn ; t1 , · · · , tn ) = f (x1 , · · · , xn ; y1 , · · · , yn ; t1 , · · · , tn )∂ 2n F (x1 , · · · , xn ; y1 , · · · , yn ; t1 , · · · , tn ).
=∂x1 · · · ∂xn ∂y1 · · · ∂yn
15

(10)

When we have two complex random processes x(t) and y(t):x(t) = xr (t) + i xi (t),y(t) = y r (t) + i y i (t),where real parts xr (t) and y r (t), and imaginary parts xi (t) and y i (t), are allreal random processes, we introduce joint probability distributions and joint
probability densities of the two complex random processes.
For example, the first order joint probability distribution of the complex
random processes x(t) and y(t) at times t1 and t2 , respectively, is
F (x; y; t1 ; t2 ) = F (xr , xi ; y r , y i ; t1 ; t2 )= P {xr (t1 ) ≤ xr ; xi (t1 ) ≤ xi ; y r (t2 ) ≤ y r ; y i (t2 ) ≤ y i },(11)

and the corresponding joint probability density isf (x; y; t1 ; t2 ) = f (xr , xi ; y r , y i ; t1 ; t2 )∂ 4 F (xr , xi ; y r , y i ; t1 ; t2 )=.
∂xr ∂xi ∂y r ∂y i

(12)Expectation (or Ensemble Average)
Expectation (or ensemble average) ηz (t) of a complex random processz(t) = x(t) + iy(t) at time t is defined as:ηz (t) = hz(t)i =where the symbol halso as E{ }.

Z∞ Z∞[x(t) + iy(t)] f (x; y; t) dxdy,

(13)

−∞−∞

i, which stands for the expectation, is often denoted

Autocorrelation
Autocorrelation Rzz (t1 , t2 ) of a complex random process z(t) = x(t) +iy(t) at times t1 and t2 is defined as:
Rzz (t1 , t2 ) = hz(t1 ) z ∗ (t2 )i=Z∞ Z∞ Z∞ Z∞z(t1 )z ∗ (t2 ) f (x1 , x2 ; y1 , y2 ; t1 , t2 ) dx1 dx2 dy1 dy2 , (14)−∞−∞−∞−∞

wehere the symbol ()∗ stands for the complex conjugate.
16Hereafter, we will usually omit sufficies such as zz, for simplicity, when
we express autocorrelations. Thus, autocorrelations Rxx (t1 , t2 ), Ryy (t1 , t2 ),Rzz (t1 , t2 ), and so on, will be all denoted simply as R(t1 , t2 ), except for cases,
when we wish to explicitly specify the random processes under consideration.
Following general properties hold for autocorrelations.
• R(t2 , t1 ) = R∗ (t1 , t2 ).
• R(t, t) = h| z(t) |2 i ≥ 0,i.e., real and positive.

• Positive definite, namely,n XnX

i=1 j=1ai a∗j R(ti , tj ) ≥ 0

for any numbers ai (i = 1, 2, · · · , n).

Proof :

0 ≤ h|nX
i=1

ai z(ti ) |2 i =• An inequality:

n X
n
X

i=1 j=1ai a∗j hz(ti ) z ∗ (tj )i =

n X
n
X

ai a∗j R(ti , tj ).

i=1 j=1| R(t1 , t2 ) |2 ≤ R(t1 , t1 ) R(t2 , t2 ).

(15)

Proof :
1. For any complex random variables v and w, we haveh| v | | w |i2 ≤ h| v |2 i h| w |2 i.
Proof :
Since we always haveh(s | v | + | w |)2 i = s2 h| v |2 i + 2sh| v | | w |i + h| w |2 i ≥ 0,
for any real variable s, the discriminant of the above quadratic
equation with respect to s must be smaller than or equal to 0, i.e.,h| v | | w |i2 − h| v |2 i h| w |2 i ≤ 0,
and, hence,h| v | | w |i2 ≤ h| v |2 i h| w |2 i.

17

2. For any complex random variables v and w, we have
| hv wi | ≤ h| v | | w |i.
Proof :Let us first prove a general statement that, for any complex random variable A = a + ib, where a and b are real random variables,
we have| hAi | ≤ h| A |i.

Let us denote the probability density of A as f (a, b). Then | hAi |and h| A |i are expressed as
Z Z| hAi | = |=A f (a, b) dadb |lim lim |∆a→0 ∆b→0

XX

A f (a, b)∆a∆b |,

(16)

and
Z Z
h| A |i =| A | f (a, b) dadb==

lim lim∆a→0 ∆b→0

lim lim∆a→0 ∆b→0

XXXX| A | f (a, b)∆a∆b| A f (a, b)∆a∆b |,(17)

where we replaced the integrtions by the infinite summations,which are performed in the same way in both equations (16) and(17), and we used a property of the probability density f (a, b) inequation (17), that it is always real and greater than or equal tozero.
Now, for any complex numbers B and C, we have
| B + C |≤| B | + | C |,since
|B+C | ==qq(B + C)(B + C)∗ =q| B |2 +B ∗ C + BC ∗ + | C |2

| B |2 +2 | B | | C | cos Φ+ | C |2 ,where we introduced an angle Φ satisfying
B ∗ C =| B | | C | eiΦ ,is always smaller than | B | + | C |, because cos Φ ≤ 1. This
relation is easily extended to the sum of arbitrary number n ofcomplex numbers B1 , B, · · ·, Bn , i.e.,
|

n
X
i=1

Bi |≤

18

n
X
i=1

| Bi |,

because
|

n
X
i=1

Bi |=| B1 +

n
X
i=2

Bi |≤| B1 | + |≤| B1 | + | B2 | + |

n
X
i=3

n
X
i=2

Bi |=| B1 | + | B2 +

n
X
i=3

Bi |

Bi |= · · · ≤| B1 | + | B2 | + · · · + | Bn | .

Applying the above relation to the summations of equations (16)and (17), we confirm that | hAi | ≤ h| A |i.
This implies that | hv wi | ≤ h| v | | w |i, since√√ √| v w |= vwv ∗ w ∗ = vv ∗ ww ∗ =| v | | w | .
3. From 1. and 2. above, we obtain| hv wi |2 ≤ h| v | | w |i2 ≤ h| v |2 i h| w |2 i,i.e.,| hv wi |2 ≤ h| v |2 i h| w |2 i.

(18)

If we adopt here v = z(t1 ) and w = z ∗ (t2 ), then we prove that
| R(t1 , t2 ) |2 ≤ R(t1 , t1 ) R(t2 , t2 ).
Autocovariance
Autocovariance C(t1 , t2 ) of a complex random process z(t) at times t1
and t2 is defined as:C(t1 , t2 ) = R(t1 , t2 ) − η(t1 ) η ∗ (t2 ),(19)

where η(t) ≡ ηz (t) = hz(t)i is the expectation of z(t) at time t.
The autocovariance of z(t) is equal to the autocorrelation of z̃(t) = z(t) −η(t), i.e.,C(t1 , t2 ) = Rz̃z̃ = hz̃(t1 ) z̃ ∗ (t2 )i.
(20)
In fact,hz̃(t1 ) z̃ ∗ (t2 )i = h[z(t1 ) z ∗ (t2 ) − z(t1 ) η ∗ (t2 ) − η(t1 ) z ∗ (t2 ) + η(t1 ) η ∗ (t2 )]i= R(t1 , t2 ) − η(t1 )η ∗ (t2 ).

19

Correlation Coefficient
Correlation coefficient r(t1 , t2 ) of a complex random process z(t) at times
t1 and t2 is defined as:r(t1 , t2 ) = q

C(t1 , t2 )C(t1 , t1 ) C(t2 , t2 )

.

(21)

It is evident that• r(t, t) = 1.
Also, the absolute value of the correlation coefficient is always smaller thanor equal to 1:• | r(t1 , t2 ) |≤ 1,
since from equations (15) and (20), we have
| C(t1 , t2 ) |2 ≤ C(t1 , t1 ) C(t2 , t2 ).
Cross–Correlation
Cross–correlation Rxy (t1 , t2 ) of two complex random processes x(t) =x (t) + i xi (t) and y(t) = y r (t) + i y i (t) at times t1 and t2 , respectively, isdefined as:r

Rxy (t1 , t2 ) = hx(t1 ) y ∗ (t2 )i =Z∞ Z∞ Z∞ Z∞

x(t1 ) y ∗ (t2 ) f (xr , xi ; y r , y i ; t1 ; t2 )dxr dxi dy r dy i , (22)−∞−∞−∞−∞

using the joint probability density of x(t) and y(t), given in equation (12).
Following properties hold for cross–correlations.
∗• Rxy (t2 , t1 ) = Ryx(t1 , t2 ).
This is evident from the above definition.

• | Rxy (t1 , t2 ) |2 ≤ Rxx (t1 , t1 ) Ryy (t2 , t2 ).
This is proven by adopting v = x(t1 ) and w = y ∗ (t2 ) in equation (18).

20

Cross–Covariance
Cross–covariance Cxy (t1 , t2 ) of two complex random processes x(t) andy(t) at times t1 and t2 is defined as:
Cxy (t1 , t2 ) = Rxy (t1 , t2 ) − ηx (t1 ) ηy∗ (t2 ).

(23)

Following properties hold for cross-covariances.
• The cross–covariance of x(t) and y(t) is equal to the cross–correlationof x̃(t) = x(t) − ηx (t) and ỹ(t) = y(t) − ηy (t), i.e.,
Cxy (t1 , t2 ) = Rx̃ỹ (t1 , t2 ) = hx̃(t1 ) ỹ ∗ (t2 )i.
• | Cxy (t1 , t2 ) |2 ≤ Cxx (t1 , t1 ) Cyy (t2 , t2 ).
Cross–Correlation Coefficient
Cross–correlation coefficient rxy (t1 , t2 ) of two complex random processes
x(t) and y(t) at times t1 and t2 is defined as:rxy (t1 , t2 ) = q

Cxy (t1 , t2 )Cxx (t1 , t1 ) Cyy (t2 , t2 )

.

(24)

It is evident that for any cross–correlation coefficient we always have
• | rxy (t1 , t2 ) |≤ 1.
Normal (Gaussian) Process
As an example of the probability density introduced in equations (5), (7),and (8), we consider here probability desnity of a particularly important classof random process, namely normal (or Gaussian) process, which is known to
be a good model of signals from astronomical radio sources, as well as ofnoises produced in antenna–receiving systems or in environments.
Real random process x(t), with expectation η(t) and autocovariance C(t1 , t2 ),is called “normal (or Gaussian) process”, if, at any times t1 , t2 , · · ·, tn for any
n, random variables x(t1 ), x(t2 ), · · ·, x(tn ) are jointly normal (or Gaussian),i.e., they are characterized by following probabilty densities.
• First–order Gaussian probability density:f (x1 ; t1 ) =

1
√

σ1 2π

21

e

−

[x1 −η1 ]2
2σ 2
1

,

(25)

where η1 ≡ η(t1 ) and σ12 ≡ C(t1 , t1 ) are expactation and dispersion,respectively, of x(t) at time t1 .
• Second–order Gaussian probability density:f (x1 , x2 ; t1 , t2 )=1√

2πσ1 σ2 1 − r 2e−12(1−r 2 )(x −η )(x −η )(x1 −η1 )2(x −η )2
−2r 1 σ1 σ 2 2 + 2 2 2
σ2
σ
1 2
1
2



,

(26)

where we introduced notations: η1 ≡ η(t1 ), η2 ≡ η(t2 ), σ12 ≡ C(t1 , t1 ),σ22 ≡ C(t2 , t2 ), and correlation coefficient:r≡q

C(t1 , t2 )C(t1 , t1 ) C(t2 , t2 ).

• n-th–order Gaussian probability density:f (x1 , · · · , xn ; t1 , · · · , tn )=q1(2π)n ∆

e

− 12

n P
n
P

i=1 j=1−1[xi −ηi ] Cij[xj −ηj ],(27)

where ηi ≡ η(t1 ), ηj ≡ η(tj ), Cij ≡ C(ti , tj ) is autocovariance matrix,
Cij−1 is its inverse, and ∆ ≡ det{Cij } is its determinant.

When n = 1 and n = 2, equation (27) is reduced to equations (25)and (26). Therefore, the first– and second–order Gaussian probability
densities given in equations (25) and (26), respectively, are special casesof the more general expression of the n-th–order normal (or Gaussian)probability density given in equation (27).
Likewise, we can conceive a number of normal (or Gaussian) processes
which are jointly normal with each other.

• Two real normal processes x(t) and y(t) are called “jointly normal (orGaussian) processes”, if, at any times t1 and t2 , random variables x(t1 )and y(t2 ) are jointly normal, i.e., they are characterized by following
joint probabilty density:f (x, y; t1 , t2 )
=

1
q

2
2πσx σy 1 − rxy

e

−

1
2 )
2(1−rxy(x−ηx )(y−ηy )(y−ηy )(x−ηx )2+
−2 rxy
2
2
σx σy
σx
σy

22

2



,

(28)

where we introduced expectations of x(t1 ) and y(t2 ): ηx ≡ ηx (t1 ) andηy ≡ ηy (t2 ), dispersions of x(t1 ) and y(t2 ): σx2 ≡ Cxx (t1 , t1 ), σy2 ≡Cyy (t2 , t2 ), and cross–correlation coefficient:rxy ≡ qCxy (t1 , t2 )Cxx (t1 , t1 ) Cyy (t2 , t2 ).

Here Cxx (t1 , t1 ), Cyy (t2 , t2 ) and Cxy (t1 , t2 ) are autocovariances andcross–covariance of x(t1 ) and y(t2 ), correspondingly.
• An arbitrary number m of normal processes x(1) (t), x(2) (t), · · ·, x(m) (t)are called “jointly normal (or Gaussian) processes”, if, at any times t1 ,t2 , · · ·, tm , random variables x(1) (t1 ), x(2) (t2 ), · · ·, x(m) (tm ) are jointly
normal, i.e., they are characterized by following joint probabilty density:f (x(1) , · · · , x(m) ; t1 , · · · , tm )=q1(2π)m ∆

e

− 12

m
m P
P

i=1 j=1−1[x(i) −η(i) (ti )] C(i)(j)[x(j) −η(j) (tj )], (29)

where η(i) (t) is expectation of x(i) (t), C(i)(j) ≡ Cx(i) x(j) (ti , tj ) is cross–−1
covariance matrix, C(i)(j)is inverse matrix of C(i)(j) , and ∆ ≡ det{C(i)(j) }is determinant of C(i)(j) .
Of course, equation (28) is a special case of equation (29) with m = 2.
• In the above statement, some random variables among x(1) (t1 ), x(2) (t2 ),· · ·, x(m) (tm ) could be values of the same normal process taken at different times. In such a case, some elements of matrix C(i)(j) are autocovariances, rather than cross–covariances.
In this sence, joint probability densities of a single normal (or Gaussian)process given in equations (25), (26), and (27) can be regarded as
special cases of the joint probability density of the jointly normal (orGaussian) processes given in equation (29).
It is not difficult to confirm that the joint probability densities of jointly
normal (or Gaussian) processes given in equations (25) – (29) are consistent
with geneal properties of joint probability densities, as well as with definitionsof the expectation and the covariances, as explained in standard textbooks.
For this purpose, we can use well–known integration formulae:
Z∞

e

−a x2

dx =

−∞

23

r

π
,
a

(30)

Z∞

2

x e−a x dx = 0,

−∞
Z∞

2

x e

1
dx =
2

−a x2

−∞

(31)
r

π
,
a3

(32)

for a > 0.
For example, if we take the form of the joint normal (or Gaussian) probability density given in equation (28), we can confirm following formulae.
•

R∞

−∞

f (x, y; t1 , t2 ) dy = f (x ; t1 ):
R∞

−∞

=
=
=
=
=

f (x, y; t1 , t2 ) dy
Z∞

1
q

2
2πσx σy 1 − rxy
−∞

Z∞

1
2πσx

q

2
1 − rxy−∞q2
1 − rxy
−∞

Z∞

1

2πσx

1

q

2
2πσx 1 − rxy

σx

1
√

2π

e

e

−

e
e

e

−

1
2 )
2(1−rxy−1
2 )2(1−rxy−1
2 )2(1−rxy(x−ηx )2
2
2 σx

Z∞


h

e



(x−ηx )(y−ηy )(y−ηy )2(x−ηx )2
−2 rxy
+
2
2
σx σy
σx
σy

(x−ηx )2(x−η )−2 rxy σ x y 0 +y 0 2
2
x
σx

2 )
(1−rxy−(x−ηx )2
+
2
σx

2
y 00
2 )
2 (1−rxy





dy

dy 0

x
(y0 −rxy x−η
σx )

2

i

dy 0

dy 00

−∞

(x−ηx )2−2
2 σx

= f (x ; t1 ),(33)

in view of equation (30), where we introduced variable transformations:y0 =y − ηy,σyy 00 = y 0 − rxyx − ηx.
σx• Expectation:
hxi =
=

Z∞ Z∞

−∞ −∞
Z∞

x f (x, y; t1 , t2 ) dx dy =xf (x ; t1 ) dx =

−∞

Z∞

−∞

σx

1
√

24

Z∞

2 π −∞xexZ∞

f (x, y; t1 , t2 ) dy dx−∞(x−ηx )2
−
2
2 σx

dx

=

σx

1
√

Z∞

2 π −∞[(x − ηx ) + ηx ] eZ∞−(x−ηx )2
2
2 σx

dx

2

(x−ηx )
ηx−√=e 2 σx2 dx = ηx ,σx 2 π −∞(34)

in view of equations (30) and (31).
• Covariance:h(x − ηx ) (y − ηy )i
=

Z∞ Z∞

−∞ −∞

=
×
=
=
=

(x − ηx ) (y − ηy ) f (x, y; t1 , t2 ) dx dy1q22πσx σy 1 − rxy

Z∞ Z∞

−∞ −∞

(x − ηx ) (y − ηy ) e

σx σy
q

Z∞ Z∞

2
2π 1 − rxy
−∞ −∞
Z∞ Z∞
σx σy

2π

q

2
1 − rxy−∞ −∞σx σy rxy
q

Z∞

e

−

−

1
2 )
2(1−rxy

1

−



2

x0 y 0 e 2(1−rxy )(y−ηy )(x−ηx )(y−ηy )(x−ηx )2+
−2 rxy
2
2
σx σy
σx
σy

(x0 2 −2rxy x0 y 0 +y 0 2 )

−

1

2

y 00

2

(x00 + rxy y 00 ) y 00 e 2(1−rxy )

2
x00
2 )
2(1−rxy

dx

00

Z∞

2



dx dy

dx0 dy 02 ) y 00 2 ][x00 2 +(1−rxydx00 dy 00

2y 00 e− 2 dy 00

2
2π 1 − rxy−∞−∞√σx σy rxy q2 ) 2π = r σ σ = C (t , t ),q=
2π(1 − rxyxy x yxy 1
22
2π 1 − rxy(35)

in view of equations (30), (31), and (32), where we introduced variabletransformations:y − ηyx − ηx
,
y0 =
,
x0 =
σx
σy
x00 = x0 − rxy y 0 ,y 00 = y 0 .
In following discussions, we will mainly use general properties of expectations
and correlations, without specifying explicit forms of probability densities.
However, when necessary, we will assume jointly normal (or Gaussian) processes, and explicitly use expressions of the normal probability density given
in equations (25) – (29).
25

1.2.2

Random Processes in Linear Systems

Definition of Linear Systems
Let us consider a system of two complex functions x(t) and y(t) of timet, which are related with each other by an operator L:
y(t) = L[x(t)],(36)

where x(t) is called “input” and y(t) is called “output” of the operator L.
Such a system is called “linear system”, if the operator L satisfies
L[a1 x1 (t) + a2 x2 (t)] = a1 L[x1 (t)] + a2 L[x2 (t)],(37)

for any complex coefficients a1 , a2 and for any functions x1 (t), x2 (t).
The linear system is also called as “linear filter”, which linearly “filters”
the input x(t) to yield the output y(t).
Impulse Response
If the input of an operator L is a delta function δ(t) of time t, the output
is called “impulse response” of the operator, which we denote as h(t):h(t) = L[δ(t)].

(38)

Here, we introduce “convolution” f (t) ∗ g(t) of functions f (t) and g(t),which is defined by a following infinite integration:f (t) ∗ g(t) =Z∞−∞f (t − α) g(α) dα,(39)

where symbol “∗” stands for the operation of the convolution. Convolution
has following properties:
f (t) ∗ g(t) = g(t) ∗ f (t)becauseg(t) ∗ f (t) =Z∞−∞g(t − β) f (β) dβ =Z∞−∞(commutative),f (β) g(t − β) dβ =Z∞−∞f (t − α) g(α) dα,

where we used a transformation of the argument of the integration: α = t−βand hence dβ = −dα,26andf (t) ∗ g(−t) =Z∞−∞f (t − β) g(−β) dβ =Z∞f (t + α) g(α) dα,−∞

where we used α = −β and dβ = −dα.
Then, the output of the linear system can be represented as a convolutionof the input and the impulse response, i.e.,y(t) = x(t) ∗ h(t) =Z∞−∞x(t − α) h(α) dα.

(40)This equation is easily proven, based on the definition of the delta function,in the following way:y(t) = L[x(t)] = L[

Z∞

−∞

=

Z∞

−∞

x(β) δ(t − β) dβ] =x(β) h(t − β) dβ =

Z∞

−∞

Z∞

−∞

x(β) L[δ(t − β)] dβx(t − α) h(α) dα = x(t) ∗ h(t).

Note that L operates only on a function of time t.
Linear Systems with Random Processes as Inputs
Hereafter, we will consider linear systems having random processes as
inputs. Then, we have following general properties.
• If x(t) and y(t) are input and output of a linear system, their expectations hx(t)i and hy(t)i are also related with each other as input andoutput of the same linear system, i.e.,hL[x(t)]i = L[hx(t)i],orhx(t) ∗ h(t)i = hx(t)i ∗ h(t),orηy (t) = L[ηx (t)].
Proof :
hx(t) ∗ h(t)i = hZ∞−∞x(t − α) h(α) dαi =Z∞−∞hx(t − α)i h(α) dα= hx(t)i ∗ h(t).

27

(41)

Note that the impulse response h(t) is a deterministic function of time,and, hence, not affected by the ensemble average.
• Autocorrelation of the output:
Ryy (t1 , t2 ) = Rxx (t1 , t2 ) ∗ h(t1 ) ∗ h∗ (t2 ),or
Ryy (t1 , t2 ) =

Z∞ Z∞

−∞ −∞

Rxx (t1 − α, t2 − β) h(α) h∗ (β) dα dβ. (42)

Proof :
1. Cross–correlation of the input and the output is
Rxy (t1 , t2 ) = hx(t1 ) y ∗ (t2 )i = hx(t1 ) x∗ (t2 ) ∗ h∗ (t2 )i= hx(t1 ) x∗ (t2 )i ∗ h∗ (t2 ) = Rxx (t1 , t2 ) ∗ h∗ (t2 ),or
Rxy (t1 , t2 ) =Z∞

Rxx (t1 , t2 − β) h∗ (β) dβ.

−∞

2. Autocorrelation of the output is
Ryy (t1 , t2 ) = hy(t1 ) y ∗ (t2 )i = hx(t1 ) ∗ h(t1 ) y ∗ (t2 )i= hx(t1 ) y ∗ (t2 )i ∗ h(t1 ) = Rxy (t1 , t2 ) ∗ h(t1 ),or
Ryy (t1 , t2 ) =Z∞−∞

Rxy (t1 − α, t2 ) h(α) dα.

3. From 1. and 2. above, we have
Ryy (t1 , t2 ) = Rxx (t1 , t2 ) ∗ h∗ (t2 ) ∗ h(t1 )= Rxx (t1 , t2 ) ∗ h(t1 ) ∗ h∗ (t2 ),or
Ryy (t1 , t2 ) =

Z∞ Z∞

−∞ −∞

1.2.3

Rxx (t1 − α, t2 − β) h(α) h∗ (β) dα dβ.

Stationary Random Processes

Definitions28

• A random process z(t) is called “stationary” (or, more specifically,“wide–sense stationary”), if the expectation does not depend on time,
and the autocorrelation is a function of time difference only:
hz(t)i = η = const,
hz(t1 ) z ∗ (t2 )i = R(τ ),(43)

where τ ≡ t1 − t2 .
• Random processes x(t) and y(t) are called “jointly stationary”, if bothof them are stationary, and their cross–correlation is a function of timedifference only:hx(t1 ) y ∗ (t2 )i = Rxy (τ ),(44)
where τ ≡ t1 − t2 .
Of course, it is not easy to find actual physical processes which strictly satisfy
these conditions. For example, some of quasars or astronomical masers are
known to exihit significant time variations in yearly, or shorter, time scales.
In a practical sense, however, physical processes are well approximated by
the stationary random processes, if equations (43) and (44) are fulfilled during time scales, which are sufficient to estimate their statistical properties(see discussions on ergodicity in section 1.2.4). In this sense, many physicalprocesses can be successfully modeled as stationary random processes.
Properties
Following formulae can be easily derived, by applying general propertiesof correlations, covariances, and so on, to the particular case of the stationary
random processes as defined above.
• R(−τ ) = R∗ (τ ).
• R(0) = h| z(t) |2 i ≥ 0.
• Positive definiteness:

n X
n
X

i=1 j=1ai a∗j R(ti − tj ) ≥ 0,

• | R(τ ) |≤ R(0).
• C(τ ) = R(τ )− | η |2 ,autocovariance.
29

for any ai .

• r(τ ) = C(τ )/C(0),correlation coefficient.

• | r(τ ) |≤ 1.
∗• Rxy (−τ ) = Ryx(τ ).

• | Rxy (τ ) |2 ≤ Rxx (0) Ryy (0).
• Cxy (τ ) = Rxy (τ ) − ηx ηy∗ ,qcross–covariance.

• rxy = Cxy (τ )/ Cxx (0) Cyy (0),• | rxy (τ ) |≤ 1.
1.2.4

cross–correlation coefficient.

Ergodicity

How can we estimate various statistical properties of a random process, if
we are given with a single sample of time series only, which is an outcome ofa single trial?
• Definition.
A random process z(t) is called “ergodic” if its ensemble averages areequal to appropriate time averages.
This implies that we can estimate any statistical property ofz(t), using time average of the single sample, if the random
process is ergodic.
• Mean–ergotic process.
A random process z(t), with constant expectation:
η = hz(t)i,
is called “mean–ergotic”, if its time average tends to η as averaging
time tends to infinity:T

1 Z
z(t)dt → η
ηT =
2T
−T

as

T → ∞.

• A condition for the mean–ergotic process.
It is evident thatZT
1hηT i =
hz(t)idt = η.
2T
−T

30

Therefore, z(t) is mean–ergotic, if its “variance”, or “dispersion”, σT2
tends to 0 as T → ∞, i.e.,
σT2 ≡ h| ηT − η |2 i → 0 as T → ∞.
Since
σT2 = h| ηT − η |2 i =

T T
1 Z Z
h(z(t1 ) − η) (z(t2 ) − η)∗ i dt1 dt2
2
(2T )
−T −T

1
=
(2T )2ZT ZT

C(t1 , t2 ) dt1 dt2 ,−T −T

where C(t1 , t2 ) is the autocovariance, and we used here equation (20),the above condition is equivalent to1(2T )2

ZT ZT

−T −T

C(t1 , t2 ) dt1 dt2 → 0 as T → ∞.

• In the stationary random case.
If z(t) is a stationary random process, and, therfore, the autocovarianceis a function of time diffference τ = t1 − t2 only:C(t1 , t2 ) = C(τ ),the double integral in the above condition is reduced to a single integral:
1
2T

Z2T

−2T

|τ |
C(τ ) 1 −2T!

dτ → 0 as T → ∞,

(45)

Z2T

(46)

becauseZT ZTC(t1 , t2 ) dt1 dt2 =

−T −T

−2T

C(τ )(2T − | τ |) dτ,as we can easily see from Figure 8. In fact, in the rectangular range
of integration −T ≤ t1 ≤ T and −T ≤ t2 ≤ T in Figure 8, theautocovariance C(τ ) is constant along a line t1 − t2 = τ , and an area ofthe hatched region, put between two lines t1 −t2 = τ and t1 −t2 = τ +∆τ
is nearly equal to the area of the enclosing parallelogram, which is equalto (2T − | τ |) ∆τ , in the linear approximation with respect to small∆τ .
31

t1-t2=2T

t1

t1-t2= τ+∆τ
t1-t2= τ

2T-|τ|
T
τ+∆τ
τ

-T

0

t2

T

t1-t2=-2T-T

Figure 8: Geometry of the integration.
• Correlation–ergodic process.
A stationary random process z(t) with an autocorrelation
R(ξ) = hz(t + ξ) z ∗ (t)i,
is called “correlation–ergodic”, if the corresponding process uξ (t):uξ (t) = z(t + ξ) z ∗ (t),is mean–ergotic, i.e., ifT

1 Zuξ (t) dt → R(ξ) as T → ∞.
RT ≡
2T
−T

Similarly to the case of the mean–ergotic process, z(t) is correlation–
ergotic, if the variance, or dispersion, σR2 T ≡ h| RT − R(ξ) |2 i tends to
0 as T → ∞, or, equivalently, if an autocovarinace of uξ (t):Cuu (τ ) = Ruu (τ ) − R2 (ξ),
32

where Ruu (τ ) is an autocorrelation of uξ (t):Ruu (τ ) = huξ (t + τ ) u∗ξ (t)i = hz(t + ξ + τ ) z ∗ (t + τ ) z ∗ (t + ξ) z(t)i,
satisfies the condition in equation (45) for any ξ, i.e.,
1
2T

Z2T

−2T

|τ |
Cuu (τ ) 1 −2T!

dτ → 0 as T → ∞.

(47)

For actual physical processes, which are approximated by the stationary
random processes, it is likely that autocovariances are finite everywhere andtend to 0 when τ → ±∞. Therefore, equations (45) and (47) appear wellsatisfied in the most cases. In the followings, we assume that these equations
are fulfilled, and, hence, we can estimate both expectation and correlation of
our physical process in terms of the time averaging of a single sample.
It is known that the power or the correlation of a moderately strong
signal from an astronomical radio source, which is estimated by time averaging in a square–law detector or in a correlator, usually reaches a sufficiently
high signal–to–noise ratio, that means a small enough dispersion, after averaging during seconds to hours, depending on telescope or array sensitivity.
The detected power or correlation is usually almost time–invariant during
time–scales from hours to months. Therefore, radio astronomical data aremostly consistent with assumptions of the stationary random process andthe ergodicity.
1.2.5

Stationary Random Processes in Linear Systems

Let us consider cases when inputs of linear systems are stationary random
processes. Then, we have following properties.
• If an input x(t) in a linear system y(t) = L[x(t)] is a stationary random
process, then an output y(t) is also a stationary random process.
Proof :
1. Expectation ηy (t) of the output y(t) is constant in time, because
ηy (t) =Z∞−∞ηx (t − α) h(α) dα = ηx

33

Z∞

−∞

h(α) dα = const,

since x(t) is stationary, and hence, ηx (t) = ηx is constant in time.
Thus, we haveηy = η xZ∞h(α) dα.

(48)−∞

2. Autocorrelation Ryy (t1 , t2 ) of the output y(t) is a function of timedifference τ = t1 − t2 only, because
Ryy (t1 , t2 ) =Z∞ Z∞

Rxx (t1 − α, t2 − β) h(α) h∗ (β) dα dβ

−∞ −∞
Z∞ Z∞

=

−∞ −∞

Rxx (τ − α + β) h(α) h∗ (β) dα dβ,

since x(t) is stationary, and hence,
Rxx (t1 − α, t2 − β) = Rxx (t1 − α − (t2 − β)) = Rxx (τ − α + β).
Then, the above formula is now expressed as:
Ryy (τ ) = Rxx (τ ) ∗ h(τ ) ∗ h∗ (−τ ).

(49)• If an input x(t) in a linear system y(t) = L[x(t)] is a stationary random
process, then the input x(t) and the output y(t) are jointly stationary.
Proof :
1. We have proven above that, if the input is stationary, then the
output is also stationary, i.e., both x(t) and y(t) are stationary.
2. Cross–correlation Rxy (t1 , t2 ) of the input x(t) and the output y(t)is a function of time difference τ = t1 − t2 only, because∗

Rxy (t1 , t2 ) = hx(t1 ) y (t2 )i =
=

Z∞

−∞

Z∞

−∞

hx(t1 ) x∗ (t2 − α)i h∗ (α) dα

Rxx (τ + α) h∗ (α) dα = Rxx (τ ) ∗ h∗ (−τ ).

Thus, we have
Rxy (τ ) = Rxx (τ ) ∗ h∗ (−τ ).

34

(50)

• Likewise, we can prove an equation:
Ryy (τ ) = Rxy (τ ) ∗ h(τ ),(51)which, together with equation (50), offers another derivation of equation (49).
• If x1 (t) and x2 (t) are jointly stationary random processes, then outputs y1 (t) and y2 (t), which are obtained from x1 (t) and x2 (t) througharbitrary linear operators L1 and L2 , respectively, are also jointly stationary.
Proof :
Let the two linear operators L1 and L2 correspond to impulse responses
h1 (t) and h2 (t), respectively. Then we havey1 (t) = L1 [x1 (t)] = x1 (t) ∗ h1 (t) =

Z∞

−∞
Z∞

y2 (t) = L2 [x2 (t)] = x2 (t) ∗ h2 (t) =−∞x1 (t − α) h1 (α) dα,x2 (t − α) h2 (α) dα.

Both y1 (t) and y2 (t) are, of course, stationary, and their cross–correlation:
Ry1 y2 (t1 , t2 ) = hy1 (t1 ) y2∗ (t2 )i
=
=

Z∞ Z∞

−∞ −∞
Z∞ Z∞
−∞ −∞

hx1 (t1 − α) x∗2 (t2 − β)i h1 (α) h∗2 (β) dα dβ
Rx1 x2 (τ − α + β) h1 (α) h∗2 (β) dα dβ,

is a function of time difference τ = t1 − t2 only. This proves the joint
stationarity of y1 (t) and y2 (t), and yields
Ry1 y2 (τ ) = Rx1 x2 (τ ) ∗ h1 (τ ) ∗ h∗2 (−τ ).
1.2.6

(52)

Spectra of Stationary Random Processes

Definitions• Fourier transform S(ω) of an autocorrelation R(τ ) of a stationary random process is called “power spectrum” (or “spectral density”) of theprocess. Here, ω is an angular frequency, which is related to a linear
35

frequency ν as ω = 2πν. Thus, the power spectrum and the autocorrelation are related to each other by the Fourier– and inverse Fouriertransforms:
S(ω) =

Z∞

R(τ )e−iωτ dτ,

(53)

Z∞

(54)

−∞

1
R(τ ) =
2π

S(ω)eiωτ dω.

−∞

Hereafter, we express a Fourier transform pair by a symbol “⇔”. Then,
S(ω) ⇔ R(τ ).
• Fourier transform Sxy (ω) of a cross–correlation Rxy (τ ) of jointly stationary random processes x(t) and y(t) is called “cross–power spectrum”. Thus,
Sxy (ω) =Z∞

Rxy (τ )e−iωτ dτ,

(55)

Z∞

(56)

−∞

1
Rxy (τ ) =2π

Sxy (ω)eiωτ dω,−∞and
Sxy (ω) ⇔ Rxy (τ ).
Note that convergence of Fourier integrals in equations (53) and (55), andtherefore in their inverses in equations (54) and (56), too, is usually guaranteed, since, for actual physical processes, R(τ ) and Rxy (τ ) are mostly finite
everywhere, and tend to zero as τ → ±∞.
Properties• Power h| z(t) |2 i of a stationary random process z(t) is equal to an
integrated power spectrum over the whole frequency range:
∞

∞

−∞

−∞

Z
1 Z
S(ω) dν,h| z(t) | i = R(0) =S(ω) dω =
2π
2

(57)

where ν = ω/(2π) is a frequency, corresponding to the angular frequency ω.
36• Power spectrum S(ω) is a real function.
Proof :

Since R(−τ ) = R∗ (τ ),∗S (ω) =Z∞∗

R (τ )e

iωτ

dτ =

Z∞

R(−τ )eiωτdτ =R(τ )e−iωτ dτ = S(ω).

−∞

−∞

−∞

Z∞

∗
• For any cross–power spectrum, Sxy (ω) = Syx(ω).

Proof :∗
Since Rxy (−τ ) = Ryx
(τ ),
∗
Syx
(ω) =

Z∞

∗
Ryx
(τ )eiωτ dτ =−∞Z∞

Rxy (−τ )e

iωτ

dτ =

−∞

Z∞

Rxy (τ )e−iωτ dτ

−∞= Sxy (ω).
• A power spectrum S(ω) corresponding to a real autocorrelation R(τ )is an even function of ω (see Figure 9):S(−ω) = S(ω).

(58)

S(ω)

ω

0

Figure 9: Power spectrum is even when autocorrelation is real.
Proof :
Since, in this case, R(−τ ) = R(τ ) (the real autocorrelation is an evenfunction of τ ),S(−ω) =

Z∞−∞

R(τ )e

iωτ

dτ =

Z∞

R(−τ )e

−∞

iωτ

dτ =

Z∞

−∞

37

R(τ )e−iωτ dτ = S(ω).

• A cross–power spectrum corresponding to a real cross–correlation satisfies
Sxy (−ω) = Syx (ω),(59)
and, therefore, is Hermitian symmetric:∗
Sxy (−ω) = Sxy
(ω),

(60)

(see Figure 10).
Re Sxy(ω)

Im Sxy(ω)

ω

0

ω

0

Figure 10: Cross–power spectrum is Hermitian symmetric (i.e., real part iseven and imaginary part is odd) when cross–correlation is real.
Proof :
Since, in this case, Rxy (−τ ) = Ryx (τ ),Sxy (−ω) =Z∞

Rxy (τ )e

−∞

iωτ

dτ =

Z∞

Rxy (−τ )e

−iωτ

−∞

dτ =

Z∞

Ryx (τ )e−iωτ dτ−∞= Syx (ω),∗
and, in view of the general property Sxy (ω) = Syx(ω), we also have∗
Sxy (−ω) = Sxy(ω).

• Real autocorrelation can be described solely by the positive frequency
range of the power spectrum.
Proof :
Since S(ω) = S(−ω), in this case,1
R(τ ) =
2π

Z∞

−∞



1 
S(ω)eiωτ dω =
2π

Z0

−∞

38

S(ω)eiωτ dω +

Z∞
0



S(ω)eiωτ dω 

∞

∞

0

0

1 Z
1 Z
[S(−w)e−iωτ + S(ω)eiωτ ] dω =S(ω)[e−iωτ + eiωτ ] dω
=
2π
2π
1
π

=

Z∞
0


∞
Z
1
S(ω) cos(ωτ ) dω = <  S(ω)eiωτ dω  .

π

(61)

0

• Real cross–correlation can be described solely by the positive frequency
range of the cross–power spectrum.
Proof :∗
Since Sxy (−ω) = Sxy(ω), in this case,1
Rxy (τ ) =
2π
=
=

1
2π
1
2π

Z∞

−∞
Z∞



1 
Sxy (ω)eiωτ dω =2πZ0

Sxy (ω)eiωτ dω +

−∞

Z∞
0



Sxy (ω)eiωτ dω 

[Sxy (−w)e−iωτ + Sxy (ω)eiωτ ] dω

0

Z∞

∗
[Sxy
(ω)e−iωτ + Sxy (ω)eiωτ ] dω

0

1
=
<
π

Z∞

Sxy (ω)eiωτ dω.

(62)

0

• White noise: if the spectrum is flat throughout the whole frequency
range, then the correlation is proportional to the delta function of τ .
If S(ω) = S = const, then1
R(τ ) =
2π

Z∞

S(ω)e

iωτ

−∞

1
dω = S
2π

Z∞

eiωτ dω = S δ(τ ).

(63)

−∞

If Sxy (ω) = Sxy = const, then1
Rxy (τ ) =2π

Z∞

Sxy (ω)eZ∞eiωτ dω = Sxy δ(τ ). (64)eiωτ dω = 2πδ(τ ),

(65)

iωτ

−∞

1
dω = Sxy2π−∞

Here we used a formula
Z∞

−∞

39

which is known as one of the definitions of the delta function.
Thus, if spectra of random processes are completely flat (white), then
their correlations are non–zero, only when τ = 0.
• Convolution theorem: Fourier transform of a convolution of two functions is equal to a product of Fourier transforms of those functions, i.e.,
if a(τ ) ⇔ A(ω) and b(τ ) ⇔ B(ω), thena(τ ) ∗ b(τ ) ⇔ A(ω) B(ω).

(66)

Proof :
Z∞−∞a(τ ) ∗ b(τ ) e
=

Z∞ Z∞

−iωτ

dτ =

Z∞ Z∞

−∞ −∞

a(τ − α) b(α) e−iωτ dα dτ000a(τ 0 ) e−iωτ b(τ 00 ) e−iωτ dτ 0 dτ 00 = A(ω) B(ω),−∞ −∞where we introduced transformations τ 0 = τ − α and τ 00 = α.
• Another convolution theorem holds for a product of functions a(τ ) andb(τ ):
1
A(ω) ∗ B(ω),(67)
a(τ ) b(τ ) ⇔2π
becauseZ∞a(τ ) b(τ ) e−iωτ dτ

−∞

∞

  ∞

 ∞



Z
1 Z Z000iω 0 τ0 B(ω 00 ) eiω τ dω 00  e−iωτ dτ
A(ω ) e
dω
=
2
(2π)

=
=
=

1
(2π)2
1
2π
1
2π

−∞

−∞ −∞
Z∞ Z∞
−∞ −∞

Z∞ Z∞

−∞ −∞
Z∞
−∞


 ∞
Z
0
00
A(ω 0 ) B(ω 00 )  e−i(ω−ω −ω ) τ dτ  dω 0 dω 00−∞

A(ω 0 ) B(ω 00 ) δ(ω − ω 0 − ω 00 ) dω 0 dω 00

A(ω − ω 0 ) B(ω 0 ) dω 0 =1A(ω) ∗ B(ω),
2π

we used here the relation
Z∞

−∞

e

−iωτ

dτ =

Z∞

−∞

40

eiωτ dτ = 2πδ(ω).

• Shift theorem:
If a(τ ) ⇔ A(ω), thena(τ − τ0 ) ⇔ A(ω) e−iωτ0 ,a(τ )eiω0 τ ⇔ A(ω − ω0 ).

(68)

Proof :Z∞−∞a(τ − τ0 ) e

−iωτ

dτ =

Z∞

0

a(τ 0 ) e−iω(τ +τ0 ) dτ 0

−∞

 ∞

Z
0
0
−iωτ
0
=  a(τ )edτ  e−iωτ0 = A(ω)e−iωτ0 ,
−∞

and,
Z∞

a(τ )e

iω0 τ

e

−iωτ

dτ =

−∞

1.2.7

Z∞

−∞

a(τ )e−i(ω−ω0 )τ dτ = A(ω − ω0 ).

(69)

Spectra of Outputs of Linear Systems

Let us call a Fourier transform H(ω) of an impulse response h(t) of a linear
system as the “system function”:
H(ω) =

Z∞h(t) e−iωt dt,

(70)

−∞

or
H(ω) ⇔ h(t).
For the system function, we haveH ∗ (ω) ⇔ h∗ (−t),
because∗H (ω) =

Z∞∗h (t)e

iωt

dt =

−∞

Z∞

h∗ (−t)e−iωt dt.

−∞

Now, let us consider stationary random processes as inputs of a linear
system y(t) = L[x(t)] with the impulse response h(t) and the system function
H(ω).
41

• Expectation of the output.
ηy = η xZ∞h(α) dα = ηx H(0).

(71)

−∞

• Power spectra of the output Syy (ω) and the inputs Sxx (ω) are mutuallyrelated to each other as
Syy (ω) = Sxx (ω) | H(ω) |2 .

(72)

Proof :
In view of the convolution theorem given in equation (66), and properties of correlations,
Rxy (τ ) = Rxx (τ ) ∗ h∗ (−τ ) ⇔ Sxy (ω) = Sxx (ω) H ∗ (ω),
Ryy (τ ) = Rxy (τ ) ∗ h(τ ) ⇔ Syy (ω) = Sxy (ω) H(ω),
and, hence
Ryy (τ ) = Rxx (τ ) ∗ h(τ ) ∗ h∗ (−τ ) ⇔ Syy (ω) = Sxx (ω) | H(ω) |2 .
• Autocorrelations of the outputs:1
Ryy (τ ) = hy(t + τ ) y (t)i =
2π
∗

Z∞

−∞

Sxx (ω) | H(ω) |2 eiωτ dω,

and, in particular,∞

1 Z
Ryy (0) = h| y(t) | i =Sxx (ω) | H(ω) |2 dω.
2π
2

(73)

−∞

• If the impulse response h(t) is real, then∗H (ω) =

Z∞h(t)eiωt dt = H(−ω),

(74)

−∞

and, therefore, | H(ω) |2 is an even function of ω, because| H(−ω) |2 = H(−ω) H ∗ (−ω) = H ∗ (ω) H(ω) =| H(ω) |2 .

(75)

• Cross–power spectrum of outputs y1 (t) = x1 (t) ∗ h1 (t) and y2 (t) =x2 (t) ∗ h2 (t) of jointly stationary inputs x1 (t) and x2 (t) through two
linear systems with impulse responses h1 (t) and h2 (t).
42

As we saw earlier in equation (52), the cross–correlation Ry1 y2 (τ ) of
the outputs is expressed through the cross–correlation of the inputs
Rx1 x2 (τ ) as
Ry1 y2 (τ ) = Rx1 x2 (τ ) ∗ h1 (τ ) ∗ h∗2 (−τ ).
Therefore, the convolution theorem in equation (66) gives us the cross–power spectrum:Sy1 y2 (ω) = Sx1 x2 (ω)H1 (ω)H2∗ (ω),(76)

where Sx1 x2 (ω) ⇔ Rx1 x2 (τ ) is a cross–power spectrum of the inputs.
• Cross–correlation of the outputs.
Ry1 y2 (τ ) =hy1 (t + τ ) y2∗ (t)i =

1
2π

1
2π

Z∞

Z∞

Sx1 x2 (ω)H1 (ω) H2∗(ω)eiωτ dω,−∞

and, hence,
Ry1 y2 (0) =1.2.8

hy1 (t) y2∗ (t)i =Sx1 x2 (ω)H1 (ω) H2∗ (ω) dω.

(77)

−∞

Two Designs of Spectrometers

As an example of applications of the theory of the stationary random process,let us consider principles of two types of spectrometers which have been
widely used in the radio astronomy (Figures 11 and 12).
In the filterbank spectrometer (Figure 11), received voltage from a radiosource is equally fed to n identical analog narrow–band BPF’s (band–pass–filters), which are called “filterbank” with successive center frequences ν1 , ν2 ,· · ·, νn . Outputs of the BPF’s are squared and averaged by SQ (square–law)
detectors and resultant powers yield a spectral shape of the source at theabove frequencies.
BPF ν1SQ Detector

BPF ν2SQ Detector

BPF ν3

SQ Detector
SpectrumLO

BPF νn-1

SQ Detector

BPF νnSQ Detector

Figure 11: Basic design of a filterbank spectrometer.
43

A/D
τ
LO

τ
τ

τ
τ

Multiplier

Integrator

Multiplier

Integrator

Multiplier

Integrator

Multiplier

Integrator

Multiplier

Integrator

Multiplier

Integrator

Fourier Transformation

In the autocorrelation spectrometer (Figure 12), on the other hand, thereceived voltage is first digitized by an analog–to–digital converter (A/D),
and then equally divided into two digital signals, which are fed to n multipliers and integrators, one directly, and another with successive time delays
0, τ , 2τ , · · ·, (n − 1)τ . The resultant ‘autocorrelation’ as a function of timedelay is then Fourier transformed, and converted to a power spectrum.

Spectrum

Figure 12: Basic design of an autocorrelation spectrometer.
The principles of the two designs look quite different. Do they really
produce the same spectrum?
As far as the ergodicity holds, it is clear that the autocorrelation spectrometer must closely approximate the calculation of the power spectrum of
the input signal (received voltage) as the Fourier transform of the autocorrelation, as we have described so far.
For the filterbank spectrometer, let us consider i-th narrow–band BPF asa linear system, which has an input stationary random process x(t), whichis the received voltage in this case, an output yi (t), and an impulse responsehi (t), corresponding to a rectangular sytem function Hi (ω):
Hi (ω) =

 q

2π


∆ω


 0

ωi − ∆ω≤ ω ≤ ωi + ∆ω,22otherwise,

where ωi = 2πνi is the i-th center angular frequency, and ∆ω is the frequency
bandwidth of the BPF. If the power spectrum of the input is Sxx (ω), then,according to equation (72), the power spectrum of the output Syy (ω) is
Syy (ω) = Sxx (ω) | Hi (ω) |2 ,(Figure 13). Since, in view of the ergodicity, the time averaging in a square–44Sxx(ω)Syy(ω)

Hi(ω)
2π
∆ω

0

ω

0

∆ω

∆ω

ωi

0

ω

ωi

ω

Figure 13: Band–pass filter passes a segment of the input power spectrum.
law detector must yield the power, or the autocorrelation at τ = 0, of theoutput signal, if the averaging time is sufficiently long, we obtain1
h| yi (t) | i =
2π
2

Z∞

−∞

1
Syy (ω) dω =
2π

Z∞

−∞

Sxx (ω) | Hi (ω) |2 dω1=∆ωωi + ∆ω2Z

Sxx (ω) dω.

ωi − ∆ω
2This “power passed by a BPF” is nothing but a mean of the power spectrum of the received voltage Sxx (ω), involved in the spectral range ωi −∆ω≤ ω ≤ ωi + ∆ω. Therefore, if ∆ω is sufficiently narrow, and Sxx (ω)
22is continuous around ωi , then we approximately haveh| yi (t) |2 i ' Sxx (ωi ).
Thus two spectrometers really yield the same power spectrum of the received
voltage.
This example gives us a clear feel, that the power spectrum, defined asa Fourier transform of the autocorrelation of the input signal, is really a“spectrum of the power” of the signal.
1.2.9

Fourier Transforms of Stationary Random Processes

So far, we have considered Fourier transformation of correlations of the stationary random processes. Now, let us proceed to considerations of theFourier transformation of the stationary random processes themselves.
Assume that a Fourier integral of a random process z(t) is expressed as
Z(ω) =

Z∞z(t) e−iωt dt.

−∞

45

(78)

Since z(t) is a random process in time t, it is natural to consider that Z(ω)
is a random process in angular frequency ω, i.e., it is a function of ω, and its
value at any ω is a random variable, which may vary from trial to trial.
If we apply the inverse Fourier transform to equation (78), we would have1z(t) =
2πZ∞

Z(ω) eiωt dω,

(79)

−∞

i.e., we could express any random process in time t as a superposition of
infinite number of frequency components, which are themselves random processes in angular frequency ω.
Strictly speaking, however, we must be aware that the convergence of
the intrgrals in equations (78) and (79) is not, in general, guaranteed, since
the random processes may have finite amplitudes from the infinite past to
the infinite future. Of course, we could restrict the actual integration range
to −T < t ≤ T with sufficiently large T . In fact, durations of actual
physical processes are most likely to be shorter than the age of our Universe.
However, a too strong emphasis on this point may cause difficulties when we
require stationarity to the random processes. Special integral forms are often
introduced in the literature to assure the convergence. We will, however,just assume some kind of convergence of the above integrals, without beingheavily involved in the mathematical strictness. Instead, we will concentrate
our attentions to several simple but useful statistical relations between the
random process z(t) and its Fourier transform Z(ω).
Properties of Fourier transforms of the random processes.
• Expectation of Z(ω) is a Fourier transform of the expectation η(t) ofz(t).
Proof :Taking ensemble average of the two sides of the Fourier transformationin equation (78), we havehZ(ω)i =Z∞−∞

hz(t)i e

−iωt

dt =

Z∞

η(t) e−iωt dt.

−∞• If z(t) is a stationary random process, the expectaion of Z(ω) has adelta–function form with respect to ω.
46

Proof :Sincehz(t)i = η = const,

we havehZ(ω)i =Z∞η e−iωt dt = 2π η δ(ω),

(80)

−∞

according to equation (65).
• An autocorrelation of Z(ω), defined as hZ(ω1 ) Z ∗ (ω2 )i, is related toa two–dimensional Fourier transform Γ(ω1 , ω2 ) of an autocorrelation
R(t1 , t2 ) = hz(t1 ) z ∗ (t2 )i of z(t), which isΓ(ω1 , ω2 ) =Z∞ Z∞

R(t1 , t2 ) e−i(ω1 t1 +ω2 t2 ) dt1 dt2 ,−∞ −∞by a formula:hZ(ω1 ) Z ∗ (ω2 )i = Γ(ω1 , −ω2 ).

Proof :

(81)

From equation (78),∗hZ(ω1 ) Z (ω2 )i =
=

Z∞ Z∞

−∞ −∞

Z∞ Z∞

−∞ −∞

hz(t1 ) z ∗ (t2 )i e−i(ω1 t1 −ω2 t2 ) dt1 dt2

R(t1 , t2 ) e−i(ω1 t1 −ω2 t2 ) dt1 dt2 = Γ(ω1 , −ω2 ).

• If z(t) is a stationary random process, having a power spectrum S(ω),
we havehZ(ω1 ) Z ∗ (ω2 )i = 2π S(ω1 ) δ(ω1 − ω2 ).
(82)
Proof :

Since, in view of the stationarity of z(t), its autocorrelation hz(t1 ) z ∗ (t2 )i =R(t1 , t2 ) = R(τ ) is a function of time difference τ = t1 −t2 only. Therefore,∗hZ(ω1 ) Z (ω2 )i =
=

Z∞ Z∞

−∞ −∞
Z∞ Z∞

−∞ −∞

47

R(t1 , t2 ) e−i(ω1 t1 −ω2 t2 ) dt1 dt2R(τ ) e−iω1 τ −i(ω1 −ω2 )t2 dτ dt2

Z∞=

R(τ ) e−iω1 τdτZ∞e−i(ω1 −ω2 ) t2 dt2−∞−∞= 2π S(ω1 ) δ(ω1 − ω2 ).
• If x(t) and y(t) are jointly stationary random processes, having a cross–power spectrum Sxy (ω), a cross–correlation of their Fourier transforms:
X(ω) =

Z∞x(t) e

−iωt

dt,

and

Y (ω) =

Z∞y(t) e−iωt dt,−∞−∞

is equal tohX(ω1 ) Y ∗ (ω2 )i = 2π Sxy (ω1 ) δ(ω1 − ω2 ).

(83)

Proof :
Since the cross–correlation hx(t1 ) y ∗ (t2 )i = Rxy (t1 , t2 ) = Rxy (τ ) is afunction of time difference τ = t1 − t2 only, we have∗hX(ω1 ) Y (ω2 )i =
=
=

Z∞ Z∞

−∞ −∞
Z∞ Z∞

−∞ −∞
Z∞

Rxy (t1 , t2 ) e−i(ω1 t1 −ω2 t2 ) dt1 dt2
Rxy (τ ) e−iω1 τ −i(ω1 −ω2 )t2 dτ dt2

Rxy (τ ) e−∞−iω1 τdτZ∞e−i(ω1 −ω2 ) t2 dt2−∞= 2π Sxy (ω1 ) δ(ω1 − ω2 ).
Thus, the autocorrelation and the cross–correlation of the Fourier transformsof the stationary random processes are uniquely related to their power andcross–power spectra by equations (82) and (83). Therefore, the Fourier transforms of the stationary random processes can be regarded as useful tools forcalculating the spectra. The FX–type correlators are the realizations of thisprinciple.
Note that the expectation of Z(ω), which is the Fourier transform of astationary random process z(t), has a delta–function form with respect tothe angular frequency ω, that means not altogether constant in ω, except fora special case when hz(t)i = η = 0. Also, the RHS of equations (82) and(83) are not functions of angular–frequency difference ω1 − ω2 only, becauseof the dependence on ω1 in S(ω1 ) and Sxy (ω1 ), except for special cases of thecomplete white spectra, where S(ω) = const or Sxy (ω) = const. Therefore,48

the Fourier transforms of the stationary random processes are not wide–sense
stationary, in general, with respect to ω.

1.3

The White Fringe

1.3.1

A Simple Interferometer

A radio interferometer, in its simplest form, can be illustrated as Figure 14.

s
θ

cτg
v1
correlatorD
multiplierv2

integrator

Figure 14: A simple interferometer.
This ia a two–element interferometer consisting of idential antennas, idential receivers and a correlator, which is a combination of a multiplier andan integrator (a time–averager). We ignore here details of receiving systems, including the frequency conversion, just regarding as if the correlation
processing is performed at RF (radio frequency) band.
Important information which is derived from interferometric observations
is the geometric delay τg . For an infinitely distant point radio source, which
we assume in this simplified case, the geometric delay is expressed in a form:
D·s
D sin θ
=
,
(84)
c
c
where D is a “baseline vector” connecting reference points of two antennas,s is a “source vector” which is a unit vector directed towards the point radio
τg =

49

source, c is the light velocity, and θ is an angle of the source direction s froma plane perpendicular to the baseline vector D. For simplicity, we assume
that the same wavefront of the electromagnetic wave from an astronomical
radio source arrive at two antennas with a time delay, which is equal to thegeometric delay τg , ignoring atmospheric and other delay factors.
We assume a case when the beam centers of the two antennas are exactly
oriented towards the radio source. Also, we assume that the source direction
is close to the plane perpendicular to the baseline, i.e., θ ≈ 0, and thegeometric delay τg is within a small range around zero. Also, we ignore
effects of diurnal motion of an observed radio source, just assuming that thesource is at rest or moving very slowly.
We ignore, at this stage, any contribution of the system noise, in order toconcentrate our attention to the basic characteristics of the correlated radiosource signals only.
In summary, we assume following properties for our simple interferometer:• point–like radio source,• identical antennas,• identical receivers,• correlation at RF–band,• source diurnal motion is neglected,• no delay other than the geometric,• no system noise contribution.
1.3.2

Received Voltages as Stationary Random Processes

Let us assume that the received voltage v(t), as well as the electric field intensity E(t) of the radio wave which generates the voltage, are real stationary
random processes, satisfying the ergodicity. Here, we used a scalar function
E(t) for the electric field intensity, since any antenna can receive only one
polarization component of the electric field intensity vector E(t). Therefore,E(t) here stands for a single polarization component of E(t) in a plane perpendicular to the direction of propagation of the transversal electromagnetic
wave, which is commonly received by two antennas of the interferometer.
In actual radio astronomical observations, we usually see that the recieved
voltage oscillates around zero value and its time average is just zero, i.e.,time invariant. Also, the outputs of the correlators, which are time–averaged
50

products of the received voltages, are almost time invariant, as far as weneglect the slow intrinsic time variability of the radio source. At the sametime, the correlator outputs vary when we artificially insert different timedelays between the two voltage time series, implying that they are functionsof the time delay (i.e., of the time difference).
Therefore, the situation, which we experience in our observations, is justconsistent with the stationarity and ergodicity assumptions.
Let us consider that the received voltage v(t) and the electric field intensity E(t), generating the voltage, are related to each other by a linear systemwith a real impulse response q(t):v(t) = E(t) ∗ q(t) =Z∞−∞E(t − α) q(α) dα.

(85)

Here q(t) expresses the response of the antenna–receiver system to the incident radio wave, which, in particular, determines the frequency characteristics of the system as a BPF (band–pass–filter) passing a limited frequency
range with a bandwidth ∆ω centered at ω0 .
As we stated above, we assume that the responses of the antenna–receiversystems, in the two antennas of our simple interferometer, are identical, forsimplicity.
1.3.3

Cross–Correlation of Received Voltages

Let us denote the received voltages of the two antennas as v1 (t) and v2 (t).
Since they are generated by the same electromagnetic wave from a radiosource, but arrived at two antennas at different times due to the geometricdelay τg , we can express them through a common electric field intensity E(t)as:v1 (t) =v2 (t) =

Z∞

−∞
Z∞
−∞

E(t − τg − α) q(α) dα,E(t − α) q(α) dα,(86)

following equation (85). It is evident that v1 (t) and v2 (t) are jointly stationary random processes, because they are the outputs of linear systems (here,
we assumed identical) with the same input stationary random process E(t).
Now, let us consider that v1 (t) and v2 (t) are fed to the correlator shownin Figure 14. Since the correlation processing is the multiplication and inte51

gration of the signals, the correlator output R can be modeled as
1
R=
2T

ZT

v1 (t)v2 (t)dt.

(87)

−T

Of course, hRi = hv1 (t)v2 (t)i, as far as v1 (t) and v2 (t) are jointly stationary
random processes, and, in view of the ergodicity, R tends to hv1 (t)v2 (t)i as
the integration time T increases to the infinity:
R → hRi = hv1 (t)v2 (t)i,as

T → ∞.

(88)

Therefore, assuming that the integration time is sufficiently long, we canapproximate the output asR∼= hv1 (t)v2 (t)i = Rv1 v2 (0),(89)

where Rv1 v2 (0) is the cross–correlation of the two jointly stationary random
processes v1 (t) and v2 (t):Rv1 v2 (τ ) = hv1 (t)v2 (t − τ )i,(90)

at the time difference τ = 0.
Since v1 (t) and v2 (t) satisfy equation (86), we have
Rv1 v2 (τ ) =
=

Z∞ Z∞

−∞ −∞
Z∞ Z∞
−∞ −∞

hE(t − τg − α)E(t − τ − β)i q(α) q(β) dα dβ
REE (τ − τg − α + β) q(α) q(β) dα dβ,(91)where REE (τ ) is the autocorrelation of E(t):REE (τ ) = hE(t) E(t − τ )i.

(92)

Let us introduce the cross–power spectrum Sv1 v2 (ω) of the received voltages v1 (t) and v2 (t), which forms a Fourier transform pair with the cross–correlation Rv1 v2 (τ ), i.e., Rv1 v2 (τ ) ⇔ Sv1 v2 (ω), where ω is the angular frequency. Using the Fourier transformation equation (55), and the shift theorem given in equation (68), we haveSv1 v2 (ω) =Z∞

Rv1 v2 (τ ) e−iωτ dτ

−∞

52

=

Z∞

−∞

=
=

 ∞ ∞

Z Z

REE (τ − τg − α + β) q(α) q(β) dα dβ  e−iωτ dτ−∞−∞

 ∞ ∞ ∞


Z Z
Z

0
0

 −iωτ dτ 0 e−iωτgR
EE (τ − α + β) q(α) q(β) dα dβ e−∞ −∞−∞ ∞
Z0
REE (τ 0 ) ∗ q(τ 0 ) ∗ q(−τ 0 ) e−iωτ dτ 0  e−iωτg .
−∞Furthermore, let us introduce the power spectrum SEE (ω) of the incident
electric field intensity E(t), SEE (ω) ⇔ REE (τ ):SEE (ω) =Z∞

REE (τ ) e−iωτ dτ,

(93)

−∞

and the system function Q(ω) of the impulse responce q(t), Q(ω) ⇔ q(t):Q(ω) =

Z∞q(t)e−iωt dt.

(94)

−∞

Note that SEE (ω) is a real and even function of ω, since E(t) is a real process,and also Q∗ (ω) = Q(−ω) for the real impulse response q(t).
Then, in view of the convolution theorem in the Fourier transformation(see equations (66) and (72)), we obtain
Sv1 v2 (ω) = SEE (ω) Q(ω) Q(−ω) e−iωτg = SEE (ω) | Q(ω) |2 e−iωτg .

(95)

Applying the inverse Fourier transformation to this equation, we obtaina formula for the cross–correlation Rv1 v2 (τ ):1
Rv1 v2 (τ ) =2π

Z∞

Sv1 v2 (ω) e

iωτ

−∞

1
dω =
2π

Z∞

−∞

SEE (ω) | Q(ω) |2 eiω(τ −τg ) dω.
(96)

Taking τ = 0 in this equation, we obtain1
R∼= Rv1 v2 (0) =
2π

Z∞

−∞

SEE (ω) | Q(ω) |2 e−iωτg dω.

(97)This is an equation which gives a relation between the expectation of thecorrelator output hRi = Rv1 v2 (0) of our simple interferometer, and the spectrum of the radio wave coming from an astronomical source SEE (ω), filteredby the frequency response | Q(ω) |2 of the antenna–receiver systems.
53

As the simplest case of the frequency response | Q(ω) |2 , let us assume arectangular filter:


 G

2

| Q(ω) | = 0≤| ω |≤ ω0 + ∆ω,
if ω0 − ∆ω
2
2

(98)

otherwise,

where ω0 is the band–center angular frequency, ∆ω is the bandwidth inangular frequency, and G is a constant coefficient, as shown in Figure 15. On|Q(ω)|2

∆ω

∆ω

G

−ω 0

ω0

0

ω

Figure 15: Rectangular frequency response of the antenna–receiver system.
the other hand, we can assume, for a continuum spectrum source, that thepower spectrum of the radio wave SEE (ω) is flat, or “white–noise”, in thefilter passband:
SEE (ω) = S(ω0 ) = S(−ω0 ) = const.
(99)
In such a case, from equations (97), (98) and (99), the expectation of thecorrelator output R is expressed as:
∞



1 Z
<
hRi =
SEE (ω) | Q(ω) |2 e−iωτg dω 
π
0

=



S(ω0 ) G 
<

π

Z

ω0 − ∆ω2=

e−iωτg dω ∆ωS(ω0 ) G <e−iω0 τgπ

= 2 B S(ω0 ) Gω0 + ∆ω
2

Z2

− ∆ω
2

0

e−iω τg dω 0 sin(πBτg )cos(ω0 τg ),
πBτg

54




(100)

where < stands for the real part of the complex quantity, B = ∆ω/(2π) is
the frequency bandwidth, and ω 0 is chosen to satisfy ω = ω0 + ω 0 . In deriving
the above equation, we used a well–known integration formula:
1
2
A function of a form
1.3.4

Zx

0

e−ix dx0 = sin x.

−x

sin x
is known as “sinc function”.
xFringe Pattern Enclosed by Bandwidth Pattern
White Fringe with a Rectangular Band of 10 % of Center Frequency

normalized cross-correlation

1

0.5

0

-0.5

-1
-8

-6

-4

-2

0

2

4

6

8

bandwidth times geometric delay B τg

Figure 16: A normalized white fringe of a noise signal limited within a rectangular frequency band of width B, equal to 10 % of central frequency (solidline). Horizontal axis shows Bτg , i.e. the geometric delay multiplied by the
bandwidth. Also shown by dashed lines is the behaviour of the 1/(πBτg )term which quickly supresses the fringe amplitude with increasing τg .
Figure (16) shows the expectation of the correlator output hRi of a white
noise signal from the radio source, which is limited within a rectangularpassband of width B, according to equation (100). The vertical axis shows55

amplitude normalized by 2 B S(ω0 ) G, and the horizontal axis shows the geometric delay τg normalized by 1/B, i.e., Bτg . In this figure, the bandwidth
B is chosen to be equal to 10 % of the central frequency (B = 0.1ω0 /2π).
We again have a fringe pattern cos(ω0 τg ) enclosed by an envelope, which, inthis case, has a sinc function form, and takes the maximum value at τg = 0.
The enclosed fringe pattern, obtained from the band–limited white noise
spectrum, is called the “white fringe”.
correlation peakfringe spacing

bandwidth patternτgfringe patterncoherence interval

Figure 17: Technical terms describing the white fringe.
Figure 17 shows basic characteristics of the white fringe. The envelope,
enclosing the rapidly oscillating fringe pattern, is called “bandwidth pattern”.
A particular case of the rectangular band gives the sinc function pattern, as
we saw already. Other band shapes give different shapes of the bandwidth
pattern. But in any case, we always have the common feature, that theinterferometric fringes of finite amplitude are obtained within a limited rangeof the geometric delay, enclosed by a bandwidth pattern, as far as the noise
signal is band–limited.
Such a limited range of the geometric delay ∆τB , where the fringe pattern
has finite amplitude, is called “coherence interval”, and is roughly represented
by an equation ∆τB = 2/B, where B, in a general band–shape case, is aquantity which effectively characterizes a bandwidth.
The fringe spacing ∆τF , in terms of the geometric delay, is determinedby a condition ω0 ∆τF = 2π, therefore, ∆τF ' 1/ν0 .
The peak of the bandwidth pattern, which gives a precise observablefor the geodetic VLBI, is called “correlation peak”. Of course, the peakis the sharper, the wider the bandwidth B, and therefore the narrower thecoherence interval is (see Figure 18).

56

normalized correlation
normalized correlation
normalized correlation

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-30

-20

-10
8 GHz white fringe0
delay in nsec

10
bandwidth 8 MHz

20

30

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-30

-20

-10
8 GHz white fringe0
delay in nsec

10
bandwidth 80 MHz

20

30

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-30

-20

-10
8 GHz white fringe0
delay in nsec

10
bandwidth 800 MHz

20

30

Figure 18: White fringes with ceter frequency 8 GHz and various bandwidths:
B = 8 MHz (top), B = 80 MHz (middle), and B = 800 MHz (bottom).
Horizontal axes show geometric delays covering a range from −30 to 30 nsec.

57

1.3.5

Amplitude and Phase Spectra of the Correlated Signals

Equation (95) shows, that the cross–power spectrum Sv1 v2 (ω) of the voltage
signals v1 (t) and v2 (t), received by two antennas, is described through thepower spectrum SEE (ω) of the radio wave coming from a source, and the frequency response, assumed rectangular here, of the antenna–receiver system| Q(ω) |2 , as:Sv1 v2 (ω) = SEE (ω) | Q(ω) |2 e−iωτg .
Therefore, if we assume that the real, and even, power spectrum SEE (ω)
A(ω)

∆ω

0

-ω0

∆ω

ω

Φ(ω)

0

Θ

ω0

ω

Figure 19: Amplitude (top) and phase (bottom) spectra of the cross–power
spectrum.
is nearly constant in the receiving frequency band, the amplitude A(ω) andphase Φ(ω) of the cross–power spectrum, which we define as
Sv1 v2 (ω) = A(ω)e−iΦ(ω) ,(101)

A(ω) = SEE (ω) | Q(ω) |2 ∼= const,(102)

Φ(ω) = ωτg ,(103)

are expressed as:within the passband (see Figure 19). Note, that the phase spectrum is expressed by a straight line, crossing the origin and having an inclination tan Θ,which is equal to the geometric delay τg , i.e., tan Θ = τg , in the present simple interferometer model. This is a general feature of the phase spectra of
the continuum spectrum sources observed by interferometers.

58

1.3.6

Coherence Interval in the Sky

The coherence interval ∆τB = 2/B, in terms of the geometric delay, corresponds to a certain angular extent ∆θB in the sky (Figure 20). Following theassumption we made earlier, we consider only a region of the sky which is
close to the direction perpendicular to the baseline of an interferometer withlength D.

∆θ B

Figure 20: Coherence interval in the sky.
For a source in the sky, which is separated from a plane perpendicular tothe baseline by an angle θ, the geometric delay τg is equal toD sin θ
,
τg =
c

c τg
and, hence, θ = arcsin
,
D




(104)

where c is the light velocity. Therefore, the angular extent ∆θB isc∆τB∆θB = 2 arcsin
2D




'

c∆τB
2c
=
.
D
DB

(105)

The coherence intervals for several values of D and B are listed in Table 1.
It is evident from this table that the coherence interval in the sky is fairly
narrow for modern interferometers, especially for VLBI. Therefore, the passive observational mode, which would just “wait for” the passage of a source
through the narrow coherence interval with τg ≈ 0, is extremely ineffective
and unrealistic, except in the “classical” systems with ∼ 100 m, or shorter,
baselines and ∼ a few MHz, or narrower, bandwidths. Consequently, modern
radio interferometers are usally equipped with a special mechanism, which
compensates the delay, by time–shifting one of two received signals, so thatthe signals corresponding to the same wave front are fed to the correlation
processing simultaneously, as we will see later.

59

bandwidth B

200 kHz

2 MHz

200 MHz

2 GHz

∆τB

10 µsec

1 µsec10 nsec

1 nsec

D = 100 m> 180◦> 180◦

1.◦ 7

0.◦ 17

D = 10 km

17◦

1.◦ 7

1.0 0

6.00 2

D = 1000 km

0.◦ 17

1.0 0

0.00 620.00 062∆θB

Table 1: Coherence interval values for various baseline length D and bandwidth B.

In VLBI, it is very important to know accurate positions of the radiosources and accurate coordinates of the baseline vectors, for successful prediction and compensation of the delay, which allow us to detect the fringe
within the quite narrow coherence interval.
1.3.7

Fringe Spacing in the Sky
diurnal motion∆θ Ffringe spacing

Figure 21: Fringe spacing in the sky.
Since the fringe spacing is ∆τF = 1/ν0 , in terms of the geometric delay,
equation (104) gives the fringe spacing in the sky ∆θF , in the direction nearly
perpendicular to the baseline, as:
∆θF =

λ0
,
D

(106)

where λ0 = c/ν0 is the wave length at the central frequency ν0 of the receiving
band (see Figure 21 and Table 2).
60

central frequency ν0

100 MHz10 GHz

100 GHzwave length λ0

3m

3 cm

3 mm

D = 100 m

1.◦ 7

1.0 0

6.00 2

D = 10 km

1.0 0

0.00 620.00 062

D = 1000 km

0.00 620.00 0062

0.00 00062

Table 2: Fringe spacing values for various baseline length D and central
frequency ν0 .

The angular resolution of an interferometer is usually expressed by equation (106), since the resolution is essentially determined by the fringe spacing.
Since every radio source diurnally moves across the dense fringe patternin the sky with the very short fringe spacing, the phase of the cosine term
ω0 τg in equation (100) changes very rapidly. Therefore, the correlator outputof the signals of our simple interferometer must oscillate also very rapidly.
This would make mostly impossible to integrate the multiplier output fora duration of time, which is long enough to detect the white fringe with asufficiently high signal–to–noise ratio, since any simple time averaging (integration) of an oscillating signal results in almost zero signal only (Figure 22).
Therefore, modern radio interferometers are usually equipped with a special
mechanism to compensate (or stop) the rapid phase change, as we will see
later.
Tt

0

Figure 22: What will come out, if we integrate (time average) this?

2

A Realistic Radio Interferometer

The simple interferometer, which we discussed in the previous section, washelpful for understanding one of the most important concepts for radio inter61

ferometry, the white fringe. Nevertheless, the simple model is far from realmodern radio interferometers in following aspects.
First, radio sources are not mere points. They usually show structures,or intensity distributions, in the sky. One of the main purposes of the VLBI,
or radio interferometry in general, is to obtain images of the radio source
structures. But if the source is not point–like, the delayed voltage modelin equation (86), which was described through a single geometric delay of
a point source, must be no longer valid. Moreover, if the source structures
are extended, we have to take into account the beam patterns of element
antennas, as well.
Extended Source

cτ
g

=

D.
s0

s0

Beam
Pattern 1

Beam
Pattern 2D
Receiver 1

Receiver 2

Local
Oscillator
τiMultiplier
Integrator

Source structureand different
antenna-receivercharacteristics

Mixer

Frequency
conversion

Instrumental
Delay

Compensation ofdelay and phasechangeCorrelator

Figure 23: A realistic interferometer.
Second, the antennas and receiving systems in a radio interferometer are
not identical, in general. Therefore, the beam patterns and the frequency
responses of the antenna–receiver systems may differ from each other, unlikein the simple interferometer model.
Third, receiving systems in modern interferometers are usually based on
the superheterodyne design, and correlation processings are performed for IF(intermediate frequency) signals after the frequency conversion, but not for
RF signals, as assumed in the simple interferometer model.
62

Finally, we have to introduce the special mechanisms in order to compensate the geometric delay and the rapid phase change in the correlator
output, as we noted in the last two paragraphs of the previous section. Thisshould be done so that we can always observe a radio source whenever it isabove the horizon, but not just within a very short duration of time, while
the source happenes to pass the very narrow coherence interval.
However, we still neglect, for a while, atmospheric and instrumental delays, other than the geometric delay. Also, we ignore, as before, any contribution of the system noise, in order to concentrate our attention to thecharacteristics of the correlated radio source signals only.
Figure 23 illustrates a “more realistic” 2–element interferometer. In thefollowing subsections, we will examine effects of the source structure anddiffferent antenna–receiver characteristics, the frequency conversion, and thecompensation of the delay and the phase change, in turn.

2.1

Source Structure, Visibility and Intensity

2.1.1

Source Coherence Function

How the correlator output of an interferometer is related to the intensity (orbrightness) distribution of a radio source in the sky?
In order to answer to this question, we must consider how the electromagnetic field from a source, which is received and converted to the voltage
signal by each antenna of the interferometer, is related to the intensity distribution of the source. For this purpose, we first address ourselves to a
problem regarding properties of radio waves from an extended radio source,
namely, whether radio waves coming from different points of a radio source
are mutually correlated, or not.
Let us choose a certain direction in the source (for example, the directionof the maximum intensity), which is denoted by a unit vector s0 , as a reference direction. Then a unit vector s, pointing towards an arbitrary directionin the source, may be expressed as s = s0 + σ, and we can use the “offset
vector from the reference direction” σ ≡ s − s0 , as a vector, which is almost
confined in the celestial sphere and indicates the direction s. See Figure 24
for the geometry.
As a quantity representing the electromagnetic wave from the source, weagain choose a single polarization component of the electric field intensity
E(t), to be received by the antennas. Since the source is now extended, wedenote a component of the electric field intensity, which comes from a unit
solid angle around a direction σ, as e(σ, t). Then the incident electric field

63

s

σ

s0

σ’

s’

e(σ ,t)e(σ ’,t’)Figure 24: Radio waves coming from different directions in a radio source.
from the whole source E(t) is expressed through e(σ, t) as
E(t) =Ze(σ, t)dΩ,(107)sourcewhere dΩ is an infinitesimal solid angle element. Now, let us consider across–correlation of electric field components e(σ, t) and e(σ 0 , t0 ), comingfrom different directions σ and σ 0 , and taken at different times t and t0 :he(σ, t) e(σ 0 , t0 )i,(see Figure 24).
Assuming again that e(σ, t) and e(σ 0 , t0 ) are jointly stationary random
processes, we express the cross–correlation as a function of time differenceτ = t − t0 :γ(σ, σ 0 , τ ) = he(σ, t) e(σ 0 , t0 )i.
(108)This function γ(σ, σ 0 , τ ) is called “source coherence function”.
Using this source coherence function, we define a cross–correlation coefficient , which is called “normalized source coherence function”:
γN (σ, σ 0 , τ ) = qγ(σ, σ 0 , τ )γ(σ, σ, 0)γ(σ 0 , σ 0 , 0)

,

(109)

(here, we assume that the expectation of the electric field he(σ, t)i = 0, and,
therefore, the cross–correlation is equal to the cross–covariance). According
to the general property of the cross–correlation coefficient, the normalized
source coherence function always satisfies
0 ≤| γN (σ, σ 0 , τ ) |≤ 1.
64

Now we use following definitions.
1. Radio waves from different directions σ and σ 0 are called “completely
coherent” if | γN (σ, σ 0 , τ ) |= 1 for any τ , and “completely incoherent”
if | γN (σ, σ 0 , τ ) |= 0 for any τ .
2. A radio source is called “coherent” if radio waves from any different
directions σ and σ 0 in the source are completely coherent, and “incoherent” if the waves are completely incoherent. In all other cases, theradio source is called “partially coherent”.
If someone puts many transmission antennas in a town, and broadcasts a
TV programm, then any “poor–reception–level” problem would simply disappear. Instead, however, we would suffer from a serious “ghost” problem,since TV signals from different antennas are mutually coherent. The “ghost”
images would drastically change when we slightly shift or rotate our TV reception antenna to an extent, comparable with the wavelength of the TV
signal. Therefore, it would even become difficult to know directions of transmission antennas in the easy way, by just rotating the reception antenna
and watching the screen. But if the TV broadcasting is turned off, and only
incoherent (independent) noises are emitted from transmission antennas, itwould get much easier to know their directions.
Fortunately, most of actual astronomical radio sources are known to beincoherent, and, therefore, can be imaged relatively simply. This is becauseradio waves emitted from the source regions are just mutually independent
noises generated by random microscopic processes occuring there. Therefore,
in following discussions, we assume that radio sources are incoherent.
Then, the source coherence function must be expressed asγ(σ, σ 0 , τ ) = γ(σ, τ )δ(σ − σ 0 ),(110)

since radio waves from directions σ and σ 0 are correlated only when σ = σ 0 .
The function γ(σ, τ ), defined by equation (110), is called “self–coherencefunction”.
2.1.2

Power of Electric Field Incident from a Certain Direction

The autocorrelation of the electric field E(t) is expressed through the self–coherence function as:
hE(t)E(t0 )i ==Z ZsourceZ Z

sourcehe(σ, t)e(σ 0 , t0 )idΩdΩ0γ(σ, τ )δ(σ − σ 0 )dΩdΩ0 =65Zsourceγ(σ, τ )dΩ.

(111)

Therefore, we obtain, for a mean square value of the electric field E(t),2hE(t) i =Zγ(σ, 0)dΩ,(112)sourcewhich implies that γ(σ, 0) is a “density per solid angle” of the mean square
electric field in the direction of σ.
Consequently, if we consider a total electric field vector E(t), which comesfrom a small solid angle element ∆Ω towards a direction σ, and includes bothof the two independent polarization components, then its mean square value(or power) is related to the self–coherence function at τ = 0 as:
1
h| E(σ) |2 i = γ(σ, 0) ∆Ω,
2

(113)

where the coefficient 1 / 2 corresponds to the fact that γ(σ, 0) includes onlya single polarization component of the electric field.
2.1.3

Poynting Flux

Now we proceed to the relation between the incident electric field and thesource intensity (or brightness) distribution. First, we consider a quantity,
characterizing the incident electromagnetic wave, which we call “Poynting
flux through a cross section”.
Let us consider the Poynting vector S(σ) of an electromagnetic wave,
which comes from the same small solid angle element ∆Ω towards the direction σ, as in equation (113). Let ∆Ω be small enough that the wave iswell approximated by a superposition of monochromatic plane waves propagating along the same direction, with individual frequencies contained in a
finite bandwidth of the incident wave.
As we saw in Chapter 2, the Poynting vector S m (σ) of a monochromatic
plane wave is given, in terms of the corresponding electric field intensity
E m (σ), by an equation:
S m (σ) =1
n | E m (σ) |2 ,
Z

(114)

where Z is the intrinsic impedance of the medium, and n = −s = −(s0 + σ)
is a unit vector along a direction of the wave propagation.
The term | E m (σ) |2 , in equation (114), corresponds to a power of theelectric field, contained within an infinitesimally narrow frequency band ofa monochromatic plane wave. Such a power is equal to the power spectrumat the frequency of the band, multiplied by the bandwidth, as we discussed66

in subsection 1.2.8. On the other hand, the power | E(σ) |2 , contained in a
finite bandwidth, is simply equal to the integral of the power spectrum overthe bandwidth. Hence, we obtain the power | E(σ) |2 , by just summing up
all monochromatic plane wave components | E m (σ) |2 over the bandwidth:| E(σ) |2 =X

| E m (σ) |2 .

Meanwhile, the Poynting vector S(σ), with the finite bandwidth, is avector sum of all monochromatic plane wave components S m (σ):S(σ) =X

S m (σ).

Therefore, an equation, with the same form as the one in the monochromatic plane wave case, given in equation (114), holds also for the Poynting
vector S(σ) and the power of the electric field | E(σ) |2 , having the finite
bandwidth:
1(115)
S(σ) = n | E(σ) |2 .
Z
Let us now consider “Poynting flux through a cross section”, or simply“Poynting flux”, which we define as equal to the energy of the electromagnetic
wave passing through a certain cross section of unit area, per unit duration
s0
σ

∆Ω
s

N

n
θ S(σ)

Figure 25: Poynting flux through a cross section.
of time. If we denote a unit vector normal to the cross section as N , thePoynting flux of the wave coming from the direction σ is given by a projectionof the Poynting vector on the direction N :1(116)
S(σ) · N = | E(σ) |2 cos θ,Z67where θ is an angle between the normal unit vector N of the cross section
and the direction n of the wave propagation (see Figure 25).
Now we consider the Poynting flux S of the wave not only from a small
solid angle towards σ, but from the all source area:
X
1X
S(σ n ) · N =S=| E(σ n ) |2 cos θn ,
(117)
Z n
n
where

X

means a summation of all small solid angle elements towards direc-

ntions σ 1 , σ 2 , · · · , σ n , · · ·, covering the whole source area, and θn is an angle
between the normal to the cross section N and the direction of propagation
nn = −(s0 + σ n ).
Combining this equation with equation (113), we obtain an equation relating a mean Poynting flux to the self–coherence function:
1X
2X
hSi =
h| E(σ n ) |2 i cos θn =γ(σ n , 0) cos θn ∆Ωn ,
(118)
Z n
Z n
where ∆Ωn is a small solid angle element towards a direction σ n . If we replace
the summation, with respect to the small solid angles, by an integration, therelation between the mean Poyinting flux hSi and the self–coherence function
γ(σ, τ ) is given by:2
hSi =γ(σ, 0) cos θdΩ.
Z sourceZ(119)

If we further introduce the Fourier transform γ̃(σ, ω) of the self–coherencefunction γ(σ, τ ) in the angular frequency ω space, i.e., γ(σ, τ ) ⇔ γ̃(σ, ω):γ̃(σ, ω) =Z∞γ(σ, τ ) e−iωτ dτ,

(120)

−∞

1
γ(σ, τ ) =2π
then, we haveγ(σ, 0) =Z∞γ̃(σ, ω) eiωτ dω,−∞Z∞

γ̃(σ, ω) dν,−∞

where ν is a frequency corresponding to the angular frequency ω, i.e., ν =ω/(2π). Since Fourier transform of a real function is an even function of thefrequency, γ(σ, 0) is also given as:γ(σ, 0) = 2

Z∞
0

68

γ̃(σ, ω) dν.

(121)

Therefore, the mean Poynting flux in equation (119) is now given as:
4
hSi =
Z
2.1.4

Z∞ Z

γ̃(σ, ω) cos θ dΩ dν.

(122)

0 source

Electric Field and Radio Source Intensity — Electromagnetics and Astronomy

As we saw in Chapter 1, results of radio astronomical observations are characterized by a number of quantities, such as, “intensity” Iν , “spectral flux
density” Sν , and “power flux density” S. These quantities are phenomenologically defined in astronomy. For example, the intensity is defined as “thequantity of radiation energy incoming from a certain direction in the sky,
per unit solid angle, per unit time, per unit area perpendicular to this direction, and per unit frequency bandwidth with center frequency ν”. As wesee, no electromagnetic quantity, such as electric field intensity E, or voltage v, appears in such a phenomenological definition. Therefore, we must
precisely define a certain relationship between the electromagnetic and radioastronomical quantities, in order to describe radioastronomical results interms of the electromagnetic quantities, which we actually measure in ourradio telescopes.
Such a relationship was defined by IEEE (Institute of Electrical and Electronics Engineers) in 1977. According to the definition, the power flux density
in astronomy is equal to the time average of the Poynting vector in electromagnetics.
The power flux density S is defined in astronomy as “the quantity ofradiation energy, over the whole frequency range, incoming through a crosssection of unit area, per unit time”. This quantity is related to the spectral
flux density Sν , and to the intensity Iν as:
S=

Z∞
0

Sν dν =Z∞ Z

Iν (σ) cos θ dΩ dν,(123)

0 source

because the spectral flux density is defined as “the quantity of radiationenergy incoming through a cross section of unit area, per unit frequency
bandwidth, and per unit time”, and the intensity is defined as we saw above.
Since the power flux density is given with respect to a certain cross sectionof unit area, we interpret the “Poynting vector”, in the IEEE definition as a
Poynting flux, through the cross section.
Then, the definition of IEEE (1977) requires that the mean Poynting flux
hSi must be equal to the power flux density S:
hSi = S,
69

(124)

since, in view of the Ergodicity, the time average must be equal to the statistical mean, provided that the averaging time is sufficiently long. Therefore,
from equations (122) and (123), we have
Z∞ Z
0

4
Iν (σ) cos θ dΩ dν =ZsourceZ∞ Z

γ̃(σ, ω) cos θ dΩ dν.

(125)

0 source

Generally speaking, equal integrals do not necessarily mean equal integrands, of course. However, in our case of the stationary random signal froman incoherent source, we can equate integrands of the both sides of equation(125). In fact, the total power of the electric field intensity is simply equal
to a sum of contributions from all elements in frequency bands and spatial solid angles. To make this point more evident, let us imagine a virtual
source which emits radiation only in limited frequency and spatial solid–angle
ranges of the actual radio source. The average Poynting flux and the power
flux density from this virtual source are equal to those of the actual sourcein the respective frequency and solid–angle ranges. They must be expressed
through the self–coherence function and the intensity, just in the same forms
as those given in equations (122) and (123), but with limited frequency andsolid–angle ranges of the integrations. Since the definition of IEEE (1977)
requires their equality, for this virtual source as well, equation (125) must
hold for arbitrary frequency range ∆ν and arbitrary spatial solid–angle range∆Ω, i.e.,
Z Z

∆ν ∆Ω

4 Z Z
Iν (σ) cos θ dΩ dν =γ̃(σ, ω) cos θ dΩ dν,Z∆ν ∆Ωwhich implies that the integrands must be equal to each other. Therefore,
we have
4Iν (σ) = γ̃(σ, ω).
(126)
ZThis is a relation between the source intensity distribution and the spectrumof the self–coherence funtion, which is the density per solid angle of the powerof the incident electric field, as we saw before. Thus, we succeeded to relate
the source intensity distribution in astronomy to the incident electric field inelectromagnetics.
2.1.5

Field of View of a Radio Telescope

The incident electric field is converted to the voltage in a radio telescope antenna, and this voltage is actually processed in our detecting devices (square–
law detectors, or correlators). Then, how can we relate the received voltage
70

v(t) to the source intensity distribution? Here, we must take into accountthat an antenna collects electric fields, coming from different directions in aradio source, weighting them according to its beam pattern, specifically thevoltage reception pattern, which we saw in Chapter 2. This “field of view”
effect must be properly corrected for, prior to infering the source intensity
distribution from detected signals (Figure 26).

s

σ

s0

σ’

s’

e(σ ,t)e(σ ’,t’)Voltage reception patternv(t)Figure 26: Voltage reception pattern.
In order to consider this problem, it is convenient to move from the time
domain to the frequency domain, since, as we saw in Chapter 2, the beam
pattern of an antenna is a function of frequency (the beam width is roughly
proportional to ν −1 ).
Let us consider Fourier transforms ẽ(σ, ω) and ṽ(ω) of the single polarization component of the electric field intensity e(σ, t), coming from a unit
solid angle of direction σ, and the received voltage v(t), respectively.
ẽ(σ, ω) =ṽ(ω) =Z∞−∞Z∞e(σ, t)e−iωt dt,(127)v(t)e−iωt dt.

(128)

−∞

Since we assume that the electric field intensity e(σ, t) and the received
voltage v(t) are stationary random processes, these are examples of the71Fourier transforms of stationary random processes, which we discussed insubsection 1.2.9.
Without being involved in details of electromagnetics on wave reception,
we just assume that the received voltage at a certain frequency is proportional
to incident electric field intensity collected by the antenna beam at the same
frequency. Then, we introduce the voltage reception pattern Q(σ, ω) as afunction relating ṽ(ω) and ẽ(σ, ω) through an equation:ṽ(ω) =Zẽ(σ, ω)Q(σ, ω) dΩ.

(129)sourceNote that the voltage reception pattern thus defined has a dimension oflength (unit of voltage is V and unit of electric field intensity is Vm−1 ).
The voltage reception pattern is, in general, a complex quantity, since thereception process might be associated with some energy dissipation.
2.1.6

Power Pattern of a Receiving Antenna

Let us now establish a relationship between the voltage reception pattern,
as defined in equation (129), and the power pattern of a receiving antenna,
which we empirically introduced in Chapter 2. Although this is a topic
related to a single–dish radio telescope, following discussions will serve as
useful preparations for further considerations of interferometers.
If we denote a resistance in the circuit of the receiving system as R, thenaveraged power W and received voltage v(t) are related to each other by anequation:
hv 2 i
W =
.
(130)
R
Introducing an autocorrelation Rvv (τ ):Rvv (τ ) = hv(t) v(t0 )i,(131)

and power spectrum Svv (ω):
Svv (ω) =Z∞

Rvv (τ )e−iωτ dτ,

(132)

Z∞

(133)

−∞

1
Rvv (τ ) =2π

Svv (ω)eiωτ dω,−∞of the received voltage v(t), we reexpress equation (130) as∞

Rvv (0)2Z
W ==Svv (ω) dν.
R
R
0

72

(134)

Hence, the power per unit frequency bandwidth Wν , which satisfies
W =Z∞

Wν dν,

0is given by2
Svv (ω).
(135)
R
Now, let us express the power spectrum Svv (ω) in terms of the incident
electric field intensity. For this purpose, we first calculate the autocorrelationof the Fourier transform ṽ(ω) of the received voltage v(t). In view of equation(129), we haveWν =hṽ(ω)ṽ ∗ (ω 0 )i =Z Zsourcehẽ(σ, ω)ẽ∗ (σ 0 , ω 0 )iQ(σ, ω)Q∗ (σ 0 , ω 0 )dΩdΩ0 .

(136)Here, the cross–correlation hẽ(σ, ω)ẽ∗ (σ 0 , ω 0 )i of the Fourier transforms of
the incident electric field intensities, must be in a form:hẽ(σ, ω)ẽ∗ (σ 0 , ω 0 )i = 2πγ̃(σ, σ 0 , ω) δ(ω − ω 0 ),(137)

since we assume them as stationary random processes, which satisfy equation(83). In the right hand side of equation (137), we introduced a cross–power
spectrum γ̃(σ, σ 0 , ω) of the electric field intensities, which is the Fouriertransform of the source coherence function γ(σ, σ 0 , τ ), as defined in equation(108).
Since we assume an incoherent source, the source coherence function isexpressed through the self–coherence function γ(σ, τ ), as given in equation(110). Therefore, the cross–power spectrum must be given as:γ̃(σ, σ 0 , ω) = γ̃(σ, ω) δ(σ − σ 0 ).

(138)

Hence, equation (137) is rewritten as:hẽ(σ, ω)ẽ∗ (σ 0 , ω 0 )i = 2πγ̃(σ, ω)δ(σ − σ 0 )δ(ω − ω 0 ).

(139)

Inserting this equation to equation (136), we obtain

hṽ(ω)ṽ ∗ (ω 0 )i = 2π Zsourceγ̃(σ, ω) | Q(σ, ω) |2 dΩ δ(ω − ω 0 ).

(140)

Now, on the other hand, the autocorrelation hṽ(ω)ṽ ∗ (ω 0 )i must also be
expressed through the power spectrum Svv (ω) of the voltage as:hṽ(ω)ṽ ∗ (ω 0 )i = 2πSvv (ω)δ(ω − ω 0 ),
73

(141)

in view of equation (82), which is a general property of Fourier transformsof the stationary random processes. Thus, from equations (140) and (141),
we obtainZ
Svv (ω) =γ̃(σ, ω) | Q(σ, ω) |2 dΩ,(142)sourceor, taking into account equations (126) and (135),Wν =Z
Iν (σ) | Q(σ, ω) |2 dΩ.
2R sourceZ(143)

Equation (143) must be equivalent to the equation
1Iν (σ)Pn (σ)dΩ,Wν = A e
2 source
Z(144)

which we introduced in Chapter 2, in order to express the power per unit
bandwidth received by an antenna with a normalized power pattern Pn (σ)
and an effective aperture Ae . Noting again, that we can select arbitrary solid
angle as the source range, we can equate integrands of equations (143) and(144), to obtain
R| Q(σ, ω) |2 = Ae Pn (σ).
(145)
ZThis is the relation between the voltage reception pattern and the normalized
power pattern.
From equations (126) and (145), the power spectrum of the received voltage in equation (142) is now given as:
1Iν (σ) Pn (σ) dΩ.
Svv (ω) = Ae
4 source
Z

2.1.7

(146)

New Dimensions of Voltage and Electric Field

For further discussions, it is convenient to redefine the volatage and theelectric field intensity asv(t)√−→ v(t),Re(σ, t)√−→ e(σ, t),
Z

(147)

to eliminate constant coefficients of the resistance R and the intrinsic impedance
Z, which may cause rather complicated appearances of our equations. Then,
dimensions of the voltage and the electric field intensity change to74• voltage v• electric field intensity e[electric power]1/2(W1/2 )

[power flux density]1/2(W1/2 m−1 ),and we have
Svv (ω)−→ Svv (ω),
R
γ̃(σ, ω)−→ γ̃(σ, ω),
Z

(148)

for the power spectra of the voltage and the electric field intensity.
Let us further redefine the voltage reception pattern assZ
Q(σ, ω) −→ Q(σ, ω).
R

(149)

Then, the voltage — field relation remain in the same form as equation (129):ṽ(ω) =Zẽ(σ, ω)Q(σ, ω)dΩ,

sourceand we obtain simple expressions:Wν = 2Svv (ω),
Iν (σ) = 4γ̃(σ, ω),(150)

Ae Pn (σ) = | Q(σ, ω) |2 ,for the power per unit bandwidth, the source intensity, and the normalized
power pattern.
2.1.8

How Does A Radio Interferometer View the Universe?

A radio interferometer can be regarded as an instrument which yields a cross–correlation or a cross–power spectrum of voltages received at antennas. We
consider now the cross–correlation and cross–power spectrum of the voltages“just received”, which we will call hereafter “received voltages”. Although
actual correlation processing is performed for voltages after frequency conversion and compensation of the delay and the phase change, present discussion
is quite meaningful, since the final results are described through the spectrumof the received voltages, as we will see later.
Let us again consider a two–element interferometer, such as shown in
Figure 27. Let a baseline vector D be drawn from antenna 1 to antenna 2,75

s = s0 + σσs0τ g (σ)D
1

2

Figure 27: Geometry of a two element radio interferometer.
and let a unit vector s = s0 + σ point towards a certain direction within aradio source, where s0 shows a reference direction of the source, as before.
Then, the geometric delay, between arrivals of the same wave front fromthe direction s at two antennas, is given by
D·σ
D·s= τ g0 +
,
(151)
c
c
D · s0where c is the light velocity, τg0 ≡is a geometric delay of the wave
ccoming from the reference direction s0 , and we use the offset vector σ = s−s0
for denoting the direction s within the source. For simplicity, we ignore here
any atmospheric or instrumental delay effect, other than the geometric delay.
Now, the electric field intensities e1 (σ, t) and e2 (σ, t), with a single polarization component, which arrive at two antennas at a certain time t fromthe same direction σ, are described through a common incident field intensitye(σ, t), representing the same wave front, asτg (σ) =e1 (σ, t) = e(σ, t − τg (σ)),
e2 (σ, t) = e(σ, t).

(152)

Therefore, their Fourier transforms ẽ1 (σ, ω) and ẽ2 (σ, ω) are given byẽ1 (σ, ω) = ẽ(σ, ω)e−iωτg (σ ) ,ẽ2 (σ, ω) = ẽ(σ, ω),(153)

where ẽ(σ, ω) is the Fourier transform of e(σ, t). We used equation (68) of
the shift theorem, in deriving the upper equation of equation (153).
76

These electric field intensities are converted to voltages at the two antennas:ṽ1 (ω) =Zẽ1 (σ, ω)Q1 (σ, ω)dΩ,Zẽ2 (σ, ω)Q2 (σ, ω)dΩ,

sourceṽ2 (ω) =(154)

sourceaccording to equation (129).
Therefore, their cross–correlation is given byhṽ1 (ω) ṽ2∗ (ω 0 )i =Z Zsourcehẽ1 (σ, ω)ẽ∗2 (σ 0 , ω 0 )iQ1 (σ, ω)Q∗2 (σ 0 , ω 0 )dΩdΩ0 .

(155)
In view of equation (153), we can here express the electric field intensities at
two antennas through the common incident electric field intensity ẽ(σ, ω):hẽ1 (σ, ω)ẽ∗2 (σ 0 , ω 0 )i = hẽ(σ, ω)ẽ∗ (σ 0 , ω 0 )ie−iωτg (σ ) .

(156)

Since the electric field is regarded as a stationary random process, and theradio source is assumed to be incoherent, we have equation (139) again forhẽ(σ, ω)ẽ∗ (σ 0 , ω 0 )i:hẽ(σ, ω)ẽ∗ (σ 0 , ω 0 )i = 2πγ̃(σ, ω) δ(σ − σ 0 ) δ(ω − ω 0 ).
Therefore, the cross–correlation of the Fourier transforms of the received
voltages hṽ1 (ω) ṽ2∗ (ω 0 )i in equation (155) are now given byhṽ1 (ω) ṽ2∗ (ω 0 )i = 2π Zsourceγ̃(σ, ω) Q1 (σ, ω)Q∗2 (σ, ω)e−iωτg (σ ) dΩ δ(ω − ω 0 ).

(157)
On the other hand, since the voltages received at two antennas are jointly
stationary random processes (they originate from the same incident wave
front), the cross–correlation of their Fourier transforms and their cross–power
spectrum are related to each other by the general formula of equation (83):hṽ1 (ω) ṽ2∗ (ω 0 )i = 2πSv1 v2 (ω) δ(ω − ω 0 ).

(158)

From equations (157) and (158), we obtain
Sv1 v2 (ω) =Zγ̃(σ, ω) Q1 (σ, ω) Q∗2 (σ, ω) e−iωτg (σ ) dΩ.

source77(159)

Now, equation (150) shows that the Fourier transform γ̃(σ, ω) of the self–coherence function and the source intensity Iν (σ) are related to each otheras1γ̃(σ, ω) = Iν (σ).
4Furthermore, equation (150) also shows that the voltage reception patternQ(σ, ω) and the normalized power pattern Pn (σ) of an antenna are related
to each other as
Ae Pn (σ) =| Q(σ, ω) |2 ,where Ae is the effective aperture of the antenna. So, let us introduce here
an analog of this equation in an interferometer case:
A0 AN (σ) = Q1 (σ, ω) Q∗2 (σ, ω),(160)

where A0 is a geometrical mean of effective apertures of two antennas:
A0 =q

Ae1 Ae2 ,(161)

and AN (σ) is called “normalized power pattern of an interferometer”. Whenthe two voltage reception patterns Q1 and Q2 are real quantities (i.e., there isno phase change associated with the field → voltage conversion), or they haveequal phases, AN (σ) is described through the normalized power patterns Pn1
and Pn2 of the respective single–dish antennas as:
AN (σ) =q

Pn1 (σ) Pn2 (σ).

(162)

Taking into account equations (150), (151) and (160), and denotingωD·σ= 2πD λ · σ,
c

(163)

where D λ = D/λ is the baseline vector normalized by the wavelength λ =2πc/ω, we rewrite equation (159) in a form:
Sv1 v2 (ω) =1AN (σ) Iν (σ) e−i2πD λ ·σ dΩ.
A0 e−iωτg04sourceZ(164)

This equation gives a relationship between the cross–power spectrum of thereceived voltages and the intensity distribution of the radio source. This is
an interferometer analog of the power–spectrum — intensity relationship ina single dish antenna, given in equation (146):
Svv (ω) =1
AePn (σ) Iν (σ) dΩ.
4 sourceZ78

In closing this subsection, it is worthwhile to notice an approximation
made in our treatment of the geometric delay. In deriving equation (153)from equation (152), we implicitly assumed that the geometric delay τg isconstant in time, though, of course, the geometric delay must change in time
due to the diurnal motion of the Earth. Such a quasi–static approach is goodenough for our present purpose to understand basic features of radio interferometry, and we will continue to follow this approach hereafter. However, it
should be noted that the quasi–static approach does not allow us to properly
take account of some observable effects, such as Doppler shift in amplitude ofcross–power spectrum of a radio source due to the Earth’s rotation. Consequently, formulae we provide here lack these effects. More complete accountof this problem will require some modification in our assumption of the stationary random natutre of the received signals.
2.1.9

Complex Visibility

The cross–power spectrum of the received voltages Sv1 v2 (ω) is an observable
quantity, which can be derived from the correlator output, as we will see
in more detail, later. Equation (164) shows that the cross–power spectrum
is a product of three terms: the geometric mean of the effective apertures
A0 , the exponential term which depends on the geometric delay τg0 at thereference direction s0 of the observed source, and an integral over the solid
angle of the source. The exponential term stands for the fringe pattern,which we discussed in section 1.3. Therefore, the integral term figures as acomplex coefficient of the fringe pattern. We can derive this integral termfrom the observed cross–power spectrum, since A0 is almost constant and canbe estimated independently, and the fringe pattern is known, as far as weknow the source direction and the baseline vector with sufficient accuracies.
We call the integral, or the complex coefficient of the fringe pattern,“complex visibility”, “fringe visibility”, or just “visibility”, and denote it as
V(ω):I
V(ω) = A (σ) I (σ) e−i2πD λ ·σ dΩ.
(165)
N

ν

Here, we expanded the range of integration from the source region to thewhole sky, since we expect Iν (σ) = 0 outside the source region, and radiosources far from the antenna beam direction do not contribute to the integral,
simply because AN (σ) = 0 for those sources.
Equation (165) shows the relationship between the complex visibility andthe intensity distribution of the radio source. Amplitude | V(ω) | and phaseΦv of the complex visibility:
V(ω) =| V(ω) | eiΦv (ω) ,
79

(166)

are called “visibility amplitude” and “visibility phase”, respectively.
The complex visibility as shown in equation (165) has a dimension of theflux density (w m−2 Hz−1 ). In fact, the complex visibility can be regarded as
an analog of the effective flux density Sν received by a single–dish antenna:Sν =IPn (σ) Iν (σ) dΩ,(167)

which we introduced in Chapter 2. Thus, the visibility amplitude is also
called as “correlated flux density”.
Although equations (165) and (167) show similar forms, and the similarity
clearly comes from a common physical nature of the complex visibility andthe effective flux density, there is a big difference between these quantities,
as well. While angular resolution of a single–dish observation is determinedby the normalized power pattern of an antenna Pn (σ) in equation (167),
angular resolution of an interferometer observation is essentially determinedby e−i2πD λ ·σ term in equation (165), which represents the fringe patternof the interferometer, but not by the normalized power patterns of element
antennas figuring in AN (σ). In this sense, the complex visibility could be
regarded as the “intensity collected by a fringe pattern”, in contrast to theeffective flux density, which is the “intensity collected by an antenna beam”.
The cross–power spectrum of the received voltages is given through thecomplex visibility as:
Sv1 v2 (ω) =1
A0 e−iωτg0 V(ω).
4

(168)

This is an analog of the single–dish relation:
Svv (ω) =1
1
W ν = A e Sν ,
2
4

(169)

where Ae is the effective aperture of the antenna, and Wν is the received
power per unit frequency, as before.
The complex visibility is an important quantity of the interferometry for
infering the structure, or “image”, of the observed source, as we will discusslater.

2.2

Frequency Conversion in Radio Interferometers

It has been technically impossible to cross–correlate received voltage signals
at RF frequencies higher than ∼ 1 GHz. Therefore, in modern radio interferometers, the received RF signals have been frequency–converted to lower
IF frequencies, before the correlation processing (Figure 28).
80

So, let us now consider how cross–power spectra SvI1 vI2 (ω) of IF signals are related to the cross–power spectra Sv1 v2 (ω) of RF signals, which wehave so far discussed (see, for example, equation (168)). It is interesting toemamine, in particular, if the condition of stationary randomness of the RF

Sv v (ω)
1

v1(t)
hR1

2

Sv v (ω)

vR1(t)

hI1

R1

R2

(LNA + BPF)

vR2(t) RF signal

Local
Oscillator

Mixer

Sv v (ω)
I1

Received
signalv2(t)hR2
RF Filter

I2

vI1(t)hI2

IF Filter(IF Amp + BPF)

vI2(t) IF signal

to correlator

to correlator

Figure 28: How the cross–power spectrum SvI1 vI2 (ω) of IF signals are related
to the cross-power spectrum Sv1 v2 (ω) of received signals?
signals are conserved after the frequency conversion, where the RF signals are“mixed” (multiplied and then band–pass filtered) with sinusoidal reference
signals provided from LOs (local oscillators).
2.2.1

Response of RF Filters

The RF signals are usually amplified and band–pass filtered, before the frequency conversion, in cm– and long mm–wave observations. Therefore, wefirst take into account, in a general form, responses of the RF filters, which
are usually composed of LNAs (low noise amplifiers) and BPFs (band–passfilters). In cases, when first stage devices in receiving systems are mixers,
such as SIS mixers used in short mm–wave observations, this step must be
simply ommited.
81Let us consider that an RF filter is a linear system, whose input is thereceived voltage v(t), output is a voltage denoted as vR (t), and filter response
is described by a real impulse response hR (t).
Then, we havevR (t) = v(t) ∗ hR (t).
(170)

This means, in view of the convolution theorem given in equation (66), that
Fourier transforms ṽ(ω) and ṽR (ω) of the stationary random processes v(t)and vR (t), respectively, and system function HR (ω) of the impulse responsehR (t):
v(t) ⇔ ṽ(ω), vR (t) ⇔ ṽR (ω), hR (t) ⇔ HR (ω),(171)
satisfyṽR (ω) = ṽ(ω) HR (ω).

(172)

The system function HR (ω) here stands for effects of the LNA gain andband–pass characteristics of the BPF.
Thus, for Fourier transforms of input voltage signals v1 (t) and v2 (t), received at two antennas, and output voltages vR1 (t) and vR2 (t) of RF filters,with impulse responses hR1 (t) and hR2 (t), respectively,

we havev1 (t) ⇔ ṽ1 (ω), vR1 (t) ⇔ ṽR1 (ω), hR1 (t) ⇔ HR1 (ω),
v2 (t) ⇔ ṽ2 (ω), vR2 (t) ⇔ ṽR2 (ω), hR2 (t) ⇔ HR2 (ω),
ṽR1 (ω) = ṽ1 (ω) HR1 (ω),
ṽR2 (ω) = ṽ2 (ω) HR2 (ω).

(173)

(174)

Hence, cross–correlation of ṽR1 (ω) and ṽR2 (ω) is givn by∗∗
hṽR1 (ω) ṽR2
(ω 0 )i = hṽ1 (ω) ṽ2∗ (ω 0 )i HR1 (ω) HR2(ω 0 ).

(175)

Now, in view of the general properties of cross–correlations of Fouriertransforms of stationary random processes, given in equation (83), we have∗hṽR1 (ω) ṽR2
(ω 0 )i = 2π SvR1 vR2 (ω) δ(ω − ω 0 ),hṽ1 (ω) ṽ2∗ (ω 0 )i = 2π Sv1 v2 (ω) δ(ω − ω 0 ),(176)

where Sv1 v2 (ω) and SvR1 vR2 (ω) are cross–power spectra of input voltages v1 (t)and v2 (t), and output voltages vR1 (t) and vR2 (t), respectively, of the RF
filters. Consequently, equation (175) yields a relation:∗(ω),
SvR1 vR2 (ω) = Sv1 v2 (ω) HR1 (ω) HR2(177)

between the cross–power spectra of input and output signals of the RF filters(see Figure 28). Note that the cross–power spectrum of the received voltages
Sv1 v2 (ω) is related to the complex visibility of an observed source in equation(168).
82

2.2.2

Fourier Transform of IF Voltage Signal

Let us now proceed to the problem of the frequency conversion.
Actual frequency conversion is often performed through several steps, bysequentially shifting a spectral range, which contains observed data, fromhigher frequencies to lower frequencies, in turn. However, we assume herea simplified scheme, where single frequency conversion units down–convert
RF signals to final IF signals, at once. Basically, multi–step conversion isequivalent to a single–step conversion, as long as the total frequency shifts
and total band–pass characteristics are the same in both schemes. Therefore,
our simplified assumption will not lower generality of following discussions.

vR (t)

2 cos(ωLOt + φLO)

Mixer

LO

vm (t)hIvI (t)

Figure 29: A simple image of a frequency converter.
Let us consider a simple frequency converter, as illustrated in Figure 29.
Generally speaking, amplitude of the LO reference signal can be arbitrarily
chosen. We adopted here 2 cos(ωLO t + φLO ) as the reference signal, whereωLO is LO frequency and φLO is initial phase, since the coefficient 2 results
in a simple form of the IF signal after the down–conversion.
Then, an RF voltage signal vR (t) is multiplied with the LO reference
signal by a mixer, and a resulting signal vm (t) has a form:vm (t) = vR (t) 2 cos(ωLO t + φLO ) = vR (t) [ei(ωLO t+φLO ) + e−i(ωLO t+φLO ) ]. (178)83

Therefore, in view of the shift theorem given in equation (68), a Fouriertransform ṽm (ω) of vm (t) (i.e., vm (t) ⇔ ṽm (ω)) is given byṽm (ω) = ṽR (ω − ωLO ) eiφLO + ṽR (ω + ωLO ) e−iφLO ,(179)

where ṽR (ω) is again a Fourier transform of the random process vR (t).
An IF filter is usually composed of a BPF and an IF amplifier. So, let us
denote an impulse response and a system function of such an IF filter as hI (t)and HI (ω), respectively. By definition, they form a Fourier transformationpair:hI (t) ⇔ HI (ω).
(180)
Then, an IF voltage signal vI (t), which is an output of the IF filter, is described through the impulse response hI (t) asvI (t) = vm (t) ∗ hI (t).

(181)

Therefore, in view of the convolution theorem given in equation (66), we
obtain a Fourier transform ṽI (ω) of the IF voltage signal vI (t) (i.e., vI (t) ⇔ṽI (ω)) in a form:ṽI (ω) = ṽm (ω) HI (ω)= [ṽR (ω − ωLO ) eiφLO + ṽR (ω + ωLO ) e−iφLO ] HI (ω).
2.2.3

(182)

Cross–Correlation of Fourier Transforms of IF Signals

Let us now consider the frequency conversion in an interferometer, as schematically shown in Figure 30.
All notations in the previous subsection are used here again, except for
new sufficies 1 and 2, which distinguish signals and devices in antennas 1 and2. We assume here that reference signals with a common frequency ωLO are
fed to two mixers, based on an image of a connected–element interferometer, which is equipped with a common frequency standard. In general, thisassumption is not applicable to VLBI, as we will see later.
Now, the Fourier transforms of the IF voltage signals from antennas 1
and 2 are given byṽI1 (ω) = [ṽR1 (ω − ωLO ) eiφLO1 + ṽR1 (ω + ωLO ) e−iφLO1 ] HI1 (ω),
ṽI2 (ω) = [ṽR2 (ω − ωLO ) eiφLO2 + ṽR2 (ω + ωLO ) e−iφLO2 ] HI2 (ω).
Therefore their cross–correlation is∗hṽI1 (ω) ṽI2
(ω 0 )i = h[ṽR1 (ω − ωLO ) eiφLO1 + ṽR1 (ω + ωLO ) e−iφLO1 ]

84

(183)

Antenna 1

Antenna 2

vR1 (t)2 cos(ωLOt + φLO1)

MixervR2 (t)LO

vm1 (t)2 cos(ωLOt + φLO2)

Mixer

vm2 (t)hI1

hI2
vI1 (t)vI2 (t)Figure 30: Frequency conversion in an interferometer.
∗∗×[ṽR2(ω 0 − ωLO ) e−iφLO2 + ṽR2
(ω 0 + ωLO ) eiφLO2 ]i∗
×HI1 (ω) HI2(ω 0 )∗= [ hṽR1 (ω − ωLO ) ṽR2
(ω 0 − ωLO )i ei(φLO1 −φLO2 )∗+hṽR1 (ω + ωLO ) ṽR2
(ω 0 + ωLO )i e−i(φLO1 −φLO2 )∗+hṽR1 (ω − ωLO ) ṽR2
(ω 0 + ωLO )i ei(φLO1 +φLO2 )∗+hṽR1 (ω + ωLO ) ṽR2
(ω 0 − ωLO )i e−i(φLO1 +φLO2 ) ]∗×HI1 (ω) HI2(ω 0 ).
(184)

Since RF voltage signals vR1 (t) and vR2 (t) are assumed to be jointly stationary random processes, their Fourier transforms must satisfy∗
hṽR1 (ω) ṽR2
(ω 0 )i = 2π SvR1 vR2 (ω) δ(ω − ω 0 ),(185)

as shown in equation (83), where SvR1 vR2 (ω) is the cross–power spectrum of
RF voltages, which is given in equation (177). Therefore, we obtain∗
hṽI1 (ω) ṽI2
(ω 0 )i∗= 2π[SvR1 vR2 (ω − ωLO ) ei(φLO1 −φLO2 ) HI1 (ω) HI2(ω) δ(ω − ω 0 )∗+SvR1 vR2 (ω + ωLO ) e−i(φLO1 −φLO2 ) HI1 (ω) HI2(ω) δ(ω − ω 0 )∗+SvR1 vR2 (ω − ωLO ) ei(φLO1 +φLO2 ) HI1 (ω) HI2(ω − 2ωLO )0×δ(ω − ω − 2ωLO )∗+SvR1 vR2 (ω + ωLO ) e−i(φLO1 +φLO2 ) HI1 (ω) HI2(ω + 2ωLO )0×δ(ω − ω + 2ωLO )].

85

(186)First two terms in RHS of equation (186) are proportional to δ(ω − ω 0 ),and therefore, satisfy the condition of jointly stationary processes, given inequation (83). However, third and fourth terms in RHS of equation (186)are proportional to δ(ω − ω 0 − 2ωLO ) and δ(ω − ω 0 + 2ωLO ), respectively,
and therefore, do not satisfy the condition of equation (83). Consequently,
generally speaking, IF voltage signals after the frequency conversion may not
be jointly stationary, i.e. their cross–correlations may depend on time andLO frequency ωLO . This is not surprising, because the mixer output vm (t)in equation (178) is a product of the RF signal with a LO reference signal, aregular cosine oscillation, which is not random, nor stationary.
2.2.4

Roles of Low–Pass Filters

Nevertheless, if the cross–power spectrum of the RF signals is band–limited
within a certain frequency range, and the IF filters are designed to pass asuitable low–frequency range only, we can make the third and fourth terms
to be zero, so that resulting IF signals become jointly stationary.
In fact, if the cross–power spectrum of the RF signals is band–limited
within a range:
0 < ω1 <| ω |< ω2 ,(187)
and IF filters HI1 (ω) and HI2 (ω) are both confined within, or proportional
to, a rectangular low–pass filter HLP (ω):
HLP (ω) = 1

for | ω |≤ ωc , 0otherwise,(188)

with a cut–off frequency ωc , which satisfies conditions:
ω2 − ωLO < ωc ,ωLO − ω1 < ωc , ω <ω +ω ,
c

1

(189)

LO

then, we can eliminate the third and fourth terms in RHS of equation (186),
as illustrated in Figure 31.
Figure 31 shows real and imaginary spectral shapes of original complex
cross–power spectrum SvR1 vR2 (ω) (solid line) in RF–band, and two spectral
functions figuring in equation (186), namely, SvR1 vR2 (ω + ωLO ) (dotted line),and SvR1 vR2 (ω − ωLO ) (broken line).
86

SvR1 vR2 (ω) is shown here as Hermitian symmetric, since corresponding
cross–correlation RvR1 vR2 (τ ) is real (equation (60)). SvR1 vR2 (ω + ωLO ) has thesame spectral shape as the original SvR1 vR2 (ω), but shifted by ωLO towards
lower frequency side, while SvR1 vR2 (ω−ωLO ) also has the same spectral shape,
but shifted by ωLO towards higher frequency side.

HLP(ω+2ωLO)

Real
HLP(ω)

1−2ωLO−ω2 −ωLO −ω1−ωc0

Sv v (ω)
Sv v (ω+ωLO)
Sv v (ω−ωLO)ω1 ωLOω2

ωLO−ω1 ω2−ωLO
Imaginary

R2

R1

R2

−ω2 −ωLO −ω1

−ωc

0

ωc

3

ω

2ωLO
ω1+ωLO

2

1

−2ωLO

R2

R1

HLP(ω−2ωLO)

2

ωc

R1

3

ω1 ωLO

ω2

2ωLO

ω

Figure 31: Original complex cross–power spectrum SvR1 vR2 (ω) (solidline), and shifted–in–frequency spectra SvR1 vR2 (ω + ωLO ) (dotted line) andSvR1 vR2 (ω − ωLO ) (broken line). Boxes assigned with numbers show frequency
ranges corresponding to passbands of filters HLP (ω + 2ωLO ), HLP (ω), and
HLP (ω − 2ωLO ), respectively. Only those parts of spectra which are confined within the central box can be passed through the product of IF filters∗
HI1 (ω) HI2(ω) in equation (186).
Both spectral functions SvR1 vR2 (ω + ωLO ) and SvR1 vR2 (ω − ωLO ) consistof lower–frequency and higher–frequency parts, corresponding to negative–
frequency and positive–frequency parts of the original band–limited RF cross–power spectrum SvR1 vR2 (ω). They are confined in three frequency ranges
shown by boxes 1 , 2 , and 3 in Figure 31. These frequency ranges correspond to passbands of rectangular filters HLP (ω + 2ωLO ), HLP (ω), and
HLP (ω −2ωLO ), respectively, which are defined by equations (188) and (189).
Let us now examine contributions from 4 terms in RHS of equation (186),
one by one.
1. In the first term, the lower–frequency part of SvR1 vR2 (ω − ωLO ), which87

is inside the box 2 , can be passed through the product of IF filters∗
HI1 (ω) HI2(ω), but the higher frequency part, which is inside the box
3 , is cut off by the same filter product.
2. In the second term, the higher–frequency part of SvR1 vR2 (ω + ωLO ),which is inside the box 2 , can be passed through the product of IF∗
filters HI1 (ω) HI2(ω), but the lower frequency part, which is inside thebox 1 , is cut off by the same filter product.
3. In the third term, the entire spectrum SvR1 vR2 (ω −ωLO ) is cut off by the∗
product of IF filters HI1 (ω) HI2(ω − 2ωLO ), since the lower–frequency∗
part, which is inside the box 2 , is cut off by the filter HI2(ω − 2ωLO ),while the higher–frequency part, which is in 3 , is cut off by the filter
HI1 (ω).
4. In the fourth term, again, the entire spectrum SvR1 vR2 (ω + ωLO ) is cut∗
off by the product of IF filters HI1 (ω) HI2(ω + 2ωLO ), since the lower–
frequency part, which is inside the box 1 , is cut off by the filter HI1 (ω),
while the higher–frequency part, which is in 2 , is cut off by the filter∗
HI2(ω + 2ωLO ).
Thus, in the RHS of equation (186), the third and fourth terms are reduced
to zero, and only first 2 terms, with the higher–frequency part of SvR1 vR2 (ω +ωLO ) and lower–frequency part of SvR1 vR2 (ω − ωLO ), are left.
Therefore, equation (186) now results in:∗
hṽI1 (ω) ṽI2
(ω 0 )i= 2π[SvR1 vR2 (ω + ωLO ) e−i(φLO1 −φLO2 )+SvR1 vR2 (ω − ωLO ) ei(φLO1 −φLO2 ) ]∗× HI1 (ω) HI2(ω) δ(ω − ω 0 ).

(190)This equation has a proper form of a cross–correlation of Fourier transforms of jointly stationary random processes, which is proportional to δ(ω −ω 0 ), as given in equation (83). Consequently, the IF voltage signals vI1 (t)and vI2 (t) can now be regarded as jointly stationary random processes.
2.2.5

Relationship between RF Spectrum and IF Spectrum

The result of the previous section means that we can now define a cross–power
spectrum SvI1 vI2 (ω) of the IF voltage signals vI1 (t) and vI2 (t), according toequation (55), as we usually do for jointly stationary random processes. In

88view of equation (83), the cross–power spectrum must be related to the cross–∗correlation of the Fourier transforms hṽI1 (ω) ṽI2
(ω 0 )i by the general formula:∗hṽI1 (ω) ṽI2
(ω 0 )i = 2π SvI1 vI2 (ω) δ(ω − ω 0 ).

(191)

Equating RHSs of equations (190) and (191), we obtain
SvI1 vI2 (ω) = [SvR1 vR2 (ω + ωLO ) e−i(φLO1 −φLO2 )+SvR1 vR2 (ω − ωLO ) ei(φLO1 −φLO2 ) ]∗× HI1 (ω) HI2
(ω),

(192)

which is a relationship between a cross–power spectrum of IF voltage signals
SvI1 vI2 (ω), after the frequency conversion, and a cross–power spectrum of RF
voltage signals SvR1 vR2 (ω), before the frequency conversion.

Real

−ωLO

0

USB

LSB

upper
lower
sideband sideband

ωLOω

HI1(ω)H*I2(ω)
IF band

RF band

Imaginary
USB

−ωLO

0

LSB

ωLO

ω

HI1(ω)H*I2(ω)

Figure 32: Upper sideband (USB) and lower sideband (LSB) contributions ina cross–power spectrum of IF voltage signals, which is confined in a passband∗of the product of IF filters HI1 (ω) HI2(ω).
The IF cross–power spectrum is shown in Figure 32, where we assumed
same initial phases of local oscillators, i.e., φLO1 − φLO2 = 0 for simplicity.
89The IF spectrum is Hermitian symmetric, according to the general property
of a cross–power spectrum, which corresponds to a real cross–correlation,
as shown in equation (60). In our case, the cross–correlation of IF voltage
signals is surely real, simply because IF voltage signals themselves are real.
Since the cross–correlation can be described by the positive–frequency region of the Hermitian symmetric cross–power spectrum, as we saw in equation(62), we will pay our attension to the positive frequency region (ω ≥ 0) of the
IF spectrum. Then, the original RF cross–power spectrum, shown by solidlines in Figure 32, is down–converted to IF–band, and the upper sideband(USB) of SvR1 vR2 (ω + ωLO ) (shown by a broken line), and the lower sideband(LSB) of SvR1 vR2 (ω − ωLO ) (shown by a dotted line) are passed through the∗
product of the IF filters HI1 (ω) HI2(ω) in the positive IF frequency region,
as we see in Figure 32.
Thus, we can notice here following properties of the IF spectrum:• Spectral shape of the RF spectrum is preserved, and just shifted into
the IF–band.
• Upper sideband (USB) and lower sideband (LSB) components are su∗
perposed in the IF passband of HI1 (ω) HI2(ω).
• In the positive frequency range of the IF spectrum, ω ≥ 0, spectrum of
the LSB component is reversed, compared with the positive frequency
part of the original RF spectrum.
∗
Because of the Hermitian symmetry Sxy (−ω) = Sxy(ω) for real processes,
as shown in equation (60), we can rewrite equation (192) in a form:

SvI1 vI2 (ω) = [SvR1 vR2 (ωLO + ω) e−i(φLO1 −φLO2 )+Sv∗R1 vR2 (ωLO − ω) ei(φLO1 −φLO2 ) ]∗× HI1 (ω) HI2
(ω),

(193)

using only positive–frequency part of the RF cross–power spectrum. The
first term in the RHS of equation (193) shows USB component in the positive frequency side (ω ≥ 0) and LSB component in the negative frequency
side (ω < 0), while the second term shows LSB component in the positivefrequency side (ω ≥ 0) and USB component in the negative frequency side(ω < 0).

90

2.3

Delay Tracking and Fringe Stopping

In subsection 1.3, we saw, based on a simple interferometer model, that aradio interferometer can detect a radio source, only when the source is withinthe “coherence interval” of the white fringe, i.e., within a certain range,around zero, of the delay between arrival times of a common wave front,
from an astronomical radio source, at antennas. The delay range is roughly
equal to 2/B, where B is a frequency bandwidth of the received signal. Table
1 of subsection 1.3.6 showed that the delay range corresponds to an angularinterval in the sky, which is usually very narrow for modern interferometers,
especially for VLBIs. If we do not have any effective means to compensate thedelay, this would imply that we can observe a source, only when the source
happens to be within the very narrow region of the sky, where the geometricdelay is nearly zero, that is, of course, highly ineffective and unrealistic.
Also, we saw that the diurnal motion of a radio source causes rapid oscillation of the correlator output due to the densely spaced fringe patterns, as
schematically illustrated in Figure 21 and Table 2 of subsection 1.3.7. Suchan oscillation would make it practically impossible to integrate the multiplier output for a duration of time, sufficient to get a signal to noise ratio,
necessary for detecting the source.
Therefore, the modern interferometers must be equipped with special
mechanisms, for compensating the delay and stopping the rapid oscillationof the correlator output. They are called “delay tracking” and “fringe stopping”, respectively. Now we would like to discuss how these functions are
realized in a relaistic interferometer.
2.3.1

General Form of Cross–Power Spectrum of IF Signals

In order to clarify meanings of the delay tracking and fringe stopping in
more detail, we will first combine results of previous discussions, and derivea general description of the cross–power spectrum of IF voltage signals.
So far, we derived
1. the cross–power spectrum of received voltages:
Sv1 v2 (ω) =1
A0 e−iωτg0 V(ω),
4given in equation (168), where A0 is the geometrical mean of effective
apertures of antennas, and V(ω) is the complex visibility:
V(ω) =I

AN (σ) Iν (σ) e−i2πD λ ·σ dΩ,91given in equation (165), where AN (σ) is the normalized power patternof an interferometer, σ = s − s0 is an offset vector showing a certain
direction s in a source with respect to the reference direction s0 , Iν (σ)
is the source intensity distribution, and D λ = D/λ is a baseline vector
normalized by a wave length λ,2. the cross–power spectrum of RF voltage signals after the RF filters(amplifiers and BPFs) with system functions HR1 (ω) and HR2 (ω):∗
SvR1 vR2 (ω) = Sv1 v2 (ω) HR1 (ω) HR2(ω),

given in equation (177),
3. the cross–power spectrum of IF voltage signals after the frequency conversion:
SvI1 vI2 (ω) = [SvR1 vR2 (ωLO + ω) e−i(φLO1 −φLO2 )+Sv∗R1 vR2 (ωLO − ω) ei(φLO1 −φLO2 ) ]∗× HI1 (ω) HI2(ω),
given in equation (193), where ωLO is the LO reference frequency, φLO1
and φLO2 are initial phases of two local oscillators, and HI1 (ω) andHI2 (ω) are system functions of the IF filters.
Combining last two equations, i.e., equations (177) and (193), we have
SvI1 vI2 (ω)∗= [HR1 (ωLO + ω) HR2(ωLO + ω) Sv1 v2 (ωLO + ω) e−i(φLO1 −φLO2 )∗+HR1(ωLO − ω) HR2 (ωLO − ω) Sv∗1 v2 (ωLO − ω) ei(φLO1 −φLO2 ) ]∗× HI1 (ω) HI2
(ω).
(194)
Here, system functions of real impulse responses of IF filters must satisfy
equation (74):∗∗HI1 (ω) = HI1(−ω), and HI2(ω) = HI2 (−ω).

Therefore, introducing new system functions H1 (ω) and H2 (ω) for responsesof combined IF filters, composed of down–converted RF and original IF filters, in antenna 1 and antenna 2, respectively:
H1 (ω) = HR1 (ωLO + ω) HI1 (ω),
H2 (ω) = HR2 (ωLO + ω) HI2 (ω),
92

(195)

we can rewrite equation (194) as:
SvI1 vI2 (ω)= H1 (ω) H2∗(ω) Sv1 v2 (ωLO + ω) e−i(φLO1 −φLO2 )+H1∗ (−ω) H2 (−ω) Sv∗1 v2 (ωLO − ω) ei(φLO1 −φLO2 ) .

(196)

Hereafter, we call the complex function H1 (ω) H2∗(ω) “bandpass characteristics of combined IF filters”.
Then, inserting equation (168) into equation (196), we obtain a generalform of the cross–power spectrum of the IF voltage signals, which describes
SvI1 vI2 (ω) in terms of source intensity distribution, geometry of observation,filter responses, and parameters of the frequency conversion:
SvI1 vI2 (ω)
1= A0 {e−i[(ωLO +ω)τg0 +φLO1 −φLO2 ] V(ωLO + ω) H1 (ω) H2∗ (ω)
4+ei[(ωLO −ω)τg0 +φLO1 −φLO2 ] V ∗ (ωLO − ω) H1∗(−ω) H2 (−ω)}.
(197)
Of course, this equation satisfies the Hermitian symmetry condition:
SvI1 vI2 (ω) = Sv∗I1 vI2 (−ω),
required for real processes vI1 (t) and vI2 (t).
Representing the complex visibility in terms of visibility amplitude andvisibility phase, as introduced in equation (166), we have
V(ωLO + ω) = | V(ωLO + ω) | eiΦv (ωLO +ω) ,V ∗ (ωLO − ω) = | V(ωLO − ω) | e−iΦv (ωLO −ω) .

(198)

Also we represent the bandpass characteristics of combined IF filters in termsof its amplitude | H1 (ω) H2∗ (ω) | and phase Φb (ω):
H1 (ω) H2∗ (ω) = | H1 (ω) H2∗ (ω) | eiΦb (ω) ,H1∗ (−ω) H2 (−ω) = | H1 (−ω) H2∗ (−ω) | e−iΦb (−ω) .

(199)

Then, the cross–power spectrum of IF signals is now reduced to
SvI1 vI2 (ω)
1= A0 | V(ωLO + ω) | | H1 (ω) H2∗ (ω) | e−i[(ωLO +ω)τg0 +φLO1 −φLO2 −Φv (ωLO +ω)−Φb (ω)]
4
1
+ A0 | V(ωLO − ω) | | H1 (−ω) H2∗ (−ω) | ei[(ωLO −ω)τg0 +φLO1 −φLO2 −Φv (ωLO −ω)−Φb (−ω)] .
4
(200)
93

Again, first term in the RHS of equation (200) shows USB component in
the positive frequency range (ω ≥ 0) and LSB component in the negative
frequency range (ω < 0), while second term shows LSB component in thepositive frequency range (ω ≥ 0) and USB component in the negative frequency range (ω < 0).
USB LSB−ωLO

amplitude0

LSB USB

ωLO

ω

ωLO

ω

phase

−ωLO

0

Figure 33: Amplitudes and phases of two terms in a cross–power spectrumof IF voltage signals. Solid lines show original RF spectrum, and dotted andbroken lines correspond to first and second terms of the RHS of equation(200), respectively. For simplicity, only contribution of geometric delay isshown in phase spectrum.
If we denote amplitudes and phases of the first and second terms of the
RHS of equation (200) as Ap (ω), Φp (ω, t) and An (ω), Φn (ω, t), respectively,
we have
SvI1 vI2 (ω) = Ap (ω) e−iΦp (ω, t) + An (ω) e−iΦn (ω, t) ,
(201)
with
1
A0 | V(ωLO + ω) | | H1 (ω) H2∗ (ω) |,4
1
An (ω) =A0 | V(ωLO − ω) | | H1 (−ω) H2∗ (−ω) |,4
Ap (ω) =

94

(202)

and
Φp (ω, t) = (ωLO + ω)τg0 + φLO1 − φLO2 − Φv (ωLO + ω) − Φb (ω),Φn (ω, t) = −(ωLO − ω)τg0 − φLO1 + φLO2 + Φv (ωLO − ω) + Φb (−ω).
(203)
Here, we took into account that phases are functions of both frequency andtime, in view of the time–variable geometric delay τg0 . This shows amplitude
and phase spectra of the cross–power spectrum of IF signals, which can becontrasted to those given in equations (101), (102), and (103), for a simple
interferometer model. Again, first term in the RHS of equation (200) shows
USB component in the positive frequency range (ω ≥ 0) and LSB component
in the negative frequency range (ω < 0), while second term shows LSB
component in the positive frequency range (ω ≥ 0) and USB component in
the negative frequency range (ω < 0).
Figure 33 shows the amplitude and phase spectra of two terms of the
IF cross–power spectrum in the RHS of equation (200), which can be compared with Figure 19 for a simple interferometer. For simplicity, we showed
only contribution of geometric delay τg0 in this figure, ignoring small effectsof φLO1 , φLO2 , Φv , and Φb terms. The IF band here is assumed to be a“video–band” (or “baseband”), which includes zero frequency, or “DC (direct current)”, component with ω = 0.
2.3.2

“Expected Correlation” of IF Signals

If we had an idealized correlator, which would be capable of multiplying
two IF voltage signals, and immediately calculating a statistical mean of the
product, such a correlator would yield the cross–correlation of the IF voltage
signals RvI1 vI2 (τ ) at time difference τ = 0:
hvI1 (t) vI2 (t)i = RvI1 vI2 (0).

According to equation (197), we expect that this “instantaneous correlator output” is described by an inverse Fourier transformation of the cross–power spectrum SvI1 vI2 (ω), with τ = 0:
∞

∞

1 Z
1 Z
SvI1 vI2 (ω) dω = < SvI1 vI2 (ω) dω
RvI1 vI2 (0) =
2π
π
−∞

=

0



A0  −i(ωLO τg +φLO1 −φLO2 )0< e
4π+ ei(ωLO τg0 +φLO1 −φLO2 )

Z∞
0

Z∞
0

V(ωLO + ω) e−iωτg0 H1 (ω) H2∗(ω) dωV ∗ (ωLO − ω) e−iωτg0 H1∗ (−ω) H2 (−ω) dω  , (204)95where we used the property of a Hermitian symmetric cross–power spectrum
given in equation (74), in order to restrict range of integration to the positivefrequency side only.
H1(ω)H2*(ω)

∆ω

∆ω

G

−ω I

ω

ωI

0

Figure 34: Rectangular response of a real filter.
If we assume, similarly to what we did in the case of a simple interferometer model in Section 1.3, a real, even, and rectangular system functionsof the combined IF filters, with a gain factor G, bandwidth ∆ω = 2πB andcenter frequency ωI , as shown in Figure 34, i.e., G

H1 (ω) H2∗(ω) = H1∗ (−ω) H2 (−ω) =  0≤| ω |≤ ωI + ∆ω,
if ωI − ∆ω22otherwise,(205)

and also assume a continuum spectrum source, with almost constant complex
visibility across the frequency bandwidth,∆ω∆ω
V(ω) ∼≤| ω |≤ ωI +,(206)
= V for ωI −22
then, we obtain from equation (204)

A0 G 
RvI1 vI2 (0) =<e−i(ωLO τg0 +φLO1 −φLO2 ) V4πωI + ∆ω2Ze−iωτg0 dωωI − ∆ω2
ωI + ∆ω2+ ei(ωLO τg0 +φLO1 −φLO2 ) V ∗Z

ωI − ∆ω2

e−iωτg0 dω 
.


(207)

Since the integral in equation (207) is described in terms of the sinc function,
as we saw in equation (100):1πωI + ∆ω2Ze−iωτg0 dω = 2Be−iωI τg0ωI − ∆ω
2

96

sin(πBτg0 ),πBτg0and the complex visibility is represented in terms of its amplitude and phase:
V =| V | eiΦv ,equation (207) is further reduced to
A0 GB | V | sin(πBτg0 )2πBτg0× [cos(ωLO τg0 + ωI τg0 + φLO1 − φLO2 − Φv )+ cos(ωLO τg0 − ωI τg0 + φLO1 − φLO2 − Φv )].

RvI1 vI2 (0) =(208)

This equation is an analog of eauation (100), which we derived for a simple
interferometer model observing a continuum spectrum source. The sinc function stands for a bandwidth pattern, and two cosine terms are fringe patterns
corresponding to center frequencies of USB and LSB, respectively. Evidently,
first cosine term shows contribution of USB component, while second cosine
term shows contribution of LSB component.
Equation (208) again shows that we can detect fringe pattern of a continuum spectrum source within a narrow coherence interval, roughly representedby ∆τB = 2/B. Also, time variation of the geometric delay τg0 in the fringe
patterns causes rapid sinusoidal oscillation of RvI1 vI2 (0), which makes it almost impossible to integrate (time–average) the product of voltage signals
for a meaningful duration of time. Therefore, we definitely need the delay
tracking and fringe stopping for a realistic interferometer.
There is no widely–accepted name for the “instantaneous correlator output” RvX1 vX2 (0), where vX1 (t) and vX2 (t) could be any voltages in radio
interferometry, i.e., they could be received voltages, RF voltages, or IF voltages. This is not an actual correlator output, since any correlator output
is obtained only after integration of a product of voltages, or a multiplieroutput, for some duration of time in an integrator. This is not the productof individual voltages, either. This is a statistical expectation of the productof voltages, or the multiplier output, at a certain instance of time.
An important point here is the following. We consider that an actual
product of voltages consists of a signal component and a noise component,
and the signal component is described by this quantity: RvX1 vX2 (0). We
apply the fringe stopping, in order to stop the rapid oscillation of this signal
quantity, so that we can suppress the noise and detect the signal by time
averaging.
Hereafter, we call this quantity as an “expected correlation” in a radio
interferometer.
972.3.3

Radio Source Tracking by an Interferometer

What is needed is to compensate the geometric delay τg0 to get the same
effect, which would be obtained, if we move one of the antennas along thesurface of a sphere of diameter equal to the baseline length, such that τg0 isalways maintained equal to zero, as illustrated in Figure 35. Such compensa-

θcτg 0 = D sinθ

D

1

2

h1

h2

vR1(t)vR2(t)

τiv’R2(t) instrumentaldelay
correlator

multiplierintegrator

Figure 35: How to track a radio source by an interferometer? Insertion of an
instrumental delay, which compensates the geometric delay, in the RF–band
can do the job.
tion of the geometric delay enables an interferometer to “track” the motionof a radio source, in order to keep the source in its “interferometer beam”,
just as a single–dish radio telescope does with its antenna beam.
A time–variable delay circuit, which is inserted into the transmission system of one of the antennas at RF–band, and provides an instrumental delay98τi equal to the geometric delay τg0 , as illustrated in Figure 35, can realize, atleast theoretically, the required tracking.
Although this statement must be intuitively obvious, it would be still
useful to logically trace the effect of delay insertion, in the following way.
Insertion of an instrumental delay τi , in the RF signal transmission system
0of antenna 2, would make the RF voltage signal vR2(t), after the delay circuit,0
and its Fourier transform ṽR2 (ω), to be equal to0vR2(t) = vR2 (t − τi ),and therefore,

0ṽR2(ω) = ṽR2 (ω) e−iωτi ,(209)

where vR2 (t), and ṽR2 (ω), correspond to the RF voltage signal before theinstrumental delay. We used the shift theorem given in equation (68), in
deriving equation (209).
Therefore, taking a cross–correlation of Fourier transforms ṽR1 (ω) and0ṽR2 (ω) of RF voltage signals, we see
00∗∗hṽR1 (ω) ṽR2
(ω 0 )i = hṽR1 (ω) ṽR2
(ω 0 )i eiω τi .

(210)

Since we assumed that the RF signals vR1 (t) and vR2 (t) are jointly station0
ary random processes, vR1 (t) and vR2(t) must also be jointly stationary random processes. Hence, the cross–correlations of their Fourier transforms andcross–power spectra must be related to each other as:0∗0
0 (ω) δ(ω − ω ),hṽR1 (ω) ṽR2
(ω 0 )i = 2πSvR1 vR2∗
hṽR1 (ω) ṽR2
(ω 0 )i = 2πSvR1 vR2 (ω) δ(ω − ω 0 ),(211)

in view of equation (83). Comparing equations (210) and (211), we obtain
0 (ω) of the RF voltage
a relation between the cross–power spectrum SvR1 vR20signal vR1 (t) of antenna 1 and the delay–inserted RF voltage signal vR2(t)of antenna 2, and the cross–power spectrum SvR1 vR2 (ω) of the RF voltages
without delay insertion:
0 (ω) = Sv
SvR1 vR2(ω) eiωτi ,R1 vR2(212)

which leads to∗iωτi
0 (ω) = Sv v (ω) HR1 (ω) H
SvR1 vR2
1 2
R2 (ω) e1∗(ω),= A0 e−iω∆τg V(ω) HR1 (ω) HR24

where∆τg = τg0 − τi .
99

(213)

We used here equations (168) and (177) for expressions of the cross–power
spectra of RF voltages SvR1 vR2 (ω), and received voltages Sv1 v2 (ω).
As we see, an only different point between combined equations (168) and(177) and this equation (213) is that the geometric delay τg0 is replaced by
the delay difference ∆τg = τg0 − τi . Therefore, we can easily derive an expected correlation of an IF voltage vI1 (t) of antenna 1 and an IF voltage with00
00 (0), from equation (208),
delay inserted at RF–band vI2of antenna 2, RvI1 vI2assuming again a rectangular filter, given in equation (205), and constantvisibility over receiving bandwidth, given in equation (206). The result is
A0 GB | V | sin(πB∆τg )2πB∆τg× [cos(ωLO ∆τg + ωI ∆τg + φLO1 − φLO2 − Φv )+ cos(ωLO ∆τg − ωI ∆τg + φLO1 − φLO2 − Φv )].

00 (0)
RvI1 vI2=(214)

Thus, as long as τi = τg0 , no delay effect is left in the RF cross–power
spectrum, given in equation (213), and, therefore, also in the expected correlation of IF voltages in equation (214). This means that the insertion of an
instrumental delay, exactly equal to the geometric delay, at RF–band could
effectively reduce τg in equation (100), or τg0 in equation (208), to zero. This
would allow us to find fringes always at the center of the coherence interval,
and to integrate the product of voltage signals, as long as we wish. Or, otherwise speaking, we could realize both delay tracking and fringe stopping bya single operation, namely by the insertion of a time variable instrumental
delay, if we were allowed to do this at RF–band.
Unfortunately, it is almost impossible, in the current level of technology,
to realize any delay insertion at RF frequencies, higher than several GHz.
Therefore, the delay insertion is usually performed at IF–band.
2.3.4

Requirements to the Delay Tracking and Fringe Stopping

Before proceeding with discussions on actual realization of the delay tracking and fringe stopping, let us here estimate the ranges of variations, andacuracies required to theoretical predictions, of the geometric delay τg0 and
its time rate τ̇g0 . For simplicity, we will implicitly rely upon an image of thedelay insertion at RF–band discussed above, since this does not cause any
loss of generality in present discussions.
If we denote the baseline vector and the unit vector directed to the radiosource as D and s0 , respectively, the geometric delay τg0 and its time rate100

τ̇g0 , due to the diurnal rotation of the Earth, are given by:
D · s0,(215)c
Ḋ · s0(ω × D) · s0
τ̇g0 =
=
,
(216)
c
c
where ω is the angular velocity vector of the rotation of the Earth (|ω| '7.3 × 10−5 rad/sec) and c ' 3 × 108 m/sec is the light velocity. Of course,
equation (216) is valid for ground–based interferometers only, and not valid
for space–VLBI baselines.
For a connected–element radio interferometer (CERI) of 300 m baseline,
say, the maximum geometric delay:
τg0 =D= 1 µsec,c
is obtained when the source direction is parallel to the baseline, and the
maximum delay rate:τgmax =

ωD
= 70 psec/sec,c
is obtained when ω ⊥ D and (ω × D) k s0 .
τ̇gmax =

D
R

ω
2R

Figure 36: The maximum geometric delay (top) and maximum delay rate(bottom) of the ground–based VLBI.
For the ground–based VLBI, the maximum geometric delay and the maximum delay rate are obtained in cases shown in Figure 36. If we denote theEarth’s radius as R⊕ = 6300 km, the maximum values areR⊕= 21 msec,
c2R⊕ ω= 3.1 µsec/sec.
=cτgmax =

(217)

τ̇gmax(218)

101

If we observe at 22 GHz, equations (208) and (218) implies that the expected
correlation, defined in subsection 2.1.8, may oscillate with frequency as highas 22 GHz × 3.1 × 10−6 = 68.2 kHz, or 68,200 cycles per second(!), due
to the diurnal motion of the source across the fringe pattern. Indeed, no
meaningful integration of the multiplier output is possible, unless we stopsuch an oscillation.
The ranges of variations of geometric delay and its rate, as examined
above, and resulting variations of the phase in the cross-power spectrum of
voltage signals, such as shown in equation (200), are orders of magnitudelarger, than those caused by other delay effects (clock offset, atmosphericpropagation delay, cable delay, etc.), and by other phase terms (LO initial
phase, visibility phase, phase of bandpass characteristics, etc.). Therefore,
the delay tracking and fringe stopping are, primarily, means for compensationof geometric delay. Although some other, much smaller, effects are also taken
into account in highly precise delay tracking and fringe stopping, performedin VLBI correlators, we will postpone discussion of this problem, until we
consider correlation processings in VLBI.
We use an appropriate theoretical prediction of the time–variable geometric delay, in order to determine length and rate of the instrumental delay,
to be applied to actual delay tracking and fringe stopping. Hence, the theoretical prediction itself, for the geometric delay τg0 , has been traditionallydenoted as τi , and called “instrumental delay”.
Let us estimate accuracies required to such an instrumental delay τi andits rate τ̇i .
In order to get a white fringe near the peak of the bandwidth pattern,a difference ∆τg between the actual geometric delay τg0 and the predicted
instrumental delay τi , which we call hereafter “residual delay”:∆τg = τg0 − τi ,(219)

must be kept well smaller than the coherence interval ∆τB = 2/B, where B
is an observing bandwidth:∆τg  ∆τB =

2
.
B

(220)

Therefore, the acuuracy of the instrumental delay must be much better than
1 µsec when B = 2 MHz, and much better than 1 nsec when B = 2 GHz, as
Table 1 shows. We now compare these required accuracies with maximum
delay values τgmax , estimated above for 300 m baseline connected–element
radio interferometer (CERI), and for VLBI. Table 3 shows relative accuracy,
in terms of ∆τg /τgmax , required to the instrumental delay. The Table shows102300 m CERI

VLBI

τgmax = 1µsec

τgmax = 21 msec

1

 5 × 10−5

B = 2 MHz (∆τg  1 µsec)

 10−3

B = 2 GHz (∆τg  1 nsec)

 5 × 10−8

Table 3: Relative accuracy of ∆τg /τgmax required to the instrumental delay.

that the required accuracy is rather high, for VLBI.
On the other hand, in order to carry out integration of the multiplieroutput in a correlator, for a duration of time sufficient to get high enoughS/N ratio, we must apply an accurate theoretical model τi (t) to compensate
for time variation of the geometric delay τg0 . If integration time is τa andobserving frequency is ν, a requirement to ∆τ̇g = (τ̇g0 − τ̇i ) for successful
integration, is traditionally given by2πν∆τ̇g τa ≤ 1.

(221)

This requirement corresponds to a condition that an accumulated change ofthe phase of the cross–power spectrum of voltage signals, such as shown inequation (213), or of the phase in cosine terms of the expected correlationin equation (214), does not exceed 1 radian during the integration time τa .
Table 4 shows required accuracy of the residual delay rate ∆τ̇g , according toν = 10 GHz
ν = 100 GHz

300 m CERI

VLBI

τa = 1000 s, τ̇gmax = 70 ps/s

τa = 1 s , τ̇gmax = 3.1µs/s≤ 1.6 ×10−2 ps/s (≤ 2.3 ×10−4 )≤ 16 ps/s (≤ 5.2 ×10−6 )≤ 1.6 ×10−3 ps/s (≤ 2.3 ×10−5 ) ≤ 1.6 ps/s (≤ 5.2 ×10−7 )Table 4: Required accuracy of ∆τ̇g and relative accuracy of ∆τ̇g /τ̇gmax (inparenthes) for a 300 m baseline CERI with integration time τa = 1000 s
and maximum delay rate τ̇gmax = 70 ps/s, and a VLBI with integration timeτa = 1 s and maximum delay rate τ̇gmax = 3.1 µs/s .
equation (221), and its ratio to the maximum delay rate, estimated above,for a 300 m baseline CERI with a typical integration time τa = 1000 s, and a
VLBI with a typical hardware integration time with a VLBI correlator τa = 1
s, at two cases of observing frequency, ν = 10 GHz and ν = 100 GHz.
103The requred accuracies are fairly high, especially in VLBI. Therefore,
theoretical prediction of the instrumental delay is usually based on state–of–the–art geophysical and astronomical models of station coordinates, radiosource coordinates, and irregularities in the rotational motion of the Earth.
2.3.5

Insertion of Instrumental Delay at IF BandLet us consider a case, when a time–variable instrumental delay τi is inserted
in the IF–band signal transmission system of antenna 2, as illustrated in
Figure 37.

θcτg = D sinθ

D

LO

τi
correlator

instrumentaldelay

multiplierintegrator

Figure 37: Time–variable instrumental delay inserted at IF–band.
0
After the insertion of the delay circuit, IF voltage signal vI2(t), and its
0Fourier transform ṽI2 (ω), are changed to0vI2(t) = vI2 (t − τi ),and, therefore,
104

0
ṽI2
(ω) = ṽI2 (ω) e−iωτi ,(222)

where vI2 (t), and ṽI2 (ω), correspond to the IF voltage signal before the instrumental delay. We again used the shift theorem given in equation (68),for deriving equation (222).
Following discussions in Subsection 2.3.3, we take a cross–correlation of0
Fourier transforms ṽI1 (ω) and ṽI2(ω) of IF voltage signals, to obtain00∗∗
hṽI1 (ω) ṽI2
(ω 0 )i = hṽI1 (ω) ṽI2
(ω 0 )i eiω τi .

(223)

0
Since the IF signals vI1 (t) and vI2(t), as well as vI1 (t) and vI2 (t), are assumed
to be jointly stationary random processes, the cross–correlations of their
Fourier transforms and cross–power spectra are related to each other, by
equation (83), as:0∗0
0 (ω) δ(ω − ω ),hṽI1 (ω) ṽI2
(ω 0 )i = 2πSvI1 vI2∗
hṽI1 (ω) ṽI2
(ω 0 )i = 2πSvI1 vI2 (ω) δ(ω − ω 0 ).

(224)

Comparing equations (223) and (224), we see that the cross–power spec0 (ω) of the IF voltage signal vI1 (t) of antenna 1 and the delay–trum SvI1 vI20
inserted IF voltage signal vI2(t) of antenna 2, and the cross–power spectrum
SvI1 vI2 (ω) of the IF voltages without delay insertion, are related to each otheras:iωτi
0 (ω) = Sv v (ω) e
SvI1 vI2
.
(225)
I1 I2
Inserting equation (197) to this equation, we have0 (ω)
SvI1 vI2
1= A0 {e−i[ωLO τg0 +ω∆τg +φLO1 −φLO2 ] V(ωLO + ω) H1 (ω) H2∗ (ω)
4+ei[ωLO τg0 −ω∆τg +φLO1 −φLO2 ] V ∗ (ωLO − ω) H1∗ (−ω) H2 (−ω)},(226)

where, again, ∆τg = τg0 − τi is the residual delay. Contrary to equation(213), which describes the effect of the delay insertion at RF–band, we cannot
here completely compensate for the geometric delay τg0 , even if we insert an
instrumental delay which is exactly equal to the geometric delay τi = τg0 . In
fact, a term ωLO τg0 is still left uncompensated in the exponential functionsof equation (226).
In order to see this point more clearly, let us derive the expected corre0 (0), as we did in equation (207), assuming again a rectangularlation RvI1 vI2
filter, given in equation (205), and constant visibility over receiving bandwidth, given in equation (206). Since everything is the same as given in105equation (207), except for a coefficient of ω term in the exponential functions, namely ω∆τg instead of ωτg0 , we obtain the result of integration:
A0 GB | V | sin(πB∆τg )2πB∆τg× [cos(ωLO τg0 + ωI ∆τg + φLO1 − φLO2 − Φv )+ cos(ωLO τg0 − ωI ∆τg + φLO1 − φLO2 − Φv )],0 (0)RvI1 vI2=(227)

where ωI is the IF band center frequency, which is similar to equation (208),
except in the argument of sinc function and in the coefficient of ωI term incosine functions.
It is evident, from this equation, that insertion of an instrumental delay
τi , which is equal to the geometric delay τg0 , at IF–band, enables us todetect a white fringe at the center of the coherence interval. However, ωLO τg0
terms in arguments of cosine functions are left uncompensated, which willcause enormously rapid oscillations of the expected correlation. Therefore,
insertion of an instrumental delay at IF–band can perform the delay tracking,but not the fringe stopping.
2.3.6

Separation of Delay Tracking and Fringe Stopping Due toFrequency Conversion

The reason, why the insertion of the instrumental delay at IF–band canperform only the delay tracking, but not the fringe stopping, may be easily
understood by examining Figure 38, which illustrates a time variation of thephase spectrum of IF voltage signals shown in Figure 33, in a particular caseof a single–sideband reception of USB component.
Let us consider phase spectra in cross–power spectra of voltage signals
without delay insertion, and compare the phase spectrum in IF–band withthe one in RF band. In doing so, we ignore minor phase terms, such as LO
initial phase, visibility phase, and phase of bandpass characteristics, which
figure in equation (200), leaving only the main term due to the geometricdelay.
Then, the phase spectrum in RF–band is a straight line crossing theorigin, with inclination equal to the geometric delay τg0 , as shown in equation(103) and Figure 19 for a simple interferometer model, or in equations (168),and (177), for a more realistic interferometer.
However, after the frequency conversion, the phase spectrum in the IF–band no longer crosses the origin, since the frequency conversion shifts the106phase

phase

=

+

ω
phase

ω
phase

=
τg

0

ω
phase

+

ω

ω
phase

=
RF

phase

=
phase

IF

phase

+

ω
phase

phase

ω

phase

+
object ofdelay
trackingω

object offringestopping

Figure 38: Necessity of fringe stopping, besides delay tracking in IF–band.
We assume here a USB single–sideband reception, and show only positivefrequency range of the phase spectrum, except in the bottom panel.

107

band–limited spectrum at RF–band to IF band without changing its shape,
as shown in Figures 32, 33, and 38.
Therefore, the frequency converted phase spectrum in IF–band is now
represented by a sum of two components, namely an inclined straight lineωτg0 , crossing the origin, plus a frequency–independent phase shift equal toωLO τg0 , where ωLO is the LO frequency, as illustrated in the right panels of
Figure 38 and in equation (200). This phase shift rapidly changes in time,with the speed as high as several tens kHz (as we saw in the example of anintercontinental VLBI observation with 22 GHz), due to the diurnal variationof the geometric delay τg0 .
The insertion of the instrumental delay at the IF–band compensates forthe first component, i.e. reduces the phase slope across the frequency bandto nearly zero, and thus makes it possible to get a white fringe in the middle of the coherence interval, as evident from equation (208). However, theinstrumental delay in the IF–band does not compensate for the second component at all. Therefore, we still need to compensate for the rapid phase shift
in the second component, by somehow controlling the interferometer phase.
This operation is the “fringe stopping”, which is separated from the “delay
tracking” after the frequency conversion.
2.3.7

Actual Implementations of Delay Tracking

In early history of radio interferometry, analog delay cables, as schematicallytransmission cabledelay cables

Figure 39: Analog delay cables used for delay tracking in early history ofradio interferometry.
illustrated in Figure 39, were used for delay tracking. N delay cables withlengths τ0 , 2τ0 , 4τ0 , 8τ0 , · · ·, 2N −1 τ0 , were mechanically inserted into, andremoved from, a signal transmission cable of an interferometer, to realize
a time variable instrumental delay τi with a range of variation: τ0 ≤ τi ≤(2N − 1)τ0 . Of cource, the longest delay cable must have a length as large
as about a half of baseline length of the interferometer. Therefore, such108

analog delay cables could be used for connected–element interferometers withbaseline lengths of several hundreds meters, but definitely not for VLBIs.

input dataring buffer
readaddress

write
addressdelay
delayedoutput data

Figure 40: A “ring buffer” used for digital delay tracking.
Recently, most of radio interferometers, including VLBIs, use digital delay
circuits for delay tracking. In order to use this technique, data signal is first
digitized, and then fed to a bulk memory, which is a kind of “ring buffer”,
schematically shown in Figure 40.
Data are written to, and read from, the ring buffer at different addresses,
and these write and read addresses are incremented by one at each “clock
timing pulse” of digital circuit. An interval of succcessive clock timing pulses
corresponds to a sampling interval ts of digitization of the data. Thus, if
the difference between the write and read addresses in the ring buffer isn, an instrumental delay equal to nts is realized in this way. And further
shifting the write (or read) address periodically, once per a certain numberof clock timing pulses, we can vary this instrumental delay in time. We will
examine this digital delay tracking in more detail, when we will discuss VLBI
correlators.
2.3.8

Actual Implementations of Fringe Stopping

In connected–element interferometers, the fringe stopping is often carried out
by actively controlling the initial phases φLO1 and φLO2 of the LO reference
signals, as illustrated in Figure 41, so that
φLO1 (t) − φLO2 (t) = −ωLO τi (t) + ψLO1 − ψLO2 ,

109

(228)

θ

cτg = D sinθD

ΦLO1(t)ΦLO2(t)LOfringestopping

τicorrelator

multiplierdelay
tracking

integrator

Figure 41: Fringe stopping by active control of LO initial phases.
where ψLO1 − ψLO2 is a nearly constant part of the difference of the initial
phases of the LO signals, and τi is the instrumental delay, i.e., the theoreticalprediction of the geometric delay τg0 .
Assuming quasi–static LO initial phases, similarly to what we did in
subsection 2.1.8 for the geometric delay, we can just insert equation (228) toequation (226) for the cross–power spectrum of IF voltages, and to equation(227) for the expected correlation in the case of a continuum flat spectrum
source and rectangular filters, after the delay tracking at IF–band. Then, we
obtain the cross–power spectrum of IF voltages:
SvfI1 v0 (ω)
I2

1= A0 {e−i[ωLO ∆τg +ω∆τg +ψLO1 −ψLO2 ] V(ωLO + ω) H1 (ω) H2∗(ω)
4+ei[ωLO ∆τg −ω∆τg +ψLO1 −ψLO2 ] V ∗ (ωLO − ω) H1∗ (−ω) H2 (−ω)},(229)
and expected correlation:
RvfI1 v0 (0) =I2

A0 GB | V | sin(πB∆τg )
2
πB∆τg
110

× [cos(ωLO ∆τg + ωI ∆τg + ψLO1 − ψLO2 − Φv )+ cos(ωLO ∆τg − ωI ∆τg + ψLO1 − ψLO2 − Φv )],(230)

respectively.
Now we can stop the rapid phase changes in the cross–power spectrum,
as well as in the expected correlation, provided that we can always keep theresidual delay to be nearly zero:∆τg = τg{0} − τi ∼= 0.
Note that this active control of LO initial phases can stop the phasechanges both in USB and LSB components of the cross–power spectrum andexpected correlation of IF voltages, as evidenced by equations (229) and(230). In particular, arguments of first and second cosine terms in the RHS
of equation (230), which show USB and LSB contributions, respectively, areboth stopped by this operation.
Another method for fringe stopping, which has been applied to VLBI
digital correlators, in particular, uses multiplication of a sinusoidal functionof time to an IF voltage, as shown in Figure 42.

θcτg = D sinθDantenna 1

antenna 2

LO
vI1(t)
2 cos(ωLO τi)fringe stopping

τifvI1(t)

correlator

vI2(t)’

multiplierdelay
tracking

integrator

Figure 42: Fringe stopping by multiplication of a sinusoidal function.
111

In this method, a sinusoidal function
2 cos(ωLO τi ),(231)

where ωLO is the LO frequency, and τi is the theoretically predicted instrumental delay, is multiplied to IF voltage vI1 (t) of antenna 1.
Assuming again that ωLO τi is a quasi–static quantity, we obtain for a newf
IF voltage vI1(t) of antenna 1 after the multiplication of 2 cos(ωLO τi ):f
vI1(t) = vI1 (t)2 cos(ωLO τi ) = vI1 (t)(eiωLO τi + e−iωLO τi ),

(232)

f
and for its Fourier transform ṽI1
(ω):
f
ṽI1
(ω) = ṽI1 (ω)(eiωLO τi + e−iωLO τi ),

(233)

f
where ṽI1(ω) is a Fourier transform of vI1 (t). Then we have a cross–correlationof Fourier transforms of IF voltages:
f
0∗
0∗
hṽI1
(ω)ṽI2(ω 0 )i = hṽI1 (ω)ṽI2(ω 0 )i(eiωLO τi + e−iωLO τi ),

(234)

0
where ṽI2(ω) is the IF voltage of antenna 2 after the delay insertion. Assuming jointly stationary random processes, we obtain for a cross–powerf0
spectrum of IF voltages vI1(t) and vI2(t):−iωLO τiiω τ0 (ω)(e LO i + e),
Svf v0 (ω) = SvI1 vI2
I1 I2

(235)

0
0 (ω) is the cross–power spectrum of IF voltages v I1 (t) and v
where SvI1 vI2I2 (t),
which is given in equation (226). Then, inserting equation (226) to equation(235), we have

Svf v0 (ω) =
I1 I2

1
A0
4

×{e−i[ωLO ∆τg +ω∆τg +φLO1 −φLO2 ] V(ωLO + ω) H1 (ω) H2∗ (ω)+e−i[ωLO (τg0 +τi )+ω∆τg +φLO1 −φLO2 ] V(ωLO + ω) H1 (ω) H2∗(ω)+ei[ωLO ∆τg −ω∆τg +φLO1 −φLO2 ] V ∗ (ωLO − ω) H1∗ (−ω) H2 (−ω)+ei[ωLO (τg0 +τi )−ω∆τg +φLO1 −φLO2 ] V ∗ (ωLO − ω) H1∗ (−ω) H2 (−ω)},(236)
where ∆τg = τg0 − τi . In first and third terms in the RHS of this equation,
rapid oscillations of exponential terms are almost stopped, as long as τi ∼= τg0 .
But, in second and fourth terms in the RHS, even more rapid oscillations
with almost doubled frequency ωLO (τg˙ 0 + τ˙i ) remain. Contributions of these
rapidly oscillating terms almost disappear when the product of IF voltages
112

are integrated for some duration of time in a correlator. Therefore, we havean “effective” cross–power spectrum of IF voltages,
Svf v0 (ω) =
I1 I2

1
A0
4

×{e−i[ωLO ∆τg +ω∆τg +φLO1 −φLO2 ] V(ωLO + ω) H1 (ω) H2∗ (ω)+ei[ωLO ∆τg −ω∆τg +φLO1 −φLO2 ] V ∗ (ωLO − ω) H1∗ (−ω) H2 (−ω)},(237)
which has essentially the same form as the one given in equation (229).
Note that, in this simple method of multiplication of a sinusoidal function to an IF voltage, phase drifts are stopped again in both USB and LSB
components.

2.4

Correlator Outputs

2.4.1

Multiplier and Integrator

As we saw earlier, a key component of a correlator in an interferometer is a
device with two inputs and one output, which is composed of a multiplierand an integrator (Figure 43). The expected correlation, which is the sta-from antenna 1

from antenna 2.

multiplierintegrator

Figure 43: Multiplier and integrator as a key component of a correlator.
tistical expectation of the product of the voltage signals, must be contained
as a signal component in a multiplier output of such a correlator. Since
the expected correlation does not rapidly oscillate any more after the delay
tracking and fringe stopping, we can integrate (time–average) the multiplier113

output during some time to suppress the noise component, and detect thisnearly constant expected correlation itself with a high signal to noise ratio.
Therefore, an output of an actual correlator can be well approximated by
the expected correlation after the delay tracking and fringe stopping. The
expected correlation is easily derived from the cross–power spectrum (CPS)of the IF voltages by an inverse Fourier transformation, as we showed inequation (204).
Now we have slightly different notations for the CPS of IF voltages inequation (229) and in equation (237), depending on practical implementations of the fringe stopping, though they are essentially equivalent to eachother. We adopt, hereafter, the expression given in equation (237), which
is familiar in VLBI practice, for describing the CPS after the delay tracking
and fringe stopping.
Then, the expected correlation, after the delay tracking and fringe stopping, is given by∞

A0  −i(ωLO ∆τg +φLO1 −φLO2 ) Z
Rvf v0 (0) =< e
V(ωLO + ω) e−iω∆τg H1 (ω) H2∗(ω) dω
I1 I2
4π
0

+ ei(ωLO ∆τg +φLO1 −φLO2 )

Z∞
0



V ∗ (ωLO − ω) e−iω∆τg H1∗ (−ω) H2 (−ω) dω  .

(238)

In the special case of a continuum flat spectrum source and rectangular filters,
the expected correlation will have essentially the same form as the one given
in equation (230):
A0 GB | V | sin(πB∆τg )2πB∆τg× [cos(ωLO ∆τg + ωI ∆τg + φLO1 − φLO2 − Φv )+ cos(ωLO ∆τg − ωI ∆τg + φLO1 − φLO2 − Φv )].

Rvf v0 (0) =
I1 I2

(239)

We will use expected correlations given in equations (238) and (239) as
theoretical expressions of the correlator outputs.
In VLBI, no operation for delay tracking and fringe stopping is done,
usually, in observing stations. Instead, they are done in VLBI correlators.
Therefore, a VLBI correlator is a little more complicated than the mere
multiplier and integrator, as we will see later.

114

2.4.2Single Sideband (USB or LSB) Reception

So far, we considered a case of double sideband (DSB) reception, i.e. a casewhen we receive signals contained in both upper sideband (USB) and lower
sideband (LSB) of LO frequency in original RF–band spectrum. In this case,the USB and LSB contributions are superposed in IF–band spectrum, as we
saw earlier.
The DSB reception is not convenient for observing line spectrum sources,
since different lines in USB and LSB are mixed up in the same IF spectrum.
Also, the DSB reception causes additional complications in data calibrations.
Therefore, in VLBI, we mostly use a single sideband (SSB), i.e., either of USB
or LSB, reception.
Amplitude and phase spectra of IF voltage signals, which were shown in
Figure 33 in the DSB reception case, now take simpler forms, in the SSB
reception case, as shown in Figure 44.
USB

−ωLO

amplitude

0

LSB

USB

ωLO

ω

−ωLO

phase

−ωLO

0

amplitude

0

LSB

ωLO

ω

ωLO

ω

phase

ωLO

ω

−ωLO

0

Figure 44: Amplitude and phase spectra of IF voltage signals in USB (left)and LSB (right) receptions. For simplicity, contributions of the geometricdelay only are shown in the phase spectra.
Since, cross–power spectra (CPS) of IF voltages both in USB and LSB
receptions are Hermitian symmetric, as illustrated in Figure 44, we do not
lose generality if we confine ourselves to consider only positive frequency
range (ω ≥ 0) of the spectra.
In positive frequency range ω ≥ 0, cross–power spectra of IF voltages
before the delay tracking and fringe stopping in USB and LSB receptions are
given by115

1
A0 e−i[ωLO τg0 +ωτg0 +φLO1 −φLO2 ] V(ωLO + ω) H1 (ω) H2∗ (ω),
4
(240)
1
A0 ei[ωLO τg0 −ωτg0 +φLO1 −φLO2 ] V ∗ (ωLO − ω) H1∗ (−ω) H2 (−ω),
SvLSB
(ω) =
I1 vI2
4(241)
SvUI1SB
vI2 (ω) =which correspond to positive frequency sides (ω ≥ 0) of the USB and LSB
spectra illustrated in Figure 44.
After the delay tracking and fringe stopping, they are reduced to1
A0 e−i[ωLO ∆τg +ω∆τg +φLO1 −φLO2 ] V(ωLO + ω) H1 (ω) H2∗ (ω),
I1 I2
4
(242)
1
SvLSB
=
A0 ei[ωLO ∆τg −ω∆τg +φLO1 −φLO2 ] V ∗ (ωLO − ω) H1∗ (−ω) H2 (−ω),
f 0 (ω)
v
I1 I2
4
(243)
SvUfSB(ω) =v0in the positive frequency range ω ≥ 0.
We can rewrite equations (240) and (241), as well as equations (242) and(243), using amplitudes and phases of the complex visibility and bandpass
characteristics, which we introduced in equations (198) and (199).
Then, before the delay tracking and fringe stopping, the USB spectrumin ω ≥ 0 is given by−iΦp (ω, t)
SvUI1SB,
vI2 (ω) = Ap (ω) ewith amplitude:
1
A0 | V(ωLO + ω) | | H1 (ω) H2∗ (ω) |,Ap (ω) =4
and phase:Φp (ω, t) = (ωLO + ω)τg0 + φLO1 − φLO2 − Φv (ωLO + ω) − Φb (ω),(244)

while the LSB spectrum in ω ≥ 0 is given by

SvLSB(ω) = An (ω) e−iΦn (ω, t) ,I1 vI2
with amplitude:
1
A0 | V(ωLO − ω) | | H1 (−ω) H2∗ (−ω) |,An (ω) =4
and phase:Φn (ω, t) = −(ωLO − ω)τg0 − φLO1 + φLO2 + Φv (ωLO − ω) + Φb (−ω).
(245)
116

After the delay tracking and fringe stopping, the USB spectrum in ω ≥ 0
becomes
SvUfSB(ω) = Ap (ω) e−i∆Φp (ω, t) ,
v0
I1 I2

with amplitude:
Ap (ω) =1
A0 | V(ωLO + ω) | | H1 (ω) H2∗ (ω) |,4

and residual phase:∆Φp (ω, t) = (ωLO + ω)∆τg + φLO1 − φLO2 − Φv (ωLO + ω) − Φb (ω),(246)
while the LSB spectrum in ω ≥ 0 becomes
SvLSB= An (ω) e−i∆Φn (ω, t) ,f 0 (ω)v
I1 I2

with amplitude:
An (ω) =1
A0 | V(ωLO − ω) | | H1 (−ω) H2∗ (−ω) |,4

and residual phase:∆Φn (ω, t) = −(ωLO − ω)∆τg − φLO1 + φLO2 + Φv (ωLO − ω) + Φb (−ω),(247)
where ∆τg is the residual delay: ∆τg = τg0 − τi .
2.4.3

Correlator Outputs in Single Sideband Reception

As we discussed earlier, correlator outputs are closely approximated by the
expected correlations after the delay tracking and fringe stopping. Therfore,
theoretical expressions for the correlator outputs in USB and LSB receptions
are given byA0<[e−i(ωLO ∆τg +φLO1 −φLO2 )RvUfSB0 (0) =
v
I1 I2
4π

Z∞
0

V(ωLO +ω) e−iω∆τg H1 (ω) H2∗(ω) dω],
(248)

and
A0
RvLSB
<[ei(ωLO ∆τg +φLO1 −φLO2 )f 0 (0) =I1 vI24π

Z∞0V ∗ (ωLO −ω) e−iω∆τg H1∗ (−ω) H2 (−ω) dω],(249)

respectively.

117

If we assume the case of rectangular filters and constant visibility, theabove equations are reduced to
RvUfSB
(0) =
v0
I1 I2

A0 GB | V | sin(πB∆τg )cos(ωLO ∆τg +ωI ∆τg +φLO1 −φLO2 −Φv ),
2
πB∆τg
(250)

and
A0 GB | V | sin(πB∆τg )cos(ωLO ∆τg −ωI ∆τg +φLO1 −φLO2 −Φv ).
I1 I2
2
πB∆τg
(251)
For a more general case of an arbitrary filter shape, when the complex
visibitlity is assumed to be constant only in each of USB and LSB receiving
bandwidths, and the filters, with a passband:

RvLSB
f 0 (0) =v

ωI − ∆ω/2 ≤ | ω | ≤ ωI + ∆ω/2,
are not necessarily rectangular, we haveZ∞
0

V(ωLO + ω) e−iω∆τg H1 (ω) H2∗ (ω) dωωI + ∆ω
2Z= V(ωLO + ω)

e−iω∆τg H1 (ω) H2∗ (ω) dωωI − ∆ω2∆ω

= V(ωLO + ω) e−iωI ∆τgZ20e−iω ∆τg H1 (ωI + ω 0 ) H2∗ (ωI + ω 0 ) dω 0 ,

− ∆ω
2

(252)
in the USB reception, andZ∞0V ∗ (ωLO − ω) e−iω∆τg H1∗ (−ω) H2 (−ω) dωωI + ∆ω2= V ∗ (ωLO − ω)Z

e−iω∆τg H1∗ (−ω) H2 (−ω) dωωI − ∆ω2∆ω

= V ∗ (ωLO − ω) e−iωI ∆τg

Z2

− ∆ω
2

0

e−iω ∆τg H1∗ (−ωI − ω 0 ) H2 (−ωI − ω 0 ) dω 0 ,
(253)
118

in the LSB reception.
Now, let us introduce “bandwidth patterns” B12 :
∆ω

1
U SB
B12
(B, ∆τg ) =
4π

Z2

0

(254)

0

(255)

e−iω ∆τg H1 (ωI + ω 0 ) H2∗ (ωI + ω 0 ) dω 0 ,

− ∆ω
2
∆ω

1
LSB
B12
(B, ∆τg ) =
4π

Z2

− ∆ω
2

e−iω ∆τg H1∗ (−ωI − ω 0 ) H2 (−ωI − ω 0 ) dω 0,

for USB reception and LSB reception, respectively. The suffix ‘12’ here accentuates that this pattern comes from bandpass characteristics of receiving
systems in antennas 1 and 2. Then, if we represent complex visibilities andbandwidth patterns through their amplitudes and phases:
U

L

V(ωLO + ωI ) =| V U | eiΦ v and V(ωLO − ωI ) =| V L | eiΦ v ,

(256)

and
U

L

U SB
U
LSB
L
B12
(B, ∆τg ) =| B12| eiΦ B and B12(B, ∆τg ) =| B12| eiΦ B ,(257)

equations (248) and (249) for theoretical expressions for correlator outputs
are reduced toUU| | B12| cos(ωLO ∆τg + ωI ∆τg + φLO1 − φLO2 − ΦUv − ΦUB ),RvUfSB0 (0) = A0 | V
I1 vI2
(258)
and
L
L
RvLSB
| | B12| cos(ωLO ∆τg − ωI ∆τg + φLO1 − φLO2 − ΦLv + ΦLB ),f 0 (0) = A0 | V
v
I1 I2
(259)
for USB reception and LSB reception, respectively.

2.4.4

Fringe Amplitude and Fringe Phase

The above equations again show the white fringes, with bandwidth patterns
in | B12 | terms and fringe patterns in cosine terms.
In general, when a correlator output shows a sinusoidal fringe pattern
A cos Φ in the center of the coherence interval where ∆τg ≈ 0, we call theamplitude A of the fringe pattern the “fringe (or correlation) amplitude”, and the phase Φ of the fringe pattern the “fringe (or correlation)phase”.
119

According to the theoretical expressions of correlator outputs in equations(258) and (259), theoretical expressions of fringe amplitude AU and fringe
phase ΦU in USB reception are given byU
AU = A0 | V U | | B12|,UΦ = ωLO ∆τg + ωI ∆τg + φLO1 − φLO2 − ΦUv − ΦUB ,(260)

and theoretical expressions of fringe amplitude AL and fringe phase ΦL in
LSB reception are given byL
AL = A0 | V L | | B12|,
LΦ = ωLO ∆τg − ωI ∆τg + φLO1 − φLO2 − ΦLv + ΦLB .

(261)

In practice of interferometric observations, the fringe amplitude and fringe
phase in the correlator outputs are often called as “visibility amplitude”
and “visibility phase”. Strictly speaking, however, one must correct thefringe amplitude and fringe phase for the residual delay, bandwidth pattern,
and other effects, in order to obtain proper visibility amplitude | V | andvisibility phase Φv as defined in equations (165) and (166). Moreover, thefringe phase contains the residual delay ∆τg of the reference direction of theradio source, which could be a useful observable for high precision astrometry
and geodesy, if one can resolve the cycle ambiguity and properly correct
the atmospheric and other disturbing effects by means of a suitable phasecompensation technique, while the visibility phase in equation (166) is defined
to be free from the residual delay of the source reference direction. Therefore,
we will henceforth distinguish the terms “fringe amplitude and phase” fromthe “visibility amplitude and phase”.
In the case of the rectangular filters and constant visibility, equations(258) and (259) are of course reduced to equations (250) and (251), since inthis particular case,
V(ωLO + ωI ) = V(ωLO − ωI ) =| V | eiΦv ,
and
U SB
LSB
B12
(B, ∆τg ) = B12(B, ∆τg ) =2.4.5

GB sin(πB∆τg )
.
2
πB∆τg

(262)

Group Delay and Fringe Frequency

As we discussed so far, the delay tracking is aimed at reducing the slope of thephase spectrum in the observed bandwidth to nearly zero, thus facilitating
the fringe detection in the middle of the coherence interval. Therefore, what is120

needed for successful delay tracking is an accurate prediction or estimation of
the phase slope, i.e. frequency derivative of the phase spectrum of the cross–power spectrum of voltage signals, which is to be used as the compensating
instrumental delay. We call the frequency derivative of the phase spectrum
Φ(ω, t) the “group delay” τG :τG =∂Φ(ω, t)
,
∂ω

(263)

where Φ(ω, t) can be Φp (ω, t) of equation (244) in case of USB reception, orΦn (ω, t) of equation (245) in case of LSB reception. Thus, we need a goodprediction or estimation of the group delay for successful delay tracking.
In an ideally simple case, the group delay is nothing but the geometric delay itself (see, for example, equation (103)). In actuality, however, equations(244) and (245) show phase terms other than the contribution of the geometric delay, which may depend on the frequency. Moreover, the phase slope
may be affected by other effects as well, such as clock offsets among VLBI antennas, atmospheric propagation delays, cable delays in signal transmission
systems, and so on, which we will see in discussions specific to VLBI. Furhermore, if some propagation delay is dispersive (i.e. frequency–dependent),
then the phase slope will also depend on the frequency derivative of the dispersive delay. In general, the group delay is a quantity which includes all theeffects mentioned above.
On the other hand, the fringe stopping is an operation to compensate
the rapid time variation of the interferometer phase. This is done for the
purpose to integrate the correlation results for some duration of time, which
is sufficient to detect the white fringe with a good enough signal to noise
ratio. Therefore, what is needed for successful fringe stopping is an accurate
prediction or estimation of the time derivative of the phase spectrum of
the correlated signals, which can be used for compensating the rapid phaseshift. We call the time derivative of the phase spectrum Φ(ω, t) the “fringe
frequency” or “fringe rate”, and denote it as Fr :Fr =∂Φ(ω, t)
,
∂t

(264)

where Φ(ω, t) can be Φp (ω, t) of equation (244) in case of USB reception, orΦn (ω, t) of equation (245) in case of LSB reception, as before. Thus, we needa good prediction or estimation of the fringe frequency for successful fringe
stopping.
Again, the actual fringe frequency may include time derivatives of theclock offsets, propagation delays, cable delays, and other terms, besides thetime derivative of the geometric delay.
121

For connected–element radio interferometers (CERI), a predicted instrumental delay τi , usually based on the geometric delay model only, is accurateenough to make ∆τg0 almost zero. Therefore, the fringe is normally foundwithin the coherence interval, and the correlated signals can be integrated
for sufficiently long time, by applying the delay tracking and fringe stopping,based on the prediction.
In VLBI, accuracies required to delay tracking and fringe stopping aremuch higher than in CERI, as we saw earlier. Therefore, the theoreticalprediction alone is usually not sufficient for obtaining fringes. Hence, onemust first perform a special search for estimating the group delay and fringe
frequency values, using the observed VLBI data themselves. If the search is
successful, then the delay tracking and fringe stopping are done satisfactorily,
and one can finally detect the correlation peak and the white fringe.
This is an additional labor imposed to VLBI, compared with CERI. But
this “necessity” had led to the birth and the remarkable success of the geodetic VLBI.
2.4.6

Complex Correlator

When we achieve a complete fringe stopping, the residual delay is kept always
zero (∆τg = 0), and, therefore, the correlator outputs given in equations (258)and (259) become constants in time. For example, in case of USB reception,
we haveU
RvUfSB(0) = A0 | V U | | B12| cos(φLO1 − φLO2 − ΦUv − ΦUB ),
v0

(265)

I1 I2

where argument of cosine function (fringe phase) is constant in time.
This is not very convenient for further analysis, since, if the correlator
output is just a constant, we cannot separately obtain the fringe amplitude
and the fringe phase.
In order to get rid of this inconvenience, we use so–called complex correlators. Complex correlators are two sets of multipliers and integrators which
provide two correlator outputs of the same signals, but with fringe phases
different to each other by 90 degrees.
Figure 45 shows a design of a complex correlator which uses a quadraturephase–shift network. The quadrature phase–shift network is a special electric
circuit which changes phases of all frequency components of a signal by 90
degrees. Then, in case of the USB reception, an output of “real correlator”
Rr will have a form such as given in equation (265), i.e.,
Rr = A0 | V | | B12 | cos(φLO1 − φLO2 − Φv − ΦB ),
122

(266)

while an output of “imaginary correlator” Ri will be 90 degrees shifted inphase, i.e.,
Ri = −A0 | V | | B12 | sin(φLO1 − φLO2 − Φv − ΦB ).

(267)

Thus, it will be easy to separate fringe amplitude and fringe phase, byusing outputs of these two correlators.
from antenna 1

from antenna 2

0

+9

N
P
Q

.

Imaginary
Correlator.

multiplier

multiplierintegrator

integrator

sin

cos

Real
Correlator

Figure 45: A complex correlator using a quadrature phase-shift network.
For a more general case when fringe is almost stopped but still a small
residual delay ∆τg remains, we have, from equation (258),
Rr = A0 | V | | B12 | cos((ωLO + ωI )∆τg + φLO1 − φLO2 − Φv − ΦB ), (268)andRi = −A0 | V | | B12 | sin((ωLO + ωI )∆τg + φLO1 − φLO2 − Φv − ΦB ), (269)
for real and imaginary correlator outputs in USB reception.
Instead of the quadrature phase–shift network, we can use two LO reference signals in the frequency conversion, shifted in phase by 90 degrees.
Also, we can use multiplications of cosine and sine functions of ωLO τi in
the fringe stopping, as we will see in discussions of VLBI correlators.

1232.4.7

Projected Baseline

We discussed earlier the fringe pattern in the case when we observe a source
at a direction nearly perpendicular to the interferometer baseline. When the
delay tracking and fringe stopping are properly performed, however, we arein position to catch a source, within the coherence interval, at any directionof the sky.

λD cos θθD cos θD

Figure 46: Fringe spacing at arbitrary direction of the sky with projectedbaseline length D cos θ.
In such a case, our interferometer is equivalent to the one having a baseline
length D cos θ in Figure 46. This “effective baseline length viewed fromthe source” is called “projected baseline length”, since this is a length of abaseline projected onto a plane perpendicular to the source direction. The
fringe pattern is now determined by the projected baseline, as illustrated in
Figure 46. Therefore, the fringe pattern now varies in time, as we track asource, which moves in the sky changing the angle θ due to the diurnal motionof the Earth. In particular, the fringe spacing in the sky is now expressed as
λ
,
(270)
∆θF =
D cos θand also varies in time.
124

3

Source Structure and Correlated Flux Density

We saw earlier how a radio source structure (or, more specifically, intensity
distribution) is related to an observable of radio interferometry, the complex
visibility. Using this relationship, we can infer the radio source structure byanalyzing observed complex visibilities.

Figure 47: Helical structure of a jet ejected from a distant quasar 3C273
revealed by a VSOP (VLBI Space Observatory Program) observation.
This is the basis of an extensive field on the image synthesis of astronomical objects with radio interferometry, which has enabled us to produce
high–resolution images of astronomical objects billions of light years away
(as an example, see Figure 47). However, details of the image synthesis techniques are out of scope of this note. Here, we confine ourselves to a generalformulation of the problem and a consideration of the effect of the finite
angular size of the source on the correlator output.

125

3.1

Basic Equations of Image Synthesis

3.1.1

Complex Visibility as an Observable

A complex correlator yields the fringe amplitude and fringe phase for an
observed radio source, which enable us to derive the complex visibility of thesource. In fact, in equations (268) and (269), we gave theoretical expressions
for outputs of a complex correlator in a case of the USB reception:
Rr = A0 | V | | B12 | cos((ωLO + ωI )∆τg + φLO1 − φLO2 − Φv − ΦB ),Ri = −A0 | V | | B12 | sin((ωLO + ωI )∆τg + φLO1 − φLO2 − Φv − ΦB ).
Therefore, theoretical expressions of fringe amplitude and fringe phase are
given by equation (260):
A = A0 | V | | B12 |,andΦ = (ωLO + ωI )∆τg + φLO1 − φLO2 − Φv − ΦB ,respectively.
Consequently, if we achieve a high enough S/N rartio, and if we areallowed to calibrate the observed fringe amplitude and fringe phase for geometric mean of effective apertures A0 , the bandwidth pattern | B12 | eiΦB , theresidual delay ∆τg , and the difference in LO initial phases φLO1 − φLO2 , on
the basis of suitable measurements or estimations, we can derive the complex
visibility of the observed source:
V(ω) =| V(ω) | eiΦv (ω) ,where we denote by ω the center frequency of the RF band, i.e., ω = ωLO +ωI .
3.1.2

Visibility and Intensity in EW–NS Coordinate System

We showed in equation (165), that the complex visibility V(ω) is related tothe source intensity distribution Iν (σ) as
V(ω) =I

AN (σ) Iν (σ) e−i2πD λ ·σ dΩ,(271)

where AN (σ) is the normalized power pattern of an interferometer, Iν (σ)
is the source intensity distribution, D λ = D/λ is the baseline vector D
normalized by the wavelength λ of the observation, and σ is an offset vectorof a direction s in a radio source from a reference direction s0 , i.e., σ = s−s0 .
126The meaning of equation (271) becomes clearer if we introduce a rectangular coordinate system, whose 3-rd axis is chosen towards the reference
direction s0 of the source, and 1-st and 2-nd axes are chosen in the east (rightascension) and the north (declination) directions, respectively (Figure 48).
Taking into account that σ = s − s0 is a vector difference of two unit
vectors s and s0 , we denote components of the offset vector σ in this new
ηv

N

σ
u

ξ

s0

E

s

N
w

D

v

u

E

Figure 48: Source brightness distribution and projected baseline in an EW–
NS plane perpendicular to the reference direction s0 of the source.
coordinate system as:
σ = (ξ, η,q

1 − ξ 2 − η 2 − 1).

(272)

Also, we denote components of the baseline vector D and the normalizedbaseline vector D λ as:
D = (u, v, w),
(273)

127

and
D λ = (uλ , vλ , wλ ),(274)

respectively, in the same coordinate system, wherevwuuλ = , vλ = , and wλ = .
λλλNote that ξ and η are taken along right ascention and declination directions, respectively, in a celestial equatorial system. Note, also, that w is
related to the geometric delay of the reference direction τg0 by an equation:w = cτg0 .
Then, we describe the argument of the exponential term in the complex
visibility in equation (271) as:q

2πD λ · σ = 2π[uλ ξ + vλ η + wλ ( 1 − ξ 2 − η 2 − 1)].

(275)

Also, we describe the solid angle element dΩ in the same equation via dξ dη
as:
dξ dη
,
(276)
dΩ = √
1 − ξ2 − η2since dξ dη is the projection of dΩ in the plane perpendicular to s0 (Figure
49).

dΩ

s

0

s

dξδη
celestial
sphereFigure 49: Solid angle element dΩ and area element dξ dη in a plane perpendicular to reference direction s0 .
Then, the complex visibility in equation (271) is given in terms of ξ, η,uλ , vλ , and wλ , by
Z∞ Z∞
√
dξ dη
2
2
V(ω) =AN (ξ, η)Iν (ξ, η)e−i2π[uλ ξ+vλ η+wλ ( 1−ξ −η −1)] √.
1 − ξ2 − η2
−∞ −∞
(277)
128

This equation can be regarded as an integral equation for the intensity
distribution Iν (ξ, η) of a radio source with given complex visibility V(ω).
The purpose of the image synthesis is to get an intensity distribution (an“image”) of a radio source by solving this integral equation using observed
complex visibilities.
3.1.3

Approximation of a Celestial Sphere by a Tangent Plane

It is not easy to solve the integral equation (277) in its general form. However,
if we have a narrow range of mapping, where ξ  1 and η  1, and second
order terms ξ 2 and η 2 can be neglected, the equation becomes much simpler.
This narow–range condition corresponds to an approximation of a smallpart of the celestial sphere, where we would like to draw an image map of
a radio source, by a tangent plane at the reference direction s0 (see Figure
50).

small rangeof mapping
plane tangent
to sphere at s0s0celestial
spheres

Figure 50: Celestial sphere can be approximated by a tangent plane for asmall range of mapping.
Let us then examine conditions under which ξ 2 and η 2 terms can be
actually neglected.
ξ 2 and η 2 terms appear in two places of equation (277), namely in√dξ dη
−i2π[uλ ξ+vλ η+wλ ( 1−ξ 2 −η 2 −1)]
√
,
and
e
.
1 − ξ2 − η2129dξ dη
, we can neglect ξ 2 and η 2 , provided only
1 − ξ2 − η2that | ξ | 1 and | η | 1.
In the second term, which can be approximated by
In the first term: √q122−i2π[uλ ξ + vλ η + wλ ( 1 − ξ 2 − η 2 − 1)] ∼ −i2π[uλ ξ + vλ η − 2 wλ (ξ + η )]
e
,
=e
(278)
we must be a little careful. Here, ξ 2 and η 2 terms appear in a phase of aperiodic function, for which only a residue of 2π is meaningful. Therefore,
it is not always possible to neglect a term in a phase even when it is muchsmaller than other phase terms. A phase term can be safely neglected only
when it is absolutely small as a phase (for example, smaller than 0.1 radian,say). If we require this “ 0.1 radian” condition, then ξ 2 and η 2 in equation(278) are negligible whenπwλ (ξ 2 + η 2 )  0.1 radian.

(279)

Let us now examine how this condition could be satisfied. Let us denote a
θf
(ξ,η)
w

λ

Dλ

Figure 51: Length of wλ (left) and a range of mapping (right).
full range of our intended image map (or, range of mapping) as θf . Then,obviouslyθfξ 2 + η 2 ≤ ( )2 ,2(see Figure 51). Therefore, the condition in equation (279) is satisfied if
πwλ (

θf 2
) ≤ 0.1 radian.
2

Adopting as a maximum value for wλ :wλmax = Dλ =
130

D
,
λ

in view of Figure 51, we see that the condition in equation (279) is readilysatisfied if
π

θf
2

!2

0.4 λ
D
≤ 0.1, and therefore θf2 ≤,λπ Dwhich is approximately equivalent to
1
θf ≤
3

s

λ
.
D

(280)

Thus, the tangent plane approximation, where ξ 2 and η 2 are neglected, isadequate, if our range of mapping θf satisfies the condition given in equation(280).
It is still left to us to examine whether equation (280) is a realistic condition in actual mappings of radio source images.
Angular resolution θr of an interferometric observation is roughly givenby the minimum fringe spacing:
θr ≈

λ
.
D

Since number of grid points (“pixels”) required for an image mapping is
proportional to (θf /θr )2 , a load of numerical processing becomes heavier, as
the range of mapping θf becomes much wider than the angular resolution θr .
Moreover, it is usually meaningless to select a too wide range of mapping,since a radio source, which is much extended than the fringe spacing of aninterferometer θr , tends to be “resolved out” and becomes “invisible” for theinterferometer, as we will see in more detail in later discussions.
Therefore, if we select a range, which is 30 times as large as the angularresolution, as a realistic range of mapping:θf ≈ 30 × θr ≈ 30 ×λ,Dthen, the condition of equation (280) for the tangent–plane approximation
becomes
s
λ
1 λ
30 ≤
,
(281)
D
3 D
or
s

λ
1
≤ ,
D
90

and, hence,

λ1≤≈ 0.◦ 007 ≈ 25 arcsec.
D
8100
131

(282)

As we saw in Table 2, such a condition on the fringe spacing is generally
satisfied in modern interferometers, except for m–wave or cm–wave CERI
arrays of relatively short baselines. In VLBI, as far as our range of mappingis ≤ 30 × (λ/D), we can safely use the tangent–plane approximation. Figure
47 gives an example, where the angular resolution is shown as an elliptical“restoring beam” in the right edge.

Figure 52: 15 GHz map of a quasar 3C279 observed with VLBA (Wajima
and Iguchi, private communication in 2005).
Another example of a beautiful VLBI image of a quasar 3C279 observed
at 15 GHz with VLBA (Wajima and Iguchi, private communication in 2005)
also covers a range of mapping which is smaller than 30 times of the angularresolution (the full width half maximum (FWHM) of an interferometer beam
is shown by an ellipse in the left bottom corner).
For typical CERI and VLBI, we have:• CERI of λ = 1 cm (30 GHz) and D = 2 km:λ= 5 × 10−6 radian = 1 arcsec,
D
132

1
3

s

λ
= 7 × 10−4 radian = 140 arcsec.
D• VLBI of λ = 1.35 cm (22 GHz) and D = 2300 km:
λ= 6.0 × 10−9 radian = 1.2 milliarcsec (mas),Ds
1 λ= 2.6 × 10−5 radian = 5.3 arcsec.
3 D• VLBI of λ = 2.3 mm (129 GHz) and D = 500 km:
λ= 4.6 × 10−9 radian = 1.0 milliarcsec (mas),Ds
1 λ= 2.3 × 10−5 radian = 4.7 arcsec.
3 D
In all above examples, the condition in equation (281) is well satisfied.
Important exceptions, where the tangent–plane approximation is not applicable, are wide–field mappings of maser sources in massive star–forming
regions. They sometimes cover sky ranges of several tens arcseconds, yet individual maser features, of which they consist, are compact at milliarcsecond(mas) level, and thus well detectable with intercontinental VLBIs. In sucha case, the range of mapping θf could be far larger than 30 θr . Therefore,
one must properly take into account sphericity of the map surface in suchwide–field mappings.
3.1.4

Interferometer is a Fourier Transformer

Under the tangent plane approximation, equation (277) is reduced toV(ω) = V(uλ , vλ ) =Z∞ Z∞

AN (ξ, η)Iν (ξ, η)e−i2π(uλ ξ+vλ η) dξ dη,

(283)

−∞ −∞

where we reexpressed V(ω) as V(uλ , vλ ), stressing dependence of the visibility
on “normalized projected baseline components” uλ and vλ .
Then, it is evident that the complex visibility V(uλ , vλ ), as given in equation (283), is a two–dimensional Fourier transform of the source brightness
distribution Iν (ξ, η) at “spatial frequency” values uλ and vλ .
Consequently, we can regard a radio interferometer as a “Fourier Transformation Device”, which yields, at the correlator output, a two–dimensional
133

Fourier component of the source brightness distribution on the sky, at spatial frequencies corresponding to two components of the normalized projectedbaseline, uλ and vλ . The Fourier transformation relation described in equation (283) is called “van Cittert–Zernike Theorem”.
Naturally, the source brightness distribution is restored from the measured visibilities by the inverse Fourier transformation:
AN (ξ, η)Iν (ξ, η) =

Z∞ Z∞

−∞ −∞

V(uλ , vλ )ei2π(uλ ξ+vλ η) duλ dvλ ,(284)

and then by correcting the normalized power pattern of the interferometer
AN (ξ, η). This is the basis of the astrophysical imaging of the radio source
structure by means of radio interferometry under the tangent plane approximation. This is important also for geodetic and astrometric VLBI, when wecalibrate the effect of the source structure on the measured delay.
Of course, it is impossible to perform a complete inverse Fourier transformation, unless a continuous distribution of the visisbility is available over
the whole spatial frequency domain, the uv–plane. Nevertheless, a better
restoration is achieved, with suitable image processing software, when thevisibilities are sampled at many uλ , vλ points, which nearly homogeneously
cover the uv–plane. It is therefore important to have a good coverage ofuv–plane for radio interferometry observations aimed at astrophysical source
imaging.

3.2

uv–Coverage

The Earth’s diurnal rotation changes the direction and length of a projectedbaseline of a ground–based interferometer in the uv–plane, and thus helps
us to sample visibilities at different uλ , vλ points. Since the interferometerbaseline rotates around a nearly fixed spin axis of the Earth, the trajectoryof the projected baseline is a circle, if the observed source is located at thecelestial pole. The trajectory is a linear oscillation when the source is at theequator, and is an ellipse, for all other directions in between the celestial
equator and the 2 celestial poles. For two stations 1 and 2 on the surfaceof the Earth, we have two baseline vectors with opposite directions D 12 andD 21 = −D 12 connecting these two stations. Therefore, in general, we have
two point–symmetric arcs of ellipses on the uv–plane for a pair of stations(Figure 53).
Configurations of radio interferometer arrays are designed to achieve goodcoverages of the uv–plane, from the trajectories of projected baselines formedby their antennas.
134

N
P
v

u

E

Figure 53: If we place an end of a baseline vector at the center of the rotating
Earth, by a parallel translation, a tip of the vector draws an ellipse, in general,
viewed from a radio source. EW and NS components of the projected baseline
are the u and v components. Since we have two vectors of equal lengths andopposite directions which connect two stations, we have two point–symmetric
elliptical trajectories.
3.2.1

uv–Trajectories

Figure 54 shows geometry of the diurnal rotation of a baseline vector D.
We use an Earth–fixed Cartesian X, Y, Z coordinate system with Z axis
directed towards North Pole, X axis directed towards Greenwich Meridian,and Y axis which completes a right–handed system. Components of thebaseline vector D in this system are denoted by DX , DY , and DZ . Directionof an observed radio source is given by the declination δ and Greenwich hourangle H of the source with respect to this Earth–fixed coordinate system.
First, let us look at the baseline vector from a direction of the North Pole(bottom–left panel of Figure 54). We see two sets of rectangular axes X, Y
and S, E in the equatorial plane, which is now perpendicular to our line ofsight. S axis here is chosen along a plane containing both the North Pole axis
and the radio source direction, which we call the “PS–plane”. We then consider S and E components of the baseline vector D in the equatorial plane.
E component is just equal to the u component, i.e. the EW–component of
the baseline vector viewed from the radio source (top–left panel of Figure54). As for S component of the baseline D, we denote it as DS . Since S,
E and X, Y axes are inclined to each other by the Greenwich hour angle
H, the u and DS components and DX and DY components of the baseline
vector D are related to each other by equations:u = DX sin H + DY cos H,
135

(285)

N

S

δ

P
DZ

Z

DY

DS
DZ

u

E

v

al

N

tor
i

X

Eq

ua

H
DX

Y

e

D

pla
n

v

P

P

DS

DY

Y
u
E

H
DX
S

X (Greenwich Meridian)Figure 54: Geometrical relationship between DX , DY , and DZ components ofa baseline vector D and u, v coordinates. Here, X, Y, Z coordinate system is
an Earth–fixed right–handed Cartesian system with Z axis towards the North
Pole P and X axis towards Greenwich Meridian. H and δ are Greenwich
hour angle and declination of an observed radio source. The Earth–fixedbaseline vector is shown as viewed from the radio source (top, left), viewed
from west (top, left), and viewed from the North Pole (bottom).

136

and
DS = DX cos H − DY sin H.

(286)

Next, when we look at the baseline vector D along the E axis from thewest side (top–right panel of Figure 54), we see equatorial S axis and polar
Z axis in the PS–plane, which is now perpendicular to the line of sight. The
radio source direction is offset from the equatorial plane by the declination δ.
Therefore, the component of the baseline vector D in an axis perpendicular tothe source direction, which is nothing but the NS–component of the baseline
vector viewed from the radio source (top–left panel of Figure 54), i.e. the v
component, is related to the DS and DZ components by an equation:v = −DS sin δ + Dz cos δ.

(287)

Combining equations (286) and (287), we obtainv = −(DX cos H − DY sin H) sin δ + DZ cos δ,(288)

for v component. Equations (285) and (288) allow us to calculate a uv–
trajectory of a radio source with varying Greenwich hour angle H.
From equations (285) and (288), we obtain an equation of ellipse:u2 +(v − DZ cos δ)22= DX+ DY2 ,
sin2 δ

(289)

which clearly shows that uv–trajectories are really ellipses.
In view of the point symmetry property mentioned above, a set of u andv:
u = −DX sin H − DY cos H,
v = (DX cos H − DY sin H) sin δ − DZ cos δ,(290)which satisfies an equation of another ellipse:
u2 +(v + DZ cos δ)22= DX+ DY2 ,
2
sin δ

(291)

is also a valid uv–trajectory of the same pair of stations.
When we calculate actual uv–trajectory of a radio source, we must take
into account that the source can be observed by an interferometer baseline
only when the source is “mutually visible” from two stations of the baseline,
i.e. when the source is within lower and upper elevation limits of radio telescopes at the two stations. This condition is roughly formulated as follows.
137

P
Z
s

z

r

δ

Y

H
X

Figure 55: Zenith distance z of a radio source direction s at a station in a
direction r. Also shown is an Earth–fixed Cartesian coordinate system with
Z–axis towards the North Pole P and X–axis towards Greenwich Meridian.
H and δ are Greenwich hour angle and declination of the source, respectively.
If X, Y , Z coordinates of a station is R = (RX , RY , RZ ), and if weapproximate the Earth by a sphere, then a unit vector r oriented towards
zenith direction of the station is roughly given by
r=
where
R=

q

R
,
R

2
RX
+ RY2 + RZ2 .

On the other hand, components of a unit vector s towards a radio source are
given in the X, Y , Z coordinates bys = (cos δ cos H, − cos δ sin H, sin δ),(see Figure 55), where H is the Greenwich hour angle, and δ is the declinationof the source.
Therefore, cosine of the zenith distance z, i.e. the angle between thezenith direction r and the source direction s, is approximately given bycos z = r · s =RX cos δ cos H − RY cos δ sin H + RZ sin δ
.
R

(292)

Thus, if upper and lower elevation limits of a radio telescope at a station
are Eu and El , respectively, then the radio source is visible with the telescope
138

when
sin Eu ≤ cos z ≤ sin El .

(293)

Actual uv–trajectories can be calculated in terms of equations (285),(288), and (290), when the condition of equation (293) is satisfied at bothends of baselines.

Figure 56: uv–trajectories of a combined KVN and VERA arrays, for a sourceat δ = +60◦ .
Figure 56 shows an example of the uv–coverage for the combined VERA
and KVN VLBI arrays thus calculated, for a source located at +60◦ declination.
3.2.2

Synthesized Beams

Let us consider a point source with flux density Sν located at the reference
direction s0 (ξ = 0 and η = 0). In this case, the intensity distribution,
which we call here the “true intensity” and denote as Iνtrue (ξ, η), in order to139

distinguish it from “synthesized” one to be introduced below, is given byIνtrue (ξ, η) = Sν δ(ξ) δ(η),(294)

where δ(x) is the delta function.
For simplicity, we assume that the normalized power pattern of the interferometer is unity, i.e. AN (ξ, η) = 1, everywhere in our range of mapping.
Then, equation (283) gives the complex visibility which is constant at every
uλ , vλ point:V(uλ , vλ ) = Sν = const.
(295)
In an actual interferometer array, we can sample the complex visibility
only along uv–trajectories. Therefore, the “synthesized intensity distribution” calculated by equation (284):
Iνsynt (ξ, η) = SνZ∞ Z∞

ei2π(uλ ξ+vλ η) du dv

along uv

,

(296)

−∞ −∞

does not reproduce the point but yields some extended distribution. This
“response of a synthesis interferometer array to a point source” is called the“synthesized beam” or the “dirty beam” of the array. This is analogous
to the beam formation of a single dish antenna with a strange “aperture
illumination” along elliptical arcs.
If we approximate the two–dimensional integration by a summation onmeshes, equation (296) is reduced toIνsynt (ξ, η) = Sν
= Sν

XX
u

v

XX
u

= 2Sν

ei2π(uλ ξ+vλ η) ∆uλ ∆vλalong uv

[ei2π(uλ ξ+vλ η) + e−i2π(uλ ξ+vλ η) ]∆uλ ∆vλ

v

XX
u

cos(2π[uλ ξ + vλ η])∆uλ ∆vλvalong uv1

along uv1

,

(297)

where “along uv” means summation along uv–trajectories composed of point–
symmetric elliptical arcs given in both equations (289) and (291), whereas“along uv1” means summation along uv–trajectories given by equation (289)
only. We can further replace the two–dimensional summation on the uv–
plane by a one–dimensional summation along the elliptical uv–trajectories
with equal intervals of Greenwich hour angle ∆H. Since an interval in Greenwich hour angle ∆H corresponds to a length interval in the uv–plane
v
u
u
t

duλ
dH

!2

dvλ
+
dH
140

!2

∆H,

we obtain from equation (297)Iνsynt (ξ, η) ∝

X
H

v
u
u
cos(2π[u (H)ξ+v (H)η]) t
λ

λ

duλ
dH

!2

dvλ
+
dH

!2

∆H

along uv1

.

(298)

Synthesized Beam of KVN+VERA
Dec = +60 deg
Freq = 22.0 GHz

normalized beam

1
0.8
0.6
0.4
0.2
0
-0.2
2
1.5

2.5

2

1.5

1

0.5 0
-0.5
RA offset (milliarcsec)

-1

-1.5

-2

-2
-2.5 -2.5

-1
-1.5

2.5

1
0.5
0
-0.5Dec offset (milliarcsec)

Figure 57: Normalized synthesized beam of the combined KVN and VERAVLBI arrays for a radio source at +60◦ declination with observing frequency
22 GHz.
uλ , vλ and their derivatives with respect to H in equation (298) are readily
calculated for a radio source with declination δ from equations (285) and(288), namely we haveDX sin H + DY cos H
,
λ

(299)

DZ
DX cos H − DY sin H
sin δ +cos δ,
λ
λ

(300)

uλ =
vλ = −
andduλ
DX cos H − DY sin H=,dH
λDX sin H + DY cos H
dvλ=sin δ.
dH
λ
141

(301)
(302)

Calculating these quantities with varying H while the source is mutuallyvisible at two ends of baselines, inserting them into equation (298), anddividing the results by a maximum value, we obtain the synthesized beam
normalized by its maximum value:
Iνsynt (ξ, η).
Iνsynt max
Figure 57 shows an example of the normalized synthesized beam for the
combined KVN and VERA VLBI arrays for a radio source located at +60◦
declination observed at 22 GHz.

3.3

Correlated Flux Density of a Source with Gaussian
Intensity (Brightness) Distribution

How is the correlated flux density related to the size of an observed radiosource? Let us consider this problem for a simple case, when the source hasa circular Gaussian intensity (brightness) distribution (Figure 58):2

2

STν − ξ Θ+η2s ,eIν (ξ, η) =πΘ2swhere STν is the total flux density (note that

(303)

Z∞ Z∞

2

e

− ξ +η2Θs2dξ dη = πΘ2s ),

−∞ −∞

Iν(ρ)

Θ0

ρ

Figure 58: Circular Gaussian intensity distribution Iν (ρ) with ρ =√ 2
ξ + η2.

and Θs is a parameter characterizing the source size. This parameter Θs isrelated to the half–power width of the Gaussian distribution Θ0 by a formula:Θ0 ∼Θs = √= 0.60Θ0 .
2 ln 2
142(304)

Then, in view of equation (283), the complex visibility of the source isgiven byST νV(uλ , vλ ) =
π Θ2s

Z∞ Z∞

e

−ξ

2 + η2Θ2se−i2π(uλ ξ+vλ η) dξ dη−∞ −∞= STν e−π Θs (uλ + vλ ) = STν e−[π Θs (
2

2

2

2

D cos θ
λ

)]

2

(305)

(see Figure 59 for geometry of the projected baseline), where we took AN (ξ, η) ∼=1, for simplicity, assuming that the source is observed at beam centers of antennas. In deriving equation (305), we used an integration formula:Z∞2e−x − i a x dx =√a2

π e− 4 .

−∞

s

0

θ

D
uλ

D cos θλv

λFigure 59: Projected baseline length:qθu2λ + vλ2 = D cos.
λThus, for a source with the circular Gaussian intensity distribution, wehave
2
D cos θ• correlated flux density = | V(uλ , vλ ) | = STν e−[π Θs ( λ )] ,• visibility phase = 0.
This means that the correlated flux density | V(uλ , vλ | is reduced to thehalf of the total flux density STν when"

D cos θ
πΘs (
)
λ
143

#2

= ln 2,(306)

or, in view of equation (304), when the half–power width of the source distribution Θ0 is equal toΘ0 =correlated fluxSTν

2 ln 2 λλ∼.
= 0.44π D cos θD cos θcorrelated flux
point source

STν

Θ0

D cos θ

0

(307)

Θ0
0.44 λ/Dcosθ
curve for given baseline0

D cos θ / λ
0.44 /Θ0
curve for given source size

Figure 60: Correlated flux density of a source with Gaussian intensity distribution, as a function of the source size (left), and as a function of the
projected baseline length (right).
Figure 60 shows the correlated flux density of the source, with a circular
Gaussian intensity distribution, as a function of the half–power width Θ0 of
the source brightness distribution, assuming a fixed baseline length D (leftpanel); and as a function of the projected baseline length normalized by thewave length, D cos θ/λ, assuming a fixed source size (right panel), according
to equation (305). This simple example shows that the source appears weak
to an interferometer, or “resolved out”, when its size becomes comparablewith or larger than the fringe spacing λ/(D cos θ). In other words, interferometers are sensitive to the sources with angular sizes comparable to or
smaller than their fringe spacings.
It is usual practice in VLBI to infer the size of a source by measuring its
correlated flux densities at different projected baseline lengths, and fitting
the results to a Gaussian model, as shown in the right panel of Figure 60.
Figure 61 shows an example (Wajima and Iguchi, private communication in2005).

144

Figure 61: Visibility amplitude (top) and phase (bottom) versus uv–distancealong a position angle (PA) 60◦ plot of a quasar 3C279 observed with VLBA
at 15 GHz (Wajima and Iguchi, private communication in 2005). Visibility
amplitude (top) decreases with uv–distance (projected baseline length) in arange from 0 to 70 × 106 λ, similarly to the curve in the right panel of Figure
60. However, visibility amplitude becomes almost flat in uv–distance range
from 70 × 106 to 400 × 106 λ, showing existence of a compact component
which is not resolved even on the longest baseline 400 × 106 λ ' 8000 km.
This Figure can be compared with an image map of the same source shownin Figure 52.

145

Figure 62: VLBI can observe very bright, compact objects only.

3.4

What Can VLBI Observe?

The above discussions show that a VLBI (or an interferometer, in general),with baseline length D and observing wavelength λ, can detect only compact
sources with characteristic angular diameters Θ0 comparable to or smaller
than λ/D, i.e.,λΘ0 ≤ .
D
If the brightness temperature of a source is TB , then the intensity Iν andthe flux density Sν are given by:
Iν =

2kTB
,
λ2

and Sν =2kTB πΘ20
πkTB
≤
,
λ2
4
2D 2

where k = 1.381 × 10−23 JK−1 is the Boltzmann constant. Note that the last
expression does not depend on the wavelength.
Now, if the minimum detectable flux density by VLBI is Sνmin , we can
impose a lower limit to the brightness temperature, TBmin , for the detectablesource:2D 2
Sνmin .
(308)
TBmin ∼=πk
If D = 2300 km and Sνmin = 0.02 Jy, then TBmin ∼= 5 × 107 K (!).
146

Such a high lower limit of the brightness temperature precludes detectionof almost all thermal sources, such as stars and molecular clouds with VLBI,
leaving only very bright, compact non–thermal sources, e.g., AGNs, masers,
and pulsars (see Figure 62), to be detected.

4

Signal–to–Noise Ratio in Radio Interferometry

So far, we approximated the result R of the multiplication and integration
in a correlator by the cross–correlation RU1 U2 (0) of the voltage signals U1 (t)and U2 (t) at input points of the correlator. In view of the ergodicity, thisapproximation should be good enough when the integration time is sufficiently long. Also, one can prove that the mathematical expectation of R isequal to RU1 U2 (0), if U1 (t) and U2 (t) are jointly stationary random processes.
However, the actual correlator output, with some limited integration time, isinevitably accompanied with thermal noise fluctuations. Therefore, we havesome finite signal–to–noise–ratio in the correlator output, which determines
detectability of the radio source signal, i.e. the sensitivity (Figure 63).

signaltsignalt

Figure 63: Signal–to–noise ratio of the correlator output determines the sensitivity of an interferometer.
We will see in the followings that a statistical theory of the dispersion of147

the time–averaged product of random processes gives a theoretical expression
for the signal–to–noise ratio S/N of the correlator output.

4.1

Statistical Model of a Correlator Output

4.1.1

Signal + Noise Inputs to a Correlator

Let us consider that the voltages from two antennas U1 (t) and U2 (t), which
are fed to input points of a correlator, consist of signals V1 (t), V2 (t) andfrom antenna 1

from antenna 2.

U1=V1+N1

multiplierintegrator

U2=V2+N2

RV1V2 + Noise

Figure 64: Input voltages to a correlator consist of signals and noises.
noises N1 (t), N2 (t) (see Figure 64):
U1 (t) = V1 (t) + N1 (t),U2 (t) = U2 (t) + N2 (t).

(309)

We assume that the delay tracking and fringe stopping were already successfully applied to the voltages U1 (t) and U2 (t), so that we can integrade themultiplier output for an appropriate interval of time without losing signal
power.
4.1.2

Integration of the Multiplier Output

If we denote an instantaneous output of the multiplier as y(t):y(t) = U1 (t)U2 (t),(310)

an output of the integrator (i.e., correlator output) z(t) is related to the input
y(t) by a linear system of time averaging with an impulse response a(t):z(t) = y(t) ∗ a(t) =

Z∞

−∞

148

y(t0 )a(t − t0 ) dt0 ,(311)

where symbol ‘∗’ stands for a convolution, as before.
If an input y(t) is a constant y in time, the time averaging result must be
equal to the constant y. Therefore, the impulse response a(t) must satisfy:Z∞a(t) dt = 1.

(312)

−∞

For example, if we take a simple running mean:
1z(t) =
2T

t+TZ

y(t0 ) dt0 ,

(313)

t−T

then the impulse response a(t) is given bya(t) =

4.1.3





1
2T


 0

for − T ≤ t ≤ T,otherwise.

Statistical Expectation of the Correlator Output

We assume that the noise components from two antennas are not correlatedwith each other, and are not correlated with signal components, either. Then,
we havehN1 (t) N2 (t)i = 0, hVi (t) Nj (t)i = 0 for i = 1, 2 and j = 1, 2,(314)

where h i stands for the ensemble average, as before. From equation (314),
we obtain
hy(t)i = hU1 U2 i = h[V1 (t) + N1 (t)] [V2 (t) + N2 (t)]i = hV1 (t) V2 (t)i.

(315)

In next subsection, we will see that the multiplier output y(t) = U1 (t)U2 (t)is a stationary random process, provided that U1 (t) and U2 (t) are jointly stationary random processes obeying the zero–mean jointly normal (or Gaussian) probability density. If so,hy(t)i = const,
and, therefore,hz(t)i =

Z∞

−∞

0

0

0

hy(t )ia(t − t ) dt = hy(t)iZ∞−∞a(t − t0 )dt0 = hy(t)i,(316)

in view of equation (312). Equations (315) and (316) show that the statistical
expectation of the correlator output in radio interferometry does not containsystem–noise contribution unlike in the single–dish radio telescope case.
1494.1.4

Dispersion of the Correlator Output

As we saw in the discussion of the correlation ergodicity in Subsection 1.2.4,
a correlator output z(t) with finite integration time should be associated withnoise–induced fluctuations around the expectation:z(t) = hz(t)i + noise = hV1 (t) V2 (t)i + noise.

(317)

This fluctuation determines detection limit of radio sources as illustrated in
Figure 64.
The magnitude of this fluctuation can be estimated by the dispersion σz2of the correlator output z(t), as we saw in the discussion of the correlation
ergotic processes. The dispersion is given byσz2 = h[z(t) − hz(t)i]2 i= h[z 2 (t) − 2z(t) hz(t)i + hz(t)i2 ]i= hz 2 (t)i − hz(t)i2 ,(318)

and, therefore, the standard deviation is given byσz =qhz 2 (t)i − hz(t)i2 .

(319)

This is a measure of strength of the noise in the correlator output.
On the other hand, strength of the signal in the correlator output is givenby the theoretical expressions of the fringe amplitude AU and AL shown inequations (260) and (261) for USB and LSB receptions, respectively, for acontinuum spectrum source.
Therefore, signal–to–noise ratio SN R for a continuum spectrum sourceis given byA(320)
SN R = ,σzwhere A is either AU or AL , depending on a choice of the single sideband
actually received.
In following subsections, we will calculate the standard deviation of thecorrelator output σz , in order to get an explicit expression of the signal–to–noise ratio in terms of physical parameters characterizing the radio source
and antenna–receiving systems.

4.2

Useful Formulae Related to the Correlator Output

In calculating the standard deviation of the correlator output σz , we will use
following formulae relevant to the present problem.
150

4.2.1

Fourth Order Moment Relation

Let x1 , x2 , · · ·, xn are zero–mean random variables obeying to the jointly
normal (or Gaussian) probability density given in equation (29):f (x1 , · · · , xn ) = q

1(2π)n ∆

e

− 12

n P
n
P

−1
xi Cijxj

i=1 j=1,(321)

where expectation ηi of the random variable xi figuring in equation (29) iszero by assumption for all i = 1, 2, · · · , n; Cij ≡ hxi xj i is a covariance matrix,
Cij−1 is its inverse matrix, and ∆ ≡ det{Cij } is its determinant. In general,
we denote an inverse matrix by a symbol ( )−1 and a determinant by det ( ).
A very useful formula is known for such jointly normal random variables
xi (i = 1, · · · , n):hxi xj xk xl i = hxi xj i hxk xl i + hxi xk i hxj xl i + hxi xl i hxj xk i,(322)

which says that a fourth order statistical moment (a term “n–th order statistical moment” is used here to denote a statistical expectation of a productof n random variables) hxi xj xk xl i is decomposed as a sum of products ofsecond order statistical moment (correlations) hxi xj i.
Proof :
1. Let us reformulate equation (321), using matrix notation of a row vector
X and its transpose X T :




x
 1 



 x2 
X =  . ,
 . 
 . 



and X T = (x1 x2 · · · xn ),xnwhere ( )T denotes a transpose matrix. Then we havef (X) = q1(2π)n ∆1Te− 2 X C

−1 X

,

(323)

where C is the covariance matrix which is a positive–definite symmetricmatrix with Cij = hxi xj i as ij–element.
151

2. Fourier transform of a probability density is called “characteristic function”. A characteristic function F (Ω) of the joint probability densityf (X) in equation (323), where Ω is a row vector of angular frequenciesΩT = (ω1 ω2 · · · ωn ), is given by
F (Ω) =

Z∞

−∞

= q

···

Z∞

−∞

1
(2π)n ∆TT

f (X) e−i Ω X dx1 · · · dxn = he−i Ω X i
Z∞

−∞

···

Z∞

1

T

e− 2 X C−1 X − iΩT X−∞

dx1 · · · dxn .
(324)

3. According to the linear algebra, a symmetric matrix can be diagonalized by a suitable orthogonal transformation. So, let T be an orthogonal
matrix, which diagonalizes the symmetric matrix C −1 :


λ
 1



T T C −1 T = 





0

···

0
..
.

λ2 · · ·.. . .
.
.

0

0



0 

0 

,
.. 
. 



· · · λn(325)

where λ1 , λ2 , · · · , λn are eigen values of the matrix C −1 .
Since T is an orthogonal matrix, by definition we have
T T T = T T T = I,

T −1 = T T ,det T = 1,(326)

where I is a unit matrix:








1 0 ··· 0  0 1 ··· 0 ,I = . . .
.
 . .
. . ..  . .
and, therefore,

0 0 ··· 1TT

T C T = (T C

−1

T)

−1

λ−1
 1




=




152

0
..
.

0

0

···

···
λ−1
2
..
..
.
.
0

···



0 

0 

,
.. 
. 


λ−1
n



(327)

and
∆ = det C = det T T det C det T = det(T T C T ) =1.
λ1 λ2 · · · λ n(328)

4. If we introduce a new row vector Y :Y T = (y1 y2 · · · yn ),which is obtained by an orthogonal transformation of X with T :Y = T T X,
then we have

T

X C

−1

∂(x1 x2 · · · xn )= det T = 1,∂(y1 y2 · · · yn )∂xi
= Tij ,∂yj

X = T Y,TT

X=Y T C

−1

TY =

n
X

λj yj2 ,j=1ΩT X = Ω T T Y =

n X
n
X

ωi Tij yj .

(329)

i=1 j=1

Therefore, the characteristic function F (Ω) given in equation (324) is
reduced to
n
n X
n
X
1 X
2
λj y j − i
ωi Tij yj
Z∞ −
Z∞
1
2 j=1
i=1
j=1
q
F (Ω) =e···dy1 · · · dyn .
(2π)n ∆ −∞ −∞(330)
If we further introduce new variables:
zj =

s

λj
yj
2

(j = 1, · · · , n),then we have
F (Ω) = q
−

n
X

1

Z∞

···

ωi Tijs(2π)n ∆ −∞(zj2 + i

n
X

Z∞

−∞

2
zj ) sλj

2ndz1 · · · dzn ,λ1 · · · λ n
s
n
n
X
X
2
2
(zj + i
ωi Tij
zj )
Z∞ −
Z∞
1
λj
j=1
i=1
√
=
e
···
dz1 · · · dzn .
π n −∞ −∞

e

j=1

i=1

(331)
153

5. Applying the integration formula:Z∞2e−x − i a x dx =√a2

π e− 4 ,−∞we reduce equation (331) toF (Ω) = e

n
n
X
1 X
−
ωi Tij
4 j=1 i=1

s

2
λj

!2

n X
n X
n
1
1 X
Tkj ωkωi Tij−2 i=1 j=1 k=1
λj

= e
1 T
= e− 2 Ω C Ω .

(332)

6. From equations (324) and (332), we have1TT

he−i Ω X i = e− 2 Ω C Ω .

(333)

Expanding exponential functions in both sides of this equation into the
Taylor series, we have111The−iΩ X i = 1 − h(ΩT X)2 i + i h(ΩT X)3 i + h(ΩT X)4 i − · · · ,
2!
3!
4!
1 T
1
1
1
(334)
e− 2 Ω C Ω = 1 − ΩT C Ω + ( ΩT C Ω)2 − · · · .
2
2! 2
Terms with fourth order of ωi ’s in the above equations are
n X
n X
n X
n
1
1 X
T
4
hxi xj xk xl iωi ωj ωk ωl ,h(Ω X) i =4!
4! i=1 j=1 k=1 l=1

(335)

and
n X
n X
n X
n
1 1 X
1 1 T
Cij Ckl ωi ωj ωk ωl( Ω C Ω)2 =2! 2
2! 4 i=1 j=1 k=1 l=1

=

n X
n X
n X
n
1 X
(Cij Ckl + Cik Cjl + Cil Cjk ) ωi ωj ωk ωl ,4! i=1 j=1 k=1 l=1(336)

respectively. Right hand sides of equations (335) and (336) are equal
to each other in view of equation (333). Consequently, coefficients ofωi ωj ωk ωl in these equations, i.e., hxi xj xk xl i and Cij Ckl + Cik Cjl +Cil Cjk , must be equal to each other, too, since both of them are invariant in any substitution of i, j, k, l. Thus, we havehxi xj xk xl i = Cij Ckl + Cik Cjl + Cil Cjk .
Since Cij = hxi xj i, this completes the proof of equation (322).
154

(337)

4.2.2Multiplier Output as a Stationary Random Process

If two voltage inputs U1 (t) and U2 (t) of a correlator are jointly stationary processes and if they obey the zero–mean jointly normal (or Gaussian) probability density such as given in equation (321), then their producty(t) = U1 (t) U2 (t) is also a stationary random process.
Proof :
1. Expectation of y(t) is a constant in time. In fact,hy(t)i = hU1 (t)U2 (t)i = RU1 U2 (0),(338)

where RU1 U2 (τ ) = hU1 (t) U2 (t − τ )i is a cross–correlation of jointly
stationary random processes U1 (t) and U2 (t), and, therefore, hy(t)i does
not depend on time t.
2. Autocorrelation of y(t) is a function of time difference only. In fact, inview of equation (322), we havehy(t) y(t − τ )i = hU1 (t) U2 (t) U1 (t − τ ) U2 (t − τ )i
= hU1 (t) U2 (t)i hU1 (t − τ ) U2 (t − τ )i
+hU1 (t) U1 (t − τ )i hU2 (t) U2 (t − τ )i
+hU1 (t) U2 (t − τ )i hU2 (t) U1 (t − τ )i
= RU2 1 U2 (0) + RU1 U1 (τ ) RU2 U2 (τ ) + RU1 U2 (τ ) RU1 U2 (−τ ),(339)
where RU1 U1 (τ ) = hU1 (t) U1 (t − τ )i and RU2 U2 (τ ) = hU2 (t) U2 (t − τ )iare autocorrelations of input voltages U1 (t) and U2 (t). This equationshows that hy(t) y(t − τ )i is a function of time difference τ only.
Thus, we proved that the multiplier output y(t) is a stationary random process.
Since the integrator output (i.e., correlator output) z(t) is an output of alinear system of time averaging, given in equation (311), with the input y(t)which was shown to be a stationary random process, we conclude that z(t)is also a stationary random process.
Hereafter, we will denote autocorrelations of the multiplier output y(t)and the integrator output z(t) as functions of time difference τ :Ryy (τ ) = hy(t) y(t − τ )i,
Rzz (τ ) = hz(t) z(t − τ )i.
155

(340)

4.2.3Time Averaging Operator

Since the multiplier output y(t) and the integrator output z(t) are related
to each other through the impulse response a(t) of the operator of time
averaging, as shown in equation (311):
z(t) = y(t) ∗ a(t),their autocorrelations are related to each other as:
Rzz (τ ) = Ryy (τ ) ∗ a(τ ) ∗ a(−τ ),(341)

in view of equation (42).
Therefore, if we denote power spectra of y(t) and z(t) as Syy (ω) andSzz (ω):Syy (ω) ⇔ Ryy (τ ),Szz (ω) ⇔ Rzz (τ ),where symbol ⇔ stands for a Fourier transformation pair, and introduce asystem function A(ω) of a(t):A(ω) ⇔ a(t),then we have, in view of equation (72),Szz (ω) = Syy (ω) | A(ω) |2 .
Since
A(ω) =

Z∞(342)

a(t) e−iωt dt,−∞

and, from equation (312),

Z∞a(t) dt = 1,−∞

the system function A(ω) must always satisfy an equation:
A(0) = 1.

(343)

Note that, for the case of a simple running mean, the system function
takes the sinc function form:Tsin(ωT )1 Z −iωtedt =A(ω) =
.
2T
ωT

(344)

−T

In general, A(ω) can be regarded as a low–pass filter with a narrow passband of about ±2π/T , as evident from the above simple case. If the integral
time T is 1 s, for example, the passband is as narrow as ±1 Hz.
156

4.2.4

Power Spectrum of Multiplier Output

Let us now describe the power spectrum of the multiplier output Syy (ω)
through power and cross–power spectra of voltages, which are the inputs toour correlator.
From equations (339) and (340), we have
Ryy (τ ) = RU2 1 U2 (0) + RU1 U1 (τ ) RU2 U2 (τ ) + RU1 U2 (τ ) RU1 U2 (−τ ).

(345)

Therefore, Fourier transformation of this equation gives us the power spectrum Syy (ω). Using convolution theorem in Fourier transformation, given inequation (67), for products of functions of τ , we obtain
Syy (ω) =Z∞

Ryy (τ )e−iωτ dτ−∞= 2πRU2 1 U2 (0) δ(ω)
1
1
SU1 U2 (ω) ∗ SU∗ 1 U2 (ω), (346)+ SU1 U1 (ω) ∗ SU2 U2 (ω) +2π
2πwhere δ(ω) is the delta function of angular frequency ω, and SU1 U1 (ω),
SU2 U2 (ω), and SU1 U2 (ω) are power and cross–power spectra of input voltages:
SU1 U1 (ω) ⇔ RU1 U1 (τ ),SU2 U2 (ω) ⇔ RU2 U2 (τ ),SU1 U2 (ω) ⇔ RU1 U2 (τ ).
Here we used a relation:Z∞e−iωτ dτ = 2πδ(ω),−∞

which was given in equation (65), and for Fourier transform of a real function
RU1 U2 (−τ ):Z∞

RU1 U2 (−τ )e

−iωτ

dτ =

−∞

Z∞

RU1 U2 (τ )eiωτ dτ

−∞

=

 ∞
∗
Z

RU U (τ )e−iωτ dτ  = S ∗
1

−∞

157

2

U1 U2 (ω).

4.2.5

Power and Cross–Power Spectra of Input Voltages

Power and cross–power spectra of input voltages figuring in equation (346)
can be described as follows.
Assuming, as before, that signal and noise are not correlated with eachother, and noises from different antennas are not correlated, either, we express autocorrelations and cross–correlation of input voltages U1 (t) and U2 (t)in terms of signal (V ) and noise (N ) components:
RU1 U1 (τ ) = RV1 V1 (τ ) + RN1 N1 (τ ),RU2 U2 (τ ) = RV2 V2 (τ ) + RN2 N2 (τ ),RU1 U2 (τ ) = RV1 V2 (τ ),

(347)

where
RV1 V1 (τ ) = hV1 (t) V1 (t − τ )i,
RV2 V2 (τ ) = hV2 (t) V2 (t − τ )i,
RV1 V2 (τ ) = hV1 (t) V2 (t − τ )i,are auto– and cross–correlations of signal voltages V1 (t) and V2 (t), while
RN1 N1 (τ ) = hN1 (t) N1 (t − τ )i,
RN2 N2 (τ ) = hN2 (t) N2 (t − τ )i,are autocorrelations of noise voltages N1 (t) and N2 (t).
Therefore, the power and cross–power spectra are obtained by Fouriertransformation of equations (347) as
SU1 U1 (ω) = SV1 V1 (ω) + SN1 N1 (ω),
SU2 U2 (ω) = SV2 V2 (ω) + SN2 N2 (ω),
SU1 U2 (ω) = SV1 V2 (ω),
where
SV1 V1 (ω) ⇔ RV1 V1 (τ ),SV2 V2 (ω) ⇔ RV2 V2 (τ ),SV1 V2 (ω) ⇔ RV1 V2 (τ ),are power and cross–power spectra of signal voltages, while
SN1 N1 (ω) ⇔ RN1 N1 (τ ),SN2 N2 (ω) ⇔ RN2 N2 (τ ),are power spectra of noise voltages.
158

(348)

4.2.6

Antenna Temperature and System Noise Temperature

We saw in equation (169), that a power spectrum Svi vi (ω) of a received signal
voltage vi (t) (i = 1, 2) at RF band, which is just received by i–th antenna,and has not yet gone through a receiving system, is given in terms of effective
flux density Sν of an observed source and efective aperture of i–th antenna
Aei by1
Svi vi (ω) = Aei Sν (i = 1, 2).
4
Since the effective flux density and the antenna temperature TAi (ω), characterizing a received power from a radio source per unit bandwidth, are related to each other by an equation:
1
k TAi (ω) = Aei Sν ,2
as we saw in Chapter 2, where k = 1.381 × 10−23 JK−1 is Boltzmann’s constant, the power spectrum of the received signal voltage is described throughthe antenna temperature as:Svi vi (ω) =1
k TAi (ω)
2(i = 1, 2).

(349)

Since the frequency conversion preserves the power spectrum of the received signal voltage, we can easily derive from equation (349) a power spectrum SVi Vi (ω) of an IF signal voltage Vi (t) at an input of a correlator. Indeed,
if we denote the system function of the receiving system as Hi (ω), as given
in equation (195), and if we assume for definiteness upper sideband (USB)reception, we have
SVi Vi (ω) =1
k TAi (ωLO + ω) | Hi (ω) |22(for ω ≥ 0, i = 1, 2),(350)

where ω is frequency in IF band and ωLO is local oscillator frequency of the
frquency conversion. Here we showed only positive frequency side of thepower spectrum, which is an essentially even function of frequency.
On the other hand, the noise component is described through the input equivalent system noise temperature TSi (ω), which is supposed to passthrough the same receiving system as the radio source signal does. Thus, inthe USB reception and in the positive frequency range, we have
SNi Ni (ω) =1k TSi (ωLO + ω) | Hi (ω) |2
2

159(for ω ≥ 0, i = 1, 2).

(351)

Therefore, from equations (348), (350), and (351), the power spectrum of
the input voltage Ui (t) is given by
SUi Ui (ω) =1k [TAi (ωLO +ω)+TSi (ωLO +ω)] | Hi (ω) |22(for ω ≥ 0, i = 1, 2),(352)

in the USB reception and in the positive frequency range.
The cross–power spectrum of two input signal voltages Vi (t), (i = 1, 2) after delay tracking and fringe stopping is given in equation (242), for the caseof the USB reception and in the positive frequency range of the Hermitiansymmetric cross–power spectrum. Therefore, taking into account also equation (348), the cross–power spectrum of the input voltages Ui (t), (i = 1, 2)
in the positive frequency range (ω ≥ 0) is given by
SU1 U2 (ω) = SV1 V2 (ω) =1
A0 e−i(φLO1 −φLO2 ) V(ωLO + ω) H1 (ω) H2∗(ω), (353)4where A0 is geometric mean of effective apertures of antennas, φLO1 − φLO2 isdifference of initial phases of local oscillators, and we put ∆τg = 0 in equation(242), assuming complete delay tracking and fringe stopping.
In a simple case of a point–like radio source with normalized power
pattern of the interferometer AN (σ) = 1 at centers of antenna beams,
the complex visibility of the source is real and equal to its flux density,
V(ωLO + ω) = Sν , as given in equation (295), and, therefore,1
1q
1 q
Ae1 Ae2 Sν = k TA1 (ωLO + ω) TA2 (ωLO + ω).
A0 V(ωLO + ω) =
4
4
2
(354)
In this case, we have, in ω ≥ 0
SU1 U2 (ω) =1 q
k TA1 (ωLO + ω) TA2 (ωLO + ω) e−i(φLO1 −φLO2 ) H1 (ω) H2∗ (ω).
2
(355)

4.3

Sensitibity of a Radio Interferometer

4.3.1

Standard Deviation Due to the Noise

Now we are ready to calculate the standard deviation σz of the correlator
output z(t) in equation (319), and then the signal–to–noise ratio in equation(320).
In equation (319):qσz = hz 2 (t)i − hz(t)i2 ,160

the first term hz 2 (t)i in square root is given by inverse Fourier transformation
at τ = 0 of the power spectrum Szz (ω), which is related to Syy (ω) given inequation (346) through equation (342). Therefore, we have1hz (t)i =
2π
2

=

1
2π

Z∞ h

Z∞

−∞

1
Szz (ω) dω =
2π

Z∞

−∞

Syy (ω) | A(ω) |2 dω

2πRU2 1 U2 (0) δ(ω)−∞

1
1
SU1 U1 (ω) ∗ SU2 U2 (ω) + SU1 U2 (ω) ∗ SU∗ 1 U2 (ω) | A(ω) |2 dω
+
2π
2π
2
= RU1 U2 (0)


1
+
(2π)2

Z∞ h

−∞

i

SU1 U1 (ω) ∗ SU2 U2 (ω) + SU1 U2 (ω) ∗ SU∗ 1 U2 (ω) | A(ω) |2 dω,(356)

where A(ω) is the system function of time averaging, and we used equation(343) for deriving the first term. On the other hand, second term hz(t)i2 isequal to RU2 1 U2 (0), since, in view of equations (317) and (338), we havehz(t)i = RU1 U2 (0).
Therefore, the first term of the right hand side of equation (356) is compensated by hz(t)i2 , and hence we have
σz2 =

∞
i
1 Z h
∗
(ω)
| A(ω) |2 dω.
S
(ω)
∗
S
(ω)
+
S
(ω)
∗
S
U1 U1
U2 U2
U1 U2
U1 U2
(2π)2
−∞

(357)
If we use explicit forms of convolution integrals, equation (357) is reduced to
σz2 =

1
(2π)2

Z∞ Z∞

−∞ −∞

[SU1 U1 (ω − ω 0 ) SU2 U2 (ω 0 )i+SU1 U2 (ω − ω 0 ) SU∗ 1 U2 (ω 0 ) | A(ω) |2 dω 0 dω.

(358)

Since A(ω) is a very narrow–band low–pass filter around ω = 0, as we
saw earlier, SU1 U1 (ω − ω 0 ) and SU1 U2 (ω − ω 0 ) can be replaced by SU1 U1 (−ω 0 ) =SU1 U1 (ω 0 ) and SU1 U2 (−ω 0 ) = SU∗ 1 U2 (ω 0 ), and we can take them out of theintegration with respect to ω. Thus we obtain,
σz2 =

1
(2π)2

Z∞ h

0

0

SU1 U1 (ω ) SU2 U2 (ω ) + SU∗ 1 U2 (ω 0 ) SU∗ 1 U2 (ω 0 )

−∞

161

i

dω

0

Z∞

−∞

| A(ω) |2 dω.
(359)

Taking into account that SU1 U1 (ω 0 ) is an even, and SU1 U2 (ω 0 ) is a Hermitiansymmetric, functions, we can further reduce equation (359) to
∞

σz2 =

1 Z
{SU1 U1 (ω 0 ) SU2 U2 (ω 0 )
2π 2
0

0

0

+< [SU1 U2 (ω ) SU1 U2 (ω )]} dω

0

Z∞

−∞

| A(ω) |2 dω.

(360)

Now, assuming the USB reception, for definiteness, we apply equations(352) and (355) for spectra in positive frequency range (ω ≥ 0) of input
voltages to obtain
k2
8π 2

σz2 =

Z∞ n
0

(TA1 + TS1 ) (TA2 + TS2 ) | H1 (ω 0 ) |2 | H2 (ω 0 ) |2o+< [(TA1 TA2 ) e−2i(φLO1 −φLO2 ) H1 (ω 0 ) H2∗ (ω 0 ) H1 (ω 0 ) H2∗ (ω 0 )] dω 0
×

Z∞

−∞

| A(ω) |2 dω,(361)

where we omitted argument ωLO + ω in TAi and TSi (i = 1, 2) for simplicity.
Since, in actual observations of most of radio sources, antenna temperatures are much smaller than system noise temperatures: TAi  TSi (i = 1, 2),
we ignore terms with TAi , compared with terms with TSi (i = 1, 2). Then,assuming a flat noise spectrum, we obtain an equation:
k
σz2 =

2

∞

∞

Z
TS 1 TS 2 Z
0 2
0 2
0
|
H
(ω
)
|
|
H
(ω
)
|
dω
| A(ω) |2 dω,
1
2
8π 2

(362)

−∞

0

which describes the dispersion of noise fluctuations in the correlator output.
4.3.2

Signal in the Correlator Output

Now, if we assume a continuum spectrum source and USB reception, thesignal in the correlator output is given by the theoretical expression of thefringe amplitude A in equation (260):
A = A0 | V | | B12 |,where A0 is the geometric mean of effective apertures of element antennas,| V | is the visibility amplitude, and | B12 | is the amplitude of the bandwidth
162

pattern defined in equation (254). In the simple case of the point–like source,
for which the visibility V is equal to the flux density Sν , we have
A0 | V |=q

A e 1 A e 2 Sν = 2 kq

TA 1 TA 2 ,and, therefore,A = 2k
4.3.3

q

TA1 TA2 | B12 | .

(363)

Signal–to–Noise Ratio of the Correlator Output

Consequently, the signal–to–noise ratio is given by
SN R =
= 8π

s

A
σz

1
TA 1 TA 2| B12 |
s
s
.
∞
R
R∞
TS 1 TS 2
2
0
2
0
2
0
| A(ω) | dω
2 | H1 (ω ) | | H2 (ω ) | dω
−∞

0

(364)

Let us first assume simple rectangular filters which are given in positivefrequency range (ω ≥ 0) by:2| Hi (ω) | =



 Gi

 0

if ωI − ∆ω≤ ω ≤ ωI + ∆ω,22otherwise,(365)

where i = 1, 2, Gi is a gain factor in a receiving system of i—th antenna, ωI
is a center IF frequency of the filter passband, and ∆ω = 2πB is an angularfrequency bandwidth, corresponding to a frequency bandwidth B.
In this case, the coefficient
G of the bandwidth pattern given in equation√(262) is equal to G = G1 G2 . Therefore, the amplitude of the bandwidth
pattern | B12 | is reduced to√√G1 G2 B sin(πB∆τg )G1 G2 B=
(for ∆τg = 0),
(366)
B12 =
2
πB∆τg
2
where we assumed complete delay tracking and fringe stopping with residual
delay ∆τg = 0. On the other hand, we obviously haveZ∞0

| H1 (ω 0 ) |2 | H2 (ω 0 ) |2 dω 0 = 2π G1 G2 B.

163

(367)

Therefore, we obtain
s

2

R∞
0

8π | B12 |=| H1 (ω 0 ) |2 | H2 (ω 0 ) |2

√

4π B.

(368)

dω 0

For more general case of non–rectangular filters, we will adopt this equation (368) as a definition of bandwidth B.
On the other hand, if we assume the simple running mean of equation(313) for the time averaging, then an integration time τa is equal to theinterval of averaging τa = 2 T . In this case, equation (344) shows that thesystem function of time averaging A(ω) has a simple sinc function form:
A(ω) =sin(ω T ).
ωT

Therefore,
Z∞

−∞

2

| A(ω) | dω =and, hence, we haveZ∞−∞

sin2 (ω T )
π
2π
dω
=
=
,
ω2 T 2
T
τa

1
s

∞
R

−∞

=

| A(ω) |2

dω

r

τa
.
2π

(369)

For more general case of non–running–mean averaging, we will adopt thisequation (369) as a definition of integration time τa .
Inserting equations (368) and (369) to equation (364), we obtain an equation for the signal–to–noise ratio SN R in radio interferometry:A=
SN R =σzs

TA 1 TA 2 q
2 B τa .
TS 1 TS 2(370)

This equation (370) is very similar to the one for the single dish telescope,√
which we mentioned in Chapter 2, except for an extra factor 2 in 2Bτa .
The difference comes from the fact that, in case of the single baseline interferometry, the system noise contributions from 2 antennas are independent,
and, therefore, not correlated, unlike in the single dish telescope case.
4.3.4

Additional Remarks on the SN R Formula

Although we derived equation (370) assuming a point–like and continuum
spectrum source in a case of USB reception, the equation can be applied tomore general cases, if we164• multiply a ratio of correlated
flux density | V | and total flux densityqSTν , i.e., | V | /STν , to TA1 TA2 for an extended source,• take line width instead of the filter bandwidth for B for a line spectrum
source,• take LSB values of TAi and TSi (i = 1, 2) in LSB reception.
An extra “efficiency factor” ηc , typically ηc ∼= 0.5 ∼ 0.9, is multiplied
to the right hand side of equation (370) in case of digital data processing,in order to take into account losses associated with digitization and digital
logics, as we will discuss later. Then, the signal–to–noise ratio SN R is usually
given by:s
TA 1 TA 2 q
SN R = ηc
2 B τa .
(371)
TS 1 TS 2
4.3.5

A Simple Interpretation of the√B τa Factor

noise
signalτanoise
signalω
Figure 65: Schematic diagram explaining how the signal–to–noise ratio of astationary signal with a continuum spectrum is improved by√ increasing the
integration time τa and the bandwidth B, in proportion to B τa .
√Figure 65 gives a qualitative explanation why we have a factor B τain a formula of signal–to–noise ratio. In fact, we can increase the numberof independent measurements, by increasing the integration time τa and the
bandwidth B for a stationary source with a continuum spectrum,
thus im√√proving the signal–to–noise ratio in proportion to τa and B.
165

4.3.6

Formulae for SN R and Minimum Detectable Flux Density

If correlated flux density of an observed source is Sν , and effective apertures,
aperture diameters, and aperture efficiencies of antennas are Ae1 , Ae2 , L1 ,L2 , and ηA1 , ηA2 , respectively, we can rewrite equation (371) in a form:
SN R = ηcorπ
SN R = ηc8kq

A e 1 A e 2 Sν q

2k

√

q

TS 1 TS 2

2 B τa ,η A 1 η A 2 L 1 L 2 Sν q
q

TS 1 TS 2(372)

2 B τa .

(373)

Or, by introducing system equivalent flux densities (SEFDs) of the antennas,
SEF D1 =we have

2 k T S1
,
Ae1

and

SEF D2 =2 k T S2
,
Ae2

q
Sν
SN R = ηc √2 B τa .
SEF D1 SEF D2(374)

For example, if we observe a source with correlated flux density 0.2 Jy,
with antenna diameters of L1 = L2 = 20 m, TS1 = TS2 = 120 K, ηA1 = ηA2 =0.5, B = 256 MHz, ηc = 0.88, and τa = 100 sec, then we expect to obtain asignal–to–noise ratio SN R ∼= 19, which is clearly detectable.
On the other hand, if we denote a limiting SN R value, necessary for
detection of a source by an interferometer, as (SN R), we can derive fromequation (373) the minimum correlated flux density Sνmin detectable by theinterferometer:Sνmin = (SN R)q

TS 1 TS 28k√.
√πηc ηA1 ηA2 L1 L2 2 B τa(375)

For VLBI, SN R ∼ 6 − 7 is usually adopted as the detection threshold. If
we adopt SN R = 7, and assume again L1 = L2 = 20 m, TS1 = TS2 = 120
K, ηA1 = ηA2 = 0.5, B = 256 MHz, ηc = 0.88 and τa = 100 sec, then we
obtain Sνmin = 0.074 Jy. We can observe many radio sources with such asensitivity. In particular, almost all extragalactic continuum spectrum radiosources regularly observed in international geodetic VLBI observations arereadily detected, since their flux densities are mostly Sν > 0.1 Jy.

166

References
Papoulis, A., 1984, Probability, Random Variables, and Stochastic Processes, 2nd Edition, McGraw-Hill (New York).
Thompson, A.R., Moran, J.M., and Swenson, G.W., Jr., 2001,
Interferometry and Synthesis in Radio Astronomy, John Wiley& Sons (New York).

167