BASIC SIGNAL THEORY
Since light is a wave, and radio signals are a form of light, it is useful to review – both mathematically and graphically - how waves add.








If we apply the explanation above to interferometers, we can learn a lot about how the VSRT
works. In what we will call a “simple interferometer”, two collectors gather waves of the same
frequency and add them together. The simple interferometer can be arranged to obtain
destructive or constructive interference.
The VSRT is not a simple interferometer – it has two collectors, but the frequencies collected by
each are changed slightly before adding them together. This frequency shift has a practical
purpose: microwave frequencies we are detecting are on the order of 1010 Hz and the equipment
needed to measure these frequencies is prohibitively expensive. By shifting the frequencies
before adding them with the VSRT, we instead measure the resulting beat frequency, which is on
the order of 106 Hz, which can be done with our less expensive equipment. The frequency
difference depends on the particular units and may need to be adjusted if the beat frequency is
not between and about 15 and 1000 kHz.
A byproduct of this frequency shift is that when you use the VSRT with a single CFL (compact
fluorescent light bulb) source – which is a source of microwave radiation - you cannot get
destructive or constructive interference, though with two CFL sources you can. Uncovering why
this is the case is an excellent way to explore how the VSRT works, but does involve quite a bit
of math, outlined below. You can, however, safely skip ahead to the assembly of the instrument
and come back to this section later if you want to learn more about how the VSRT integrates
signals.
Useful trigonometric identities for the following calculations are summarized in Table 1.



CASE 1: A SIMPLE INTERFEROMETER
We define a simple interferometer as two collectors, distance D apart, looking at a source that is
an angle from the normal. We assume that light is traveling at c and is coming from a single
source that is far enough away that the wave fronts are parallel, as show in Figure 6. We also
assume that the signals have the same frequency.
Figure 6: Simple interferometer
Because the source is not necessarily directly overhead, the wave received by the collector on the
left has to travel an additional distance x, and is delayed by t. Using the figure above, the basic
definitions of , , and we can express this additional distance x
as
Select values for x are ideal for constructive interference, and others are ideal for destructive
interference. Note that as long as D, , and c are constant, so is t.
Conditions for Destructive Interference
Applying trig identity 1 from table 1 to our original expression for , we find that
Putting in this form allows us to see that the sine term depends on t and the cosine term
does not – it depends on t, a constant, instead. As a result, there are values of t for which the
cosine term – and therefore - will always be zero, the condition for destructive interference.
The term will be zero when the argument is some odd multiple of /2 (recall that /2 in
radians is equivalent to 90˚) or when:
where n=0,1,…

Solving for t we find
If we plug this expression for into our expression for x, we find
In other words, we will have destructive interference when the two sine waves arrive an odd
number of half wavelengths apart, equivalent to being of phase by 180˚.
Conditions for Constructive Interference
In our expression for
we will have constructive interference when the term is a maximum. This occurs
when
or when
.
Therefore, when the distance x is an even multiple of half wavelengths (or an integer number of
wavelengths, ), we will have constructive interference.
Interestingly, since , the distance D between the collectors AND the location of the
source are important factors that affect what kind of interference occurs. Therefore, with a simple
interferometer observing at a given frequency, we could in principle simply separate the
collectors (affecting D) or change the source direction ( ) to produce constructive or destructive
interference.

CASE 2: VSRT WITH A SINGLE LIGHT SOURCE
The VSRT is NOT a simple interferometer because it adds waves with slightly different
frequencies. The VSRT has two collectors, or Low Noise Block Downconverter Feeds (LNBFs).
Each LNBF (A and B), downconverts the collected signal frequency by respectively.
For example, an LNBF signal at 12 GHz is mixed with a local oscillator at 11.25 GHz so that it
is downconverted to the difference ( ) of the two frequencies, 750 MHz. As a result of
these shifts, the signals that exit the LNBFs take the following form:
If we combine SA and SB, we have
Applying trig identity 1 listed in table 1, we find that
.
The terms highlighted in yellow are where differs from from the simple
interferometer. The sine term still depends on t, but the cosine term now also depends on t.
Therefore, unless , the cosine term varies with time. Note that if then
.
Therefore, because the LNBFs convert the incoming frequencies down to slightly different
output frequencies, the VSRT does not behave exactly as the simple baseline interferometer
described in CASE 1. So what kind of signal does the VSRT put out?
After the frequency shifted signals are combined into , the signal is passed through a
Square Law Detector, where the signal is squared:
Using trig identities 2 and 3 from table 1, we find:
There are several terms in this expression and since this signal is integrated over time, it is useful
to investigate which terms contribute most to the overall signal. The order of magnitude of each
of the terms is listed in Table 2.

Table 2
Term Typical Values Order of
Magnitude
and
-
x/c = 10 m/
Recall that . If we plug these values in and only keep the dominant factors, we
find
or
.
As a result of this simplification, the arguments for the cosine terms are either a high (1011) or a
low (107) angular frequency. If we plot sample cosines with high (in red) and low frequencies (in
blue) on the same plot, we get Figure 7:
Figure 7: Comparison of waves with large and small frequencies
Notice that over time, the high frequency (red) curve oscillates many more times around zero
than does the low frequency (blue) curve over the same period of time. Therefore, the high
frequency plot will average out to zero faster than the low frequency.

Given that the high frequency terms will average out to zero faster over time, any term
containing a high frequency in the expression for will go to zero and we are left with
.
We can interpret this expression as a constant plus a sinusoidal term that depends on the beat
frequency between the two oscillators. Again, this beat frequency is on the order of
106 Hz, within the range of our existing electronics. Note that since the cosine term depends on
time, there is no time when you can get destructive or constructive interference using the VSRT
and a single CFL source.
Table 3 summarizes the similarities and differences between a simple interferometer and a VSRT
with a single source.

Interestingly, we can get constructive and destructive interference with the VSRT, but we have to
use two CFLs.
CASE 3: VSRT WITH TWO CFL SOURCES
If we now use the VSRT with two CFLs, source1 and source2, the registered signal is the sum of
4 sine waves:
where the notably different terms are representing time delays associated with different
locations of source 1 or 2.
Using trig identity 1 listed in table 1, this expression simplifies to
Again, the only terms that are different from are the delay times .
If we square the signal and time average the ter***ike we did in CASE 2, the expression for
reduces to
.
Using trig identity 4 from table 1, we simplify further so that
Notice the similarities between and from CASE 1.
Most notably both have a term that does not depend on time, introducing an opportunity for
constructive and destructive interference. These similarities are summarized in Table 4.

Table 4: Summary of how a VSRT with two CFLs can emulate a simple interferometer
Simple Interferometer with a Single
Source
 single source infinitely far away at
an angle (source position
measured as angular deviation from
normal)
 two collectors a distance D apart
VSRT with Two CFLs
 Two sources infinitely far away at an angle arriving with time
delays
 two collectors a distance D apart,
 Collectors adding signals with DIFFERENT frequencies, where one
is offset by and the other by
Signals are simply added Signals are frequency shifted then added
Summed Wave:
Summed Wave:
Maximum destructive interference
( ) happens when:
- i.e. when
the waves are out of phase by 180˚,
or an odd number of half
wavelengths apart
Maximum constructive interference
happens when:
, i.e. when the light
waves are in phase, or spaced apart by
an integer number of wavelengths apart
Maximum destructive interference happens when:
- i.e. when the
wavelengths are out of phase by 180 ˚, or an odd number of half
wavelengths apart
Maximum constructive interference happens when:
, i.e. when the wavelengths are in
phase, or an integer number of wavelengths apart
In summary: If we now look at two sources with the VSRT, we find that the VSRT acts like the
simple interferometer discussed in CASE 1. The expression for CASE 3 is most
similar to the expression for CASE 1. The cosine term at the end of the expression for
does not depend on t, so it is possible for us to have constructive and destructive
interference. Instead of t, we now have the difference in the time delays between the two
sources, , so once again, as in CASE 1, the interference depends on the location of the
sources and the distance between the collectors. (See also VSRT memo 34 (Rogers, VSRT
Memo #34, 2008) using an alternate analysis using the concept of the correlation of signals in
radio interferometry. From this viewpoint it is shown how the square law detector provides the
correlation of the signals from two LNBFs.)

WORKS CITED
NAIC - Arecibo Observatory. (n.d.). Retrieved from www.naic.edu
NRAO/AUI - VLA. (n.d.). Retrieved from www.nrao.edu
Rogers, A. E. (2008, March 14). VSRT Memo #34. Retrieved August 10, 2009, from
http://www.haystack.mit.edu/edu/undergrad/VSRT/VSRT_Memos/034.pdf
Rogers, A. E., & Pratap, P. (2009, January 30). VSRT Memo #47. Retrieved August 4, 2009,
from http://www.haystack.mit.edu/edu/undergrad/VSRT/VSRT_Memos/047.pdf
VLA, Wiki. (n.d.). Retrieved August 4, 2009, from Wiki:
http://en.wiki.org/wiki/Very_Large_Array
VSRT_Team. (n.d.). VSRT Software. Retrieved from
http://www.haystack.mit.edu/edu/undergrad/srt/SRT%20Software/SRT_software_exec.html.

VSRT ASSEMBLY MANUAL
Dr. Martina B. Arndt
Department of Physics
Bridgewater State College (MA)
Based on work by
Dr. Alan E.E. Rogers
MIT‟s Haystack Observatory (MA)
With contributions from:
Robert Schweitzer „09
Bridgewater State College (MA)
August, 2009

THE VSRT: ASSEMBLY
The Very Small Radio Telescope (VSRT) is an inexpensive, portable, easily assembled
instrument made out of off the shelf parts. Included in this section are an equipment list and
detailed assembly instructions along with a few trouble shooting tips. A complete series of
Memos related to the VSRT are available online as well (VSRT_Team, VSRT Memos).
EQUIPMENT: HARDWARE LIST
The VSRT is made up of Ku 12 GHz Low Noise Block Downconverter Feeds (LNBFs – i.e. two
recycled Direct TV receivers), power supplies, inline amplifiers, a diode detector and a USB
video frame grabber to digitize the data for processing in a PC. Table 1 lists the full part list
(Rogers & Pratap, VSRT Memo #47, 2009).
When the VSRT is fully assembled, without the satellite dishes, and attached to a computer, it
looks something like the set up in Figure 1.
Compact Fluorescent
lamp source Active feeds
forming
interferometer
baseline
DC power supply
Frame
grabber
Square law
detector
DC power
injector
Inline
amplifiers
Power
splitter
Laptop displaying data from
frame grabber
Display of
fringes spectrum
Voltmeter
monitor of
power
Fringe amplitude vs
time
Az/El of sun and
satellites
Figure 1: Components of the VSRT interferometer

ASSEMBLY
Once you have collected and identified all the parts for the VSRT, you can follow the schematic
in Figure 2 to assemble it. Several of the steps are detailed below, along with photos to help
guide you. Note that you will need to do some soldering.
Following are figures and instructions to assemble specific parts of the schematic shown in
Figure 2.

TIPS ON USING THE INTERFEROMETER AND TROUBLESHOOTING
Note that many memos have been published that address various aspects of the VSRT
(VSRT_Team, VSRT Memos). These are an excellent resource for troubleshooting as well.
Beat frequency is confused with the low frequency noise
In most cases the difference in local oscillator frequencies for the 2 LNBFs will be in
an acceptable range between about 15 and 1000 kHz. If this is not the case and the
frequencies are so close that the beat frequency is confused with the low frequency
noise you have 2 options.
a. Purchase another LNBF
b. Make a frequency adjustment as described in VSRT memo #23 (Rogers,
VSRT Memo #23, 2007).
If you can’t get “fringes” at all run the following tests.
c. Hook up a voltmeter to the detector output and check that each LNBF give a
reasonable power level by connecting one LNBF at a time.

d. With one LNBF at a time check that the power increases when the CFL is
brought close. If you can run the setup outside point the LNBF at the sky and
observe the power. Compare your results with the typical readings given in
VSRT memo #17 (Rogers, VSRT Memo #17, 2007).
e. Make sure the cable lengths from the LNBFs to the power combiner are equal.
f. Check that the L.O. difference is not too close to zero. Look close to the zero
frequency end of the spectrum for the “beat”. Heat one LNBF which will
cause its L.O. frequency to change to see if you can move a beat out of the
noise at the low end of the spectrum.
Antenna pointing.
When trying to point a dish at the Sun or other source use a mirror stuck to the center of
the dish as a pointing guide. A flat 38mm×38mm mirror from Edmund Scientific part
number 3052323 (Scientific, 2009) is a good choice.
Square Law detector
The detector will only take the square of the waveform over a relatively narrow range. In most
cases the inline amplifier which comes with the Radio Shack power injector plus one additional
amplifier will provide about the right level (between 30 to 250 mV see VSRT memo #7 (Rogers,
VSRT Memo #7, 2007) ) as measured with a voltmeter with the load of the frame grabber
connected. If you are not in the square law range, experiments for which you measure levels will
be in error while those which rely on looking for a null will not be affected.
1] Alignments
The beam patterns of the LNBFs should be measured and taken into account when necessary.
2] Delay compensation
To satisfy the “white fringe” condition the path lengths should be close to one half of the inverse
bandwidth or better. In some experiments it may be necessary to make the paths more equal by
adjusting the cable lengths to one LNBF or using an optional filter to narrow the I.F. Bandwidth.