Radio Telescope

of 76 /76
DESIGN AND CONSTRUCTION OF A RADIO TELESCOPE FOR UNDERGRADUATE RESEARCH by Christopher Stathis A senior thesis submitted to the faculty of Ithaca College in partial fulfillment of the requirements for the degree of Bachelor of Science Department of Physics Ithaca College May 2011

Embed Size (px)


Radio Telescope tutorial

Transcript of Radio Telescope

Page 1: Radio Telescope




Christopher Stathis

A senior thesis submitted to the faculty of

Ithaca College

in partial fulfillment of the requirements for the degree of

Bachelor of Science

Department of Physics

Ithaca College

May 2011

Page 2: Radio Telescope

Copyright c⃝ 2011 Christopher Stathis

All Rights Reserved

Page 3: Radio Telescope



of a senior thesis submitted by

Christopher Stathis

This thesis has been reviewed by the senior thesis committee and the depart-ment chair and has been found to be satisfactory.

Date Dr. Bruce Thompson, Advisor

Date Dr. Matthew Price, Advisor

Date Dr. Matthew C. Sullivan, Senior Thesis Instructor

Date Dr. Beth Ellen Clark Joseph, Chair

Page 4: Radio Telescope




Christopher Stathis

Department of Physics

Bachelor of Science

Radio telescopes provide a practical and economical alternative to optical ob-

servatories for astrophysics research and education at primarily undergraduate

physics and astronomy institutions. We have developed an inexpensive radio

telescope capable of observing at Ku Band and L Band frequencies. The

telescope consists a three-stage superheterodyne receiver on custom circuit

boards mounted on a 3 meter parabolic antenna. Software for data collection

and hardware interfacing has been developed in MATLAB. We expect the

telescope to be capable of drift continuum observations of the Sun and Milky

Way at 3cm wavelengths. It can also be easily adapted to measure spectral

emission of neutral hydrogen and OH masers at 1.5 GHz frequencies. I present

our design methods for the radio frequency receiver and printed circuit boards

as well as a discussion of viable celestial sources of radio emission.

Page 5: Radio Telescope


The completion of this project would not have been possible without gen-

erous support from the faculty and students in the Ithaca College community,

especially my advisors Dr. Matthew Price and Dr. Bruce Thompson, whose

wonderful encouragement and infinite patience made the monumental task of

preparing a research thesis manageable and rewarding. I’d also like to thank

Dr. Dan Briotta for his cooperation as the gatekeeper of Ford Observatory,

Dr. Luke Keller for sharing his time and his connections at Cornell, and my

Senior Thesis II instructor Dr. Matthew C. Sullivan for his unwaveringly high


A number of financial contributions also helped to make this radio telescope

a reality. I extend my gratitude to Kathy Schaufler for her donation of the

satellite dish that has become our antenna. Special thanks also go to Orbital

Research, specifically their CEO Mike Stevens, for his donation of a bias tee

that is being used to supply power to our receiver. I also acknowledge the

DANA program for a stipend which enabled me to work on the telescope full

time over the Summer of 2009.

Finally, thanks to Jennifer Mellott for her help and patience in the labora-

tory, Nate Porter ’11 and Tori Roberts ’12 for helping to retrieve the satellite

dish, Jon White ’11 for helping with illustrations, and Laura Spitler of Cor-

nell University for sharing her knowledge and her access to radio electronics


Page 6: Radio Telescope
Page 7: Radio Telescope


Table of Contents vii

List of Figures ix

1 Introduction 11.1 A Brief History of Radio Astronomy . . . . . . . . . . . . . . . . . . 2

1.1.1 Jansky Merry-Go-Round . . . . . . . . . . . . . . . . . . . . . 21.1.2 Reber Sky Survey . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Important Discoveries By Radio Astronomers . . . . . . . . . . . . . 41.2.1 Non-Thermal Radiation . . . . . . . . . . . . . . . . . . . . . 41.2.2 Cosmic Microwave Background . . . . . . . . . . . . . . . . . 61.2.3 Galactic Supermassive Black Hole . . . . . . . . . . . . . . . . 7

1.3 Modern Uses For Radio Telescopes . . . . . . . . . . . . . . . . . . . 7

2 Theory 92.1 The Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . . 92.2 Quantifying Radio Observations . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Flux Density of a Radio Source . . . . . . . . . . . . . . . . . 112.2.2 Blackbody Temperature . . . . . . . . . . . . . . . . . . . . . 13

2.3 Radiation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Cosmic Sources of Radio . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.1 The Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.2 Neutral Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.3 The Milky Way . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Radiometer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.1 Superheterodyne Model . . . . . . . . . . . . . . . . . . . . . 272.5.2 Frequency Mixing . . . . . . . . . . . . . . . . . . . . . . . . . 282.5.3 Square Law Detection . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Radiometer Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 30


Page 8: Radio Telescope

viii Contents

3 Design and Implementation 353.1 Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.2 Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Superheterodyne Receiver . . . . . . . . . . . . . . . . . . . . . . . . 363.2.1 Ku Band Low Noise Block . . . . . . . . . . . . . . . . . . . . 373.2.2 L Band Frequency Mixer and Local Oscillator . . . . . . . . . 423.2.3 Filter-Amplifier Stages . . . . . . . . . . . . . . . . . . . . . . 433.2.4 Detector Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.5 Printed Circuit Board Design . . . . . . . . . . . . . . . . . . 48

4 Results and Testing 534.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Receiver Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Conclusions and Future Work 575.1 First Expected Observations . . . . . . . . . . . . . . . . . . . . . . . 575.2 Longterm Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Appendix A Matlab Script 61

Bibliography 62

Page 9: Radio Telescope

List of Figures

1.1 Data taken by Jansky’s Merry-Go-Round. . . . . . . . . . . . . . . . 31.2 Photograph of the Reber parabolic antenna. . . . . . . . . . . . . . . 51.3 Plot of the Cosmic Microwave Background. . . . . . . . . . . . . . . . 7

2.1 Diagram of the electromagnetic spectrum. . . . . . . . . . . . . . . . 102.2 Plot of the solid angle. . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 The blackbody energy distribution. . . . . . . . . . . . . . . . . . . . 142.4 Diagram of a bremsstrahlung interaction. . . . . . . . . . . . . . . . . 162.5 Diagram of synchrotron radiation. . . . . . . . . . . . . . . . . . . . . 212.6 Plot of synchrotron observations demonstrating the spectral index. . . 242.7 Radio spectrum of the Sun. . . . . . . . . . . . . . . . . . . . . . . . 322.8 Block diagram of a superheterodyne receiver. . . . . . . . . . . . . . . 33

3.1 Block diagram of the complete radio telescope. . . . . . . . . . . . . . 383.2 Schematic of the RF receiver circuit. . . . . . . . . . . . . . . . . . . 393.3 Schematic of the IF detector circuit. . . . . . . . . . . . . . . . . . . . 403.4 Diagram of the LNB multiplexer. . . . . . . . . . . . . . . . . . . . . 423.5 Bode plot of the image rejection mixer. . . . . . . . . . . . . . . . . . 453.6 The Sallen-Key filter topology. . . . . . . . . . . . . . . . . . . . . . . 463.7 Plot of experimental gain for 4-pole Butterworth filter. . . . . . . . . 473.8 Heating profile for reflow soldering. . . . . . . . . . . . . . . . . . . . 493.9 PCB schematic of the RF receiver board. . . . . . . . . . . . . . . . . 503.10 PCB schematic of the detector circuit. . . . . . . . . . . . . . . . . . 51

4.1 Results of testing the receiver with a microwave source. . . . . . . . . 554.2 Measurement of the intrinsic noise of receiver PCBs. . . . . . . . . . . 56


Page 10: Radio Telescope
Page 11: Radio Telescope

Chapter 1


Before the 20th century, astronomers possessed a limited toolset with which to observe

the sky. The whole of our knowledge concerning objects in space was derived from

the visible light we observed. Methods for detecting infrared light were eventually

developed, but large gaps in our understanding still remained. Since Maxwell’s work

in the 1860s, scientists knew that electromagnetic radiation could occur at any wave-

length, yet the efforts of physicists such as Nikola Tesla to observe unconventional

frequencies of radiation coming from space were consistently met with failure [1].

The first person to successfully observe radio frequency light from space, Karl Jan-

sky, did so by accident. His discoveries led to a revolution in astrophysics. Through

World War II and into the 1960s, research in radio astronomy exploded, and the

results challenged fundamental theories about the nature of the universe. The newly

discovered abundance of radio emission in the universe did not agree with scientists’

understanding of radiation. Radio telescopes also provided the means to detect new

and exotic classes of celestial bodies. Radio astronomy has been responsible for many

important discoveries since its inception, such as the measurement of the cosmic mi-

crowave background and its anisotropy. Today, astrophysicists use radio telescopes


Page 12: Radio Telescope

2 Chapter 1. Introduction

for a variety of applications, including monitoring the sun for sunspots, studying the

atmosphere and ionosphere of planets in the solar system, and seeking out extrater-

restrial intelligence.

1.1 A Brief History of Radio Astronomy

1.1.1 Jansky Merry-Go-Round

In 1931, Bell Laboratories commissioned a project to study the causes of poor signal

quality in shortwave radio communications, specifically those due to fluctuating iono-

spheric and atmospheric conditions. A young American physicist named Karl Guthe

Jansky was hired to carry out the research. To monitor the air for these noise signals,

Jansky constructed a directional antenna spanning 30 meters in diameter. He steered

the device by rotating it along a circular track, which earned it the nickname of ”Jan-

sky’s Merry-Go-Round.” He detected and categorized three types of noise signals in

his first months of observations - lightning strikes, automobile ignition sparks, and

a slowly oscillating hiss. A sample of Jansky’s data, recorded by pen and paper, is

depicted in Fig 1.1. By mid 1932, this slow oscillation was the only noise that Jansky

could not explain, and it became the focus of his research. By holding the antenna

in one place for several weeks at a time, Jansky determined that the source drifted

across the sky with a steady period of 24 hours. Based on this, he hypothesized that

the Sun was the source of the signal. To confirm this, he sought to observe it during

the total solar eclipse of August 31, 1932 [2]. Evidence that the signal changed when

the Sun was blocked would confirm this hypothesis. Jansky’s data from this solar

eclipse revealed no measurable effects. This forced Jansky to seek out other possible

explanations. The astrophysicists who he consulted about this problem realized that

the precise period of oscillation was 23 hours and 56 minutes, the length of a sidereal

Page 13: Radio Telescope

1.1. A Brief History of Radio Astronomy 3

Figure 1.1 A plot produced by Jansky’s directional antenna. Jansky at-tributed the pulses seen in this image to electrical storms and automobileignition sparks, but could not immediately explain the slow background os-cillation. This was later determined to be emitted by the center of the MilkyWay galaxy. (Source: [3])

day. This showed that the noise source must lie somewhere in space. In his pursuit

of atmospheric noise observations, Jansky had serendipitously produced the first evi-

dence that measurable radio frequency signals are emitted by celestial bodies [3]. He

continued to monitor this signal, and in 1933 concluded that it was strongest when

his antenna was pointed at the constellation Sagittarius. Optical telescope images

of the same region of the sky confirmed that the source of the radiation was at the

center of the Milky Way Galaxy [2]. This result was well received by astronomers,

but did little to answer the question that Jansky was tasked by Bell to answer. As a

result, his project was discontinued in 1934. While he made efforts to pursue further

research on his own time, illness and financial problems prevented him from making

significant progress [4].

1.1.2 Reber Sky Survey

It was not until 1937 that an amateur radio astronomer, Grote Reber, made efforts to

continue Jansky’s work. In the midst of the Great Depression, Reber applied to work

at Bell Labs alongside Jansky, but was turned away. In response, he constructed his

Page 14: Radio Telescope

4 Chapter 1. Introduction

own radio antenna in his backyard - a parabolic dish 9 meters in diameter, not unlike

the antennas used for modern radio telescopes today. An image of the telescope is

depicted in Fig 1.2. Reber mounted his dish on a tilting stand so he could steer it along

the altitudinal direction. He developed several receivers for the telescope operating

at a variety of frequencies. The one to produce the most useful data operated at 160

MHz [5]. Reber used this telescope to perform a survey of the 160 MHz radio sky

in 1941, earning him publications in several astrophysical journals [5]. The survey

raised questions about the nature of the universe, specifically drawing attention to the

abundance of low frequency radiation emitted in space [6]. Reber reached celebrity

status in the astrophysics community for his work, leading major observatories to

adopt their own radio astronomy research programs.

1.2 Important Discoveries By Radio Astronomers

1.2.1 Non-Thermal Radiation

The abundance of low frequency radiation in space demonstrated by Reber’s survey

came as a surprise. Scientists at the time believed that radiation in space occurred

only as blackbody radiation. They suspected that given the high temperature of stars,

most radiation in space would be observed at frequencies around the visible spectrum

where their blackbody radiation is most intense [3]. The Reber sky survey was the

first empirical result to suggest that this was not the case. Blackbody radiation

does not account for the intensities of radio frequency radiation Reber observed [7].

Some other previously unknown mechanism was responsible. The study of these

unexplained emission modes led to the observation of synchrotron radiation, thermal

bremsstrahlung, atomic and molecular spectral emission, and other modern physical


Page 15: Radio Telescope

1.2. Important Discoveries By Radio Astronomers 5

Figure 1.2 A photograph of the parabolic radio antenna used by Grote Reberto conduct a survey of the radio sky in 1941. The telescope was 9 metersin diameter and featured an altitudinal steering mechanism. Reber’s resultsdrove scientists to pursue further radio astronomy research throughout themid-20th century. (Source: National Science Foundation (public domain))

Page 16: Radio Telescope

6 Chapter 1. Introduction

1.2.2 Cosmic Microwave Background

Most radiation at radio frequencies is non-thermal in nature. RF blackbody radiation

is present in the universe but has intensity peaks at cold temperatures (less than 50

Kelvin) [7]. As a result, blackbody radiation emitted by hot objects like stars is

negligibly weak at RF - their spectrum is dominated by other mechanisms. However,

theoretical models predicted the temperature of the interstellar medium (ISM) to be

sufficiently cold for blackbody peaks to occur at a low frequency. The temperature

of the ISM could be determined by measuring this peak. In 1955, French astronomer

Emile Le Roux performed a sky survey at a frequency of 900 MHz and discovered

that the ISM emits a nearly isotropic, persistent background radiation peaking at a

temperature of 3K ± 2K [8]. Ten years later, American astronomers Penzias and

Wilson confirmed this measurement to a greater degree of precision, and earned the

Nobel Prize in Physics in 1978 for their efforts [9]. The significance of the thermal

radiation in empty space, referred to as the cosmic microwave background radiation

(CMBR), is in its implications to cosmology. Its existence supports the Big Bang

theory which has become the standard model for the first moments of the universe.

The model states that while the universe was sufficiently hot, it consisted of free

electrons and nuclei which continuously absorbed and re-emitted photons. As the

universe expanded, the temperature fell to a point where electrons and protons could

bind to form atoms. These atoms did not absorb all photons in the way that free

particles did, so this transition left behind a collection of photons which had been

emitted by particles but not re-absorbed [8]. The energy observed in the CMBR is a

direct observation of these photons which continue to propogate through space. An

image of the CMBR is depicted in Fig 1.3.

Page 17: Radio Telescope

1.3. Modern Uses For Radio Telescopes 7

Figure 1.3 An image of the cosmic microwave background radiation pro-duced by the Wilkinson Microwave Anisotropy Probe in 2010. The cosmicmicrowave background provides strong evidence for the Big Bang theory. Ofcurrent interest to cosmologists is the anisotropic distribution of the radia-tion, which is clearly seen here. (Source: [10])

1.2.3 Galactic Supermassive Black Hole

The radio source at the center of the Milky Way first detected by Jansky is known as

Sagittarius A*. Radio and X-ray observations of this region suggest that the source

has an angular diameter of 37 microarcseconds and is positioned near a supermassive

black hole. Gravitational lensing models confirm that the radio source is not the center

of the black hole itself, but the blackbody radiation produced by matter heating to

extreme temperatures as it accelerates near the event horizon [11]. The discovery

that our galaxy has a black hole at its center was made possible by radio astronomy.

1.3 Modern Uses For Radio Telescopes

A variety of research questions are being addressed by radio astronomers today. Cos-

mologists are interested in the ansiotropy of the cosmic microwave background due

Page 18: Radio Telescope

8 Chapter 1. Introduction

to its implications to the general theory of relativity and the early history of the

universe. For example, the observed anisotropy of the CMBR was used to determine

that the Milky Way galaxy is traveling at 22 km/s with respect to the rest frame of

the ISM [8]. More applied uses for radio astronomy include the observation of sunspot

population at the Solar and Heliospheric Observatory, as well as the study of Earth’s

ionosphere, currently being conducted by the National Astronomy and Ionosphere

Center at Arecibo Observatory. Astronomers are also using radio to observe distant

objects such as quasars, pulsars, and supernova remnants by interferometry at obser-

vatories like the Very Large Array, which consists of 27 independent antennas [12].

Page 19: Radio Telescope

Chapter 2


2.1 The Electromagnetic Spectrum

Radio frequency radiation is a specific type of electromagnetic (EM) wave. EM waves

have a frequency proportional to the total energy that they carry. This is described

by the Planck-Einstein equation [7]

E = hν (2.1)

where E is the total energy in joules, h is Planck’s constant, and ν is the frequency in

Hertz. The full range of possible frequencies makes up the electromagnetic spectrum.

This spectrum is divided into categories as seen in Fig 2.1. The radio spectrum is

divided further into several bands. The designations developed by the Institute of

Electrical and Electronics Engineers (IEEE) are depicted in Table 2.1.


Page 20: Radio Telescope

10 Chapter 2. Theory

Frequency (Hz)

















































Wavelength (m)

Figure 2.1 Diagram of the electromagnetic spectrum, with frequency in-creasing from left to right. Radio and microwave frequency radiation makeup the lowest frequencies, and as a result of Eq 2.1, the lowest energies.

Table 2.1 IEEE frequency band designations for the radio spectrum.

Band Frequency Range Nomenclature

HF 3 to 30 MHz High Frequency

VHF 30 to 300 MHz Very High Frequency

UHF 300 to 1000 MHz Ultra High Frequency

L 1 to 2 GHz Long Wave

S 2 to 4 GHz Short Wave

C 4 to 8 GHz Compromise between S, X

X 8 to 12 GHz Crosshair (WWII Fire Control)

Ku 12 to 18 GHz Under Kurz

K 18 to 27 GHz Kurz

Ka 27 to 40 GHz Above Kurz

V 40 to 75 GHz -

W 75 to 110 GHz -

mm 110 to 300 GHz mm Wavelengths

Page 21: Radio Telescope

2.2. Quantifying Radio Observations 11

2.2 Quantifying Radio Observations

2.2.1 Flux Density of a Radio Source

A radiation source in the sky has some apparent size described by its solid angle Ω.

The solid angle is a two dimensional angle that resolves an area segment of a sphere.

This is analogous to a one-dimensional planar angle which resolves the arc length of

a segment of a circle. The solid angle in spherical coordinates is

dΩ = sin θdθdϕ (2.2)

The unit of the solid angle is the steradian (sr.) Integration over all solid angles gives

the surface area of the sphere of radius r, as seen in Fig 2.2. Radiating bodies can be

described according to their emitted power per unit area, or flux. The basic unit of

radiated energy, the monochromatic intensity I(ν), is defined as the flux per unit solid

angle at a specific frequency. It has units of J m−2s−1Hz−1sr−1. Radio astronomers

most commonly characterize sources in terms of their monochromatic flux density,

which is the monochromatic intensity integrated over the solid angle.

S =∫

I(ν)dΩ (2.3)

The flux density S has units of Wm−2Hz−1, but in radio astronomy the conventional

unit of flux density is the Jansky [13], defined as 10−26Wm−2Hz−1.

Measuring the flux through a solid angle of the sky is the fundamental task of a

radio telescope. The resolution of the telescope is described by the size of this solid

angle, called the beamwidth. A high resolution corresponds to a small beamwidth.

Astronomers use calibration techniques to determine the beamwidth of a telescope so

that the flux density of a source can be calculated from a measurement of flux.

Page 22: Radio Telescope

12 Chapter 2. Theory



yFigure 2.2 The solid angle is a two-dimensional angle which describes thesurface area of a section of a sphere. Integration over all solid angles givesthe area of a sphere of radius r. The size of this solid angle is also referredto as the beamwidth of the telescope. Telescopes with a solid angle of Ω canresolve features of celestial bodies that appear larger than Ω in the sky.

Page 23: Radio Telescope

2.2. Quantifying Radio Observations 13

2.2.2 Blackbody Temperature

Prior to the mid 20th century, astrophysicists believed that all radiated energy from

celestial bodies was due to blackbody radiation. This is a specific mechanism of

emission in which the monochromatic intensity is dependent on the temperature of

the body being studied. The intensity emitted at a given frequency ν by an ideal

blackbody is expressed mathematically by Planck’s Law, given as [7]

Bν(T ) =2hν3


exp( hνkT

− 1)(2.4)

This quantity is known as the monochromatic intensity because it refers to the inten-

sity at one frequency. A plot of the energy emitted by a blackbody as a function of

frequency for a variety of temperatures is shown in Fig 2.3.

Since RF radiation is of low frequencies, hν << kT except in the case that T is

very small. For sufficiently large values of T, the exponential term in Eq 2.4 reduces




kT− 1

)≈ hν


The regime where hν << kT is known as the Rayleigh-Jeans regime because Planck’s

Law reduces to the Rayleigh-Jeans Law [7]

Bν(T ) =2ν2kT


Monochromatic intensity increases linearly with temperature within this regime. When

hν ∼ kT , the Rayleigh-Jeans Law breaks down and Planck’s Law must be used for

accurate calculations. At Ku band frequencies, this breakdown occurs near a tem-

perature of 1 K. This means that for blackbodies of a temperature above 1 K the

Rayleigh-Jeans Law is sufficient for predicting their radiation spectrum at Ku Band.

The temperature of a blackbody is calculated by solving Eq 2.6 for T , which yields

T =Bν(T )c2


Page 24: Radio Telescope

14 Chapter 2. Theory

Figure 2.3 Log-log plot of the energy distribution produced by blackbodyradiation. Notice that the intensity goes as the square of the frequency withinthe Rayleigh-Jeans Limit, which breaks down at frequencies far outside theradio spectrum. The point of maximum intensity is a function of the black-body temperature, given by Wien’s Displacement Law, which increases infrequency as the temperature increases. Source: [14]

Page 25: Radio Telescope

2.3. Radiation Mechanisms 15

This equation will not necessarily hold to great degrees of precision for real objects

since it is derived only for ideal blackbodies. However, it describes the temperature

an ideal blackbody would have if its monochromatic intensity is measured as Bν . This

temperature is known as the brightness temperature and is widely used to discuss the

radio emission of real sources [13].

2.3 Radiation Mechanisms

Comparing empirical observations of intensity at RF to the intensities predicted by

the Rayleigh-Jeans Law reveals some discrepencies. For example, if the Sun were con-

sidered a perfect blackbody, the Rayleigh-Jeans law implies that it has a temperature

of 100,000 K at a frequency of 1.4 GHz. Measurements of the Sun at visible frequen-

cies give a temperature of approximately 6000 K [15]. This discrepency implies that

blackbody radiation is not the only source of radiation emitted by the Sun. Most of

these other sources of radiation were not well understood until the mid 20th century.

2.3.1 Bremsstrahlung

Radiation from stars such as the sun and insterstellar ionic clouds such as those found

in the Milky Way is often a result of charged particles such as electrons experienc-

ing acceleration due to the presence of ions. This interaction is known as thermal

bremsstrahlung or free-free radiation. In the most common case it consists of elec-

trons interacting with hydrogen ions in a plasma. A region of space occupied by a

high density of hydrogen ions is called an HII region [3].

Consider a single electron interacting with a single H+ ion. The particles feel a

force governed by Coulomb’s Law. In collisions, strong forces are generated which

typically produce radiation in the X-ray frequencies. The energy of X-ray photons

Page 26: Radio Telescope

16 Chapter 2. Theory





Figure 2.4 Diagram of a bremsstrahlung interaction. Electrons acceleratingdue to the presence of ions emit radiation according to Larmor’s formula.These interactions are characterized by the impact parameter b which de-scribes the minimum distance between the charges throughout the interac-tion.

is much higher than that of radio photons. For this reason we are only interested in

weak interactions, i.e., an electron passing by an H+ ion. The forces on the particle

during an interaction are described in terms of the distance between the particle and

the ion, l, and the impact parameter, b. The impact parameter is defined as the

minimum value of l throughout the interaction. When b = l, the particle feels the

strongest force, as shown in Fig 2.4. The force felt by the particle can be separated

into components parallel and perpendicular to its velocity

F∥ =1



l2sinϕ (2.8)

F⊥ =1



l2cosϕ (2.9)

In either case the acceleration causes the particles to emit radiation according to

Larmor’s formula [16]

P =µ0q



Page 27: Radio Telescope

2.3. Radiation Mechanisms 17

where P is the radiated power, q is the charge, and v is the acceleration of the particle.

A complete investigation of bremsstrahlung should consider the radiation from both

forces, however it is known that the parallel component produces radiation outside of

the radio range, so for the purpose of this discussion we study only the perpendicular

component [14]. Bremmstrahlung from one interaction is better described as an

event rather than a sustained signal. A single pulse of radiation is produced by the

electron, which we approximate as a gaussian distribution [14]. Fourier analysis of

this distribution suggests that the maximum frequency of radiation resulting from

the interaction is [14]

ω ≈ v


which lies well into the infrared range in most cases.

Now consider a region of space populated by a hydrogen ion plasma. The average

velocity of the electrons depends on their average energy, or the temperature of the

plasma. The predicted range of values for the impact parameter b depends on the

density of the plasma. Assuming the region is in thermal equilibrium, the electrons

and ions have equal average kinetic energy, but since the ions are many orders of

magnitude more massive, we assume they remain stationary. The monochromatic

intensity of radiation can be written in terms of the emission coefficient, which is

defined as

ϵν =δIνδr


or the total flux per unit volume per unit solid angle at a frequency ν. It has units

of J m−3s−1Hz−1sr−1. From this we write

Ib = 4πϵν (2.13)

where ϵν is the emission coefficient and the factor of 4π is a result of integration over


Page 28: Radio Telescope

18 Chapter 2. Theory

For a specific range of impact parameters bmin → bmax, and electron and ion

densities given by Ne and Ni, this emission coefficient is given by [14]

ϵν =π2q6NeNi









Detailed derivations of this coefficient can be found in advanced thermodynamics

texts such as ref [17].

Ultimately we are interested in the spectral index for bremsstrahlung distributions.

The spectral index is a unitless quantity that describes the frequency dependence of

the total flux emitted by a source. It is the slope of the intensity plotted as a function

of frequency in a log-log plot. (Unfortunately there is no accepted sign convention

for the spectral index so a positive index could imply a positive or negative slope

depending on the context. Throughout this report a positive spectral index is taken

to imply a positive slope.) We assume that the particles which make up the source

have a power law energy distribution [18]

N(E)dE ∝ E−δdE (2.15)

where N is the number of particles per unit volume with energies between E and

E + dE, and δ is a positive constant.

To calculate the spectral index we will need the maximum and minimum values

of the impact parameter that give rise to radio frequency radiation. The impulse on

the passing electron due to the field of the ion is

∆P =∫

Fdt =1


∫qEdt (2.16)

Again we are only interested in the perpendicular component of force, so we take the

perpendicular component of the electric field E⊥ = E cos θ. Recalling Eq 2.9, we may


∆P =1

4πϵ0q2∫ ∞


cos3 ϕ

b2dt (2.17)

Page 29: Radio Telescope

2.3. Radiation Mechanisms 19

By applying a change of variables dt = b/v sec2 ϕδϕ, this becomes

∆P =q2


∫ π/2

−π/2cosϕdϕ =



Since the momentum transfer is inversely proportional to the impact parameter, we

can solve for the minimum value of b in the case where ∆P is as large as possible.

This occurs for a perfectly elastic interaction where the change in momentum is twice

that of the initial momentum of the electron, or

∆Pmax = 2mev (2.19)

From this we see that the minimum value of the impact parameter is [14]

bmin =q2


and recalling Eq 2.11,

bmax =v


From these results we can calculate how much of the radiated energy will be re-

absorbed by the plasma (so not transmitted to Earth), described by the absorption

coefficient. The absorption coefficient κν is given by Kirchoff’s Law [14]

κν =ϵν

Bν(T )(2.22)

Substituting for ϵν and applying the Rayleigh-Jeans Law for Bν(T ), we have

κν =1



T 3/2









or simply

κν ∝ 1


within the Rayleigh-Jeans regime. With this value of the absorption coefficient, we see

that the flux density goes precisely like the flux of a blackbody, with a spectral index

Page 30: Radio Telescope

20 Chapter 2. Theory

of α = 2. This is why astronomers call this interaction “thermal bremsstrahlung.” As

frequency increases beyond 10 GHz, the flux density decreases slowly with frequency

and has a spectral index of α = −0.1 in HII regions [14].

2.3.2 Synchrotron Radiation

Synchrotron radiation makes up most of the radiation in space which cannot be ac-

counted for as blackbody radiation or bremsstrahlung [13]. It is produced by electrons

in strong magnetic fields moving at relativistic speed. Scientists first observed it in

the laboratory as an unwanted source of energy loss in a circular electron accelera-

tor [3]. The trajectory of an electron in a uniform, static magnetic field forms a helix.

Particles undergoing circular motion are constantly accelerated in the direction per-

pendicular to the motion. This is illustrated in Fig 2.5. The angular velocity of this

circular motion ωC is called the angular cyclotron frequency. The angle between the

direction of velocity and the magnetic field line is the pitch angle θ. Cyclotron fre-

quency is a function of the field strength B, the charge q, the mass m0. Accounting

for relativistic speed, it is given as [14]

ωC =qB


, νC =qB



where γ = 1/√

1 − (v/c)2 is the Lorentz factor in special relativity. The synchrotron

radiation of one particle is sharply peaked at a specific frequency. For this reason our

model assumes that all synchrotron radiation produced by this particle will occur at

one frequency called critical frequency ν, which for ultrarelativistic particles is given

as [14]

ν = γ2νC (2.26)

Radiation from a particle moving in this manner will be observed as a series of pulses

occurring at a regular interval. This is because synchrotron radiation will only be

Page 31: Radio Telescope

2.3. Radiation Mechanisms 21





Charged particle

(proton or electron)

Magnetic Field

Figure 2.5 An illustration of synchrotron radiation. Charged particles movein a spiral pattern when subject to a strong magnetic field. The resultingacceleration causes them to emit radiation at the cyclotron frequency.

Page 32: Radio Telescope

22 Chapter 2. Theory

detected when the direction of the particle’s velocity is in the observer’s line of sight.

For the same reasons it is also sharply polarized.

We are ultimately interested in the power radiated by a celestial body due to

synchrotron radiation. Relativistic electrodynamics [16] describes the energy lost by

a single relativistic particle moving in a spiral trajectory as

P =q4B2v2γ2


sin2 θ (2.27)

for a specific pitch angle θ. This can be rewritten in terms of the magnetic energy

density [16]

UB =B2


and in terms of the Thomson cross-section, a constant defined as

σT =




so we are left with

P = 2σTβ2γ2cUB sin2 θ (2.30)

where β = v/c. To describe an ensemble of particles of synchrotron radiation we

are interested in the value of their spectral index. To calculate this we will need the

emission coefficient. We can write

ϵνdν ≈ −dE

dtN(E)dE (2.31)

since the power emitted is the time derivative of the emitted energy. In this case the

energy E is the energy of the individual relativistic electrons, given by E ≈ γmec2.

Substituting these expressions and applying some clever algebra we arrive at

ϵν ∝ B(δ+1)/2ν(δ−1)/2 (2.32)

From this the spectral index can be extracted

P (ν) ∝ ν(−δ+1)/2 (2.33)

Page 33: Radio Telescope

2.4. Cosmic Sources of Radio 23

The exponent on the frequency ν is the spectral index. The value of δ for RF syn-

chrotron emission between 1 GHz and 100 GHz in our galaxy has been empirically

measured as 2.4 [19]. In this frequency regime, the spectral index has a value of −0.7,

as shown in Fig 2.6. This result shows that the spectrum of a synchrotron source

follows a different distribution than the blackbody curve, which has a spectral index

of 2 since in the Rayleigh-Jeans regime the energy is proportional to ν2. Furthermore,

the flux from a synchrotron source is a relatively strong function of frequency, so we

can expect the strongest levels of synchrotron radiation to be emitted at the lower

frequencies of the power law regime - between 1 and 10 GHz.

2.4 Cosmic Sources of Radio

The strongest sources of RF radiation as observed from Earth are the Sun and the

Milky Way. Both of these sources produce a complicated mixture of radiation mech-

anisms for which there are a variety of models. Their emission can be measured

even with the most insensitive radiometers, so they are suitable targets for the first

measurements taken with a new telescope. In addition to the Sun and Milky Way,

the spectral line of neutral hydrogen molecules is easily accessible. This is due to the

abundance of hydrogen throughout the universe.

2.4.1 The Sun

Solar RF radiation is classified into four components: “quiet Sun” emission due to

plasma bremsstrahlung, a slowly-varying component of synchrotron radiation and

bremsstrahlung due to the magnetic fields trapped by sunspots, synchrotron radia-

tion from electrons in short-lived noise storms, and wideband synchrotron emission

from short radio bursts associated with solar flares [15]. The range and relative inten-

Page 34: Radio Telescope

24 Chapter 2. Theory

Figure 2.6 Summary of observational synchrotron radiation data demon-strating the negative spectral index throughout most of the radio spectrum.This shows that synchrotron radiation is fundamentally different than black-body radiation and follows a different distribution of energy. Source: [20]

Page 35: Radio Telescope

2.4. Cosmic Sources of Radio 25

sities of these signals are summarized in Figure 2.7. At Ku Band, the slowly-varying

component which depends on sunspot population is accessible. Monitoring this com-

ponent is a technique used by NASA to determine the population of sunspots. They

take flux measurements between 3 and 14 GHz daily [15]. Sunspot activity is cur-

rently of particular interest because recently the sunspot count has been lower than

models predicted [21]. Monitoring changes in sunspot activity is an ideal task for

a small radio telescope operating in the GHz bands. It is also well-suited for the

detection of radio bursts, which are easy to identify because their intensity is many

orders of magnitude higher than that of the quiet Sun.

Like many other celestial bodies, the Sun appears larger at lower radio frequencies

than it does in the visible spectrum. This means that even for a low-resolution radio

telescope, the beamwidth is smaller than the size of the source, and it is possible to

measure features rather than just continuum emission. For example, even a small

radio telescope has the capability to determine if the upper hemisphere of the Sun

emits stronger than the lower hemisphere.

2.4.2 Neutral Hydrogen

In 1944, H. C. van de Hulst theorized the existence of a 1420 MHz spectral line

which was experimentally confirmed from observations of the Milky Way by Ewan

and Purcell in 1951 [20]. Quantum mechanics predicts energy emission by atoms

in the radio frequencies by hyperfine splitting. This splitting arises from magnetic

dipole interactions between the nucleus and the electron. The hyperfine splitting

of hydrogen atoms in the ground state produces this 1420 MHz radiation. Given a

proton spin of s = 12

and an electronic angular momentum quantum number l = 12

it can be shown that the energy difference between the hyperfine energy levels in

Page 36: Radio Telescope

26 Chapter 2. Theory

hydrogen is given in atomic units by

δE =4






gpα2 (2.34)

where µ is the reduced mass of the electron, gp = 5.5883 is the Lande factor of the

proton, and α is the fine structure constant. Complete derivationis of this energy

can be found in quantum mechanics texts such as ref [22]. Using this relation the

frequency of the emitted energy can be calculated as 1420 MHz, which corresponds

to a wavelength of 21 cm. For a hydrogen atom this transition occurs with an average

frequency of 2.9 × 10−15 Hz. This may seem a rare occurrence, however due to the

abundance of hydrogen in the universe it is relatively common. The center of the

Milky Way as well as HII regions and the Sun are rich in neutral hydrogen.

2.4.3 The Milky Way

The view of the Milky Way galaxy from Earth is obscured by high amounts of dust

opaque at visible wavelengths but transparent at RF [13]. As a result, radio tele-

scopes can probe deeper areas of the galaxy than optical telescopes. Similar to the

Sun, radio emission from the Milky Way is divided into three components: ther-

mal bremsstrahlung and spectral emission from HII plasmas, synchrotron radiation

emitted by supernova remnants, and spectral emission from an abundance of neutral

hydrogen (1.4 GHz) and hydroxyl (1.7 GHz) [23]. Monitoring the radio emission of

the Milky Way is an effective way to measure the galactic rotation curve. A simple

model to predict the rotation of the galaxy is [24]

ω =



where M is the mass of the galaxy concentrated at the center, G is the gravitational

constant, and R is the distance from the central mass. From some arbitrary point on

Page 37: Radio Telescope

2.5. Radiometer Design 27

the disc of the galaxy, we will observe the maximum angular velocity when it is in

the direction of our line of sight. As a result, the observed frequency of radiation will

be redshifted. This is especially useful when looking at spectral emission, since the

precise frequency of radiation with no redshift is known and can be compared to the

measured frequencies. When monitoring the neutral hydrogen emission of the galaxy

over time, it has been shown that the frequency of highest flux density varies by ± 5

MHz [24]. Using these data we may determine how the galaxy rotates with respect

to the Solar System.

2.5 Radiometer Design

Radio astronomers are fortunate enough to work with signals of a sufficiently low

frequency such that they can be manipulated using electronics rather than optics.

The radio telescope collects radiation not using lenses and mirrors, but with circuitry.

The radio receiver is the most important part of the radio telescope and represents

the most significant design challenge. The most efficient methods for dealing with

high frequency radio signals using electronics are well understood, so most telescope

radiometers are similar in their functions. However, the methods for implementing

these functions vary widely.

2.5.1 Superheterodyne Model

The top-level organization of most radio telescope receivers follows a common design

format. A block diagram of a typical superheterodyne receiver can be seen in Fig

2.8. The RF signal from the antenna is amplified using a low noise amplifier (LNA)

which in some systems is cryogenically cooled to reduce noise. This amplified signal

is converted, or modulated, to a lower intermediate frequency (IF) by mixing the

Page 38: Radio Telescope

28 Chapter 2. Theory

incoming RF signal with an artificial waveform supplied by a local oscillator. This

IF signal is filtered using a narrowband band-pass filter. A square law diode rectifier

converts the filtered IF signal to a voltage proportional to its power. The output

voltage can be sent to a waveform analyzer or converted to a digital signal using a

voltage-to-frequency converter and finally processed by a computer for data collection.

2.5.2 Frequency Mixing

Radio frequency signals are cumbersome to process so the superheterodyne receiver

employs a mixer to convert the RF signal to a lower IF. This allows modules to be

designed for one specific IF despite the wide bandwidth of RF signals received by the

antenna, which is desirable because signals incident on radio antennas are difficult

to amplify efficiently at their native frequencies [25]. Signal processing, especially

using analog methods, is similarly diffcult. Converting the input to an IF simplifies

transmission as well, since high quality cable is needed to transmit RF signals with-

out significant power loss, especially over long distances. Frequency downconversion

solves all of these issues but not without introducing some design challenges.

A frequency mixer, or multiplier, outputs a superposition of signals consisting of

the sum and difference of the two input frequencies, each with a phase shift of 90

degrees relative to the input. In a radio telescope receiver, one of these signals is the

RF input signal (i.e. the raw signal measured by the telescope antenna) and the other

signal is an artificial waveform produced by a local oscillator. Local oscillator circuits

can be constructed using crystals, voltage-controlled oscillators (VCOs), or digital-

analog converters (DACs.) VCOs and DACs provide the advantage of tunability -

astronomers who wish to target a specific input frequency to examine, such as a

spectral emission line, would require the ability to tune their local oscillator with a

reasonable degree of precision. Using a DAC generally requires a microcontroller.

Page 39: Radio Telescope

2.5. Radiometer Design 29

The behavior of a frequency multiplier is clearly illustrated by the trigonometric

identity [25]

sinA sinB =1

2[cos(A−B) − cos(A + B)] (2.36)

Only the lower, subtracted frequency component is desired - the summed component

is called the image. The image is a potential source of high frequency noise for the

IF section of the receiver. Isolating the RF image and the LO from the IF signal is

an important aspect of receiver design. Filters usually immediately follow frequency

mixers to eliminate the image signal.

2.5.3 Square Law Detection

A square law detector is in essence a half-wave diode rectifier with a low pass filter.

A simple half wave rectifier consists of a diode with a resistor to ground in parallel

with its input. For large signals, a capacitor placed in parallel with the output will

hold the output voltage near the peak voltage of the AC input. For smaller input

signals, the behavior is more complicated.

The current-voltage curve of a diode is described by the Schockley Diode Equation


I = IS(expVD


− 1) (2.37)

where IS is the reverse bias saturation current, VD is the voltage across the diode,

n is a unitless coefficient called the quality factor, and VT = kT/q is the thermal

voltage. Generally the behavior of diode detectors is separated into three regimes

governed by this current-voltage curve. For high VD (nominally +20 dBm and above)

the output voltage of a detector goes linearly with the input voltage, as described. At

less than +20 dBm, we approach the regime where VD ≈ VT and the proportionality

of the output to the input is hard to predict. Below a nominal -20 dBm, we enter

Page 40: Radio Telescope

30 Chapter 2. Theory

the square law region. Small-signal analysis shows the behavior of the diode in this

regime. Consider a diode connected to a simple RC low-pass filter. Since the I-V

relationship of the diode is inherently exponential, we can expand it in a power series

of vD.

iD = c1vD + c2v2D + c3v

3D... (2.38)

Suppose that the small-signal input voltage is sinusoidal so vD = A cos(ωt). Then

applying the appropriate trigonometric identities, we have

iD = c1A cos(ωt) +c2A

2(1 + cos(2ωt))


c3A3(cos(3ωt) + 3 cos(ωt))

4... (2.39)

which simplifies to a constant proportional to A2 plus terms containing harmonics of

cos(ωt). The low-pass filter connected to the output of the diode will filter out these

sinusoidal terms, so they become negligible. In the context of small-signal analysis,

vout = iDRL, so the output of the square law detector is

V0 = αRLV2i (2.40)

where α is some constant. To express it in a way that is more applicable to radio


V0 ∝ Pi (2.41)

where Pi is the input power. Notice that this only describes how the output is

proportional to the input and so we cannot calculate the absolute input power from

the output. These calculations are made possible by telescope calibration.

2.6 Radiometer Calibration

The behavior of radiometer output is more commonly discussed as a temperature

rather than a voltage or power level. The so-called noise temperature is defined by

P = kBTsBn (2.42)

Page 41: Radio Telescope

2.6. Radiometer Calibration 31

where P is the power delivered by a noise source, kB is Boltzmann’s constant, Ts is the

noise temperature and Bn is the bandwidth of the signal in Hz [3]. The noise figure, a

common specification on RF electronic components, describes how the intrinsic noise

of the component degrades the quality of the signal being passed through it. It is

defined as the ratio of the signal-to-noise ratios in decibels, or [3]

F = 10 log(SNRin



The first step toward calibrating a radiometer is to determine its intrinsic noise tem-

perature which describes its noise output when no input is present. It would be

convenient if we could point the telescope at something which produces absolutely no

radio frequency radiation, however such things do not exist, so calibration becomes

a more complicated process. To do this we compare the behavior when the telescope

is pointed at sources of known flux density. Most often the Sun is used as a refer-

ence and is compared to the observed temperature of the ”cold sky”, or some region

of space that does not have a strong radio emitter. The noise temperature of the

antenna can be written as

Tsys =Ssrc

2KB(Ssrc/Ssky − 1)− Tsky (2.44)

where Ssrc is the known flux density of a measured source and Tsky is the measured

temperature of a cold point in the sky. For example, if the Sun were to be used as a

calibration source, we would first calculate the ratio of the observed flux density of

the sun and the observed flux density of some quiet point in the sky.

Page 42: Radio Telescope

32 Chapter 2. Theory

Figure 2.7 A plot of the radio frequency radiation from the Sun as a functionof wavelength. Frequency is given along the top in MHz. At Ku Band (10 to12 GHz) we can measure quiet Sun bremmstrahlung as well as synchrotronradiation arising from sunspots and intense radio bursts. Source: [15]

Page 43: Radio Telescope

2.6. Radiometer Calibration 33

Low Noise












Radio Signal IF Signal DC Voltage

Figure 2.8 A functional block diagram of a typical superheterodyne receiver.A low noise amplifier amplifies the input signal before it is converted down toan intermediate frequency and filtered using band-pass filters. The filteredoutput is passed through a diode detector which converts the AC signal to aDC power level.

Page 44: Radio Telescope

34 Chapter 2. Theory

Page 45: Radio Telescope

Chapter 3

Design and Implementation

The radio telescope described in this paper was designed to be flexible and inexpensive

without sacrificing too much radiometer sensitivity. Less than $700 was spent on the

project, almost all of which was spent on developing the receiver. The telescope can

only operate in drift mode at this time due to lack of automated motor control.

3.1 Antenna

The telescope antenna is a 3m parabolic dish previously used to receive a satellite

internet connection, provided free of charge by a donor in Spencer, NY. The telescope

is planted outside the Clinton Ford Observatory at Ithaca College. We plan to mount

it such that it points toward the equator and both the Sun and the center of the Milky

Way will pass through the beamwidth. This is sufficient for simple flux measurements

of these accessible sources.


Page 46: Radio Telescope

36 Chapter 3. Design and Implementation

3.1.1 Resolution

The resolution of an ideal parabolic antenna can be modeled by [3]

R = 1.22λ

D, (3.1)

where D is the diameter of the dish and λ is the wavelength of radiation being ob-

served. A smaller value of R corresponds to a more narrow beamwidth. Objects with

smaller angular size in the sky can be resolved by telescopes with better resolutions.

At Ku band (λ ≈ 3 cm), then, the resolution of our 3 meter parabolic antenna is

about 0.01 rad or about 35 arcminutes. This is comparable to the apparent size of

the Sun.

3.1.2 Gain

The gain of a parabolic antenna is modeled by

G =π2d2

λ2eA (3.2)

where G is the d is the diameter of the dish, λ is the wavelength observed, and eA is

a term which describes the efficiency of the dish. Typically, parabolic antennas range

in efficiency values from 0.5 to 0.75 [3]. Assuming that our dish is perfectly efficient,

we expect a gain of about 120 dB, but realistically the gain probably lies somewhere

near 80 dB. Calibration will determine a specific value.

3.2 Superheterodyne Receiver

The receiver developed for this project is an adaptation of the analog version of

Haystack Observatory’s Small Radio Telescope (SRT). A newer, digital version of the

SRT is also commercially available, and complete schematics for both versions as well

Page 47: Radio Telescope

3.2. Superheterodyne Receiver 37

as some limited documentation are freely downloadable on the MIT website [24].

These schematics served as the basis for our circuit designs. The IF filters and

amplifiers used here remain largely unchanged, with only a few component values

changed to improve efficiency. The local oscillator and power supply circuits have been

redesigned to suit the desired functionality, and some extra amplifiers for enhanced

drive capability have been added. Since the SRT is designed to receive L Band signals

and we have engineered our telescope for Ku Band, we have purchased a Ku Band

Low Noise Block (LNB) which downconverts a Ku Band input signal to L Band.

The complete receiver receives a Ku Band input which is downconverted to L

Band at the feed horn, transmitted to an RF Receiver board also placed at the horn

which mixes the L band signal down to audio frequencies, and finally transmitted via

coaxial cable to a second circuit board which holds additional filter-amplifiers and a

square law detector. The output of this second board goes to a computer for data

collection. For complete schematics of the final design, images of the PCB layout,

and photographs of the circuit boards, see Figs 3.3 and 3.2. A block diagram of the

complete system can be seen in Fig 3.1.

3.2.1 Ku Band Low Noise Block

Lumped-element circuits, even with the smallest surface-mount components, are not

well-behaved at frequencies above 10 GHz. It is possible to construct such circuit

boards, however it requires the precise impedance control of microstrip board traces

to prevent signal loss due to reflections. This type of impedance control dramatically

increases the price of PCB fabrication to the point that it was not feasible for this

project. As a far more affordable alternative, we sought to purchase a commercial

block converter.

Low Noise Blocks (LNBs), as they are referred to in the satellite communications

Page 48: Radio Telescope

38 Chapter 3. Design and Implementation

Ku Band

L Band

L Band

100 MHz-DC 0°

100 MHz-D

C 90°


Audio Audio DC

Figure 3.1 Block diagram of the complete telescope. The Ku Band (11GHz) input signal undergoes two frequency modulations and several stagesof amplification and filtering before being rectified to a DC voltage by thesquare law detector. The output of the square law detector is connected toa computer for data collection.

Page 49: Radio Telescope

3.2. Superheterodyne Receiver 39

Figure 3.2 Schematic of the RF receiver circuit, which connects directly tothe output of the LNB. It houses the frequency mixers, local oscillator, andphase detecting filters.

Page 50: Radio Telescope

40 Chapter 3. Design and Implementation

Figure 3.3 Schematic of the IF detector circuit, which houses the 4-poleButterworth filter and the square law detector.

Page 51: Radio Telescope

3.2. Superheterodyne Receiver 41

industry, consist of a high gain amplifier, a frequency mixer for down-conversion, and

sometimes a phase-locked loop for stability control. They are designed to receive high

frequency signals from satellites at the antenna, usually at C band, Ku band, or more

rarely X band. It amplifies these signals and converts them to an IF suitable for

transmission along common coaxial lines. Thus, they serve the exact same purpose

as the first stages of a superheterodyne receiver.

LNBs can be found for a wide range of prices. Aside from designated frequency

input, they are specified for gain, overall noise figure, phase noise, voltage stand-

ing wave ratio, and local oscillator stability. The most inexpensive LNBs are noisy

and have typical voltage standing-wave ratios (VSWR) of 3:1 or worse. VSWR is a

specification that describes how much power is typically lost due to the reflections

of electromagnetic waves propogating through the device. Perfectly designed devices

have uniform impedance throughout and produce no reflections, so the VSWR is 1:1.

Inexpensive LNBs also allow for drifting of the local oscillator frequency by 1 MHz

or higher. These devices are sufficient for a satellite TV receiver, which receives high

amplitude signals and does not pass them through much further amplification before

they reach their destination. However, for a radio telescope, too much intrinsic noise

at the antenna can render the entire system unusable. The smallest noise signal will

be amplified by each successive amplifier stage in the receiver and can become large

enough to obscure telescope measurements. As a result we chose to purchase a high

quality industrial-grade LNB with lower noise figures. The LNB was purchased from

Orbital Research (model LNB1075S-500P-WF65) for $280. It specifies a noise figure

of 0.6 dB with a gain of 65 dB.

Generally, LNBs do not have a separate connector port for a power supply. It is

expected that the power supply voltage will be multiplexed with its output signal as

shown in Fig 3.4. In typical communications applications this is accomplished with a

Page 52: Radio Telescope

42 Chapter 3. Design and Implementation




10 MHz Ref

DC In To Modem






DC Power

RF Signal

DC PowerRF Only


(SMA) (F-Female)



Figure 3.4 A multiplexer is the conventional method of supplying power toan LNB. A DC power signal is connected to the multiplexer which passespower to the LNB and receives the RF signal from the LNB through thesame port. The multiplexer passes this RF signal to the modem (the receivercircuit board) but blocks the DC power through this port. As a result thereceiver must be powered by other means.

multiplexer, called a bias tee, which is built in to commercial satellite modems. The

manufacturers of the LNB were kind enough to provide me with a bias tee, usually

priced at $100, free of charge. For inputs the bias tee takes the LNB power supply

and its output signal. It also has a port for a 10 MHz reference signal for external

local oscillator control, but our LNB does not have this feature. The bias tee allows

the power supply signal to flow into the input of the LNB but blocks it from flowing

through its output, which goes to the rest of the receiver.

3.2.2 L Band Frequency Mixer and Local Oscillator

As part of the telescope receiver, the LNB amplifies the raw Ku band signal by a

nominal 65 dB and downconverts it to L band. This L band signal is a suitable input

for the printed circuit boards we have developed. The local oscillator signal on this

board is generated by a voltage controlled oscillator (VCO), Minicircuits JTOS-1550.

This chip was chosen because the control pin can rest at its minimum voltage and

a suitable signal will still be produced. It has a frequency output of between 1150

and 1550 MHz depending on the control voltage, ranging from 0.5 V to 20 V. A

Page 53: Radio Telescope

3.2. Superheterodyne Receiver 43

potentiometer has been placed on the tuning pin so that the LO frequency can be

tuned if desired. The LO signal emitted by the VCO is amplified by a nominal 6

dBm using a monolithic amplifier before being passed to a passive 90 power splitter,

Minicircuits QCN-19D. This splitter outputs two signals, one in phase with the LO

and one 90 out of phase. The splitter provides no amplification so the outputs are

each at roughly half the power level of the input.

The L band signal transmitted from the output of the LNB is independently mixed

with these two signals using a a pair of Minicircuits RMS-11F+ frequency multipliers.

The purpose of splitting the RF signal in this way is so that they can be filtered using

phase detector circuits, which provide a high level of isolation from the unwanted

image signals.

3.2.3 Filter-Amplifier Stages

The purpose of the circuits following the RMS-11F+ frequency mixers are to greatly

amplify the signal and filter it to a pass-band with corner points at 10 KHz and 70

KHz. The outputs of the frequency mixers are first passed to identical non-inverting

amplifiers constructed using high speed op-amps, Analog Devices 797ARZ. These op-

amps produce a gain of 20 dB. Following both of these amplifiers are all-pass filters,

also called a phase detector. The all-pass filter, as its name suggests, passes signals

of all frequencies (within the limits of the op-amp) with unity gain, and changes only

the phase of the output. The phase shift produced by the all-pass filter can be written

in radians as

ϕ = π − 2 arctan(ωRC) (3.3)

The capacitance values in our circuit are 10 nF for the in-phase signal and 1500 pF for

the out of phase signal. The resistance values are the same for both - 1K. The outputs

of these all-pass filters are combined into one signal - when they are in phase, the

Page 54: Radio Telescope

44 Chapter 3. Design and Implementation

maximum amplitude will be produced. When they are out of phase, the total signal

will be greatly attenuated. From this, we can see how the phase detector filters out

unwanted frequencies. At 40 KHz, the center of the passband, the in-phase signal is

shifted by 138 while the quadrature-phase signal, already 90 out of phase, is shifted

by an additional 43 for a total phase shift of 133. The result is a superposition

of signals which are only 5 out of phase. As the input frequency moves away from

the center passband, the output signals becomes further out of phase until reaching a

90 phase shift at DC and total deconstructive interference (180 phase shift) at high

frequencies. Testing these circuits using artificial sine wave inputs produced good

results, yielding unity gain at all tested frequencies for the individual phase detectors

and a maximum amplitude mixed output near 40 KHz as seen in Fig 3.5.

The blocked frequencies were found to decrease by roughly 15 to 20 dB per decade

but attenuated to zero quickly.

Two more unity gain filter-amplifiers reside on the IF circuit board. Together

these make a 4-pole Butterworth filter. This filter is designed to improve the flatness

of the passband between 10 KHz and 70 KHz. A common way to describe a filter is

by its transfer function. The transfer function, denoted H(s), is defined as the ratio

of the laplace transform of the output voltage to the laplace transform of the input


H(s) =Y (s)




The laplace transform converts a function in the time domain to one in the frequency

domain, thus, the transfer function is a time-independent expression for the filter’s

complex gain as a function of frequency [26]. The transfer function of this topology

is given in general as [27]

H(s) =Z3Z4

Z1Z2 + Z4(Z1 + Z2) + Z3Z4


Page 55: Radio Telescope

3.2. Superheterodyne Receiver 45

20 30 40 50 60 70 80 900.15









Frequency (KHz)





Figure 3.5 Plot of the gain of the image rejection mixer built from two all-pass filters with different input reactances. The pass band is observed to benear 40 KHz as anticipated.

Page 56: Radio Telescope

46 Chapter 3. Design and Implementation

Figure 3.6 The band pass filter used in the audio section of our super-heterodyne receiver consists of two Sallen-Key filters in series, which havethe topology pictured above. (Source: Wikimedia Commons)

where the impedances Z are specified according to the diagram in Fig 3.6. Specifically,

this Butterworth filter is two Sallen-Key filters placed in series. The Sallen-Key

topology has a cutoff frequency at [27]

fC =1



so these filters have a frequency cutoff at roughly 25 KHz (Q300 in Fig 3.10) and

60 KHz (Q301). Gain is approximately equal to +53 dB within this pass band, and

decreases at a nominal 80dB per decade outside of the pass band. Bench tests with

this circuit confirm the passband suggested by the theory as shown in Fig 3.7.

3.2.4 Detector Circuit

The receiver has two outputs, one on the output of a square law diode rectifier and

another right before the input to the detector. The AC output can be used to trou-

bleshoot the detector, and is also useful if one wants to bypass the square law detector

Page 57: Radio Telescope

3.2. Superheterodyne Receiver 47

20 30 40 50 60 70










Frequency (Hz)


n (d


Figure 3.7 A plot of the measured gain of the 4-pole Butterworth filter onthe IF circuit board. The experimental gain is depicted in blue, and a plotconstructed from an LTSpice simulation is depicted in red. This filter consistsof two Sallen-Key filters placed in series, with the first (Q300) providing low-pass filtering and the second (Q301) providing high-pass filtering. Thesefilters are designed to provide a pass band between 25 KHz and 60 KHz.

Page 58: Radio Telescope

48 Chapter 3. Design and Implementation

entirely. Converting the AC signal to a DC voltage is done in hardware with the square

law detector but it is equally viable to do it using software.

The square law detector is designed using a germanium schottky diode. Schottky

diodes have a nominal forward voltage of 0.2 V [26]. A resistor network connected to

the power supply biases the anode and cathode of the schottky diode such that when

the input amplitude is zero, the diode rests near the edge of saturation. A 0.1 uF

capacitor rectifies the output signal into a DC voltage. Since this signal is weak, an

amplifier has been inserted on the output. This amplifier is a non-inverting op-amp

based amplifier with a nominal voltage gain of 50.

3.2.5 Printed Circuit Board Design

Our receiver is split into two circuit boards, one which houses the radio frequency

components and initial amplifiers, and another which completes the filtering and

houses the diode detector. This was done so that the RF circuits would be small

enough to fit in the feed horn of the antenna with the LNB. Transmitting the L band

signal from the LNB over any long distance could cause power loss due to reflections.

Additionally, the RF circuits are such that small-scale surface mount components

must be used to prevent radiative power loss. In the IF board the signals are on the

order of 50 KHz so this is not a concern.

The free CAD software suite ExpressPCB was used to draw the schematics and

PCBs for the receiver. The PCB drawings for the RF and IF boards can be seen in

Fig 3.9 and 3.10 respectively.

SMT soldering is done using a process called reflow soldering, where instead of

using a traditional soldering iron to heat connections, the board is placed in a reflow

oven and heated using a specified heat profile. Solder paste is used to secure compo-

nents to their pads before heating. When the paste reaches temperatures above 200

Page 59: Radio Telescope

3.2. Superheterodyne Receiver 49





50Preheat Soaking Reflow

60 120 180 240


Temp (C)

Time (s)Time (s)

Figure 3.8 A plot of the suggested heat profile for successful reflow solderingusing 64Sn/36Pb solder paste. This heat profile was used to solder SMTcomponents to our PCBs using a common commercial frying pan.

C the metals inside melt and the connection is made. The solder paste used here is

64% tin and 36% lead. It is specified as a free-flowing solder paste, which is more

diluted than a solder paste specified for restricted flow. Free-flowing solder paste is

hard to work with when making very small connections. For example, one would not

choose a free-flowing paste to solder a 64-pin microcontroller. For our purposes it

proved easy to work with. Lacking a true reflow oven, we opted to use a frying pan

instead. In an informal study by Sparkfun of inexpensive SMT soldering methods

this was shown to be the best. The heating profile used for soldering is depicted in

Fig 3.8.

Page 60: Radio Telescope

50 Chapter 3. Design and Implementation

Figure 3.9 PCB layout schematic of the RF receiver circuit which receivesthe signal from the LNB at the antenna. Our PCBs were designed usingExpressPCB software. Most components are surface-mounted and were sol-dered by reflow soldering. The green represents the ground plane on thebottom of the board while the red represents traces and plane connectionson the top of the board. The silkscreen is in yellow.

Page 61: Radio Telescope

3.2. Superheterodyne Receiver 51

Figure 3.10 PCB layout schematic of the IF detector circuit which is po-sitioned on the ground near the computer used for data collection. Mostcomponents here are through-hole and were soldered by hand. The colorscheme is the same as in the RF board.

Page 62: Radio Telescope

52 Chapter 3. Design and Implementation

Page 63: Radio Telescope

Chapter 4

Results and Testing

4.1 Data Collection

Collection of raw radio telescope data is done using Matlab software. The receiver

output signal is connected to a NIDAQ DaqCard 6024E interface. This interface is

connected to a laptop PC through PCMCIA which runs the Matlab script. It has a

maximum sample rate of 200 KHz and 12 bits of resolution. At the maximum voltage

input range of 10 V this corresponds to a resolution of 2.4 mV. The MATLAB script

monitors the voltage at the DaqCard port at the given sample rate and stores it in a

vector with a timestamp. This is more than sufficient for the expected outputs. The

code for this software can be found in Appendix A.

4.2 Receiver Testing

Rigorous bench testing of the complete telescope receiver requires a suitable test

input signal. The ideal input signal for testing is a Ku band noise source, however

any RF signal generator operating near Ku band would suffice. Commercial RF


Page 64: Radio Telescope

54 Chapter 4. Results and Testing

generators and noise sources are expensive and were not available for this project, so

an alternative method of testing was necessary. A 10.5 GHz microwave transmitter

and an RF frequency meter were obtained for this purpose. We have also obtained

a commercial frequency meter that is capable of measuring signals up to 3 GHz in

frequency. This allows us to characterize the signals flowing through the L band

portion of our circuitry.

The 10.5 GHz signal from the microwave horn lies outside the specified bandwidth

of the LNB, and so we expect the filters in the LNB to attenuate the signal. Also

note that the majority of the power from this signal will not pass through the filter-

amplifiers since its frequency is too high. For this reason, power measurements are

not especially useful, however we may still draw conclusions about the behavior of

the receiver by examining the output. The frequency meter reports a signal at the

output of the LNB centered near 850 MHz, which is in agreement with the expected

intermediate frequency specified in the LNB datasheet. We also observe this signal

at the input to the RMS-11F frequency mixers. The output of the mixers produces

a signal which appears to be centered near 200 MHz. Given that the local oscillator

operates at about 1100 MHz, 200 MHz is approximately the difference of the two

inputs, so this is a reasonable result. From this we conclude that the receiver is

correctly mixing the L band input and the local oscillator signals.

At the audio output of the receiver we observe the filtered signal via PCMCIA

interface as recorded by the Matlab software. Manually waving the microwave horn

across the beam of the LNB produces a noise-like signal that is much stronger than the

receiver’s background noise. Around the edges of the beam, the measured intensity

is decreased because only part of the microwave signal emitted from the horn is

absorbed. When the horn is placed directly perpendicular to the LNB, the maximum

amplitude is produced. We have chosen to check this signal at the audio output

Page 65: Radio Telescope

4.2. Receiver Testing 55

3 3.5 4 4.5−500











Time (s)





Figure 4.1 Demonstration of the receiver collecting Ku band radio wavesand converting them to an amplified audio frequency signal. This outputis the result of passing 15 mW of microwave radiation at 10.5 GHz directlyin front of the beam of the LNB and recording the output via PCMCIAinterface. At the maximum amplitude of the output we see a voltage of 420mV corresponding to a 50 Ω-matched power of 3.6 mW. Notice that despitethe large amounts of gain in the receiver, the output is lower in power thanthe original input signal - this is due to the fact that the input frequency liesoutside the pass band of the LNB.

Page 66: Radio Telescope

56 Chapter 4. Results and Testing

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2−8









Time (s)





Figure 4.2 With no power supplied to the LNB the intrinsic noise of thereceiver is observed to be on the order of 2 to 4 mV, corresponding to less than1 µW of RMS power. The resolution limitations of our computer interfaceis evident. While this does not reveal the noise temperature of the overallsystem since the noise of the LNB is not taken into account, it does showthat the filter-amplifiers generate very little noise, as expected.

rather than at the output of the square law detector because the signal is strong, and

lies outside of the square law region of the diode, which is near -20 dBm. This shows

that even with the frequency band mismatch, a 15 mW signal incident on the LNB

antenna is enough to drive the detector well beyond its intended input range, which

results in non-linearity. Astronomical signals will be significantly weaker and we do

not expect to have this problem when taking real measurements.

Page 67: Radio Telescope

Chapter 5

Conclusions and Future Work

We have successfully developed an antenna and receiver system which collects and

measures Ku band radio waves. Based on laboratory testing we find that the receiver

is sufficiently sensitive such that the telescope will be capable of detecting astronom-

ical sources of radio such as the Sun and the Milky Way. Due to time constraints,

astronomical data has not yet been collected with the telescope at the time of this

writing. A mechanism for the mounting of the antenna is currently in development.

We expect to begin calibrating the telescope and conducting simple measurements

when the antenna has been successfully mounted.

5.1 First Expected Observations

The antenna mount will allow us to steer the beam of the telescope and aim it at

specific sources of radio. Since the Sun is low in the sky in the winter months this will

be necessary for solar measurements. Once the telescope is mounted and the receiver

is properly set up for data collection at Ford Observatory, we will be able to calibrate

the system using the process described in Section 2.6. This process should include a


Page 68: Radio Telescope

58 Chapter 5. Conclusions and Future Work

measurement of the total flux from the Sun, since that is the most accessible radio

source in the sky. This can be compared against the flux measured by professional

solar observatory such as SOHO. For a hot reference source, the simplest method

would be to point the dish at the observatory building or at the ground if possible.

If the calibration of the telescope and measurements of the Sun are successful, the

next target for measurement should be the Milky Way. Positioning of the telescope

beam is not crucial in this case since the galaxy covers a large portion of the night

sky and it is likely that some part of it would pass through the beam at some point

in the day. A common method of calibrating smaller radio telescopes is to cool a slab

of foam to liquid nitrogen temperatures and point the beam toward it.

5.2 Longterm Goals

If it is found that the telescope receiver is sensitive enough to observe more elusive

sources of radio or distinguish between the Sun’s levels of radio activity due to radio

bursts, as we expect it may be, an automated and computerized steering system will

have to be developed. This constitutes a significant longterm mechanical engineering

project and may also be expensive. Little research has been done on steering mecha-

nism designs thus far since we have mainly been concerned with the viability of drift


Given a computerized steering mechanism, we would be able to conduct some sim-

ple research tasks with the telescope. This includes monitoring the Sun for increased

sunspot activity which manifests itself as a stronger flux in the radio range. The most

effective way to monitor for these changes is to program the telescope to follow the

Sun across the sky and record the data over several days (or weeks.) We may also

measure the galactic rotation curve of the Milky Way by similar tracking methods.

Page 69: Radio Telescope

5.2. Longterm Goals 59

The possibility of observing at L band can also be explored. The primary moti-

vation for observing at L band is to measure emission from neutral Hydrogen. Most

smaller radio telescopes operate in this band, so the expected results are also well-

documented. Converting the receiver from Ku band to L band can be accomplished

simply by replacing the Ku band LNB with a low noise, high gain amplifier since the

second stage of the receiver is designed to receive L band signals. However, since our

LNB also acts as the feed, a custom feed horn would have to be developed.

Page 70: Radio Telescope

60 Chapter 5. Conclusions and Future Work

Page 71: Radio Telescope

Appendix A

Matlab Script

Below is the Matlab script used to collect data from the output of the receiver. It

has been adapted from a script written by Dr. Bruce Thompson for PHYS 360,

Intermediate Physics Laboratory, at Ithaca College.

function[data,t]=GetAnalogData(deviceName, NofChannels, SampleRate, NofSamples);

\% function to get analog data via the National Instruments DAQ card

\% input:

\% NofChannels - the number of channels to aquire, sampling will be

\% sequential from channel 0 to N-1

\% SampleRate - the sample rate in Hz

\% NofSamples - total number of samples to obtain per channel


\% output:

\% data - data samples in NofChannels columns

\% t - times of the samples in data

\% set up analog IO


Page 72: Radio Telescope

62 Appendix A. Matlab Script

daq = daqhwinfo(’nidaq’)

device = deviceName;


daq = daqhwinfo(ai)


addchannel(ai, c);

set(ai, ’InputType’,’SingleEnded’);

set(ai, ’SampleRate’, SampleRate);

set(ai, ’SamplesPerTrigger’,NofSamples);

set(ai, ’TriggerType’,’Manual’);

\%initiate the device


\% show parameters and wait for start keypress


sprintf(’NChan: \%2i SRate: \%f NSamples: \%6i Time: \%f\n’, ...

NofChannels, SampleRate, NofSamples, TT);



\% Clear the memory upon completion


clear ai;

Page 73: Radio Telescope


[1] R. Nelson, “Communicating with Mars: The experiments of Tesla and

Hodowanec,” Proceedings of the Borderland Sciences Research Foundation, May


[2] K. Jansky, “Electrical disturbances apparently of extraterrestrial origin,” Pro-

ceedings of the IRE, vol. 21, p. 1387, 1933.

[3] R. L. Wilson, Tools of Radio Astronomy. Springer-Verlag Telos, December 1996.

[4] K. Jansky, “Directional studies of atmospherics at high frequencies,” Proceedings

of the IRE, vol. 20, 1920.

[5] G. Reber, “A play entitled the beginning of radio astronomy,” RAS Canada,

vol. 82, 1988.

[6] G. Reber, “Cosmic static,” Proceedings of the IRE, vol. 28, no. 68, 1940.

[7] J. Bernstein, Modern Physics. Prentice Hall, April 2000.

[8] H. Kragh, Cosmology and Controversy: The Historical Development of Two The-

ories of the Universe. Princeton University Press, 1999.

[9] R. W. Wilson, “The cosmic microwave background radiation,” in Nobel Lecture,

Bell Laboratories, December 1978.


Page 74: Radio Telescope

64 Bibliography

[10] G. H. et al, “Three-year Wilkinson microwave anisotropy probe (WMAP1) ob-

servations: Temperature analysis,” Astrophysical Journal Supplement, vol. 170,

pp. 288–314, 2007.

[11] F. Melia, The Galactic Supermassive Black Hole. Princeton University Press,


[12] M. Schmidt, “3c 273 : A star-like object with large red-shift,” Nature, vol. 197,

March 1963.

[13] J. D. Kraus, Radio Astronomy. Cygnus-Quasar Books, 2 ed., 1986.

[14] J. J. Condon and S. M. Ransom, Essential Radio Astronomy. National Radio

Astronomy Observatory, September 2010.

[15] C. U. K. D. E. Gary, Solar and Space Weather Radiophysics: Current Status and

Future Developments. Springer, 1 ed., 2004.

[16] D. J. Griffiths, Introduction to Electrodynamics. Benjamin Cummings, 3 ed.,


[17] R. K. Pathria, Statistical Mechanics. Butterworth-Heinemann, 2 ed., 1996.

[18] F. G.-S. Bernard Burke, An Introduction to Radio Astronomy. Cambridge Uni-

versity Press, December 1996.

[19] D. Casadei and V. Bindi, “The origin of cosmic ray electrons and positrons,”

The Astrophysical Journal, vol. 612, pp. 262–267, May 2004.

[20] J. Binney, The Physics of Quantum Mechanics: An Introduction. Cappella

Archive, 2008.

Page 75: Radio Telescope

Bibliography 65

[21] I. O’Neill, “Where are the sunspots?.”

are-the-sunspots-are-we-in-for-a-quiet-solar-cycle/, June 2008. Universe Today.

[22] B. H. Bransden, Quantum Mechanics. Benjamin Cummings, 2 ed., 2000.

[23] J. Crovisier and J. Dickey, “The spatial power spectrum of galactic neutral hy-

drogen from interferometric observations of the 21cm emission line,” Astronomy

and Astrophysics, vol. 122, no. 282, 1983.

[24] W. Johnson, “Classroom experience with the small radio telescope,” Proceedings

of the American Astronomical Society, 2004.

[25] J. B. Hagen, Radio Frequency Electronics: Circuits and Applications. Cambridge

University Press, 2 ed., 2009.

[26] B. Razavi, Introduction to Microelectronics. Wiley, 2006.

[27] P. Horowitz and W. Hill, The Art of Electronics. Cambridge University Press,

2 ed., July 1989.

Page 76: Radio Telescope

66 Bibliography