Link Characteristics and Performance

Telecommunications Satellite Communications

Courseware Sample 86312-F0

Order no.: 86312-10 First Edition Revision level: 07/2015

By the staff of Festo Didactic

Printed in Canada All rights reserved ISBN 978-2-89640-419-3 (Printed version) ISBN 978-2-89747-103-3 (CD-ROM) Legal Deposit – Bibliothèque et Archives nationales du Québec, 2014 Legal Deposit – Library and Archives Canada, 2014

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Safety and Common Symbols

The following safety and common symbols may be used in this manual and on the equipment:

Symbol Description

DANGER indicates a hazard with a high level of risk which, if not avoided, will result in death or serious injury.

WARNING indicates a hazard with a medium level of risk which, if not avoided, could result in death or serious injury.

CAUTION indicates a hazard with a low level of risk which, if not avoided, could result in minor or moderate injury.

CAUTION used without the Caution, risk of danger sign , indicates a hazard with a potentially hazardous situation which, if not avoided, may result in property damage.

Caution, risk of electric shock

Caution, hot surface

Caution, risk of danger

Caution, lifting hazard

Caution, hand entanglement hazard

Notice, non-ionizing radiation

Direct current

Alternating current

Both direct and alternating current

Three-phase alternating current

Earth (ground) terminal

Symbol Description

Protective conductor terminal

Frame or chassis terminal

Equipotentiality

On (supply)

Off (supply)

Equipment protected throughout by double insulation or reinforced insulation

In position of a bi-stable push control

Out position of a bi-stable push control

IV © Festo Didactic 86312-10

Table of Contents

Preface ...... XI About This Manual ...... XIII To the Instructor ...... XV List of Equipment Required ...... XVII

Introduction Characterizing a Communications Link ...... 1

DISCUSSION OF FUNDAMENTALS ...... 1 Communications inks ...... 1 Measurements and sources of error ...... 1 Instrument uncertainties ...... 2 Measurement inaccuracies ...... 2 Environmental factors ...... 2

Exercise 1 Power Gain and Antenna Parameters ...... 3

DISCUSSION ...... 3 Power gain ...... 3 Antennas ...... 5 Field regions ...... 8 Solid angles ...... 8 Radiation intensity ...... 10 Directive gain and directivity ...... 11 Power density and the inverse-square law ...... 12 Radiation pattern and beamwidth ...... 14 Beam solid angle ...... 19 Shortcut for calculating the beam solid angle ...... 20 Efficiency ...... 20 Effective aperture ...... 21 Antenna gain ...... 22 Gain of a pyramidal horn antenna ...... 23 Electrical size ...... 25 Gain of a parabolic antenna ...... 26 Shortcut for calculating the gain of a dish antenna from the frequency and the diameter ...... 26 The relationship between gain and beamwidth ...... 28 Shortcut for calculating the beamwidth of a dish antenna ...... 28 Shortcut for calculating the gain of a dish antenna from the beamwidth ...... 28 General rules for aperture antennas ...... 28 Measuring antenna gain ...... 29 Polarization ...... 29 The Satellite Communications Training System ...... 30 Horn antenna polarization ...... 30 Frequencies used in the system ...... 32 Power Sensors ...... 32 Power measurements ...... 33

PROCEDURE ...... 34 System startup ...... 34 Connection Diagrams ...... 35 Power gain ...... 36 Aligning the antennas ...... 38 Feed-line loss ...... 39 Gain of the repeater ...... 44 Gain of Down Converter 2 ...... 49 Antenna gain ...... 50 Gain of the large-aperture horn antenna at the uplink frequency ...... 51 Gain of the small-aperture horn antenna versus downlink frequency ...... 54 Gain of the large-aperture horn antenna versus downlink frequency ...... 58 Effective aperture and efficiency ...... 62 Beamwidth and radiation pattern ...... 62 3 dB beamwidth ...... 64 Radiation pattern ...... 65 Polarization isolation ...... 69

Exercise 2 Losses, Radiated Power and Receiver Input Power ...... 71

DISCUSSION ...... 71 Effective isotropic radiated power of a directional antenna ... 71 Friis transmission equation ...... 72 Free-space loss ...... 73 Notes on free-space loss ...... 73 Atmospheric attenuation and path loss ...... 75 Feeder loss ...... 75 Pointing loss ...... 76 Polarization mismatch loss ...... 76 Receiver input power ...... 77

PROCEDURE ...... 78 System startup ...... 78 Preliminary measurements ...... 79 Free-space loss ...... 79 Set up for measuring free-space loss ...... 80 Notes on measuring the transmitted and received power ...... 82 Determining the effective isotropic radiated power ...... 82 Measuring the receiver input power ...... 84 Calculating the free-space loss ...... 86 Receiver input power ...... 92 Polarization mismatch loss ...... 94

VI © Festo Didactic 86312-10 Table of Contents

Exercise 3 Noise and the Link Budget ...... 101

DISCUSSION ...... 101 Noise in communications systems ...... 101 Thermal noise characteristics ...... 102 Amplitude distribution ...... 103 Frequency distribution ...... 105 Half-power and equivalent noise bandwidths ...... 107 Additive white Gaussian noise (AWGN) ...... 108 Thermal noise power ...... 109 Noise power example ...... 111 Signal-to-noise ratio ...... 111 Noise figure ...... 112 Noise figure example ...... 114 Noise temperature ...... 117 Noise figure versus noise temperature ...... 119 Composite noise figure and noise temperature of cascaded elements ...... 120 Noise figure and noise temperature example ...... 121 Noise figure and noise temperature of an attenuator or a feed line ...... 126 Antenna noise temperature ...... 127 System noise temperature ...... 128 Composite temperature of the antenna and feed line ...... 130 Link performance and receiver figure of merit ...... 131 Link budget ...... 132 Link budget example ...... 132 Analog and digital figures of merit ...... 135

PROCEDURE ...... 136 System startup ...... 136 Noise figure ...... 137 Set up for measuring noise figure ...... 137 Measuring received carrier power and RF front end gain ...... 138 Measuring the gain of the preamplifier ...... 139 Measuring the receiver output noise power ...... 141 Calculating the receiver noise figure and effective input noise temperature...... 143 System noise temperature ...... 143 Link budget ...... 145

Appendix A Glossary of New Terms ...... 153

Appendix B Setting Up the Satellite Communications Training System . 157 Set up the modules ...... 158 Align the antennas ...... 162 Connect the power supplies ...... 162 USB connections to the Telemetry and Instrumentation Add-On ...... 163

Appendix C Care of Microwave Cables ...... 165

Appendix D Using the Telemetry and Instrumentation Add-On ...... 167 Virtual Instruments ...... 167 Data Generation/Acquisition Interface ...... 167 Spectrum Analyzer Interface...... 168 Digital Inputs ...... 169 Digital Outputs ...... 169 USB Connectors ...... 169 Virtual Instrument package ...... 170 Using the Binary Sequence Generators ...... 171 Symbols used in the manuals ...... 171 Generator Settings ...... 172 Digital Output Settings and connections ...... 173 Using the Waveform Generator ...... 175 Symbol used in the manuals ...... 175 Settings and connections ...... 175 Using the Oscilloscope ...... 176 Symbols used in the manuals ...... 176 Settings and connections ...... 177 Using the Spectrum Analyzer ...... 178 Symbol used in the manuals ...... 178 Settings and connections ...... 178 Using the True RMS Voltmeter / Power Meter ...... 181 Symbols used in the manuals ...... 181 Settings and connections ...... 181 Using the Bit Error Ratio Tester ...... 182 Symbol used in the manuals ...... 183 Settings and connections ...... 183

Appendix E Using Conventional Instruments ...... 185 Instrument symbols and terms ...... 185 Signal levels in the Satellite Communications Training System ...... 185 Power Sensors ...... 187 Spectrum analyzer ...... 188

VIII © Festo Didactic 86312-10 Table of Contents

Appendix F Noise Measurement Using a Spectrometer ...... 189 How a spectrum analyzer works ...... 189 Sensitivity ...... 189 Detector mode...... 190 Trace averaging ...... 192 Resolution bandwidth ...... 193 Corrections to the measured noise level ...... 194 Averaging ...... 194 Equivalent noise bandwidth ...... 195 Preamplifier gain ...... 195 Summary ...... 195

Index of New Terms ...... 197 Bibliography ...... 201

Preface

Since the Soviet Union shocked the western world by launching the first artificial satellite, SPUTNIK I, on October 4, 1957, the science of satellites and satellite communications has undergone an amazing evolution. Today satellites play an essential role in global communications including telephony, data networking, video transporting and distribution, as well as television and radio broadcasting directly to the consumer. They fulfill critical missions for governments, the military and other organizations that require reliable communications links throughout the world, and generate billions of dollars annually in revenue for private enterprise.

Communications satellites offer several important advantages over other types of long-range communications systems: the capability of direct communication between two points on earth with only one intermediate relay (the satellite), the ability to broadcast or collect signals and data to or from any area ranging up to the entire surface of the world, and the ability to provide services to remote regions where ground-based, point-to-point communications would be impractical or impossible.

One of the greatest advantages of satellite communications systems is the ratio of capacity versus cost. Although satellites are expensive to develop, launch and maintain, their tremendous capacity makes them very attractive for many applications. INTELSAT I, launched in 1965, had a capacity of only 240 two-way telephone channels or one two-way television channel, and an annual cost of $32 500 per channel. Since then, the capacity and lifetime of communication satellites have increased tremendously resulting in a drastic reduction in the cost per channel. Communications satellites now have capacities sufficient for several hundred video channels or tens of thousands of voice or data links.

In addition to applications designed specifically for communications purposes, satellites are used extensively for navigation systems, scientific research, mapping, remote sensing, military reconnaissance, disaster detection and relief and for many other applications. All of these applications, however, require at least one communications link between the satellite and one or more earth stations.

The Satellite Communications Training System is a state-of-the-art training system for the field of satellite communications. Specifically designed for hands- on training, the system covers modern satellite communication technologies including analog and digital modulation. It is designed to use realistic satellite uplink and downlink frequencies at safe power levels and to reflect the standards commonly used in modern satellite communications systems.

The Orbit Simulator provides interactive visualization of satellite orbital mechanics and coverage, and the theory behind antenna alignment with geostationary satellites. The optional Dish Antenna and Accessories provides hands-on experience in aligning a typical antenna with real geostationary satellites.

We invite readers of this manual to send us their tips, feedback, and suggestions for improving the book.

Please send these to [email protected]. The authors and Festo Didactic look forward to your comments.

XII © Festo Didactic 86312-10

About This Manual

Manual Objective

When you have completed this manual, you will be familiar with the main concepts and parameters that characterize microwave links and link performance. These concepts and parameters apply to both terrestrial and satellite RF links.

You will be familiar with the parameters that characterize antennas and with the different losses in a microwave link. You will also be familiar with the origins and characteristics of noise and with the noise parameters used to characterize components and systems.

Finally, you will be familiar with the analysis of an RF link known as a link budget, a mathematical model of the link that predicts its performance.

The objective of this manual is to demonstrate the concepts that determine link a performance and not to make precise measurements, since precise measurements require specialized equipment and ideal conditions.

Many calculations will be required as you carry out the exercise procedures. b Although tables are provided to facilitate these calculations, you are encouraged to use a spreadsheet program or a mathematics application such as Matlab or Scilab for these calculations.

Description

This Student Manual is divided into several units each of which covers one topic. Each unit begins with an Introduction presenting important background information. Following this are a number of exercises designed to present the subject matter in convenient instructional segments. In each exercise, principles and concepts are presented first followed by a step-by-step, hands-on procedure to complete the learning process.

Each exercise contains: x A clearly defined Exercise Objective x A Discussion Outline listing the main points presented in the Discussion x A Discussion of the theory involved x A Procedure Outline listing the main sections in the Procedure x A step-by-step Procedure in which the student observes and measures the important phenomena, including questions to help in understanding the important principles. x A Conclusion x Review Questions

In this manual, all New Terms are defined in the Glossary of New Terms. In a addition, an Index of New Terms is provided at the end of the manual.

Prerequisite

As a prerequisite to this course, you should have performed the exercises in the manual Principles of Satellite Communications, part number 86311-00. This manual contains many tips on using the Satellite Communications Training System and the optional Telemetry and Instrumentation Add-On. Most of these tips are not repeated in the present manual.

Systems of units

Units are expressed using the SI system of units.

Safety considerations

Safety symbols that may be used in this manual and on the equipment are listed in the Safety Symbols table at the beginning of the manual.

Safety procedures related to the tasks that you will be asked to perform are indicated in each exercise.

Make sure that you are wearing appropriate protective equipment when performing the tasks. You should never perform a task if you have any reason to think that a manipulation could be dangerous for you or your teammates.

When studying communications systems, it is very important to develop good safety habits. Although microwaves are invisible, they can be dangerous at high levels or for long exposure times. The most important safety rule when working with microwave equipment is to avoid exposure to dangerous radiation levels.

The radiation levels in the Satellite Communications Training System are too low to be dangerous. The highest power level in the system is at the RF OUTPUT of the Earth Station Transmitter and is typically 5 dBm (approximately 3.2 mW) at 11 GHz. The maximum power density that can be produced by the Satellite Communications Training System using the supplied equipment is approximately 0.13 mW/cm2, well below all Canadian, American, and European standards for both microwave exposed workers and the general public. c For more detailed information, refer to Safety with RF fields in Exercise 1-1 of the manual Principles of Satellite Communications, part number 86311-00.

XIV © Festo Didactic 86312-10

To the Instructor

You will find in this Instructor Guide all the elements included in the Student Manual together with the answers to all questions, results of measurements, graphs, explanations, suggestions, and, in some cases, instructions to help you guide the students through their learning process. All the information that applies to you is placed between markers and appears in red.

Accuracy of measurements

The numerical results of the hands-on exercises may differ from one student to another. For this reason, the results and answers given in this manual should be considered as a guide. Students who correctly performed the exercises should expect to demonstrate the principles involved and make observations and measurements similar to those given as answers.

Sample Exercise Extracted from the Student Manual and the Instructor Guide

Exercise 3

Noise and the Link Budget

EXERCISE OBJECTIVE When you have completed this exercise, you will be familiar with noise in communications systems and with the characteristics of noise. You will also be familiar with various parameters concerning noise and how they are calculated. You will see how a link budget is calculated to include all the gains, losses and noise in the system.

DISCUSSION OUTLINE The Discussion of this exercise covers the following points:

Noise in communications systems Thermal noise characteristics Amplitude distribution. Frequency distribution. Half-power and equivalent noise bandwidths Additive white Gaussian noise (AWGN) Thermal noise power Noise power example. Signal-to-noise ratio Noise figure Noise figure example. Noise temperature Noise figure versus noise temperature. Composite noise figure and noise temperature of cascaded elements Noise figure and noise temperature example. Noise figure and noise temperature of an attenuator or a feed line. Antenna noise temperature System noise temperature Composite temperature of the antenna and feed line. Link performance and receiver figure of merit Link budget Link budget example. Analog and digital figures of merit.

DISCUSSION Noise in communications systems

Unwanted electrical signals are present in every electrical system. These unwanted signals are referred to as noise.

Noise arises from a variety of sources. Man-made sources of noise include carriers from transmitters other than those you wish to receive. This noise is referred to as interference. Natural sources of noise include the atmosphere, the sun, and the ground around a receiving antenna.

Noise limits the performance of a communications system because it tends to mask the desired signal. In an analog system, noise reduces the quality of the receiver output signal. In a digital system, noise limits the ability of the demodulator to make correct symbol decisions, which leads to errors in the received bits. For this reason, the ratio of signal power to noise power (or a

similar measurement) is an important factor in evaluating the performance of a communications link.

John Bertrand Johnson was Much of the potential noise in a communication system can be reduced or a Swedish-born American eliminated by careful design. However, there is one natural source of noise that electrical engineer and can never be eliminated. This is called thermal noise, Johnson noise, or physicist. Johnson first Johnson-Nyquist noise and is caused by the random, thermal agitation of an measured thermal noise at extremely large number of charged carriers (electrons) in electrical conductors. Bell Telephone Laboratories in 1926. He concluded that Not all random noise in a system is thermal noise. Shot noise, for example arises thermal noise is intrinsic to from the quantized nature of current flow. This and other random phenomena all electrical components, produce noise whose characteristics are similar to thermal noise. The combined even in the absence of an effect of all these random phenomena is often treated as if it were all caused by applied voltage. thermal noise.

Harry Nyquist, another Swedish-born American Thermal noise characteristics electrical engineer, also worked as Bell Labs. In Figure 43 shows an example of thermal noise as it could be displayed on an 1927, Nyquist provided a oscilloscope. mathematical explanation of thermal noise. 4

0 Time Amplitude (V) Ͳ1

Ͳ2

Ͳ3

Ͳ4

Figure 43. Noise trace (time-domain representation) of 1 V rms thermal noise.

Because thermal noise is a random process, its instantaneous value (voltage or current) is constantly changing and it is impossible to predict what the value will be at a given time. Nonetheless, several properties of the noise are stable over time and these can be used to characterize the noise.

Amplitude distribution

From one instant to the next, the amplitude of unfiltered thermal noise may change very little or it may swing widely from one extreme value to another. These values can be analyzed statistically. If the noise is sampled (that is, the instantaneous noise amplitude is measured at regular intervals) and the observed range of amplitudes is divided into small ranges, it is possible to

102 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

determine how many samples fall within each of the ranges. The results can be plotted as a histogram. This histogram gives an idea of the statistical distribution of different amplitudes in the noise.

Figure 44 shows the same noise trace as Figure 43 but rotated 90°. The sampled amplitudes are shown as dots. The vertical lines divide the possible amplitudes into a number of equally-spaced ranges. The histogram below the trace shows the relative number samples that fall within each amplitude range.

Amplitude Time Number of samples

Amplitudes ranges Figure 44. Noise trace and amplitude histogram of 1 V rms thermal noise.

Although this histogram was generated from a small number of samples, it reveals some important characteristics of thermal noise. Amplitudes near 0 V occur more frequently than amplitudes with a large positive or negative voltage. We can say that small amplitudes are more probable than large amplitudes. In addition the histogram seems to be relatively symmetrical and is centered on 0 V.

If an extremely large number of samples are taken of the noise, and if the amplitude interval of the histogram is made extremely small, the histogram tends toward a continuous curve called the probability density function. In the case of thermal noise, the probability density function has the familiar bell-curve shape of the Gaussian, or normal, probability density function shown in Figure 45. Thermal noise results from contributions of an almost infinite number of independent sources. For this reason, it can be described as a Gaussian random process.

The standard deviation ߪ indicates how much the values deviate from the mean value. For thermal noise, the standard deviation is equal to the rms value. Figure 45 represents the amplitude distribution of a 1 V rms noise signal with no dc offset, as shown in Probability density Figure 43 and Figure 44.

In a normal distribution, approximately 68% of the values are within one Value (e.g. amplitude in V) standard deviation from the mean; approximately 95% Figure 45. Gaussian (normal) probability density function (࢓ࢋࢇ࢔ ൌ ૙, ࣌ൌ૚). of the values lie within two standard deviations; and In theory, the domain of a probability density function ranges from െλ to ൅λ. 99.7% of the values within However, the function approaches zero for values far from the mean value. three standard deviations from the mean. The probability density function shows the relative likelihood for a random variable, such as a voltage, to take on a given value. The probability that the variable will fall within a particular range is given by the area under the density function between the limits of the specified range. For example, in Figure 45, the probability that a single voltage sample will fall between -1 V and +1 V is equal to the area under the curve between -1 and +1. In Figure 45, this area is approximately 0.68. Since the probability that any value will lie somewhere under the curve is 1.0 (100%), the total area under the curve is equal to one.

Frequency distribution

Johnson showed that thermal noise power is constant throughout the frequency spectrum. For this reason, thermal noise is referred to as white noise. Just as white light contains equal amounts of all frequencies within the visible band of electromagnetic radiation, white noise contains equal amounts of all frequencies. More precisely, thermal noise contains an equal amount of noise power per unit bandwidth at all frequencies from dc to about 1000 GHz. For this reason, it’s power spectrum can be represented as a straight, horizontal line, as shown in Figure 46a extending from 0 Hz to a very high frequency.

104 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

ܰ଴ Spectral density 0 0 Frequency a) White noise spectrum

B Magnitude

0 0 Frequency b) Brickwall band-pass filter transfer function

ܰ଴

Spectral Density 0 0 Frequency c) Filtered noise spectrum

Figure 46. Power spectrum of unfiltered and filtered white noise.

The symbol for noise power The noise power per unit bandwidth is referred to as the noise power spectral density (N-zero). Its units are W/Hz or dBm/Hz. spectral density ܰ଴ (N-zero) ܰ଴ should not be confused with the symbol for output noise Although white noise contains a vast range of frequencies, harmful noise power power ܰ௢ (N-o). is that which occurs in the bandwidth of the communications channel or the bandwidth of the receiver. The channel along with the receiver input stage (as well as the IF stage of a spectrum analyzer) limit the bandwidth, that is, they act as a band-pass filter.

Figure 46b shows the magnitude transfer function of an ideal band-pass filter. This ideal filter is called a brickwall filter because it allows all frequencies within its bandwidth ܤ to pass unaffected and completely blocks all other frequencies.

If white noise is applied to the input of the brickwall filter, the spectrum of the noise at the output resembles Figure 46c. This is essentially a rectangular slice of the input noise power with power spectral density ܰ଴ within the filter bandwidth and zero elsewhere.

Regardless of whether the If ܰ଴ at the input of a network (an electrical component, circuit, or device in question is a com- communications channel) is known, and if the network acts as an ideal (brickwall) ponent, a cascade of com- filter with bandwidth ܤ, then the noise power ܰ at the output is simply the power ponents, or an entire sys- spectral density multiplied by the bandwidth: tem, we will refer to the device as a network. ܰൌܰ଴ܤ (43)

The symbols ܤ, ܤܹ and ܹ where ܰ is the output noise power are often used for band- ܰ଴ is the noise power spectral density width. ܤ is the (ideal rectangular) bandwidth

Half-power and equivalent noise bandwidths

An ideal rectangular filter has uniform frequency response within its bandwidth and zero response outside its bandwidth, as shown in Figure 46b. Unfortunately, this ideal response is unattainable in real systems. Instead of stopping abruptly, the frequency response rolls off gradually. For this reason, many different definitions of bandwidth are used, the most common of which is the familiar half- power bandwidth or 3 dB bandwidth. This is the interval between frequencies at which the gain of the system has dropped to half power (3 dB below) the peak value.

The 3 dB bandwidth of a practical filter is shown in blue in Figure 47. Since the power is proportional to the square of the voltage, the half-power point occurs at ඥͳʹΤ ൌ ͲǤ͹Ͳ͹ of the peak amplitude.

The equivalent noise band- Although the 3 dB bandwidth is very useful when dealing with signals, it does not width also called the noise give accurate values when dealing with noise. This is due to the fact that there is equivalent bandwidth, the no direct relationship between the amount of noise power a filter passes and its noise-power equivalent 3 dB bandwidth. bandwidth, or simply the noise bandwidth. Instead, the equivalent noise bandwidth ܧܰܤ is used. This is defined as the bandwidth of a fictitious, ideal rectangular filter (brickwall filter) with the same Common symbols for band-center gain as the actual system, which would pass exactly as much white equivalent noise bandwidth noise power as the actual system does. Both of these bandwidths are illustrated are ܧܰܤ, ܧܰܤܹ, and ܤே in Figure 47.

1.1 1 0.9 0.8 3 dB 0.7 0.707 (half-power) bandwidth 0.6 Practical filter 0.5 Ideal Filter

Amplitude [V] 0.4 0.3 0.2 Equivalent noise 0.1 bandwidth 0

Figure 47. Half-power bandwidth and equivalent noise bandwidth.

106 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

For most analog filters and band-limited circuits, the equivalent noise bandwidth is slightly larger than the half-power bandwidth. Since a spectrum analyzer uses a narrow band-pass filter, the difference between equivalent noise bandwidth and half-power bandwidth must be taken into consideration when measuring noise power with this type of instrument.

Since the narrow band-pass filter in a spectrum analyzer allows the instrument a to resolve closely-spaced frequency components, its 3 dB bandwidth is referred to as the resolution bandwidth. The ENB of this filter may be greater than its 3 dB bandwidth. This is explained in Appendix F Noise Measurement Using a Spectrometer. It is important to read this appendix before beginning the Procedure of this exercise.

The mathematical definition of equivalent noise bandwidth is:

ͳ ஶ (44) ܧܰܤ ൌ නܲሺ݂ሻ݂ ܲ௠௔௫ ଴

where ܧܰܤ is the equivalent noise bandwidth ܲ௠௔௫ is the maximum value of the power transfer function ஶ නܲሺ݂ሻd݂ is the area under the curve of the power transfer function ଴

Additive white Gaussian noise (AWGN)

The generally accepted model for thermal noise in communication channels is described as follows: x The noise is additive. The noise adds positive and negative random values to the amplitude of the signal.

Transmitted Received signal Signal

Channel Noise x The noise is white, that is, its power spectral density is flat. In theory, white noise has unlimited bandwidth and all frequencies are represented equally. In practice, white noise has a uniform spectral density within the bandwidth of interest. x The noise has a Gaussian, or normal probability density because it results from the agitation of a very large number of independent atomic particles. This distribution arises when a large number of independent sources contribute additively to the end result, as long as the contribution of each is small compared to the sum. With this type of distribution, instantaneous noise voltages near the mean value are more probable than voltages much higher or lower than the mean value.

This model for thermal noise is often referred to as additive white Gaussian noise (AWGN).

Thermal noise power

As previously mentioned, all components and circuits generate thermal noise. As its name implies, the amount of thermal noise generated depends on the temperature of the device. A representation, or electrical model, of a thermal noise source is shown in Figure 48a. The source (a resistor) is shown connected to a load (the input of an amplifier).

Thermal noise Matched source load (resistance) (amplifier)

a) Thermal noise source and matched load

Noiseless ܴ resistor ܴ Noise source with no resistance

b) Equivalent circuit

Figure 48. Network consisting of a matched noise source and load (an amplifier).

Since the resistor has two terminals, where the current into one is identical to the current out of the other, it referred to as a one-port element.

The RMS thermal noise voltage ܸ௡ǡ௥௠௦ produced by the resistor of resistance ܴ at a temperature ܶ (in kelvin) is:

(45) ܸ௡ǡ௥௠௦ ൌ ξͶ݇ܶܤܴ

where ݇ is Boltzmann’s constant ή ݇ ൌ ͳǤ͵ͺ ൈ ͳͲିଶଷ ൤ ൨ ή ൌ െʹʹͺǤ͸ ൤ ൨ ή ൌ െͳͻͺǤ͸ ൤ ൨ ή ܶ is the temperature in kelvin (Ͳι ൌ ʹ͹͵Ǥͳͷ ሻ ܤ is the system equivalent noise bandwidth in Hz ܴ is the resistance in ohms.

108 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

Figure 48b shows an electrically equivalent model of the circuit in Figure 48a. It consists of a noiseless resister ܴ in series with an ideal noise source which as no resistance.

Maximum power is delivered when the impedance of the source and of the load are matched (in this case, when the resistance of the load equals that of the source). The input noise voltage delivered to the impedance-matched load is one half the source noise voltage. The input noise power ܰ௜ delivered to the amplifier, expressed in watts and in dBm, is given by:

ଶ ൫ܸ Τʹ൯ (46) ܰ ൌ ௡ǡ௥௠௦ ௜ሾሿ ܴ Ͷ݇ܶܤܴ ȳ ൌ ൤ ήή ή ൨ Ͷܴ ή ȳ ൌ݇ܶܤሾሿ

ܾ݇ܶ ܰ ൌͳͲ ௜ ሾሿ ͲǤͲͲͳ

where ݇ is Boltzmann’s constant ܶ is the temperature in K ܤ is the system equivalent noise bandwidth in Hz a All logarithms in this manual are base 10 logarithms.

Equation (46) shows that the power delivered by a thermal noise source to the impedance-matched load depends only on the temperature and the bandwidth. It is independent of the impedance. The input noise power spectral density ܰ଴௜ is therefore:

ܰ (47) ܰ ൌ ௜ ൌ݇ܶሾ Τሿ ଴௜ ܤ ݇ܶ ൌͳͲ ሾȀ ሿ ͲǤͲͲͳ

The noise power spectral density ܰ଴ at any point in a circuit is simply the noise power contained in a bandwidth of 1 Hz. With white noise, the noise power spectral density is constant over a vast range of frequencies. The noise power at any point can be therefore determined from the noise power spectral density and the bandwidth using Equation (43).

Sometimes it is necessary to determine the noise power in some bandwidth ܤଶ when the noise power in a different bandwidth ܤଵ is known. This is easily done using Equation (48).

ܰ஻మ ൌܰ଴ܤଶ (48)

ܰ஻భ ൌ ܤଶ ܤଵ

ܤଶ ൌܰ஻భ ܤଵ

Expressed in dBm, this gives the very useful formula:

ܤଶ (49) ܰ஻మ ሾሿ ൌ ܰ஻భ ሾሿ ൅ ͳͲ ܤଵ

Noise power example

Thermal noise in a circuit with a bandwidth of ͷͲ has a power of െͶͺ. What would be the noise power if the bandwidth were reduced to ͳͲ ?

ͳͲ ܰ ൌܰ ൅ͳͲ ͳͲ ͷͲ ͷͲ ൌെͶͺെ͹ ൌ െͷͷ

What in the noise power spectral density?

ͳ ܰ ൌܰ ൅ͳͲ൬ ൰ ଴ ͷͲ ͷͲ ൈ ͳͲ଺ ൌ െͶͺ െ ͹͹ ൌ െͳʹͷȀ

Signal-to-noise ratio

The signal-to-noise ratio (SNR or S/N) is a measure that compares the level of a signal to the level of background noise. It is defined as the power ratio between a signal and the background noise. It can be expressed as a ratio or in decibels.

ܲ ܵ (50) ܴܵܰ ൌ ௦௜௚௡௔௟ ൌ ܲ௡௢௜௦௘ ܰ ܵ ܴܵܰ ሺሻ ൌ ͳͲ ܰ

where ܵ is the average power of the signal ܰ is the average power of the noise

110 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

Since the noise power ܰ is proportional to the bandwidth, the bandwidth is assumed to be that of the circuit, unless a different bandwidth is specified. A SNR higher than 1 (> 0 dB) indicates that there is more signal than noise.

An ideal amplifier would amplify the noise and the signal at its input equally without adding any noise of its own. The signal-to-noise ratio at the output would therefore be the same as at the input.

Unfortunately, every component and network adds some noise of its own. To characterize this added noise, it is not sufficient to compare the output noise power to the input noise power because the network may amplify or attenuate both the input noise and signal while adding noise of its own. For this reason, the noise figure is used.

Noise figure

Since a receiving system must process very weak signals in the presence of noise, it is important to be able to characterize the ability of the system to process these low-level signals. One very useful parameter is the noise figure, defined in the 1940’s by Harald T. Friis. This parameter is suitable for characterizing an entire receiving system as well as individual system components such as amplifiers and mixers. It is also used to characterize any number of components in cascade. Because the networks shown in Figure 49 have both an input and an output, they are referred to as two-port elements.

Input Output Input Output

a) Two-port element b) Cascaded two-port elements

Input Output

c) Earth Station Receiver Down Converters

Figure 49. Examples of networks.

Friis defined the noise figure ܨ of a network as the degradation or decrease of the signal-to-noise ratio as the signal passes through the network:

ܵ Τܰ (51) ܨൌ ௜ ௜ ܵ௢Τܰ௢

where ܵ௜Ȁܰ௜ is the signal-to-noise ratio at the input ܵ௢Ȁܰ௢ is the signal-to-noise ratio of the output

Like gain, the noise figure can be expressed either as a unitless ratio or in decibels. When expressed as a ratio, as in Equation (51), it is sometimes referred to as the noise factor. When expressed in decibels, the noise figure is usually written as ܰܨ:

ܰܨሾୢ୆ሿ ൌͳͲܨ (52) ܵ Τܰ ൌͳͲ ௜ ௜ ܵ௢Τܰ௢

ൌͳͲܵ௜Τܰ௜ െͳͲܵ௢Τܰ௢

ൌܴܵܰ௜ሾୢ୆ሿെܴܵܰ௢ሾୢ୆ሿ

In this manual, the term “noise figure” is the term generally used in phrases a referring to the degradation in signal-to-noise ratio with no units being implied. Where necessary, the term “noise factor ܨ” is used for the ratio and the term “noise figure ܰܨ” is used for the value in decibels.

When the gain ܩ, the input noise power ܰ௜ and the output noise power ܰ௢ in decibels are known, the noise figure in decibels can be calculated directly:

ܵ௜Τܰ௜ (53) ܰܨሾୢ୆ሿ ൌͳͲ ܵ௢Τܰ௢ ܵ ܰ ൌͳͲ൬ ௜ ൈ ௢൰ ܵ௢ ܰ௜ ͳ ܰ ൌͳͲ൬ ൈ ௢൰ ܩ ܰ௜

ൌܰ௢ሾୢ୆ሿെܰ௜ ሾୢ୆ሿ െܩሾୢ୆ሿ

The noise figure of a system does not depend on the type of modulation used. Since the noise power is proportional to the bandwidth (Equation (43)), the same bandwidth must be considered at the input and the output of the network when determining ܰ௢ and ܰ௜. For this reason, the noise figure is independent of bandwidth.

The noise figure is a parameter that expresses the noisiness of a two-port network or device (that is, one with an input and an output). Since the noise figure represents a degradation, or decrease, in signal-to-noise ratio, a low noise figure is desirable. The higher the noise figure, the more noise is added by the network in consideration.

112 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

Noise figure example

Figure 50 provides an example to illustrate the use of the noise figure. Power [dBm] Power [dBm]

Frequency [GHz] Frequency [GHz]

(a) At the input of an amplifier (b) At the output of an amplifier

Figure 50. Signal and noise levels as seen on a spectrum analyzer.

The signal at the input of the amplifier is at -40 dBm, 30 dB above the noise floor (-70 dBm), as displayed by a spectrum analyzer with a certain resolution bandwidth. The gain of the amplifier is 20 dB. The amplifier amplifies both the input signal and the input noise by 20 dB and adds 5 dB of noise. As a result, the signal at the output is only 25 dB above the noise floor. The noise figure of this amplifier is therefore:

ܰܨ ൌ ܴܵܰ௜ െܴܵܰ௢ ൌ ͵Ͳ െ ʹͷ ൌͷ

Alternatively, using Equation (53):

ܰܨ ൌ ܰ௢ െܰ௜ െܩ ൌ െͶͷ ൅ ͹Ͳ െ ʹͲ ൌͷ

The noise factor can be written as shown in Equation (54).

ܵ Τܰ (54) ܨൌ ௜ ௜ ܵ௢Τܰ௢ ܵ Τܰ ൌ ௜ ௜ ܩܵ௜Τሺܩܰ௜ ൅ܰ௔ሻ ܩܰ ൅ܰܽ ൌ ௜ ܩܰ௜

where ܩ is the gain of the network ܵ௜ is the signal power at the input ܵ௢ is the signal power at the output ܵ௢ ൌܩܵ௜ ܰ௜ is the noise power at the input ܰ௔ is the noise power added by the network ܰ௢ is the noise power at the output ܰ௢ ൌܩܰ௜ ൅ܰ௔

Equation (54) shows that the noise ܰ௔ added by the network can be considered separately from the input noise ܰ௜. This is illustrated in Figure 51.

ܩ ൌ ͳͲͲ ܰ௢ ൌܩܰ௜ ൅ܰ௔ ܰ௜ ൌͳɊ ܰ௔ ൌͳͲɊ ൌ ͳͳͲɊ

(a) Noisy amplifier

ܰ௜ ൌͳɊ ܩ ൌ ͳͲͲ ܰ௢ ൌܩሺܰ௜ ൅ܰ௔௜ሻ Noiseless ൌ ͳͳͲɊ

ܰ௔௜ ൌͲǤͳɊ

(b) Equivalent circuit

Figure 51. Noise in amplifiers.

Instead of a noisy amplifier, as in Figure 51a, the amplifier can be considered to be ideal (noiseless) but with an additional, equivalent noise source ܰ௔௜ ൌܰ௔Τܩ at the input, as shown in Figure 51b. The two noise sources are uncorrelated and therefore simply add together. Models a and b are electrically equivalent.

Equation (55) shows the noise factor expressed using ܰ௔௜.

ܩܰ ൅ܰܽ (55) ܨൌ ௜ ܩܰ௜ ܩሺܰ ൅ܰ ሻ ൌ ௜ ௔௜ ܩܰ௜ ܰ ൅ܰ ൌ ௜ ௔௜ ܰ௜ ܰ ൌͳ൅ ௔௜ ܰ௜

where ܰ௔௜ is the noise power added by the network, referred to the input (ܰ௔௜ ൌܰ௔Τܩ)

An ideal network that contributes no noise would have ܰ௔௜ ൌͲ and a noise factor of 1 (a 0 dB noise figure).

Equation (54) and Equation (55) show that the noise figure is not an absolute measure of noise. Rather, it expresses the noisiness of a network relative to the input noise. The noise added by the network is fixed and does not change with

114 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

the input signal. As a result, when more noise is present at the input, the contribution of the added noise to the total noise is less significant.

In order to make comparisons between different devices using the noise figure, a reference value for the input noise source must be chosen. Friis suggested that the noise figure be defined for an input noise source at a standard temperature ܶ଴ ൌ ʹͻͲ (16.8°C or 62.3°F). The suggestion was later adopted by the IEEE as part of its standard definition of noise figure.

Equation (56) shows that the input noise spectral density from a resistance at temperature ܶ଴ is:

ܰ଴ ൌ݇ܶ଴ (56) ൌͳǤ͵ͺൈͳͲିଶଷ ൈ ʹͻͲ ൌͶǤͲͲൈͳͲିଶଵȀ ൌͶǤͲͲൈͳͲିଵ଼Ȁ ൌ െͳ͹Ͷ Ȁ

where ݇ is Boltzmann’s constant

To determine the output noise power under these standard conditions, a standard matched noise source (resistive load at ܶ଴ ൌ ʹͻͲ) is connected to the input of the network and the noise power at the output is measured.

The definition of noise factor adopted by the IEEE (Institute of Electrical and Electronics Engineers) is therefore:

ܵ Τܰ (57) ܨൌ ௜ ௜ǡ௥௘௙ ܵ௢Τܰ௢ ܵ Τ݇ܶ ܤ ൌ ௜ ଴ ܩܵ௜Τሺܩ݇ܶ଴ܤ൅ܰ௔ሻ ܩ݇ܶ ܤ൅ܰ ൌ ଴ ௔ ܩ݇ܶ଴ܤ

where ܰ௜ǡ௥௘௙ is the noise from a matched resistance at ܶ଴ ܰ௔ is the noise added by the network, referred to the output

By substituting ܩܰ௔௜ for ܰ௔, this can be expressed as:

ܩሺ݇ܶ ܤ൅ܰ ሻ (58) ܨൌ ଴ ௔௜ ܩ݇ܶ଴ܤ ݇ܶ ܤ൅ܰ ൌ ଴ ௔௜ ݇ܶ଴ܤ ܰ ൌͳ൅ ௔௜ ݇ܶ଴ܤ

where ܰ௔ is the noise power added by the network, referred to the output ܰ௔௜ is the noise power added by the network, referred to the input

Since the noise figure is defined for a reference temperature of 290 K, a different value is used when the source temperature ܶ is not 290 K. This value is called the operational noise figure (or factor) or effective noise figure (or factor) and is related to the noise factor by Equation (59).

ܶ (59) ܨ ൌͳ൅ ଴ ሺܨെͳሻ ௢௣ ܶ

where ܨ௢௣ is the operational noise factor ܶ is the source temperature ܶ଴ is 290 K ܨ is the noise factor

Note, however, that for small deviations in source temperature, the difference between ܨ and ܨ௢௣ may not be significant. For source temperatures between approximately 260 K and 330 K, the difference between ܨ and ܨ௢௣ in decibels is less than 0.5 dB and it is less than 0.25 dB for source temperatures between 280 K and 300 K.

Noise temperature

Equation (46) showed that the power delivered by a thermal noise source at the input of an impedance-matched network depends only on the temperature (in kelvin) and the bandwidth: ܰ௜ ൌ݇ܶܤ.

In Equation (55), the noise power added by a network ܰ௔ is considered separately and is expressed as an equivalent input noise ܰ௔௜ ൌܰ௔Τܩ.

The definition of noise figure considers an impedance-matched source resistance at temperature ܶ଴ ൌ ʹͻͲ producing an input noise power ܰ௜ ൌ݇ܶ଴ܤ. Figure 52a shows a network (an amplifier) where the input noise is produced by a source at temperature ܶ଴ and the noise added by the network is ܰ௔.

Figure 52b shows an electrically equivalent model where the network is noiseless. The input noise is produced by a source at temperature ܶ଴ and an equivalent input noise ܰ௔௜ is produced by an additional source at temperature ܶ௘. The two noise sources are uncorrelated and therefore simply add together.

116 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

Gain ܩ ܰ ൌ݇ܶܤ ܰ ൌܩܰ ൅ܰ ௜ ଴ Noise ܰ௔ ௢ ௜ ௔

a) Noisy amplifier

ܰ௜ ൌ݇ܶ଴ܤ Gain ܩ ܰ௢ ൌܩሺܰ௜ ൅ܰ௔௜ሻ Noiseless ൌܩܰ௜ ൅ܰ௔ ܰ௔௜ ൌ݇ܶ௘ܤ

b) Equivalent circuit

Figure 52. Equivalent circuits illustrating effective input noise temperature ࢀࢋ.

The temperature ܶ is called the effective input noise temperature and is The temperature ܶ௘is often ௘ referred to as the effective always measured in kelvin. This is defined as the temperature of a fictitious noise noise temperature or just source at the input of a two-port network or amplifier that would result in the the noise temperature and same output noise power, when connected to a noise-free network or amplifier, sometimes as simply the as that of the actual network or amplifier connected to a noise-free source. “temperature,” even though it is not the physical tem- Note that in Figure 52, if the input noise source temperature was absolute zero perature. (0 K), then there would be no noise at the input of the amplifier (ܰ௜ ൌͲ). In this case, the only noise at the output would be the noise ܰ௔ produced by the amplifier where ܰ௔ ൌܩܰ௔௜ ൌܩ݇ܶ௘ܤ.

Equation (55) then can be rewritten as:

ܰ (60) ܨൌͳ൅ ௔௜ ܰ௜ ݇ܶ ܤ ൌͳ൅ ௘ ݇ܶ଴ܤ ܶ ൌͳ൅ ௘ ܶ଴

where ܨ is the noise factor ܶ௘ is the effective input noise temperature in kelvin ܶ଴ is 290 K

Equation (60) shows that the noisiness of a network can be modeled as if it were caused by an additional noise source operating at the effective noise temperature ܶ௘. Powers from uncorrelated sources are additive so noise temperatures are also additive. A network that adds no noise at all has ܨൌͳ and ܶ௘ ൌͲ.

If the noise figure is known, the effective noise temperature is given by:

ܶ௘ ൌܶ଴ሺܨെͳሻ (61)

The noise figure and the effective noise temperature both represent the noise performance of devices. Figure 53 shows the relationship between the noise figure, the noise factor, and the noise temperature in K.

5.0

4.5

4.0

3.5

3.0 [dB] 2.5 ܰܨ ܨ (noise factor) 2.0

Noise figure or factor 1.5

1.0

0.5

0.0 0 50 100 150 200 250 300 350 400 450 500

Effective noise temperature ܶ௘ (K)

Figure 53. Noise figure ࡺࡲ [dB] and ࡲ (noise factor) versus effective noise temperature ࢀࢋ.

Noise figure versus noise temperature

Although the noise figure and the effective noise temperature are both parameters that represent the noisiness of a network, there are cases where one is easier to use than the other. In terrestrial applications, the physical temperature is usually close to 290 K (16.85°C or 62.33°F). Since the noise figure is defined for this temperature, the noise figure is the parameter usually used for terrestrial applications. Noise figures for these applications typically fall in the range of 1 to 10 dB.

For satellite communications however, receivers typically have low noise figures, typically between 0.5 and 1.5 dB. Comparing two devices with low noise figures would require specifying the noise figure to several decimal places. In this case, it is usually more convenient to use the effective noise temperature since the numbers are larger. Besides, with satellite communications, the reference temperature of 290 K may not be appropriate. When using the noise temperature, no reference temperature is required.

Another you reason for using noise temperature instead of noise figure is that noise figure only applies to two-port devices (devices with both an input and an output) whereas noise temperature also applies to one-port devices (devices with only an output), such as antennas.

118 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

Composite noise figure and noise temperature of cascaded elements

Friis developed formulas for determining the noise factor and the noise temperature of two-port elements connected in cascade. Each element may be a single component or a more complex network.

ܩଵ ܩଶ ܩଷ ܨଵ ܨଶ ܨଷ ܶ௘ଵ ܶ௘ଶ ܶ௘ଷ

Figure 54. Noise factors and effective input temperatures of elements in cascade.

Equation (62) shows the Friis formula for the noise factor of cascaded elements:

ܨଶ െͳ ܨଷ െͳ ܨସ െͳ ܨ௡ െͳ (62) ڮ ଵ ൅ ൅ ൅ܨൌܨ ௡ିଵܩڮ ଶܩଵܩ ଷܩଶܩଵܩ ଶܩଵܩ ଵܩ

where ܨ is the composite noise factor of the cascade th ܨ௝ is the noise factor of the j element or stage th ܩ௝ is the gain of the j element or stage ݊ is the number of elements or stages

Friis’s formula for effective input noise temperature is similar:

ܶ௘ଶ ܶ௘ଷ ܶ௘ସ ܶ௘௡ (63) ڮ ௘ ൌܶ௘ଵ ൅ ൅ ൅ܶ ௡ିଵܩڮ ଶܩଵܩ ଷܩଶܩଵܩ ଶܩଵܩ ଵܩ

where ܶ௘ is the composite effective input temperature of the cascade th ܶ௘௝ is the effective input temperature of the j element or stage th ܩ௝ is the gain of the j element or stage ݊ is the number of elements or stages

These equations show that the first stage in a network is most susceptible to add noise to the system, since the noise figure and noise temperature of the subsequent stages are reduced by the product of the gains of the preceding stages. It is therefore important that the first stage in the receiver have as low noise figure as possible, as well as high gain. This is why the first active component in a receiver is a low-noise amplifier (LNA).

Noise figure and noise temperature example

Figure 55a shows a network made up of two amplifiers in series. Figure 55b shows an electrically equivalent circuit. All impedances are matched (those of the sources, of the amplifiers, and of the load at the output). ܰܨଵ and ܰܨଶ are the noise factors of amplifier 1 and 2, respectively.

Amplifier 1 Amplifier 2

Signal ܵ௜ ܵଵ ൌܩଵܵ௜ ܩ ܵ௢ ܩଵ ଶ Input noise ܰܨ ܰ௜ ܰܨଵ ܰଵ ൌܩଵܰ௜ ൅ܰ௔ ଶ ܰ௢ (Noise source at ܶ଴)

Matched Matched source load

a) Two-stage network

Amplifier 1 Amplifier 2

Signal ܵ ௜ ܵଵ ܩଵ ܵଵ ܩଶ ܵ௢ Input noise ܰ௜ ܰܨଵ ܰଵ ܰܨଶ ܰଵ ܰ௢ Added noise ܰ ௔௜ଵ ܰ௔௜ଶ

Matched Matched Matched source Matched load load source

b) Equivalent circuit

Figure 55. Two-stage network and equivalent circuit for determining noise figure.

The input noise and signal sources are shown in series. As these signals are uncorrelated, they simply add together.

For this example, we will assign values to each parameter and calculate the result.

Amplifiers 1 and 2 are identical; each amplifier has a gain ܩ ൌ ͳͲ and a noise figure ܰܨ ൌ ͵. In linear terms, the gain ܩൌͳͲ and the noise factor ܨൌʹ for each amplifier. The signal at the input of the network ܵ௜ ൌ െͶͲ.

We wish to calculate the noise figure of the entire network. The noise source at the input must therefore be at temperature ܶ଴ ൌ ʹͻͲ since this is the temperature required by the definition of noise figure.

In the equivalent circuit, the first amplifier is represented showing three sources at its input: ܵ௜, ܰ௜, and ܰ௔௜ଵ.

120 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

ܵ௜ represents the signal: ିସ଴Τ ଵ଴ ܵ௜ ൌͳͲ mW ൌ ͲǤͲͲͲͳ mW

ܰ௜ represents the noise from the source at 290 K (for convenience, we will calculate noise in units of ݇ܶ଴ܤ):

ܰ௜ ൌ݇ܶ଴ܤ

ܰ௔௜ଵ represents the noise added by amplifier 1, referred to its input. Since the noise factor ܨൌʹ, from Equation (55):

ܰ௔௜ଵ ൌܰ௜ሺܨെͳሻ

ൌ݇ܶ଴ܤሺʹെͳሻ

ൌ݇ܶ଴ܤ

The signal at the output of amplifier 1 (and at the input of amplifier 2) is:

ܵଵ ൌܩܵ௜ ൌ ͲǤͲͲͳ

The noise at the output of amplifier 1 (and at the input of amplifier 2) is:

ܰଵ ൌܩሺܰ௜ ൅ܰ௔௜ଵሻ

ൌ ʹͲ݇ܶ଴ܤ

The second amplifier can also be represented as having three sources at its input: ܵଵ, ܰଵ, and the noise added by amplifier 2: ܰ௔௜ଶ ൌ݇ܶ଴ܤ.

The noise added by amplifier 2 is the same as the noise added by amplifier a one, since they both have the same noise figure.

The signal at the output of amplifier 2 is:

ܵ௢ ൌܩܵଵ ൌ ͲǤͲͳ

The noise at the output of amplifier 2 is:

ܰ௢ ൌܩሺܰଵ ൅ܰ௔௜ଶሻ

ൌͳͲሺʹͲ݇ܶ଴ܤ൅݇ܶ଴ܤሻ

ൌ ʹͳͲ݇ܶ଴ܤ

The signal-to-noise ratio at the input of the network is: ͲǤͲͲͲͳ ܵ௜Τܰ௜ ൌ ݇ܶ଴ܤ

The signal-to-noise ratio at the output of the network is: ͲǤͲͳ ܵ௢ൗܰ௢ ൌ ʹͳͲ݇ܶ଴ܤ

Therefore, the total noise factor of the entire network is:

ܵ௜Τܰ௜ ܨ் ൌ ܵ௢Τܰ௢ ͲǤͲͲͲͳ ʹͳͲ݇ܶ ܤ ൌ ൈ ଴ ͲǤͲͳ ݇ܶ଴ܤ ൌʹǤͳ

From Equation (62), the theoretical noise factor for this network is:

ሺܨଶ െͳሻ ܨ் ൌܨଵ ൅ ܩଵ ͳ ൌʹ൅ ͳͲ ൌʹǤͳ

The noise figure of the entire network is:

ܰܨ் ൌͳͲܨ் ൌ ͵Ǥʹʹ

This example illustrates why the noise figure of a network depends mostly on the noise figure of the first stage of the network. According to the definition of noise figure, the noise at the input of the network must be equal to ݇ܶ଴ܤ in order to calculate the noise figure of the network. The noise added by this first stage can be significant in proportion to its input noise. The deterioration of the signal-to- noise ratio in the first stage is therefore significant.

Since the first stage of the network amplifies this noise by its gain, and adds noise of its own, the noise at the input of the second stage is much greater than ݇ܶ଴ܤ. Hence the noise added by the second stage is less significant, in proportion to its input noise, than the noise added by the first stage is to its input noise. Therefore, the deterioration of the signal-to-noise ratio in the second stage is much less than in the first stage.

With satellite communications, the received signal is very weak because of the great distance between the transmitter and the receiver. For this reason, the noise figure of the receiver must be very low. What changes in the circuit of Figure 55 would be most effective in reducing the total noise figure?

Table 26 shows the effect of certain changes that could be made in the components of the circuit of this example in an attempt to reduce the total noise figure. It also shows the effective input noise temperature of the entire network for each case.

122 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

Three parameters are shown in the table: ܩଵ, ܰܨଵ, and ܰܨଶ. The Example column shows the values already used in the example and which yield a total noise figure of 3.22 dB. The various changes and the results of each change are shown using columns of different colors.

Table 26. Effect of changing noise figure and gain.

Parameter Example Changes / Results

Gain ܩଵ 10 dB 50 dB

Noise figure 1 ܰܨଵ 3 dB 0.5 dB 0.5 dB

Noise figure 2 ܰܨଶ 3 dB 0.5 dB 0.5 dB

Total noise ࡺࡲ 3.22 dB 3.01 dB 3.03 dB 0.86 dB 0.53 dB figure ࢀ

Effective input noise ࢀࢋ 319 K 290 K 293 K 64 K 38 K temperature

From Equation (62), we know that the gain of the last stage of the network has no effect on the total noise figure, so no change in ܩଶ is shown in the table.

In the green column, the gain ܩଵ of the first stage is changed from 10 dB to 50 dB (a 10 000-fold increase in gain). As a result, the total noise figure of the network becomes 3.01 dB, a reduction of approximately 0.2 dB. The improvement is small because increasing the gain of the first stage only reduces the noise contribution of the second stage.

In the purple column, the noise figure of the second amplifier is reduced from 3 dB to 0.5 dB. As a result, the total noise figure is reduced by approximately 0.2 dB, again a small improvement.

In the blue column, the noise figure of the first amplifier is changed from 3 dB to 0.5 dB. This causes the total noise figure to drop to 0.86 dB, an improvement of 2.36 dB.

The orange column of Table 26 shows the effect of reducing the noise figure of the second stage as well as that of the first stage. Although this would yield a small additional improvement of 0.33 dB, this improvement may not be enough to justify the cost of two low-noise amplifiers.

The bottom row of Table 26 shows the effective input noise temperature ܶ௘ that corresponds to each total noise figure ܰܨ். As already mentioned, for low noise devices, it is often more convenient to use the effective input noise temperature, rather than the noise figure.

It is only when evaluating the noise figure of a network that the noise source is considered to be at the standard temperature of 290 K. Once the noise figure is known and the effective input noise temperature calculated, the noisiness of the entire network is equal to that of one noise-free amplifier with a resistance connected to its input whose temperature is equal to the physical temperature ܶ௦ of the source, and an additional resistance whose temperature ܶ௘ is equal to the effective input noise temperature of the network. This is illustrated in Figure 56.

Amplifier 1 Amplifier 2

Signal ܵ௜ ܵ ܩଵ ܩଶ ௢ ܰܨଵ ܰܨଶ ܰ௢ Input noise ܰ௜

Matched Matched Source Load a) Two-stage network

Signal ܵ௜

ܩ் ܵ௢ Input noise ܰ௜ ൌ݇ܶ௦ܤ Noiseless ܰ௢

Added noise ܰ௔௜ ൌ݇ܶ௘ܤ

Matched Matched Source Load

b) Equivalent circuit

Figure 56. Network and an equivalent one-stage circuit showing the physical source temperature ࢀ࢙ and the effective input noise temperature ࢀࢋ.

124 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

Noise figure and noise temperature of an attenuator or a feed line

Figure 57 shows networks that include a passive, lossy device, such as an attenuator or a feed line. Amplifier Attenuator

Loss ܮ௔௧௧

a) Attenuator

Amplifier Feed line

Loss ܮி

a) Feed line

Figure 57. Networks including an attenuator and a feed line.

Figure 58 shows a typical attenuator circuit with a matched input and output impedance.

Series resistance

Input shunt Output shunt

Figure 58. Typical attenuator circuit.

Attenuators and feed lines are both passive, impedance matched devices. The device loss ܮ is defined as:

ܲ (64) ܮൌ ௜ ܲ௢ ͳ ൌ ܩ

where ܲ௜ is the input power ܲ௢ is the output power ܮ is the loss of the attenuator or line (ܮ൐ͳ) ܩ is the gain of the attenuator or line (ܩ൏ͳ)

The noise figure for an attenuator or feed-line is defined the same way as for an active component, that is, considering an input noise source at the standard temperature of 290 K. When connected to a matched load, the input noise and the attenuator or feed-line together appear to the matched load as a simple noise

source at 290 K, providing that no components are higher than this temperature. The noise at the output of the attenuator (or line) is therefore ܰ௢ ൌ݇ܶ଴ܤ.

If the input signal power to the attenuator or feed-line is ܵ௜, then the noise factor is:

ܵ Τܰ (65) ܨൌ ௜ ௜ ܵ௢Τܰ௢ ͳ ܰ ൌ ൈ ௢ ܩ ܰ௜ ݇ܶ ܤ ൌܮ ଴ ݇ܶ଴ܤ ൌܮ

where ܨ is the noise figure of the attenuator or feed-line ܮ is the loss of the attenuator or feed-line ܮൌͳΤ ܩ

Equation (65) shows that noise figure of an attenuator or of a lossy line is equal to its loss ܮ. This is because the signal-to-noise degradation results from the signal being attenuated while the noise level remains fixed.

Only the noise factor is The effective input noise temperature of a passive, lossy device with loss ܮ is defined in reference to a related to its physical temperature: standard temperature of 290K. The effective input ܶ௘௅ ൌܶ௅ሺܮെͳሻ (66) noise temperature is not related to a standard tem- where is the effective input noise temperature of a passive, lossy device perature. ܶ௘௅ ܶ௅ is the physical temperature of the lossy device ܮ is the device loss

See also Equation (61). Note that, when the temperature ܶ௅ of the lossy device is 290 K, the effective input noise temperature is:

ܶ௘௅ ൌܶ଴ሺܮെͳሻ (67)

ൌܶ଴ሺܨെͳሻ

where ܨ is the noise figure of the lossy device

Antenna noise temperature

Random noise is present at the terminals of a receiving antenna. A small part of this noise is thermal noise generated due to the antennas ohmic resistance. The remainder is noise the antenna picks up from other sources, both natural and man-made. The total amount of this noise can be represented as though it were thermal noise generated by a fictitious resistance at a temperature known as the effective antenna noise temperature ܶ஺ or antenna noise temperature, or simply antenna temperature.

126 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

The antenna noise temperature is the effective temperature ܶ஺ such that the noise ܰ஺ at the terminals of the antenna over a bandwidth ܤ is:

ܰ஺ ൌ݇ܶ஺ܤ (68)

where ܰ஺ is the noise produced by the antenna over the bandwidth ܤ is the bandwidth ݇ is Boltzmann’s constant

The amount of noise produced by an antenna depends on what the antenna “sees.” An antenna pointed at a clear sky picks up radiation that has been absorbed and released by molecules in the atmosphere. The antenna temperature depends on the frequency and the elevation angle of the antenna. The presence of clouds and especially of rain will increase the antenna temperature. The noise contribution from the sky is represented by the temperature ܶ௦௞௬.

In addition, the radiation from the ground may be captured by the side lobes of the antenna and to some extent by the main lobe is the elevation angle is small. The noise contribution from the ground is represented by the temperature ܶ௚௥௢௨௡ௗ.

Neglecting the small amount of noise due to the antenna’s ohmic resistance, the total antenna noise temperature is given by ܶ஺ ൌܶ௦௞௬ ൅ܶ௚௥௢௨௡ௗ.

System noise temperature

Figure 59 shows a receiving system consisting of an antenna, a feed line, and a receiver. The antenna noise temperature is ܶ஺. The physical temperature of the feed line is ܶி. The loss ܮி of the feed line is equal to the reciprocal of its gain ܩி. The receiver has an effective input noise temperature ܶ௘ோ௫.

Composite noise temperature ܶ௖௢௠௣ ൌܶ௘ி ൅ܮܶ௘ோ௫ Antenna Receiver ܶ஺ ܶ௘ோ௫ Feed line at ܶி ܮி ൌͳΤ ܩி ܶ௘ி ൌܶிሺܮி െͳሻ

System noise System noise temperature ܶଵ (referred temperature ܶଶ(referred to antenna output) to receiver input)

Figure 59. System noise temperature.

From Equation (66) since the feed line acts as an attenuator with loss ܮி, its effective noise temperature is ܶ௘ி ൌܶிሺܮி െͳሻ.

The feed line and the receiver can be considered as two cascaded elements. The composite noise temperature ܶ௖௢௠௣ of these two elements is determined by applying Equation (63):

ܶ௘ோ௫ (69) ܶ௖௢௠௣ ൌܶ௘ி ൅ ܩி

ൌܶ௘ி ൅ܮிܶ௘ோ௫

where ܶ௖௢௠௣ is the composite noise temperature of the feed line and receiver ܶ௘ோ௫ is the effective input noise temperature of the receiver ܶ௘ி is the effective noise temperature of the feed line ܮி is the loss of the feed-line

The system noise temperature ܶ or system temperature is similar in concept to the effective input noise temperature of a network. It is the temperature in kelvin of a fictitious resistance, placed at a specified point in a system made up of ideal (noiseless) components that would result in the same output noise power as produced by the actual system.

This fictitious resistance is usually considered to be located at the input of the receiver. One can say in this case that this is the “system temperature referred to the input of the receiver.”

Transmitter designers often Imagine first of all that this fictitious resistance is placed at the output of the prefer to refer the system antenna. In Figure 59, the system noise temperature referred to the output of the noise temperature to the antenna is ܶଵ. Since the antenna acts as an noise source, the noise temperature output of the antenna. of the entire system ܶଵ, referred to the output of the antenna, is simply equal to the sum ܶଵ ൌܶ஺ ൅ܶ௖௢௠௣:

ܶଵ ൌܶ஺ ൅ܶ௖௢௠௣ (70)

ൌܶ஺ ൅ܶ௘ி ൅ܮிܶ௘ோ௫

ൌܶ஺ ൅ܶிሺܮி െͳሻ ൅ܮிܶ௘ோ௫

where ܶଵ is the system noise temperature referred to the antenna output ܶ஺ is the antenna noise temperature ܶ௖௢௠௣ is the composite noise temperature of the feed line and receiver ܶ௘ி is the effective noise temperature of the feed line ܶி is the physical temperature of the feed line (often close to ܶ଴) ܮி is the loss of the feed-line ܶ௘ோ௫ is the effective input noise temperature of the receiver

Receiver designers often Now imagine that the fictitious resistance is placed at the input of the receiver, prefer to refer the system rather than at the output of the antenna. The temperature of this fictitious noise temperature to the resistance is the system temperature ܶଶ referred to the input of the receiver. input of the receiver. How does the system temperature ܶଶ, referred to the receiver input, compare to the system temperature ܶଵ, referred to the antenna output? Referring to Figure 52 on page 118, recall that noise ܰ௔ added by an amplifier with gain ܩ can be represented by a noise source ܰ௔௜ at the input of a noiseless amplifier where ܰ௔ ൌܩܰ௔௜ ൌܩ݇ܶ௘௜ܤ. The effective noise temperature ܶ௘௜ of the amplifier is referred to the input of the amplifier. If the noise source was located at the output of the amplifier, rather than at the input, the added noise could be represented

128 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

by ܰ௔ ൌ݇ܶ௘௢ܤ, where ܶ௘௢ is the noise temperature referred to the output. It follows that ܶ௘௢ ൌܩܶ௘௜.

This also holds if the amplifier is replaced by a feed line whose gain ܩி ൌͳΤ ܮி. The system temperature ܶଶ, referred to the output of the feed line (the input of the receiver) is therefore:

ܶଶ ൌܩிܶଵ (71) ܶ ൌ ଵ ܮி

ܶ஺ ͳ ൌ ൅ܶி ൬ͳ െ ൰൅ܶ௘ோ௫ ܮி ܮி ൌܶ

where ܶଶ ൌܶ is the system noise temperature referred to the receiver input ܶଵ is the system noise temperature referred to the antenna output ܩி is the gain of the feed line (less than 1) ܮி is the loss of the feed-line ܮி ൌͳΤ ܩி ܶ஺ is the antenna noise temperature ܶி is the feed-line physical temperature ܶ௘ோ௫ is the receiver effective noise temperature

Both system noise temperatures ܶଵ and ܶଶ take into account the noise produced by the antenna, the feed line, and the receiver. The noise temperature ܶଶ is called the system noise temperature ܶ at the receiver input.

Composite temperature of the antenna and feed line

In many terrestrial systems, the antenna temperature ܶ஺ and the feed line physical temperature ܶி are both close to 290 K. When ܶ஺ ൌܶி, this is also their composite temperature, regardless of the feed-line loss:

ܶ஺ ͳ ܶ௖௢௠௣ǡ௔௡௧௘௡௡௔௔௡ௗ௙௘௘ௗ௟௜௡௘ ൌ ൅ܶி ൬ͳ െ ൰ ܮி ܮி

ܶ஺ ͳ ൌ ൅ܶ஺ ൬ͳ െ ൰ ܮி ܮி

ൌܶ஺ ൌܶி

Link performance and receiver figure of merit

The performance of an individual link is evaluated as the ratio of the received carrier power ܥ to the noise power spectral density ܰ଴. The ratio ܥܰΤ ଴ has units of hertz (Hz).

The received carrier power ܥ is simply the power received ܲோ௫ at the input of the receiver. From Equation (42) on page 77:

ܥൌܲோ௫ (72)

ܲ ܩ ͳ ܩ ൌ൬ ்௫ ் ൰൬ ൰ቆ ோ ቇ ܮఏ்ܮி்௫ ܮிௌܮ஺ ܮఏோܮிோ௫ܮ௣௢௟

ͳ ൌܧܫܴܲ൬ ൰ܩሾሿ ܮ

where ܥൌܲோ௫ is the carrier power (power at receiver input)

்ܲ௫ is the power at the transmitter output

ܩ் is the gain of the transmitting antenna

ܮఏ் is the transmitting antenna pointing loss

ܮி்௫ is the transmitter feed-line loss

ܮிௌ is the free-space loss

ܮ஺ is the atmospheric attenuation

ܩோ is the gain of the receiving antenna

ܮఏோ is the receiving antenna pointing loss

ܮிோ௫ is the receiver feed-line loss

ܮ௣௢௟ is the antenna polarization mismatch loss ܧܫܴܲ is the effective isotropic radiated power ܮ is the path loss ܩ is the composite receiving gain

The noise power spectral density ܰ଴ at the receiver input is proportional to the system noise temperature ܶ referred to the receiver input:

ܰ଴ ൌ݇ܶ (73)

where ݇ is Boltzmann’s constant

Therefore, the carrier power to the noise power spectral density at the receiver input is:

ܥ ͳ ܩ ͳ (74) ൌܧܫܴܲ൬ ൰൬ ൰ ܰ଴ ܮ ܶ ݇

The carrier power to the noise power spectral density ܥܰΤ ଴ is the key parameter in evaluating the performance of a link.

130 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

Although Equation (74) was derived by determining the values at the receiver input, the ܥܰΤ ଴ ratio is independent of the point chosen in the receiving chain as long as the carrier power and the system noise power spectral density are calculated (not measured) at the same point. This is because the system temperature ܶ is calculated as though all components are ideal (noiseless) and all of the noise in the system results from a single, fictitious noise source at the receiver input.

Because the components of the receiver are not ideal, each component adds noise. For this reason, the ܥܰΤ ଴ ratio can only be directly measured at the output of the RF front end (at the input of the demodulator).

The ratio of the composite receiving gain to the system noise temperature ܩȀܶ is called the figure of merit of the receiving equipment.

Link budget

The objective of link performance analysis is to determine whether the communication system will provide the expected performance. The link budget compares the actual ܥܰΤ ଴ ratio to the ratio required to give the expected performance. The actual ܥܰΤ ଴ ratio should exceed the required ܥܰΤ ଴ ratio by a certain margin.

Performance analysis is carried out by calculating each of the terms of Equation (74) separately.

Link budget example

The following example analyzes the performance of the downlink between a satellite repeater (the transmitter) in a geostationary satellite and an earth-station receiver. The values used in this example are given in the tables below.

First, the ܧܫܴܲ is calculated:

ܧܫܴܲ ൌ ்ܲܩ் ܲ ܩ ൌ ்௫ ் ܮఏ்ܮி்௫

ܧܫܴܲሾௗ஻௠ሿ ൌ்ܲ௫ሾሿ ൅ܩ் െܮఏ் െܮி்௫ሾሿ

The transmitter power in this example is 10 W. The earth station is within the 3 dB beamwidth of the downlink antenna. The values for this example are shown in Table 27.

Table 27. Calculating the EIRP.

Item Symbol Value Value (dB)

Transmitter power ்ܲ௫ 10 W 40 dBm

Transmitter antenna gain ܩ் 35 dBi

Maximum transmitter antenna pointing loss ܮఏ் 3 dB

Transmitter antenna feed-line loss ܮி்௫ 1 dB

Effective isotropic radiated power ܧܫܴܲ (12 589 W) 71 dBm

When necessary, both the absolute value and the value in decibels are given a in the example tables. Values in parentheses are shown for information only; conversion into these values is not necessary for the calculations.

The path loss ܮ is calculated for a range of 40 000 km, a downlink frequency of 11 GHz, and an atmospheric attenuation of 0.3 dB.

ܮൌܮிௌܮ஺ Ͷߨܴ݂ ଶ ൌ൬ ൰ ܮ ܿ ஺

ܮሾௗ஻ሿ ൌͳͲܮிௌ ൅ܮ஺ሾሿ

Table 28. Calculating the path loss.

Item Symbol Value Value (dB)

Distance ܴ 40 000 km

Frequency ݂ௗ 11 GHz

20 Free-space loss ܮிௌ 3.4 x 10 205.3 dB

Atmospheric attenuation ܮ஺ 0.3 dB

Path loss ܮ 205.6 dB

The composite receiver gain ܩ is calculated using the values in Table 29, assuming there is no polarization mismatch loss:

ܩ ܩൌ ோ ܮఏோܮிோ௫ܮ௣௢௟

ܩሾௗ஻ሿ ൌܩோ െܮఏோെܮிோ௫െܮ௣௢௟ሾሿ

132 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Discussion

Table 29. Calculating the composite receiver gain.

Item Symbol Value Value (dB)

Receiver antenna gain ܩோ (100 000) 50 dB

Receiver antenna pointing loss ܮఏோ 0.5 dB

Receiver antenna feed-line loss ܮிோ௫ 1.26 1 dB

Polarization mismatch loss ܮ௣௢௟ 0 dB

Composite receiving gain ࡳ 70 795 48.5 dB

The earth station antenna points to a clear sky; its temperature ܶ஺ is 65 K. The feed line temperature ܶ௘ி, however, is 290K. The noise figure ܨோ௫ of the receiver is 1 dB. The receiver antenna feed-line loss ܮிோ௫ is given in Table 29.

The system noise temperature ܶ is calculated using:

ܶ௘ோ௫ ൌ ሺܨோ௫ െͳሻܶ଴

ிೃೣሾሿ ൌቆͳͲ ଵ଴ െ ͳቇ ʹͻͲ

ܶ஺ ͳ ܶൌ ൅ܶி ൬ͳ െ ൰൅ܶ௘ோ௫ ܮி ܮி

Table 30. Calculating system noise temperature.

Item Symbol Value Value (dB)

Antenna temperature ܶ஺ 65 K

Feed line temperature ܶி 290 K

Receiver noise figure ܨோ௫ 1.26 1 dB

Receiver noise temperature ܶ௘ோ௫ 75.4 K

System noise temperature ࢀ 186.8 K

The figure of merit of the receiver is ܩܶΤ . The link performance is characterized by the value ܥܰΤ ଴:

ܥ ͳ ܩ ͳ ሾ ሿ ൌ ܧܫܴܲ൬ ൰൬ ൰ ܰ଴ ܮ ܶ ݇

ܥ ܩ ሾ ሿ ൌ ܧܫܴܲ െ ܮ ൅ െ ݇ሾሿ ܰ଴ ܶ

Table 31. Calculating figure of merit and ࡯ࡺΤ ૙.

Item Symbol Value Value (dB)

Figure of merit ܩܶΤ (380.2 K-1) 25.8 dB K-1

1.38E-23 -198.6 dBm Boltzmann’s constant ݇ W K-1 Hz-1 K-1 Hz-1

Link performance figure ࡯ࡺΤ ૙ 89.8 dBHz

The link budget says a great deal about the overall system design and performance. When the system is being designed, a minimal ܥܰΤ ଴ value is calculated that will provide the desired performance, and a link margin is added. The system is then designed to meet this value while making trade-offs between the system cost and various other constraints.

For example, if the system described above required a ܥܰΤ ଴ value of 83 dB, then a system built respecting parameters given in the tables would have a link margin of 6.8 dB.

Analog and digital figures of merit

An important figure of merit for an analog system is the average signal-to-noise ratio ܵܰΤ or ܴܵܰ at the input of the demodulator.

As mentioned before, the ܥܰΤ ଴ ratio is independent of the point chosen in the receiving chain as long as the carrier power and the system noise power spectral density are calculated at the same point. Since ܰ଴ is the noise power in the bandwidth of 1 Hz, the ܥܰΤ ଴ can be converted into a signal-to-noise ratio by multiplying ܰ଴ by the bandwidth ܤ of the receiver:

ܵ ܥ (75) ൌ ܰ ܰ଴ܤ

where ܵ is the signal power at the demodulator input ܰ is the noise power at the demodulator input ܥ is the carrier power at a certain point in the receiver chain ܰ଴ is the noise power spectral density at the same point ܤ is the bandwidth of the receiver

In digital communication systems, the figure of merit ܧ௕Τܰ଴ is usually used. Consider a digital signal that conveys information one bit of the time at a bit rate ܴ௕. Each bit has a duration ܶ௕ ൌͳΤ ܴ௕. The energy per bit is therefore ܧ௕ ൌ ܵܶ௕. The relation between ܧ௕Τܰ଴ and ܵܰΤ is:

ܧ ܵܶ (76) ௕ ൌ ௕ ܰ଴ ܰ଴ ܵ ܤ ൌ൬ ൰൬ ൰ ܰ ܴ௕

where ܧ௕ is the energy per bit ܴ௕ is the bit rate

134 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Procedure Outline

Equation (76) shows that the ratio ܧ௕Τܰ଴ is a normalized expression of the signal-to-noise ratio.

a Measuring the ratio ܧ௕Τܰ଴ is beyond the scope of this manual.

PROCEDURE OUTLINE The Procedure is divided into the following sections:

System startup Noise figure Set up for measuring noise figure. Measuring received carrier power and RF front end gain. Measuring the gain of the preamplifier. Measuring the receiver output noise power. Calculating the receiver noise figure and effective input noise temperature. System noise temperature Link budget

Before carrying out the following exercise Procedure, be sure to read a Appendix F Noise Measurement Using a Spectrometer on page 187.

PROCEDURE System startup

1. If not already done, set up the system and align the antennas visually as shown in Appendix B.

2. Make sure that no hardware faults have been activated in the Earth Station Transmitter or the Earth Station Receiver.

Faults in these modules are activated for troubleshooting exercises using DIP b switches located behind a removable panel on the back of these modules. For normal operation, all fault DIP switches should be in the “O” position.

3. Turn on each module that has a front panel Power switch (push the switch into the I position). After a few seconds, the Power LED should light.

4. If you are using the optional Telemetry and Instrumentation Add-On: x Make sure there is a USB connection between the Data Generation/Acquisition Interface, the Virtual Instrument, and the host computer, as described in Appendix B. x Turn on the Virtual Instrument using the rear panel power switch.

If the TiePieSCOPE drivers need to be installed, this will be done b automatically in Windows 7 and 8. In Windows XP, the Found New Hardware Wizard will appear (it may appear twice). In this case, do not connect to Windows Update (select No, not this time and click Next). Then select Install the software automatically and click Next. x Start the Telemetry and Instrumentation application. In the Application Selector, do not select Work in stand-alone mode.

If the Telemetry and Instrumentation application is already running, exit b and restart it. This will ensure that no faults are active in the Satellite Repeater.

Noise figure

Set up for measuring noise figure

5. Position the Earth Station Receiver, its antenna, and the spectrum analyzer so that you can easily connect this antenna to either to the RF INPUT of the Earth Station Receiver or to the input of the spectrum analyzer without putting tension on the cable.

During this exercise, you will connect the antenna at the Earth Station a Receiver to the RF INPUT of the Earth Station Receiver and then directly to the input of the spectrum analyzer. It is important to be able to make these connections without putting tension on the cable as this could move the antenna or change its orientation.

6. Make the connections shown in Figure 60 (you will require the RF Amplifier and the spectrum analyzer, but do not connect them now). In this exercise, the external RF Amplifier will be used as a spectrum analyzer preamplifier when measuring the noise power, as described in Appendix F. The RF Amplifier and the spectrum analyzer will be connected later as required. a In this exercise, the RF Amplifier is referred to as the “preamplifier.”

RF OUTPUT Digital Up Converter Up Converter Modulator 1 2 I Q I Q Large-Aperture Earth Station Transmitter Horn Antenna I OUTPUT Q OUTPUT (Uplink) to to I INPUT Q INPUT RF OUTPUT Small-Aperture Satellite Small-Aperture Horn Antenna Horn Antenna (Uplink) Repeater (Downlink)

Long cable Earth Station Receiver

Down Converter Down Converter Preamplifier 2 1 (RF Amplifier) Large-Aperture Horn Antenna (Downlink) Spectrum Analyzer

Figure 60. Connections for an unmodulated carrier.

7. On the Earth Station Transmitter and the Earth Station Receiver, select a Channel not being used by another system in the same laboratory.

136 © Festo Didactic 86312-10 Exercise 3 – Noise and the Link Budget Procedure

On the Earth Station Transmitter, make the following adjustments:

Data Source...... Sampler Scrambler ...... Off Clock & Frame encoder ...... Off

The signal at the output of the Digital Modulator is an unmodulated carrier. The RF OUTPUT signal from the transmitter is the same unmodulated carrier shifted up to the uplink frequency.

8. Align the antennas and optimize the alignment (see Aligning the antennas on page 38). Make sure the knob on the mast of the antenna at the Earth Station Receiver is tight.

Adjust the Gain control on the Earth Station Receiver so that the green Level LED is lit.

Measuring received carrier power and RF front end gain

In this section, you will measure the carrier power at the input of the receiver and at the output of the RF front end (the IF 1 OUTPUT) with the Gain control set to maximum. This will allow you to calculate the maximum gain on the RF front end, which consists of Down Converter 2 and Down Converter 1.

9. Taking care not to change the orientation of the antenna, disconnect the cable at the RF INPUT of the Earth Station Receiver and connect it to the input of the spectrum analyzer, as shown in Figure 61. The antenna should not move at all (i.e. its orientation should not change). If it does move, reconnect it to the Earth Station Receiver and realign the antennas.

Earth Station Receiver

Down Converter Down Converter Preamplifier 2 1 (RF Amplifier)

Spectrum Analyzer

Figure 61. Measuring ࡯࢏. As the signal received by the antenna is very weak, no attenuation will be b required on the spectrum analyzer. Use averaging, if necessary, to reduce fluctuations. Use a marker or a cursor to read the power of the spectral line.

Enter the measured input carrier power ܥ௜ in the first column of Table 32.

Table 32. Maximum gain of the receiver RF front end.

࡯࢏ ࡯࢕ ࡳࡾ࢞ ൌ࡯࢕ െ࡯࢏ [dBm] [dBm] [dB]

Table 32. Maximum gain of the receiver RF front end.

࡯࢏ ࡯࢕ ࡳࡾ࢞ ൌ࡯࢕ െ࡯࢏ [dBm] [dBm] [dB]

-45.5 1.57 47.1

You can expect the carrier at the IF 1 OUTPUT of the receiver to be approximately 45 dB above the signal received by the antenna. On the spectrum analyzer, select the appropriate scale and input attenuation level required in order to avoid overloading the instrument (if you are using the Telemetry and Instrumentation Add-On, connect the 20 dB Attenuator to the Frequency Converter and add 20 dB to the displayed level).

Being very careful not to move the antenna, disconnect the cable from the input of the spectrum analyzer and connect it to the RF INPUT of the Earth Station Receiver, as shown in Figure 62. Use the remaining long cable to connect the IF 1 OUTPUT to the spectrum analyzer.

Earth Station Receiver

Down Converter Down Converter Preamplifier 2 1 (RF Amplifier)

Spectrum Analyzer

Figure 62. Measuring ࡯࢕.

On the Earth Station Receiver, turn the Gain control to the maximum position and measure the carrier power ܥ௢ at the IF 1 OUTPUT using the required settings and attenuation. Enter this value into Table 32.

Calculate the gain of the receiver and enter this into Table 32.

Measuring the gain of the preamplifier

10. The gain of the preamplifier (RF Amplifier) is approximately 20 dB. However, it is preferable measure its gain more precisely.

On the Earth Station Receiver, reduce the Gain control until the peak on the spectrum analyzer is approximately -20 dBm (considering any attenuation used). Do not change the Gain setting for the rest of this section. Note the power ܲ௜௡ in the first column of Table 33.

Table 33. Preamplifier gain calculation.

ࡼ࢏࢔ ࡼ࢕࢛࢚ ࡳ࢖࢘ࢋ ൌࡼ࢕࢛࢚ െࡼ࢏࢔ [dBm] [dBm] [dB]

Table 33. Preamplifier gain calculation.

ࡼ࢏࢔ ࡼ࢕࢛࢚ ࡳ࢖࢘ࢋ ൌࡼ࢕࢛࢚ െࡼ࢏࢔ [dBm] [dBm] [dB]

-19.7 -0.1 19.6

11. Disconnect the spectrum analyzer from the receiver.

Power off the receiver and disconnect the power cable on the rear panel of the receiver.

Connect the RF amplifier directly to the IF 1 OUTPUT of the receiver, as shown in Figure 63, using a male SMA-SMA adapter. Tighten the connectors so that they are snug but not over tightened. A small wrench for this purpose is provided in the Accessories.

RF Amplifier

RF Amplifier output

Cable to receiver power input (on rear panel)

Cable from receiver power supply

Figure 63. RF Amplifier connected to the IF 1 OUTPUT of the receiver.

The RF Amplifier is powered by the receiver power supply as shown in Figure 63. Connect the power cable from the receiver power supply to the connector on the bottom of the RF Amplifier. Plug the power cable from the RF Amplifier into the power connector on the rear panel of the receiver.

Power on the receiver.

12. Connect the spectrum analyzer as shown in Figure 64.

Earth Station Receiver

Down Converter Down Converter Preamplifier 2 1 (RF Amplifier)

Spectrum Analyzer

Figure 64. Measuring preamplifier gain.

Note the power of the peak displayed on the spectrum analyzer in the second column of Table 33. Calculate the gain of the preamplifier.

Measuring the receiver output noise power

13. Disconnect the cable at the RF INPUT of the Earth Station Receiver and connect a 50 load (a matched load) in its place.

Earth Station Receiver

Down Converter Down Converter Preamplifier 50 2 1 (RF Amplifier)

Spectrum Analyzer

Figure 65. Measuring ࡺ࢕.

Power off the Earth Station Transmitter. This will reduce extraneous noise.

On the Earth Station Receiver, turn the Gain control to the maximum position and measure the noise level at the output of the preamplifier using

the spectrum analyzer. Enter this value as ܰ࢖࢘ࢋǡ࢕ in Table 34.

As mentioned in Appendix F, it is preferable to use a wide span on the a spectrum analyzer as this collects more noise power.

Fill in the other rows of Table 34 using the appropriate units.

Most spectrum analyzers display the resolution bandwidth corresponding to b the selected span. If this is not the case, consult the user’s manual of the spectrum analyzer. The Averaging correction depends on the type of averaging used on the spectrum analyzer. For video averaging or video filtering, this is +2.5 dB. For power (RMS) averaging, it is 0 dB. For FFT spectrum analyzers, it is also 0 dB.

For most conventional spectrum analyzers, the Equivalent noise bandwidth correction required may range from approximately -0.2 dB to -0.5 dB. If this value is not specified in the user documentation, a value of -0.5 dB is recommended. For FFT spectrum analyzers, this correction is 0 dB. The virtual Spectrum Analyzer of the Telemetry and Instrumentation Add-On is an FFT spectrum analyzer.

Table 34. Measuring noise power at IF 1 OUTPUT.

Measured preamplifier output noise power ࡺ࢖࢘ࢋǡ࢕

Preamplifier gain ࡳ࢖࢘ࢋ

Spectrum analyzer Span

Resolution bandwidth ࡾ࡮ࢃ for this Span

Averaging correction ࡯࢕࢘࢘ࢇ࢜ࢍ

Equivalent noise bandwidth correction ࡯࢕࢘࢘ࡱࡺ࡮

Corrected receiver output noise power in RBW ࡺࡾ࡮ࢃǡ࢕ ൌࡺ࢖࢘ࢋǡ࢕ െࡳ࢖࢘ࢋ ൅࡯࢕࢘࢘ࢇ࢜ࢍ ൅࡯࢕࢘࢘ࡱࡺ࡮ Receiver output noise power spectral density ࡺ૙࢕ ൌࡺࡾ࡮ࢃǡ࢕ ൅૚૙ܔܗ܏ሺ૚ࡾ࡮ࢃΤሻ

Table 34. Measuring noise power at IF 1 OUTPUT.

Using the Telemetry and Instrumentation Add-On:

Measured preamplifier output noise power ࡺ࢖࢘ࢋǡ࢕ -49.8 dBm

Preamplifier gain ࡳ࢖࢘ࢋ 19.6 dB Spectrum analyzer Span 5 MHz/div. Resolution bandwidth ࡾ࡮ࢃ for this Span 302 734 Hz

Averaging correction ࡯࢕࢘࢘ࢇ࢜ࢍ 0 dB

Equivalent noise bandwidth correction ࡯࢕࢘࢘ࡱࡺ࡮ 0 dB Corrected receiver output noise power in RBW -69.4 ࡺࡾ࡮ࢃǡ࢕ ൌࡺ࢖࢘ࢋǡ࢕ െࡳ࢖࢘ࢋ ൅࡯࢕࢘࢘ࢇ࢜ࢍ ൅࡯࢕࢘࢘ࡱࡺ࡮ Receiver output noise power spectral density -124.2 dBm/Hz ࡺ૙࢕ ൌࡺࡾ࡮ࢃǡ࢕ ൅૚૙ܔܗ܏ሺ૚ࡾ࡮ࢃΤሻ

Using a conventional spectrum analyzer and power averaging (example):

Measured preamplifier output noise power ࡺ࢖࢘ࢋǡ࢕ -39.0 dBm

Preamplifier gain ࡳ࢖࢘ࢋ 19.6 dB Spectrum analyzer Span 300 MHz Resolution bandwidth ࡾ࡮ࢃ for this Span 3 MHz

Averaging correction ࡯࢕࢘࢘ࢇ࢜ࢍ 0

Equivalent noise bandwidth correction ࡯࢕࢘࢘ࡱࡺ࡮ -0.5 dB Corrected receiver output noise power in RBW -59.1 dBm ࡺࡾ࡮ࢃǡ࢕ ൌࡺ࢖࢘ࢋǡ࢕ െࡳ࢖࢘ࢋ ൅࡯࢕࢘࢘ࢇ࢜ࢍ ൅࡯࢕࢘࢘ࡱࡺ࡮ Receiver output noise power spectral density -123.9 dBm/Hz ࡺ૙࢕ ൌࡺࡾ࡮ࢃǡ࢕ ൅૚૙ܔܗ܏ሺ૚ࡾ࡮ࢃΤሻ

Using a conventional spectrum analyzer and video averaging (example):

Measured preamplifier output noise power ࡺ࢖࢘ࢋǡ࢕ -41.5 dBm

Preamplifier gain ࡳ࢖࢘ࢋ 19.6 Spectrum analyzer Span 300 MHz Resolution bandwidth ࡾ࡮ࢃ for this Span 3 MHz

Averaging correction ࡯࢕࢘࢘ࢇ࢜ࢍ 2.5 dB

Calculating the receiver noise figure and effective input noise temperature

14. Using the receiver gain ܩோ௫ from Table 32, the receiver input noise power spectral density ܰ଴௜ from Equation (56), and the receiver output noise power spectral density ܰ଴௢ from Table 34 (all in dB), calculate the noise figure ܰܨோ௫ of the Earth Station Receiver.

ݔሾሿܴܩோ௫ ൌ ܰͲ݋ሾሿ െܰͲ݅ሾሿ െܨܰ ൌ െͳʹͶǤʹ െ ሺെͳ͹Ͷሻ െ Ͷ͹Ǥͳ ൌʹǤ͹

Calculate the noise factor ܨோ௫ of the receiver and the equivalent input noise temperature ܶ௘ோ௫.

ேிΤ ଵ଴ ܨோ௫ ൌͳͲ ൌͳͲ଴Ǥଶ଻ ൌͳǤͺ͸

ܶ௘ோ௫ ൌ ሺܨோ௫ െͳሻܶ଴ ൌ ሺͳǤͺ͸ െ ͳሻʹͻͲ ൌ ʹͶͻ

System noise temperature

15. Make sure the Earth Station Transmitter is turned off.

While observing the spectrum analyzer display, remove the 50 ȳ load at the RF INPUT of the receiver and connect the antenna. Is the noise level at the IF 1 OUTPUT different with the antenna connected rather than the 50 ȳ load?

No, there was no change in the displayed noise level.

What can you conclude about the noise temperature of the antenna and cable together as compared to the noise temperature of the 50 ȳ load?

The noise temperature of the antenna and cable together is the same as that of the 50 ȳ load, that is, ܶ଴.

16. Considering that the antenna temperature ܶ஺ and the feed cable temperature ܶி are both 290K, calculate the system temperature ܶ referred to the input of the receiver.

ܶ஺ ͳ ܶൌ ൅ܶி ൬ͳ െ ൰൅ܶ௘ோ௫ ܮி ܮி ʹͻͲ ͳ ൌ ൅ ʹͻͲ ൬ͳ െ ൰ ൅ ʹͶͻ ܮி ܮி ൌ ʹͻͲ ൅ ʹͶͻ ൌ ͷ͵ͻ

To check your calculation, recall the definition of system temperature ܶ. Then calculate what the output noise power spectral density of the receiver would be using this system temperature. Compare the results of your calculation with the receiver output noise power spectral density determined in Table 34.

The system temperature, referred to the input of the receiver, is the temperature in kelvin of a fictitious resistance, placed at the receiver input in a system made up of ideal (noiseless) components that would result in the same output noise power as produced by the actual system.

The output noise power spectral density of such a system with a gain of 47.1 dB is calculated as follows:

ܶ ൌ ͷ͵ͻ

ܩோ௫ ൌͶ͹Ǥͳ

ܰ଴௢ ൌܩோ௫݇ܶ

ܰ଴௢ሾΤ ሿ ൌܩோ௫ሾሿ ൅݇ሾΤሿ ή ൅ͳͲܶ ൌ Ͷ͹Ǥͳ െ ͳͻͺǤ͸ ൅ ʹ͹Ǥ͵ ൌ െͳʹͶǤʹȀ

This is exactly the same receiver output noise power spectral density measured in Table 34.

Link budget

In this section, you will calculate the figure of merit of the receiver. You will also ܥܰΤ ଴ value for the downlink and compare this with the measured value.

17. Measure the distance ܴ between the downlink antennas and enter this in Table 35.

Measure the power ்ܲ௫ at the output of the repeater. If you use the Power Sensor on the repeater (via telemetry), you can measure this power directly. If you use a spectrum analyze, you must take into consideration the cable loss. Enter this power in Table 35.

Then fill in the remainder of the table. These values will be used in subsequent calculations and should be entered in the following tables as required.

Table 35. Initial values for link budget.

Channel

Downlink frequency ࢌࢊ࢕࢝࢔ GHz (from Table 1 on page 32)

Distance ࡾ m

Transmitter power ࡼࢀ࢞ dBm ࡼࢀ࢞ ൌ

Gain of small-aperture horn antenna ࡳࢀ dBi (from Table 10 on page 56)

Gain of large-aperture horn antenna ࡳࡾ dBi (from Table 12 on page 59)

Receiver noise figure ࡺࡲ dB (from Step 14 on page 143)

Receiver noise temperature ࢀࢋࡾ࢞ K (from Step 14 on page 143)

Table 35. Initial values for link budget.

Channel

Downlink frequency ࢌࢊ࢕࢝࢔ 9.0 GHz (from Table 1 on page 3232)

Distance ࡾ 2.01 m (measured)

Transmitter power ࡼࢀ࢞ -12.4 dBm ࡼࢀ࢞ ൌࡼࡼ࢕࢝ࢋ࢘ ࡿࢋ࢔࢙࢕࢘ or ࡼࢀ࢞ ൌࡼ࢙࢖ࢋࢉ࢚࢛࢘࢓ ࢇ࢔ࢇ࢒࢟ࢠࢋ࢘ ൅ࡸࡲ

Gain of small-aperture horn antenna ࡳࢀ 14.2 dBi (from Table 10 on page 56)

Gain of large-aperture horn antenna ࡳࡾ 18.4 dBi (from Table 12 on page 59)

Receiver noise figure ࡺࡲ 2.7 dB (from Step 14 on page 143)

Receiver noise temperature ࢀࢋࡾ࢞ 249 K (from Step 14 on page 143)

18. Fill in Table 36 making the necessary calculations.

Table 36. Calculating downlink EIRP (using the repeater as a transmitter).

Item Symbol Value Value (dB)

Transmitter power ்ܲ௫

Transmitter antenna gain (small horn) ܩ்

Maximum transmitter antenna pointing loss ܮఏ் 0 dB

Transmitter antenna feed-line loss ܮி்௫

Effective isotropic radiated power ܧܫܴܲ

Table 36. Calculating downlink EIRP (using the repeater as a transmitter).

Item Symbol Value Value (dB)

Transmitter power ்ܲ௫ 0.058 mW -12.4 dBm

Transmitter antenna gain (small horn) ܩ் 14.2 dBi

Maximum transmitter antenna pointing loss ܮఏ் 0 dB

Transmitter antenna feed-line loss ܮி்௫ 2.4 dB

Effective isotropic radiated power ܧܫܴܲ (0.87 mW) -0.6 dBm

ܧܫܴܲሾௗ஻௠ሿ ൌ்ܲ௫[dBm] ൅ܩ் െܮఏ் െܮி்௫[dB]

19. Calculate the path loss ܮ using:

ܮൌܮிௌܮ஺ Ͷߨܴ݂ ଶ ൌ൬ ൰ ܮ ܿ ஺

ܮሾௗ஻ሿ ൌͳͲܮிௌ ൅ܮ஺ሾௗ஻ሿ

Table 37. Calculating path loss.

Item Symbol Value Value (dB)

Distance ܴ

Frequency ݂ௗ௢௪௡

Free-space loss ܮிௌ

Atmospheric attenuation ܮ஺ 0 dB

Path loss ܮ

Table 37. Calculating path loss.

Item Symbol Value Value (dB)

Distance ܴ 2.01 m

Frequency ݂ௗ௢௪௡ 9 GHz

5 Free-space loss ܮிௌ 5.74x10 57.6 dB

Atmospheric attenuation ܮ஺ 0 dB

Path loss ܮ 57.6 dB

20. The composite receiver gain ܩ is calculated using the equation:

ܩ ܩൌ ோ ܮఏோܮிோ௫ܮ௣௢௟

When all values are expressed in decibels, the composite receiver gain ܩ is:

ܩൌܩோ െܮఏோെܮிோ௫ െܮ௣௢௟

Table 38. Calculating composite receiver gain.

Item Symbol Value Value (dB)

Receiver antenna gain ܩோ

Receiver antenna pointing loss ܮఏோ 0 dB

Receiver antenna feed-line loss ܮிோ௫

Polarization mismatch loss ܮ௣௢௟

Composite receiving gain ࡳ

Table 38. Calculating composite receiver gain.

Item Symbol Value Value (dB)

Receiver antenna gain (large horn) ܩோ 18.4 dB

Receiver antenna pointing loss ܮఏோ 0 dB

Receiver antenna feed-line loss ܮிோ௫ 1.58 2.4 dB

Polarization mismatch loss ܮ௣௢௟ 0 dB

Composite receiving gain ࡳ 39.8 16.0 dB

21. Both the antenna and the feed line are at room temperature (290 K).

The system noise temperature ܶ is calculated using:

ܶ௘ோ௫ ൌ ሺܨோ௫ െͳሻܶ଴

ிೃೣሾሿ ൌቆͳͲ ଵ଴ െ ͳቇ ʹͻͲ

ܶ஺ ͳ ܶൌ ൅ܶி ൬ͳ െ ൰൅ܶ௘ோ௫ ܮி ܮி

Table 39. Calculating system noise temperature.

Item Symbol Value Value (dB)

Antenna temperature ܶ஺

Feed line temperature ܶி

Receiver noise figure ܨோ௫

Receiver noise temperature ܶ௘ோ௫

System noise temperature ࢀ

Table 39. Calculating system noise temperature.

Item Symbol Value Value (dB)

Antenna temperature ܶ஺ 290 K

Feed line temperature ܶி 290 K

Receiver noise figure ܨோ௫ 1.86 2.7 dB

Receiver noise temperature ܶ௘ோ௫ 249 K

System noise temperature ࢀ 539 K

22. The figure of merit of the receiver is ܩܶΤ . The link performance is characterized by the value ܥܰΤ ଴:

ܥ ͳ ܩ ͳ ൌܧܫܴܲ൬ ൰൬ ൰ ܰ଴ ܮ ܶ ݇

When the values ܥܰΤ ଴, ܧܫܴܲ, ܮ, ܩܶΤ , and ݇ are expressed in decibels, the equation is:

ܥ ܩ ൌܧܫܴܲെܮ൅ െ݇ ܰ଴ ܶ

Table 40. Calculating figure of merit and ࡯ࡺΤ ૙.

Item Symbol Value Value (dB)

Figure of merit ܩܶΤ

Boltzmann’s constant ݇

Link performance figure ࡯ࡺΤ ૙

Table 40. Calculating figure of merit and ࡯ࡺΤ ૙.

Item Symbol Value Value (dB)

Figure of merit ܩܶΤ 0.074 K-1 -11.3 dB/K

Boltzmann’s constant ݇ 1.38E-23 W K-1 Hz-1 -198.6 dBm K-1 Hz-1

Link performance figure ࡯ࡺΤ ૙ 129.1 dBHz

ܥ ܩ [dB ] ൌܧܫܴܲെܮ൅ െ݇[all in dB] ܰ଴ ܶ

23. Use the values measured in Table 32 and Table 34 to determine the link performance figure ܥܰΤ ଴ at the IF 1 OUTPUT of the receiver.

Table 41. Measured link performance figure.

Carrier power dBm ࡯࢕

Noise power spectral density dBm/Hz ࡺ૙

Link performance figure dBHz ࡯ࡺΤ ૙

How does the ܥܰΤ ଴ value determined by measurement in Table 41 compare with the calculated value in Table 40?

Table 42. Measured link performance figure.

Carrier power 1.57 dBm ࡯࢕

Noise power spectral density -124.2 dBm/Hz ࡺ૙

Link performance figure 125.8 dBHz ࡯ࡺΤ ૙

The measured value and the calculated value are fairly close.

24. When you have finished using the system, exit any software being used and turn off the equipment.

CONCLUSION In this exercise, you measured the noise figure of the receiver and calculated the receiver noise temperature. Then you determined the system noise temperature referred to the input of the receiver and calculated the figure of merit ܩܶΤ . You also calculated the value ܥܰΤ ଴ for the system.

REVIEW QUESTIONS 1. Describe additive white Gaussian noise (AWGN)

Additive white Gaussian noise (AWGN) is a model for noise in communications channels. It describes noise as additive (adding positive and negative random values to the amplitude of the signal), white (having equal noise power per unit bandwidth over all frequencies of interest) and Gaussian (having a normal amplitude probability distribution).

2. What is the noise figure of a device?

The noise figure is the degradation or decrease of the signal-to-noise ratio as the signal passes through the device.

3. What is the noise temperature of a device?

The noise temperature, or more precisely, the effective input noise temperature of a device, is the source noise temperature in kelvin in a device that will result in the same output noise power, when connected to a noise- free device, as that of the actual device connected to a noise-free source

4. Define the system noise temperature.

The system noise temperature is the temperature in kelvin of a fictitious resistance, placed at a specified point in a system made up of ideal (noiseless) components that would result in the same output noise power as produced by the actual system.

5. What is the importance of the ܥܰΤ ଴ ratio of a communications link?

The ܥܰΤ ଴ ratio (carrier power to the noise power spectral density) determines the performance of a link.

Bibliography

Agilent Technologies, Spectrum Analysis Basics, Application Note 150 (5952- 0292.pdf).

Agilent Technologies, Spectrum and Signal Analyzer Measurements and Noise, Application Note (5966-4008E.pdf).

Antenna-theory.com, http://www.antenna-theory.com.

Chartrand, Mark R., Satellite Communications for the Nonspecialist, Bellingham, SPIE Press, 2004, ISBN 0-8194-5185-1.

Haslett, Christopher, Essentials of Radio Wave Propagation, Cambridge, Cambridge University Press, 2008, ISBN 978-0-511-37112-7.

Hewlett Packard, Spectrum Analysis noise figure measurements, Application Note 150-9 (5952-9229.pdf).

Maral, Gérard and Bousquet, Michel, Satellite Communications Systems, Fourth Edition, Chichester, John Wiley & Sons, 2002, ISBN 978-0-471-49654-0.

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