Communication Systems/Print Version

Current Status: 25% developed

Introduction edit

This book will eventually cover a large number of topics in the field of electrical communications. The reader will also require a knowledge of Time and Frequency Domain representations, which is covered in-depth in the Signals and Systems book. This book will, by necessity, touch on a number of different areas of study, and as such is more than just a text for aspiring Electrical Engineers. This book will discuss topics of analog communication schemes, computer programming, network architectures, information infrastructures, communications circuit analysis, and many other topics. It is a large book, and varied, but it should be useful to any person interested in learning about an existing communication scheme, or in building their own. Where previous Electrical Engineering books were grounded in theory (notably the Signals and Systems book), this book will contain a lot of information on current standards, and actual implementations. It will discuss how current networks and current transmission schemes work, and may even include information for the intrepid engineer to create their own versions of each.

This book is still in an early stage of development. Many topics do not yet have pages, and many of the current pages are stubs. Any help would be greatly appreciated.

Table of Contents edit

Introduction edit

People are prone to take for granted the fact that modern technology allows us to transmit data at nearly the speed of light to locations that are very far away. 200 years ago, it would be deemed preposterous to think that we could transmit webpages from China to Mexico in less than a second. It would seem equally preposterous to think that people with cellphones could be talking to each other, clear as day, from miles away. Today, these things are so common, that we accept them without even asking how these miracles are possible.

What is Communications? edit

Communications is the field of study concerned with the transmission of information through various means. It can also be defined as technology employed in transmitting messages. It can also be defined as the inter-transmitting the content of data (speech, signals, pulses etc.) from one node to another.

To Whom is This Book For? edit

This book is for people who have read the Signals and Systems wikibook, or an equivalent source of the information. Topics considered in this book will rely heavily on knowledge of Fourier Domain representation and the Fourier Transform. This book can be used to accompany a number of different classes spanning the 3rd and fourth years in a study of electrical engineering. Knowledge of integral and differential calculus is assumed. The reader may benefit from knowledge of such topics as semiconductors, electromagnetic wave propagation, etc., although these topics are not necessary to read and understand the information in this book.

What will this Book Cover edit

This book is going to take a look at nearly all facets of electrical communications, from the shape of the electrical signals, to the issues behind massive networks. It makes little sense to be discussing these subjects outside the realm of current examples. We have the Internet, so in discussing issues concerning digital networks, it makes good sense to reference these issues to the Internet. Likewise, this book will attempt to touch on, at least briefly, every major electrical communications network that people deal with on a daily basis. From AM radio to the Internet, from DSL to cable TV, this book will attempt to show how the concepts discussed apply to the real world.

This book also acknowledges a simple point: It is easier to discuss the signals and the networks simultaneously. For this kind of task to be undertaken in a paper book would require hundreds, if not thousands of printed pages, but through the miracle of Wikimedia, all this information can be brought together in a single, convenient location.

This book would like to actively solicit help from anybody who has experience with any of these concepts: Computer Engineers, Communications Engineers, Computer Programmers, Network Administrators, IT Professionals. Also, this book may cover all these topics, but the reader doesn't need to have prior knowledge of all these disciplines to advance. Information will be developed as completely as possible in the text, and links to other information sources will be provided as needed.

Where to Go From Here edit

Since this book is designed for a junior and senior year of study, there aren't necessarily many topics that will logically follow this book. After reading and understanding this material, the next logical step for the interested engineer is either industry or graduate school. Once in graduate school, there are a number of different areas to concentrate study in. In industry, the number is even higher.

Division of Material edit

Admittedly, this is a very large topic, one that can span not only multiple printed books, but also multiple bookshelves. It could then be asked "Why don't we split this book into 2 or more smaller books?" This seems like a good idea on the surface, but you have to consider exactly where the division would take place. Some would say that we could easily divide the information between "Analog and Digital" lines, or we could divide up into "Signals and Systems" books, or we could even split up into "Transmissions and Networks" Books. But in all these possible divisions, we are settling for having related information in more than 1 place.

Analog and Digital edit

It seems most logical that we divide this material along the lines of analog information and digital information. After all, this is a "digital world", and aspiring communications engineers should be able to weed out the old information quickly and easily. However, what many people don't realize is that digital methods are simply a subset of analog methods with more stringent requirements. Digital transmissions are done using techniques perfected in analog radio and TV broadcasts. Digital computer modems are sending information over the old analog phone networks. Digital transmissions are analyzed using analog mathematical concepts such as modulation, SNR (signal to noise ratio), Bandwidth, Frequency Domain, etc... For these reasons, we can simplify both discussions by keeping them in the same book.

Signals and Systems edit

Perhaps we should divide the book in terms of the signals that are being sent, and the systems that are physically doing the sending. This makes some sense, except that it is impossible to design an appropriate signal without understanding the restrictions of the underlying network that it will be sent on. Also, once we develop a signal, we need to develop transmitters and receivers to send them, and those are physical systems as well.

Systems Approach edit

It is a bit confusing to be writing a book about Communication Systems and also considering the pedagogical Systems Approach. Although using the same word, they are not quite the same thing.

This approach is almost identical to the description above (Signals & Systems) except that it is not limited to the consideration of signals (common in many university texts), but can include other technological drivers (codecs, lasers, and other components).

In this case we give a brief overview of different communication systems (voice, data, cellular, satellite etc.) so that students will have a context in which to place the more detailed (and often generic) information. Then we can then zoom in on the mathematical and technological details to see how these systems do their magic. This lends itself quite well to technical subjects since the basic systems (or mathematics) change relatively slowly, but the underlying technology can often change rapidly and take unexpected terns.

I would like to suggest that the table of contents in this book be rearranged to reflect this pedagogical approach: Systems examples first, followed by the details.

Why would anyone want to study (tele)communications? edit

Telecommunications is an alluring industry with a provocative history filled with eccentric personalities: Bell, Heavyside, Kelvin, Brunel and many others. It is fraught with adventure and danger: adventure spanning space and time; danger ranging from the remote depths of the ocean floor to deep space, from the boardrooms of AT&T to the Hong Kong stock exchange.

Telecommunications has been heralded as a modern Messiah and cursed as a pathetic sham. It has created and destroyed empires and institutions. It has proclaimed the global village while sponsoring destructive nationalism. It has come to ordinary people, but has remained largely in the control of the ‘media’ and even 'big brother'. Experts will soon have us all traveling down a techno-information highway, destination — unknown.

Telecommunications has become the lifeblood of modern civilization. Besides all that, there’s big bucks in it

About This Book edit

There are a few points about this book that are worth mentioning:

  • The programming parts of this book will not use any particular language, although we may consider particular languages in dedicated chapters.

This page will attempt to show some of the basic history of electrical communication systems.

Chronology edit

1831 Samuel Morse invents the first repeater and the telegraph is born

1837 Charles Wheatstone patents "electric telegraph"

1849 England to France telegraph cable goes into service—and fails after 8 days.

1850 Morse patents "clicking" telegraph.

1851 England-France commercial telegraph service begins. This one uses gutta-percha, and survives.

1858 August 18 - First transatlantic telegraph messages sent by the Atlantic Telegraph Co. The cable deteriorated quickly, and failed after 3 weeks.

1861 The first transcontinental telegraph line is completed

1865 The first trans-Atlantic cable goes in service

1868 First commercially successful transatlantic telegraph cable completed between UK and Canada, with land extension to USA. The message rate is 2 words per minute.

1870 The trans-Atlantic message rate is increased to 20 words per minute.

1874 Baudot invents a practical Time Division Multiplexing scheme for telegraph. Uses 5-bit codes & 6 time slots—90 bps max. rate. Both Western Union and Murray would use this as the basis of multiplex telegraph systems.

1875 Typewriter invented.

1876 Alexander Graham Bell and Elisa Grey independently invent the telephone (although it may have been invented by Antonio Meucci as early as 1857)

1877 Bell attempts to use telephone over the Atlantic telegraph cable. The attempt fails.

1880 Oliver Heaviside's analysis shows that a uniform addition of inductance into a cable would produce distortionless transmission.

1883 Test calls placed over five miles of under-water cable.

1884 - San Francisco-Oakland gutta-percha cable begins telephone service.

1885 Alexander Graham Bell incorporated AT&T

1885 James Clerk Maxwell predicts the existence of radio waves

1887 Heinrich Hertz verifies the existence of radio waves

1889 Almon Brown Strowger invents the first automated telephone switch

1895 Gugliemo Marconi invents the first radio transmitter/receiver

1901 Gugliemo Marconi transmits the first radio signal across the Atlantic 1901 Donald Murray links typewriter to high-speed multiplex system, later used by Western Union

1905 The first audio broadcast is made

1910 Cheasapeake Bay cable is first to use loading coils underwater

1911 The first broadcast license is issued in the US

1912 Hundreds on the Titanic were saved due to wireless

1915 USA transcontinental telephone service begins (NY-San Francisco).

1924 The first video signal is broadcast

1927 First commercial transatlantic radiotelephone service begins

1929 The CRT display tube is invented

1935 Edwin Armstrong invents FM

1939 The Blitzkrieg and WW II are made possible by wireless

1946 The first mobile radio system goes into service in St. Louis

1948 The transistor is invented

1950 Repeatered submarine cable used on Key West-Havana route.

1956 The first trans-Atlantic telephone cable, TAT-1, goes into operation. It uses 1608 vacuum tubes.

1957 The first artificial satellite, Sputnik goes into orbit

1968 The Carterphone decision allows private devices to be attached to the telephone

1984 The MFJ (Modification of Final Judgement) takes effect and the Bell system is broken up

1986 The first transAtlantic fiber optic cable goes into service

Claude Shannon edit

Harry Nyquist edit

Communications Basics edit

It is important to know the difference between a baseband signal, and a broad band signal. In the Fourier Domain, a baseband signal is a signal that occupies the frequency range from 0 Hz up to a certain cutoff. It is called the baseband because it occupies the base, or the lowest range of the spectrum.

In contrast, a broadband signal is a signal which does not occupy the lowest range, but instead a higher range, 1 MHz to 3 MHz, for example. A wire may have only one baseband signal, but it may hold any number of broadband signals, because they can occur anywhere in the spectrum.

Wideband vs Narrowband edit

in form of frequency modulation. wideband fm has been defined as that in which the modulation index normally exceeds unity.

Frequency Spectrum edit

A graphical representation of the various frequency components on a given transmission medium is called a frequency spectrum.

Consider a situation where there are multiple signals which would all like to use the same wire (or medium). For instance, a telephone company wants multiple signals on the same wire at the same time. It certainly would save a great deal of space and money by doing this, not to mention time by not having to install new wires. How would they be able to do this? One simple answer is known as Time-Division Multiplexing.

Time Division Multiplexing edit

Time-Division Multiplexing (TDM) is a convenient method for combining various digital signals onto a single transmission media such as wires, fiber optics or even radio. These signals may be interleaved at the bit, byte, or some other level. The resulting pattern may be transmitted directly, as in digital carrier systems, or passed through a modem to allow the data to pass over an analog network. Digital data is generally organized into frames for transmission and individual users assigned a time slot, during which frames may be sent. If a user requires a higher data rate than that provided by a single channel, multiple time slots can be assigned.

Digital transmission schemes in North America and Europe have developed along two slightly different paths, leading to considerable incompatibility between the networks found on the two continents.

BRA (basic rate access) is a single digitized voice channel, the basic unit of digital multiplexing.


North American TDM edit

The various transmission rates are not integral numbers of the basic rate. This is because additional framing and synchronization bits are required at every multiplexing level.


In North America, the basic digital channel format is known as DS-0. These are grouped into frames of 24 channels each. A concatenation of 24 channels and a start bit is called a frame. Groups of 12 frames are called multiframes or superframes. These vary the start bit to aid in synchronizing the link and add signaling bits to pass control messages.


S Bit Synchronization edit

The S bit is used to identify the start of a DS-1 frame. There are 8 thousand S bits per second. They have an encoded pattern, to aid in locating channel position within the frame.


This forms a regular pattern of 1 0 1 0 1 0 for the odd frames and 0 0 1 1 1 0 for the even frames. Additional synchronization information is encoded in the DS-1 frame when it is used for digital data applications, so lock is more readily acquired and maintained.

For data customers, channel 24 is reserved as a special sync byte, and bit 8 of the other channels is used to indicate if the remaining 7 bits are user data or system control information. Under such conditions, the customer has an effective channel capacity of 56 Kbps.

To meet the needs of low speed customers, an additional bit is robbed to support sub-rate multiplexer synchronization, leaving 6 x 8 Kbps = 48 Kbps available. Each DS-0 can be utilized as:

• 5 x 9.6 Kbps channels or
• 10 x 4.8 Kbps channels or
• 20 x 2.48 Kbps channels.

In the DS-2 format, 4 DS-1 links are interleaved, 12 bits at a time. An additional 136 Kbps is added for framing and control functions resulting in a total bit rate of 6.312 Mbps.

Signaling edit

Signaling provides control and routing information. Two bits, called the A and B bits, are taken from each channel in frames 6 and 12 in the multiframe. The A bit is the least significant bit in each channel in frame 6, and the B bit is the least significant bit in each channel in frame 12. This provides a signaling rate of 666 2/3 bps per channel.

The quality of voice transmission is not noticeably affected when 2% of the signal is robbed for signaling. For data, it may be a different story. If the data is encoded in an analog format such as FSK or PSK, then robbing bits is of no consequence, but if the data is already in digital form, then robbing bits results in unacceptable error rates. It is for this reason that in North America, a 64 Kbps clear channel cannot readily be switched through the PSTN. This means that data customers are limited to 56 Kbps clear channels. This simple condition has a profound effect on the development of new services such as ISDN. In most facilities, the A and B bits represent the status of the telephone hook switch, and correspond to the M lead on the E&M interface of the calling party.

ESF edit

CCITT has modified the North American digital hierarchy for the deployment of ISDN, by means of recommendation G.704. ESF consists of 24 DS-0 channels in a frame, but groups them into a 24-frame multiframe instead of the usual 12-frame multiframe.

The S bit is renamed the F bit, but only 1/4 of them are used for synchronization. This is possible because of improvements in frame search techniques and allows more signaling states to be defined.

Bit robbing is still used for signaling over an ESF link, but with the advent of ISDN, it will not be permitted. Instead, channel 24 is used to support a D channel.


Typical T1 CPE Application edit

The large telecom carriers are not the only ones who deploy high-speed TDM facilities. In many cases, heavy users of voice or data services can reduce their transmission costs by concentrating their numerous low speed lines on to a high speed facility.

There are many types of T1 multiplexers available today. Some are relatively simple devices, while others allow for channel concatenation, thus supporting a wide range of data rates. The ability to support multiple DS-0s allows for easy facilitation of such protocols as the video teleconferencing standard, Px64.


Multiplexers edit

Multiplexing units are often designated by the generic term Mab where a is input DS level and b is the output DS level. Thus, an M13 multiplexer combines 28 DS–1s into a single DS–3 and an M23 multiplexer combines 7 DS–2s into a single DS–3.


ZBTSI edit

ZBTSI (zero byte time slot interchange) is used on DS–4 links. Four DS-1 frames are loaded into a register, and renumbered 1–96. If there are any empty slots [all zeros], the first framing bit is inverted and all blank slots are relocated to the front of the frame. Channel 1 is then loaded with a 7-bit number corresponding to the original position of the first empty slot. Bit 8 used to indicate whether the following channel contains user information or another address for an empty slot.

If there is a second vacancy, bit 8 in the previous channel is set, and the empty slot address is placed in channel 2. This process continues until all empty positions are filled.

The decoding process at the receiver is done in reverse. Borrowing 1 in 4 framing bits for this system is not enough to cause loss of synchronization and provides a 64 Kbps clear channel to the end-user.

European TDM Carriers edit

European systems were developed along slightly different principles. The 64 Kbps channel is still the basic unit, but signaling is not included in each channel. Instead, common channel signaling is used. In a level 1 carrier, channels 0 and 16 are reserved for signaling and control. This subtle difference means that European systems did not experience the toll fraud and 56 k bottlenecks common to North American systems, and they experience a much larger penetration of ISDN services.


Zero Substitutions edit

In order to prevent transmission systems from loosing lock on the data stream, it is necessary to avoid long strings of zeros. One of the most effective ways of doing this is to replace the zeros with a predetermined code. This substitution must be done in such a way that the receiver can identify it and strip it off before passing the data stream to the client.

AMI provides a simple means of detecting substitutions. In the normal course of events, alternate marks are inverted. Therefor, deliberately inducing a bipolarvariation at the transmitter can alert the receiver of a substitution. However, a single violation is indistinguishable from a transmission error. Consequently, some additional condition must also occur.

There are two common methods to create a second condition:

· Create a second bipolar violation in the opposite direction, within a specified time. This has the effect of keeping the average signal level at zero.
· Count the number of marks from the last substitution to predict the next type of violation

B6ZS edit

B6ZS (binary six zero substitution) is used on T2 AMI transmission links.

Synchronization can be maintained by replacing strings of zeros with bipolar violations. Since alternate marks have alternate polarity, two consecutive pulses of the same polarity constitute a violation. Therefore, violations can be substituted for strings of zeros, and the receiver can determine where substitutions were made.

Since the last mark may have been either positive (+) or negative (-), there are two types of substitutions:


These substitutions force two consecutive violations. A single bit error does not create this condition.


B8ZS edit

This scheme uses the same substitution as B6ZS.


B3ZS edit

B3ZS is more involved than B6ZS, and is used on DS–3 carrier systems. The substitution is not only dependent on the polarity of the last mark, but also on the number of marks since the last substitution.


HDB3 edit

HDB3 (high density binary 3) introduces bipolar violations when four consecutive zeros occur. It can therefore also be called B4ZS. The second and thirds zeros are left unchanged, but the fourth zero is given the same polarity as the last mark. The first zero may be modified to a one to make sure that successive violations are of alternate polarity.


HDB3 is used in Europe. Violation, or V pulses are injected after three consecutive zeros. The fourth zero is given the same polarity as the last mark. In the event of long strings of zeros occurring, a succession of single polarity pulses would occur, and a dc offset would build-up.

To prevent this, the first zero in a group of 4, may be modified to a 1. This B or balancing pulse assures that successive violations are of alternate polarity.


Block Code Substitution edit

These schemes operate on bytes rather than a bit at a time. Some transmit the signal as binary levels, but most use multi-level pulses. Some authors categorize these as line codes.

A binary block code has the designation nBmB, where n input bits are encoded into m output bits. The most common of these is the 3B4B code.


In Europe 4B3T, which encodes 4 binary bits into 3 ternary levels, has been selected as the BRA for ISDN. In North America, 2B1Q which encodes 2 binary bits into 1 quaternary level has been selected for BRA.


Some block codes do not generate multilevel pulses. For example, 24B1P or 24B25B simply adds a P or parity bit to a 24 bit block.

Benefits of TDM edit

TDM is all about cost: fewer wires and simpler receivers are used to transmit data from multiple sources to multiple destinations. TDM also uses less bandwidth than Frequency-Division Multiplexing (FDM) signals, unless the bitrate is increased, which will subsequently increase the necessary bandwidth of the transmission.

Synchronous TDM edit

Synchronous TDM is a system where the transmitter and the receiver both know exactly which signal is being sent. Consider the following diagram:

Signal A ---> |---| |A|B|C|A|B|C|   |------| ---> Signal A
Signal B ---> |TDM| --------------> |De-TDM| ---> Signal B
Signal C ---> |---|                 |------| ---> Signal C

In this system, starting at time-slice 0, every third time-slice is reserved for Signal A; starting at time-slice 1, every third time-slice is reserved for Signal B; and starting at time-slice 2, every third time-slice is reserved for Signal C. In this situation, the receiver (De-TDM) needs only to switch after the signal on each time-slice is received.

The data flow of each input connection is divided into units where each input occupies one input time slot. Each input connection has a time slot alloted in the output irrespective of the fact whether it is sending data or not.

      A -----|A3|A2|A1|---> |---| .............|C3|B3|A3|C2|B2|A2|C1|B1|A1|   |------| --->  A
             |  |  |  |                        |        |        |
      B -----|B3|B2|B1|---> |MUX| -------------|--------|--------|----------> |De-MUX| --->  B
             |  |  |  |                        |        |        |
      C -----|C3|C2|C1|---> |---|              |        |        |            |------| --->  C
                   <-->                                 <-------->
                  Bit Interval                       Frame (x seconds)

Sync TDM is inefficient when one or more input lines have no data to send. Thus, it is used with lines with high data traffic.

Sampling rate is same for all signals. Maximum sampling rate = twice the maximum frequency all the signals.

Statistical TDM edit

Synchronous TDM is beneficial because the receiver and transmitter can both cost very little. However, consider the most well-known network: the Internet. In the Internet, a given computer might have a data rate of 1kbps when hardly anything is happening, but might have a data rate of 100kbps when downloading a large file from a fast server. How are the time-slices divided in this instance? If every time slice is made big enough to hold 100Kbps, when the computer isn't downloading any data, all of that time and electricity will be wasted. If every time-slice is only big enough for the minimum case, the time required to download bigger files will be greatly increased.

The solution to this problem is called Statistical TDM, and is the solution that the Internet currently uses. In Statistical TDM, each data item, known as the payload (we used time-slices to describe these earlier), is appended with a certain amount of information about who sent it, and who is supposed to receive it (the header). The combination of a payload and a header is called a packet. Packets are like envelopes in the traditional "snail mail" system: Each packet contains a destination address and a return address as well as some enclosed data. Because of this, we know where each packet was sent from and where it is going.

The downside to statistical TDM is that the sender needs to be smart enough to write a header, and the receiver needs to be smart enough to read the header and (if the packet is to be forwarded,) send the packet toward its destination.

Link Utilization edit

Statistical multiplexing attempts to maximize the use of a communication path. The study of this is often called queuing theory. A queue is simply a line of customers or packets waiting to be served. Under most circumstances, the arrival rate is unpredictable and therefor follows a random or Poisson distribution pattern, whereas the service time is constant.

The utilization or fraction of time actually used by a packet multiplexing system to process packets is given by:


The queue length or average number of items waiting to be served is given by:

A T1 link has been divided into a number of 9.6 Kbps channels and has a combined user data rate of 1.152 Mbps. Access to this channel is offered to 100 customers, each requiring 9.6 Kbps data 20% of the time. If the user arrival time is strictly random find the T1 link utilization.
The utilization or fraction of time used by the system to process packets is given by:
A 24 channel system dedicated to DATA, can place five 9.6 Kbps customers in each of 23 channels, for a total of 115 customers. In the above statistical link, 100 customers created an average utilization of 0.167 and were easily fitted, with room to spare if they transmit on the average 20% of the time. If however, the customer usage were not randomly distributed, then the above analysis would have to be modified.

This example shows the potential for statistical multiplexing. If channels were assigned on a demand basis (only when the customer had something to send), a single T1 may be able to support hundreds of low volume users.

A utilization above 0.8 is undesirable in a statistical system, since the slightest variation in customer requests for service would lead to buffer overflow. Service providers carefully monitor delay and utilization and assign customers to maximize utilization and minimize cost.

Packets edit

Packets will be discussed in greater detail once we start talking about digital networks (specifically the Internet). Packet headers not only contain address information, but may also include a number of different fields that will display information about the packet. Many headers contain error-checking information (checksum, Cyclic Redundancy Check) that enables the receiver to check if the packet has had any errors due to interference, such as electrical noise.

Duty Cycles edit

Duty cycle is defined as " the time that is effectively used to send or receive the data, expressed as a percentage of total period of time." The more the duty cycle , the more effective transmission or reception.

We can define the pulse width, τ, as being the time that a bit occupies from within its total alloted bit-time Tb. If we have a duty cycle of D, we can define the pulse width as:




The pulse width is equal to the bit time if we are using a 100% duty cycle.

Introduction edit

It turns out that many wires have a much higher bandwidth than is needed for the signals that they are currently carrying. Analog Telephone transmissions, for instance, require only 3 000 Hz of bandwidth to transmit human voice signals. Over short distances, however, twisted-pair telephone wire has an available bandwidth of nearly 100000 Hz!

There are several terrestrial radio based communications systems deployed today. They include:

  • Cellular radio
  • Mobile radio
  • Digital microwave radio

Mobile radio service was first introduced in the St. Louis in 1946. This system was essentially a radio dispatching system with an operator who was able to patch the caller to the PSTN via a switchboard. Later, an improved mobile telephone system, IMTS, allowed customers to dial their own calls without the need for an operator. This in turn developed into the cellular radio networks we see today.

The long haul PSTNs and packet data networks use a wide variety of transmission media including

  • Terrestrial microwave
  • Satellite microwave
  • Fiber optics
  • Coaxial cable

In this section, we will be concerned with terrestrial microwave systems. Originally, microwave links used FDM exclusively as the access technique, but recent developments are changing analog systems to digital where TDM is more appropriate.

Fixed Access Assignment edit

Three basic methods can be used to combine customers on to fixed channel radio links:

  • FDMA - (Frequency division multiple access) analog or digital
  • TDMA - (Time division multiple access) three conversation paths are time division multiplexed in 6.7 mSec time slots on a single carrier.
  • CDMA - (Code division multiple access) this uses spread spectrum techniques to increase the subscriber density. The transmitter hops through a pseudo-random sequence of frequencies. The receiver is given the sequence list and is able to follow the transmitter. As more customers are added to the system, the signal to noise will gradually degrade. This is in contrast to AMPS where customers are denied access once all of the frequencies are assigned code division multiple access [digital only]

What is FDM? edit

Frequency Division Multiplexing (FDM) allows engineers to utilize the extra space in each wire to carry more than one signal. By frequency-shifting some signals by a certain amount, engineers can shift the spectrum of that signal up into the unused band on that wire. In this way, multiple signals can be carried on the same wire, without having to divy up time-slices as in Time-Division Multiplexing schemes.In analog transmission, signals are commonly multiplexed using frequency-division multiplexing (FDM), in which the carrier bandwidth is divided into subchannels of different frequency widths, each carrying a signal at the same time in parallel

Traditional terrestrial microwave and satellite links employ FDM. Although FDM in telecommunications is being reduced, several systems will continue to use this technique, namely: broadcast & cable TV, and commercial & cellular radio.

Analog Carrier Systems edit

The standard telephony voice band [300 – 3400 Hz] is heterodyned and stacked on high frequency carriers by single sideband amplitude modulation. This is the most bandwidth efficient scheme possible.


The analog voice channels are pre-grouped into threes and heterodyned on carriers at 12, 16, and 20 kHz. The resulting upper sidebands of four such pregroups are then heterodyned on carriers at 84, 96, 108, and 120 kHz to form a 12-channel group.

Since the lower sideband is selected in the second mixing stage, the channel sequence is reversed and a frequency inversion occurs within each channel.


This process can continue until the available bandwidth on the coaxial cable or microwave link is exhausted.


In the North American system, there are:

  • 12 channels per group
  • 5 groups per supergroup
  • 10 super groups per mastergroup
  • 6 master groups per jumbogroup

In the European CCITT system, there are:

  • 12 channels per group
  • 5 groups per supergroup
  • 5 super groups per mastergroup
  • 3 master groups per supermastergroup

There are other FDM schemes including:

  • L600 - 600 voice channels 60–2788 kHz
  • U600 - 600 voice channels 564–3084 kHz
  • L3 - 1860 voice channels 312–8284 kHz, comprised of 3 mastergroups and a supergroup
  • L4 - 3600 voice channels, comprised of six U600s

Benefits of FDM edit

FDM allows engineers to transmit multiple data streams simultaneously over the same channel, at the expense of bandwidth. To that extent, FDM provides a trade-off: faster data for less bandwidth. Also, to demultiplex an FDM signal requires a series of bandpass filters to isolate each individual signal. Bandpass filters are relatively complicated and expensive, therefore the receivers in an FDM system are generally expensive.

Examples of FDM edit

As an example of an FDM system, Commercial broadcast radio (AM and FM radio) simultaneously transmits multiple signals or "stations" over the airwaves. These stations each get their own frequency band to use, and a radio can be tuned to receive each different station. Another good example is cable television, which simultaneously transmits every channel, and the TV "tunes in" to which channel it wants to watch.

Orthogonal FDM edit

Orthogonal Frequency Division Multiplexing (OFDM) is a more modern variant of FDM that uses orthogonal sub-carriers to transmit data that does not overlap in the frequency spectrum and is able to be separated out using frequency methods. OFDM has a similar data rate to traditional FDM systems, but has a higher resilience to disruptive channel conditions such as noise and channel fading.

Voltage Controlled Oscillators (VCO) edit

A voltage-controlled oscillator (VCO) is a device that outputs a sinusoid of a frequency that is a function of the input voltage. VCOs are not time-invariant, linear components. A complete study of how a VCO works will have to be relegated to a more detailed section based on feedback and oscillators. This page will, however, attempt to answer some of the basic questions about VCOs.

A basic VCO has input/output characteristics as such:

v(t) ----|VCO|----> sin(a[f + v(t)]t + O)

VCOs are often implemented using a special type of diode called a "Varactor". Varactors, when reverse-biased, produce a small amount of capacitance that varies with the input voltage.

Phase-Locked Loops edit

If you are talking on your cellphone, and you are walking (or driving), the phase angle of your signal is going to change, as a function of your motion, at the receiver. This is a fact of nature, and is unavoidable. The solution to this then, is to create a device which can "find" a signal of a particular frequency, negate any phase changes in the signal, and output the clean wave, phase-change free. This device is called a Phase-Locked Loop (PLL), and can be implemented using a VCO.

Purpose of VCO and PLL edit

VCO and PLL circuits are highly useful in modulating and demodulating systems. We will discuss the specifics of how VCO and PLL circuits are used in this manner in future chapters.

Varactors edit

As a matter of purely professional interest, we will discuss varactors here.

Further reading edit

What is an Envelope Filter? edit

If anybody has some images that they can upload, it would be much better then these ASCII art things.

The envelope detector is a simple analog circuit that can be used to find the peaks in a quickly-changing waveform. Envelope detectors are used in a variety of devices, specifically because passing a sinusoid through an envelope detector will suppress the sinusoid.

Circuit Diagram edit

In essence, an envelope filter has the following diagram:

+      |      |      +
       \     (c)
vin    /R     |     vout
       \      |
-      |      |      -

Where (c) represents a capacitor, and R is a resistor. Under zero input voltage (vin = 0), the capacitor carries no charge, and the resistor carries no current. When vin is increased, the capacitor stores charge until it reaches capacity, and then the capacitor becomes an open circuit. At this point, all current in the circuit is flowing through the resistor, R. As voltage decreases, the capacitor begins to discharge it's stored energy, slowing down the state change in the circuit from high voltage to low voltage.

Positive Voltages edit

By inserting a diode at the beginning of this circuit, we can negate the effect of a sinusoid, dipping into negative voltage, and forcing the capacitor to discharge faster:

+      |      |      +
       \     (c)
vin    /R     |     vout
       \      |
-      |      |      -

Purpose of Envelope Filters edit

Envelope filters help to find the outer bound of a signal that is changing in amplitude.

(Envelope Detectors)

Modulation is a term that is going to be used very frequently in this book. So much in fact, that we could almost have renamed this book "Principles of Modulation", without having to delete too many chapters. So, the logical question arises: What exactly is modulation?

Definition edit

Modulation is a process of mixing a signal with a sinusoid to produce a new signal. This new signal, conceivably, will have certain benefits over an un-modulated signal. Mixing of low frequency signal with high frequency carrier signal is called modulation.


we can see that this sinusoid has 3 parameters that can be altered, to affect the shape of the graph. The first term, A, is called the magnitude, or amplitude of the sinusoid. The next term,   is known as the frequency, and the last term,   is known as the phase angle. All 3 parameters can be altered to transmit data.

The sinusoidal signal that is used in the modulation is known as the carrier signal, or simply "the carrier". The signal that is used in modulating the carrier signal(or sinusoidal signal) is known as the "data signal" or the "message signal". It is important to notice that a simple sinusoidal carrier contains no information of its own.

In other words we can say that modulation is used because some data signals are not always suitable for direct transmission, but the modulated signal may be more suitable.

Types of Modulation edit

There are 3 basic types of modulation: Amplitude modulation, Frequency modulation, and Phase modulation.

amplitude modulation
a type of modulation where the amplitude of the carrier signal is modulated (changed) in proportion to the message signal while the frequency and phase are kept constant.
frequency modulation
a type of modulation where the frequency of the carrier signal is modulated (changed) in proportion to the message signal while the amplitude and phase are kept constant.
phase modulation
a type of modulation where the phase of the carrier signal is varied accordance to the low frequency of the message signal is known as phase modulation.

Why Use Modulation? edit

Why use modulation at all? To answer this question, let's consider a channel that essentially acts like a bandpass filter: both the lowest frequency components and the highest frequency components are attenuated or unusable in some way, with transmission only being practical over some intermediate frequency range. If we can't send low-frequency signals, then we need to shift our signal up the frequency ladder. Modulation allows us to send a signal over a bandpass frequency range. If every signal gets its own frequency range, then we can transmit multiple signals simultaneously over a single channel, all using different frequency ranges.

Another reason to modulate a signal is to allow the use of a smaller antenna. A baseband (low frequency) signal would need a huge antenna because in order to be efficient, the antenna needs to be about 1/10th the length of the wavelength. Modulation shifts the baseband signal up to a much higher frequency, which has much smaller wavelengths and allows the use of a much smaller antenna.

Examples edit

Think about your car radio. There are more than a dozen (or so) channels on the radio at any time, each with a given frequency: 100.1 MHz, 102.5 MHz etc... Each channel gets a certain range (usually about 0.22 MHz), and the entire bayot gets transmitted over that range. Modulation makes it all possible, because it allows us to send voice and music (which are essential baseband signals) over a bandpass (or "Broadband") channel.

Non-sinusoidal modulation edit

A sine wave at one frequency can be separated from a sine wave at another frequency (or a cosine wave at the same frequency) because the two signals are "orthogonal".

There are other sets of signals, such that every signal in the set is orthogonal to every other signal in the set.

A simple orthogonal set is time multiplexed division (TDM) -- only one transmitter is active at any one time.

Other more complicated sets of orthogonal waveforms—Walsh codes and various pseudo-noise codes such as Gold codes and maximum length sequences—are also used in some communication systems.

The process of combining these waveforms with data signals is sometimes called "modulation", because it is so very similar to the way modulation combines sine waves with data signals.

Further reading edit

There is lots of talk nowadays about buzzwords such as "Analog" and "Digital". Certainly, engineers who are interested in creating a new communication system should understand the difference. Which is better, analog or digital? What is the difference? What are the pros and cons of each? This chapter will look at the answers to some of these questions.

What are They? edit

What exactly is an analog signal, and what is a digital signal?

Analog signals are continuous in both time and value. Analog signals are used in many systems, although the use of analog signals has declined with the advent of cheap digital signals. All natural signals are Analog in nature or analog signal is that signal which amplitude on Y axis change with time on X axis...
Digital signals are discrete in time and value. Digital signals are signals that are represented by binary numbers, "1" or "0". The 1 and 0 values can correspond to different discrete voltage values, and any signal that doesn't quite fit into the scheme just gets rounded off.

or digital signal is that signal which have certain or fixed value on Y axis change with time on X axis...

Digital signals are sampled, quantized & encoded version of continuous time signals which they represent. In addition, some techniques also make the signal undergo encryption to make the system more tolerant to the channel.

What are the Pros and Cons? edit

Each paradigm has its own benefits and problems

Analog systems are less tolerant to noise, make good use of bandwidth, and are easy to manipulate mathematically. However, analog signals require hardware receivers and transmitters that are designed to perfectly fit the particular transmission.
Digital signals are more tolerant to noise, but digital signals can be completely corrupted in the presence of excess noise. In digital signals, noise could cause a 1 to be interpreted as a 0 and vice versa, which makes the received data different than the original data. Imagine if the army transmitted a position coordinate to a missile digitally, and a single bit was received in error. This single bit error could cause a missile to miss its target by miles. Luckily, there are systems in place to prevent this sort of scenario, such as checksums and CRCs, which tell the receiver when a bit has been corrupted and ask the transmitter to resend the data. The primary benefit of digital signals is that they can be handled by simple, standardized receivers and transmitters, and the signal can be then dealt with in software (which is comparatively cheap to change).
Discrete Digital and Analogue

Discrete data has a fixed set of possible values.
Digital data is a type of Discrete data where the fixed value can either be 1 or 0.
Analogue data can take on any real value.

Sampling and Reconstruction edit

The process of converting from analog data to digital data is called "sampling". The process of recreating an analog signal from a digital one is called "reconstruction". This book will not talk about either of these subjects in much depth beyond this, although other books on the topic of EE might, such as A-level Physics (Advancing Physics)/Digitisation.

Further reading edit

Signals need a channel to follow, so that they can move from place to place. These Communication Mediums, or "channels" are things like wires and antennae that transmit the signal from one location to another. Some of the most common channels are listed below:

Twisted Pair Wire edit

Twisted Pair is a transmission medium that uses two conductors that are twisted together to form a pair. The concept for the twist of the conductors is to prevent interference. Ideally, each conductor of the pair basically receives the same amount of interference, positive and negative, effectively cancelling the effect of the interference. Typically, most inside cabling has four pairs with each pair having a different twist rate. The different twist rates help to further reduce the chance of crosstalk by making the pairs appear electrically different in reference to each other. If the pairs all had the same twist rate, they would be electrically identical in reference to each other causing crosstalk, which is also referred to as capacitive coupling. Twisted pair wire is commonly used in telephone and data cables with variations of categories and twist rates.

Other variants of Twisted Pair are the Shielded Twisted Pair cables. The shielded types operate very similar to the non-shielded variety, except that Shielded Twisted Pair also has a layer of metal foil or mesh shielding around all the pairs or each individual pair to further shield the pairs from electromagnetic interference. Shielded twisted pair is typically deployed in situations where the cabling is subjected to higher than normal levels of interference.

Coaxial Cable edit

Another common type of wire is Coaxial Cable. Coaxial cable (or simply, "coax") is a type of cable with a single data line, surrounded by various layers of padding and shielding. The most common coax cable, common television cable, has a layer of wire mesh surrounding the padded core, that absorbs a large amount of EM interference, and helps to ensure a relatively clean signal is transmitted and received. Coax cable has a much higher bandwidth than a twisted pair, but coax is also significantly more expensive than an equal length of twisted pair wire. Coax cable frequently has an available bandwidth in excess of hundreds of megahertz (in comparison with the hundreds of kilohertz available on twisted pair wires).

Originally, Coax cable was used as the backbone of the telephone network because a single coaxial cable could hold hundreds of simultaneous phone calls by a method known as "Frequency Division Multiplexing" (discussed in a later chapter). Recently however, Fiber Optic cables have replaced Coaxial Cable as the backbone of the telephone network because Fiber Optic channels can hold many more simultaneous phone conversations (thousands at a time), and are less susceptible to interference, crosstalk, and noise then Coaxial Cable.

Fiber Optics edit

Fiber Optic cables are thin strands of glass that carry pulses of light (frequently infrared light) across long distances. Fiber Optic channels are usually immune to common RF interference, and can transmit incredibly high amounts of data very quickly. There are 2 general types of fiber optic cable: single frequency cable, and multi-frequency cable. single frequency cable carries only a single frequency of laser light, and because of this there is no self-interference on the line. Single-frequency fiber optic cables can attain incredible bandwidths of many gigahertz. Multi-Frequency fiber optics cables allow a Frequency-Division Multiplexed series of signals to each inhabit a given frequency range. However, interference between the different signals can decrease the range over which reliable data can be transmitted.

Wireless Transmission edit

In wireless transmission systems, signals are propagated as Electro-Magnetic waves through free space. Wireless signals are transmitted by a transmitter, and received by a receiver. Wireless systems are inexpensive because no wires need to be installed to transmit the signal, but wireless transmissions are susceptible not only to EM interference, but also to physical interference. A large building in a city, for instance can interfere with cell-phone reception, and a large mountain could block AM radio transmissions. Also, WiFi internet users may have noticed that their wireless internet signals don't travel through walls very well.

There are 2 types of antennas that are used in wireless communications, isotropic, and directional.

Isotropic edit

People should be familiar with isotropic antennas because they are everywhere: in your car, on your radio, etc... Isotropic antennas are omni-directional in the sense that they transmit data out equally (or nearly equally) in all directions. These antennas are excellent for systems (such as FM radio transmission) that need to transmit data to multiple receivers in multiple directions. Also, Isotropic antennas are good for systems in which the direction of the receiver, relative to the transmitter is not known (such as cellular phone systems).

Directional edit

Directional antennas focus their transmission power in a single narrow direction range. Some examples of directional antennas are satellite dishes, and wave-guides. The downfall of the directional antennas is that they need to be pointed directly at the receiver all the time to maintain transmission power. This is useful when the receiver and the transmitter are not moving (such as in communicating with a geo-synchronous satellite).

Receiver Design edit

It turns out that if we know what kind of signal to expect, we can better receive those signals. This should be intuitive, because it is hard to find something if we don't know what precisely we are looking for. How is a receiver supposed to know what is data and what is noise, if it doesnt know what data looks like?

Coherent transmissions are transmissions where the receiver knows what type of data is being sent. Coherency implies a strict timing mechanism, because even a data signal may look like noise if you look at the wrong part of it. In contrast, noncoherent receivers don't know exactly what they are looking for, and therefore noncoherent communication systems need to be far more complex (both in terms of hardware and mathematical models) to operate properly.

This section will talk about coherent receivers, first discussing the "Simple Receiver" case, and then going into theory about what the optimal case is. Once we know mathematically what an optimal receiver should be, we then discuss two actual implementations of the optimal receiver.

It should be noted that the remainder of this book will discuss optimal receivers. After all, why would a communication's engineer use anything that is less than the best?

The Simple Receiver edit

A simple receiver is just that: simple. A general simple receiver will consist of a low-pass filter (to remove excess high-frequency noise), and then a sampler, that will select values at certain points in the wave, and interpolate those values to form a smooth output curve. In place of a sampler (for purely analog systems), a general envelope filter can also be used, especially in AM systems. In other systems, different tricks can be used to demodulate an input signal, and acquire the data. However simple receivers, while cheap, are not the best choice for a receiver. Occasionally they are employed because of their price, but where performance is an issue, a better alternative receiver should be used.

The Optimal Receiver edit

Engineers are able to mathematically predict the structure of the optimal receiver. Read that sentence again: Engineers are able to design, analyze, and build the best possible receiver, for any given signal. This is an important development for several reasons. First, it means that no more research should go into finding a better receiver. The best receiver has already been found, after all. Second, it means any communications system will not be hampered (much) by the receiver.

Derivation edit

here we will attempt to show how the coherent receiver is derived.

Matched Receiver edit

The matched receiver is the logical conclusion of the optimal receiver calculation. The matched receiver convolves the signal with itself, and then tests the output. Here is a diagram:

s(t)----->(Convolve with r(t))----->

This looks simple enough, except that convolution modules are often expensive. An alternative to this approach is to use a correlation receiver.

Correlation Receiver edit

The correlation receiver is similar to the matched receiver, instead with a simple switch: The multiplication happens first, and the integration happens second.

Here is a general diagram:

s(t) ----->(X)----->(Integrator)--->

In a digital system, the integrator would then be followed by a threshold detector, while in an analog receiver, it might be followed by another detector, like an envelope detector.

Conclusion edit

To do the best job of receiving a signal we need to know the form of the signal that we are sending. After all we can't design a receiver until after we've decided how the signal will be sent. This method poses some problems in that the receiver must be able to line up the received signal with the given reference signal to work the magic: if the received signal and the reference signal are out of sync with each other, either as a function of an error in phase or an error in frequency, then the optimal receiver will not work.

Further reading edit

Analog Modulation edit

Analog Modulation Overview edit

Let's take a look at a generalized sinewave:


It consists of three components namely; amplitude, frequency and phase. Each of which can be decomposed to provide finer detail:


Types of Analog Modulation edit

We can see 3 parameters that can be changed in this sine wave to send information:

  •  . This term is called the "Amplitude", and changing it is called "Amplitude Modulation" (AM)
  •   This term is called the "Frequency Shift", and changing it is called "Frequency Modulation"
  •  . this term is called the "Phase angle", and changing it is called "Phase Modulation".
  • The terms frequency and phase modulation are often combined into a more general group called "Angle Modulation".

The Breakdown edit

Each term consists of a coefficient (called a "scaling factor"), and a function of time that corresponds to the information that we want to send. The scaling factor out front, A, is also used as the transmission power coefficient. When a radio station wants their signal to be stronger (regardless of whether it is AM, FM, or PM), they "crank-up" the power of A, and send more power out onto the airwaves.

How we Will Cover the Material edit

We are going to go into separate chapters for each different type of modulation. This book will attempt to discuss some of the mathematical models and techniques used with different modulation techniques. It will also discuss some practical information about how to construct a transmitter/receiver, and how to use each modulation technique effectively.

Amplitude modulation is one of the earliest radio modulation techniques. The receivers used to listen to AM-DSB-C are perhaps the simplest receivers of any radio modulation technique; which may be why that version of amplitude modulation is still widely used today. By the end of this module, you will know the most popular versions of amplitude modulation, some popular AM modulation circuits, and some popular AM demodulation circuits.

Amplitude Modulation edit

Amplitude modulation (AM) occurs when the amplitude of a carrier wave is modulated, to correspond to a source signal. In AM, we have an equation that looks like this:


We can also see that the phase of this wave is irrelevant, and does not change (so we don't even include it in the equation).

AM Double-Sideband (AM-DSB for short) can be broken into two different, distinct types: Carrier, and Suppressed Carrier varieties (AM-DSB-C and AM-DSB-SC, for short, respectively). This page will talk about both varieties, and will discuss the similarities and differences of each.


Characteristics edit

Modulation Index edit

Amplitude modulation requires a high frequency constant carrier and a low frequency modulation signal.

A wave carrier is of the form  
A wave modulation signal is of the form  

Notice that the amplitude of the high frequency carrier takes on the shape of the lower frequency modulation signal, forming what is called a modulation envelope.


The modulation index is defined as the ratio of the modulation signal amplitude to the carrier amplitude.


The overall signal can be described by:


More commonly, the carrier amplitude is normalized to one and the am equation is written as:

In most experienced author books this expression is simply written as:
If the modulation index is zero ( ) the signal is simply a constant amplitude carrier.
If the modulation index is 1 ( ), the resultant waveform has maximum or 100% amplitude modulation.

Sidebands edit

Expanding the normalized AM equation:


we obtain:



  represents the carrier
  represents the lower sideband
  represents the upper sideband

The sidebands occupies on the both sides of the center frequency. They are the sum and difference frequencies of the carrier and modulation signals. In the above example, they are just single frequencies, but normally the baseband modulation signal is a range of frequencies and hence two bands are formed.

AM Modulator edit

The standard amplitude modulation equation is:


From this we notice that AM involves a multiplication process. There are several ways to perform this function electronically. The simplest method uses a switch.

Switching Modulators edit

Switching modulators can be placed into two categories: unipolar and bipolar.

Bipolar Switching Modulator edit

The bipolar switch is the easiest to visualize. Note that an AM waveform appears to consist of a low frequency dc signal whose polarity is reversing at a carrier rate.


The AM signal can be created by multiplying a dc modulation signal by

This seems complicated but, if the square wave switching function has a 50% duty cycle, this simplifies to:
This tells us that the square wave is actually composed of a series of cosines (phase shifted sines) at odd multiples of the fundamental switching frequency. Therefore, using this signal to multiply the baseband signal results in AM signals being generated at each of the odd harmonics of the switching (carrier) frequencies. Since the amplitude of the harmonics decreases rapidly, this technique is practical for only the first few harmonics, and produces an enormous amount of unwanted signals (noise).
A band pass filter can be used to select any one of the AM signals. The number of different output frequencies can be significantly reduced if the multiplier accepts sinewaves at the carrier input.
Removing the DC component from the input eliminates the carrier signal and creates DSBSC modulation.

Physically this is done by reversing the signal leads:


The process of reversing the polarity of a signal is easily accomplished by placing two switch pairs in the output of a differential amplifier. The Mc1496 Balanced Modulator is an example of such a device.

Unipolar Switching Modulator edit

As previously mentioned, an AM signal can be created by multiplying a dc modulation signal by 0 & 1.

The spectrum of this signal is defined by:

Physically this is done by turning the modulation signal on and off at the carrier rate:


A high amplitude carrier can be used to turn a diode on and off. A dc bias is placed on the modulation signal to make certain that only the carrier (not the modulation signal) can reverse bias the diode.


It may not seem obvious, but the output of this circuit contains a series of AM signals. A bandpass filter is needed to extract the desired one. Normally it is the 1st or 3rd harmonic of the fundamental. (The 1st harmonic is the fundamental.)

Collector Modulator edit

The diode switching modulator is incapable of producing high power signals since it is a passive device. A transistor can be used to overcome this limitation. A collector modulator is used for high level modulation.

Square Law Modulator edit

The voltage-current relationship of a diode is nonlinear near the knee and is of the form:

The coefficient a and b are constants associated with the particular diode.

Amplitude modulation occurs if the diode is kept in the square law region when signals combine.

Let the injected signals be of the form:
The voltage applied across the diode and resistor is given by:
The current in the diode and hence in the resistor is given by:
Which expands to:

Modulation Index Measurement edit

It is sometimes difficult to determine the modulation index, particularly for complex signals. However, it is relatively easy to determine it by critical observation.
There are two practical methods to derive the modulation index:

  1. By representing an AM wave as it is in time domain.(using maxima - minima terms.)
  2. By Trapezoidal method.

The trapezoidal oscilloscope display can be used to determine the modulation index.

AM modulation index:  

The trapezoidal display makes it possible to quickly recognize certain types of problems, which would reduce the AM signal quality.


The highest authorized carrier power for AM broadcast in the US is 50 kilowatts, although directional stations are permitted 52.65 kilowatts to compensate for losses in the phasing system. The ERP can be much higher

C-QUAM edit

The basic idea behind the C-Quam modulator is actually quite simple. The output stage is an ordinary AM modulator however; the carrier signal has been replaced by an amplitude limited vector modulator. Therefore, the limiter output is really a phase-modulated signal.


A standard AM receiver will detect the amplitude variations as L+R. A stereo receiver will also detect the phase variations and to extract L-R. It will then process these signals to separate the left and right channels.

To enable the stereo decoder, a 25 Hz pilot tone is added to the L-R channel.

AM Receivers edit

The most common receivers in use today are the super heterodyne type. They consist of:

  • Antenna
  • RF amplifier
  • Local Oscillator and Mixer
  • IF Section
  • Detector and Amplifier

The need for these subsystems can be seen when one considers the much simpler and inadequate TRF or tuned radio frequency amplifier.

TRF Amplifier edit

It is possible to design an RF amplifier to accept only a narrow range of frequencies, such as one radio station on the AM band.


By adjusting the center frequency of the tuned circuit, all other input signals can be excluded.


The AM band ranges from about 500 kHz to 1600 kHz. Each station requires 10 kHz of this spectrum, although the baseband signal is only 5 kHz.

Recall that for a tuned circuit:  . The center or resonant frequency in an RLC network is most often adjusted by varying the capacitor value. However, the Q remains approximately constant as the center frequency is adjusted. This suggests that as the bandwidth varies as the circuit is tuned.

For example, the Q required at the lower end of the AM band to select only one radio station would be approximately:
As the tuned circuit is adjusted to the higher end of the AM band, the resulting bandwidth is:

A bandwidth this high could conceivably pass three adjacent stations, thus making meaningful reception impossible.

To prevent this, the incoming RF signal is heterodyned to a fixed IF or intermediate frequency and passed through a constant bandwidth circuit.

Superheterodyne Receiver edit


The RF amplifier boosts the RF signal into the mixer. It has broad tuning and amplifies not just one RF station, but many of them simultaneously. It also amplifies any input noise and even contributes some of its own.

The other mixer input is a high frequency sine wave created by a local oscillator. In AM receivers, it is always 455 kHz above the desired station carrier frequency. An ideal mixer will combine the incoming carrier with the local oscillator to create sum and difference frequencies. .

A real mixer combines two signals and creates a host of new frequencies:

• A dc level
• The original two frequencies
• The sum and difference of the two input frequencies
• Harmonics of the two input frequencies
• Sums and differences of all of the harmonics

Since the RF amplifier passes several radio stations at once, the mixer output can be very complex. However, the only signal of real interest is the difference between the desired station carrier frequency and the local oscillator frequency. This difference frequency, also called the IF (intermediate frequency) will alway be 455 kHz. By passing this through a 10 kHz BPF (band pass filter) centered at 455 kHz, the bulk of the unwanted signals can be eliminated.

Local Oscillator Frequency edit

Since the mixer generates sum and difference frequencies, it is possible to generate the 455 kHz IF signal if the local oscillator is either above or below the IF. The inevitable question is which is preferable.

Case I The local Oscillator is above the IF. This would require that the oscillator tune from (500 + 455) kHz to (1600 + 455) kHz or approximately 1 to 2 MHz. It is normally the capacitor in a tuned RLC circuit, which is varied to adjust the center frequency while the inductor is left fixed.
solving for C we obtain  
When the tuning frequency is a maximum, the tuning capacitor is a minimum and vice versa. Since we know the range of frequencies to be created, we can deduce the range of capacitance required.

Making a capacitor with a 4:1 value change is well within the realm of possibility.

Case II The local Oscillator is below the IF. This would require that the oscillator tune from (500 - 455) kHz to (1600 - 455) kHz or approximately 45 kHz to 1145 kHz, in which case:

It is not practical to make a tunable capacitor with this type of range. Therefore the local oscillator in a standard AM receiver is above the radio band.

Image Frequency edit

Just as there are two oscillator frequencies, which can create the same IF, two different station frequencies can create the IF. The undesired station frequency is known as the image frequency.


If any circuit in the radio front end exhibits non-linearities, there is a possibility that other combinations may create the intermediate frequency. Once the image frequency is in the mixer, there is no way to remove it since it is now heterodyned into the same IF band as the desired station.

AM Demodulation edit

AM Detection edit

There are two basic type of AM detection, coherent and non-coherent. Of these two, the non-coherent is the simpler method.

  • Non-coherent detection does not rely on regenerating the carrier signal. The information or modulation envelope can be removed or detected by a diode followed by an audio filter.
  • Coherent detection relies on regenerating the carrier and mixing it with the AM signal. This creates sum and difference frequencies. The difference frequency corresponds to the original modulation signal.

Both of these detection techniques have certain drawbacks. Consequently, most radio receivers use a combination of both.

Envelope Detector edit

When trying to demodulate an AM signal, it seems like good sense that only the amplitude of the signal needs to be examined. By only examining the amplitude of the signal at any given time, we can remove the carrier signal from our considerations, and we can examine the original signal. Luckily, we have a tool in our toolbox that we can use to examine the amplitude of a signal: The Envelope Detector.

An envelope detector is simply a half wave rectifier followed by a low pass filter. In the case of commercial AM radio receivers, the detector is placed after the IF section. The carrier at this point is 455 kHz while the maximum envelope frequency is only 5 kHz. Since the ripple component is nearly 100 times the frequency of the highest baseband signal and does not pass through any subsequent audio amplifiers.

An AM signal where the carrier frequency is only 10 times the envelope frequency would have considerable ripple:
Synchronous Detector edit

In a synchronous or coherent detector, the incoming AM signal is mixed with the original carrier frequency.


If you think this looks suspiciously like a mixer, you are absolutely right! A synchronous detector is one where the difference frequency between the two inputs is zero Hz. Of in other words, the two input frequencies are the same. Let's check the math.

Recall that the AM input is mathematically defined by:

At the multiplier output, we obtain:

The high frequency component can be filtered off leaving only the original modulation signal.

This technique has one serious drawback. The problem is how to create the exact carrier frequency. If the frequency is not exact, the entire baseband signal will be shifted by the difference. A shift of only 50 Hz will make the human voice unrecognizable. It is possible to use a PLL (phase locked loop), but making one tunable for the entire AM band is not trivial.

As a result, most radio receivers use an oscillator to create a fixed intermediate frequency. This is then followed by an envelope detector or a fixed frequency PLL.

Squaring Detector edit

The squaring detector is also a synchronous or coherent detector. It avoids the problem of having to recreate the carrier by simply squaring the input signal. It essentially uses the AM signal itself as a sort of wideband carrier.

The output of the multiplier is the square of the input AM signal:

Since the input is being multiplied by the   component, one of the resulting difference terms is the original modulation signal. The principle difficulty with this approach is trying to create a linear, high frequency multiplier.

AM-DSB-SC edit

AM-DSB-SC is characterized by the following transmission equation:


It is important to notice that s(t) can contain a negative value. AM-DSB-SC requires a coherent receiver, because the modulation data can go negative, and therefore the receiver needs to know that the signal is negative (and not just phase shifted). AM-DSB-SC systems are very susceptible to frequency shifting and phase shifting on the receiving end. In this equation, A is the transmission amplitude.

Double side band suppressed carrier modulation is simply AM without the broadcast carrier. Recall that the AM signal is defined by:

The carrier term in the spectrum can be eliminated by removing the dc offset from the modulating signal:

Double Balanced Ring Modulator edit

One of the circuits which is capable of creating DSBSC is the double balance ring modulator.


If the carrier is large enough to cause the diodes to switch states, then the circuit acts like a diode switching modulator:


The modulation signal is inverted at the carrier rate. This is essentially multiplication by ±1. Since the transformers cannot pass dc, there is no term which when multiplied can create an output carrier. Since the diodes will switch equally well on either cycle, the modulation signal is effectively being multiplied by a 50% duty cycle square wave creating numerous DSBSC signals, each centered at an odd multiple of the carrier frequency. Bandpass filters are used to extract the frequency of interest.

Some IC balanced modulators use this technique, but use transistors instead of diodes to perform the switching.

Push Pull Square Law Balanced Modulator edit


This circuit uses the same principles as the diode square law modulator. Since dc cannot pass through the transformer, it would be expected that there would be no output signal at the carrier frequency.

The drain current vs. gate-source voltage is of the form:

The net drain current in the output transformer is given by:
By applying KVL around the gate loops we obtain:
Putting it all together we obtain:
From this we note that the first term is the originating modulation signal and can easily be filtered off by a high pass filter. The second term is of the form:
which is AM DSBSC.

AM-DSB-C edit

In contrast to AM-DSB-SC is AM-DSB-C, which is categorized by the following equation:


Where c is a positive term representing the carrier. If the term   is always non-negative, we can receive the AM-DSB-C signal non-coherently, using a simple envelope detector to remove the cosine term. The +c term is simply a constant DC signal and can be removed by using a blocking capacitor.

It is important to note that in AM-DSB-C systems, a large amount of power is wasted in the transmission sending a "boosted" carrier frequency. since the carrier contains no information, it is considered to be wasted energy. The advantage to this method is that it greatly simplifies the receiver design, since there is no need to generate a coherent carrier signal at the receiver. For this reason, this is the transmission method used in conventional AM radio.

AM-DSB-SC and AM-DSB-C both suffer in terms of bandwidth from the fact that they both send two identical (but reversed) frequency "lobes", or bands. These bands (the upper band and the lower band) are exactly mirror images of each other, and therefore contain identical information. Why can't we just cut one of them out, and save some bandwidth? The answer is that we can cut out one of the bands, but it isn't always a good idea. The technique of cutting out one of the sidebands is called Amplitude Modulation Single-Side-Band (AM-SSB). AM-SSB has a number of problems, but also some good aspects. A compromise between AM-SSB and the two AM-DSB methods is called Amplitude Modulation Vestigial-Side-Band (AM-VSB), which uses less bandwidth then the AM-DSB methods, but more than the AM-SSB.

Transmitter edit

A typical AM-DSB-C transmitter looks like this:

             c    cos(...)
             |       |
Signal ---->(+)---->(X)----> AM-DSB-C

which is a little more complicated than an AM-DSB-SC transmitter.

Receiver edit

An AM-DSB-C receiver is very simple:

AM-DSB-C ---->|Envelope Filter|---->|Capacitor|----> Signal

The capacitor blocks the DC component, and effectively removes the +c term.

AM-SSB edit

To send an AM-SSB signal, we need to remove one of the sidebands from an AM-DSB signal. This means that we need to pass the AM-DSB signal through a filter, to remove one of the sidebands. The filter, however, needs to be a very high order filter, because we need to have a very aggressive roll-off. One sideband needs to pass the filter almost completely unchanged, and the other sideband needs to be stopped completely at the filter.

To demodulate an AM-SSB signal, we need to perform the following steps:

  1. Low-pass filter, to remove noise
  2. Modulate the signal again by the carrier frequency
  3. Pass through another filter, to remove high-frequency components
  4. Amplify the signal, because the previous steps have attenuated it significantly.

AM-SSB is most efficient in terms of bandwidth, but there is a significant added cost involved in terms of more complicated hardware to send and receive this signal. For this reason, AM-SSB is rarely seen as being cost effective.

Single sideband is a form of AM with the carrier and one sideband removed. In normal AM broadcast, the transmitter is rated in terms of the carrier power. SSB transmitters attempt to eliminate the carrier and one of the sidebands. Therefore, transmitters are rated in PEP (peak envelope power).

With normal voice signals, an SSB transmitter outputs 1/4 to 1/3 PEP.

There are numerous variations of SSB:

  • SSB - Single sideband - amateur radio
  • SSSC - Single sideband suppressed carrier - a small pilot carrier is transmitted
  • ISB - Independent sideband - two separate sidebands with a suppressed carrier. Used in radio telephony.
  • VSB - Vestigial sideband - a partial sideband. Used in broadcast TV.
  • ACSSB - Amplitude companded SSB

There are several advantages of using SSB:

• More efficient spectrum utilization
• Less subject to selective fading
• More power can be placed in the intelligence signal
• 10 to 12 dB noise reduction due to bandwidth limiting

Filter Method edit

The simplest way to create SSB is to generate DSBSC and then use a bandpass filter to extract one of the sidebands.


This technique can be used at relatively low carrier frequencies. At high frequencies, the Q of the filter becomes unacceptably high. The required Q necessary to filter off one of the sidebands can be approximated by:


Several types of filters are used to suppress unwanted sidebands:

  • LC - Maximum Q = 200
  • Ceramic - Maximum Q = 2000
  • Mechanical - Maximum Q = 10,000
  • Crystal - Maximum Q = 50,000

In order to reduce the demands placed upon the filter, a double heterodyne technique can be used.


The first local oscillator has a relatively low frequency thus enabling the removal of one of the sidebands produced by the first mixer. The signal is then heterodyned a second time, creating another pair of sidebands. However, this time they are separated by a sufficiently large gap that one can be removed by the band limited power amplifier or antenna matching network.

Observe the spectral distribution under the following conditions:
• Audio baseband = 100 HZ to 5 KHz
• LO1 = 100 kHz
• LO2 = 50 MHz
The spectral output of the first mixer is:
If the desired sideband suppression is 80 dB, the Q required to filter off one of the sidebands is approximately:
It is evident that a crystal filter would be needed to remove the unwanted sideband.
After the filter, only one sideband is left. In this example, we’ll retain the USB. The spectrum after the second mixer is:
The Q required to suppress one of the side bands by 80 dB is approximately:
Thus, we note that the required Q drops in half.

This SSB filter technique is used in radiotelephone applications.

Phase Shift Method edit


The output from the top mixer is given by:

The output from the bottom mixer is given by:
The output of the sumer is:
which corresponds to the lower sideband.

The major difficulty with this technique is the need to provide a constant 90o phase shift over the entire input audio band. To overcome this obstacle, the Weaver or third method uses an audio sub carrier, which is phase shifted.

Weaver Method edit

The Weaver or ‘third’ method places the baseband signal on a low frequency quadrature carrier.


This has the advantage of not requiring a broadband phase shifter however; the use of four mixers makes it awkward and seldom used.


SSB Transmitter edit

AM-SSB transmitters are a little more complicated:

Signal ---->(X)---->|Low-Pass Filter|----> AM-SSB

The filter must be a very high order, for reasons explained in that chapter.

SSB Receiver edit

An AM-SSB receiver is a little bit complicated as well:

AM-SSB ---->(X)---->|Low-Pass Filter|---->|Amplifier|----> Signal

This filter doesnt need to be a very high order, like the transmitter has.

These receivers require extremely stable oscillators, good adjacent channel selectivity, and typically use a double conversion technique. Envelope detectors cannot be used since the envelope varies at twice the frequency of the AM envelope.

Stable oscillators are needed since the detected signal is proportional to the difference between the untransmitted carrier and the instantaneous side band. A small shift of 50 Hz makes the received signal unusable.

SSB receivers typically use fixed frequency tuning rather than continuous tuning as found on most radios. Crystal oscillators are often used to select the fixed frequency channels.

AM-VSB edit

Single-sideband modulation works satisfactorily for an information-bearing signal (e.g.,

speech signal) with an energy gap centered around zero frequency. However, for the spectrally

efficient transmission of wideband signals, we have to look to a new method of modulation

for two reasons:

  • Typically, the spectra of wideband signals (exemplified by television video signals and

computer data) contain significant low frequencies, which make it impractical to use

SSB modulation.

  • The spectral characteristics of wideband data befit the use of DSB-SC. However, DSBSC

requires a transmission bandwidth equal to twice the message bandwidth, which

violates the bandwidth conservation requirement.

To overcome these two practical limitations, we need a compromise method of modulation

that lies somewhere between SSB and DSB-SC in its spectral characteristics. Vestigial sideband,

the remaining modulation scheme to be considered in this section, is that compromise


Vestigial sideband (VSB) modulation distinguishes itself from SSB modulation in two

practical respects:

  • Instead of completely removing a sideband, a trace or vestige of that sideband is transmitted, hence, the name “vestigial sideband.”
  • Instead of transmitting the other sideband in full, almost the whole of this second band is also transmitted.

Transmitter edit

Here we will talk about an AM-VSB transmitter circuit.

Receiver edit

Here we will talk about an AM-VSB receiver circuit. (AM)

Frequency Modulation edit

If we make the frequency of our carrier wave a function of time, we can get a generalized function that looks like this:


We still have a carrier wave, but now we have the value ks(t) that we add to that carrier wave, to send our data.

As an important result, ks(t) must be less than the carrier frequency always, to avoid ambiguity and distortion.


Deriving the FM Equation edit

Recall that a general sinusoid is of the form:


Frequency modulation involves deviating a carrier frequency by some amount. If a sine wave is used to deviate the carrier, the expression for the frequency at any instant would be:

 instantaneous frequency
 carrier frequency
 carrier deviation
  modulation frequency

This expression describes a signal varying sinusoidally about some average frequency. However, we cannot simply substitute this expression into the general equation for a sinusoid to get the FM equation. This is because the sine operator acts on angles, not frequency. Therefore, we must define the instantaneous frequency in terms of angles.

It should be noted that the modulation signal amplitude governs the amount of carrier deviation while the modulation signal frequency governs the rate of carrier deviation.

The term   is an angular velocity (radians per second) and is related to frequency and angle by the following relationship:

To find the angle, we must integrate   with respect to time:
We can now find the instantaneous angle associated with the instantaneous frequency:
This angle can now be substituted into the general carrier signal to define FM:
The FM modulation index is defined as the ratio of the carrier deviation to modulation frequency:
Consequently, the FM equation is often written as:

Bessel's Functions edit

This is a very complex expression and it is not readily apparent what the sidebands of this signal are like. The solution to this problem requires a knowledge of Bessel's functions of the first kind and order p. In open form, it resembles:

  Magnitude of the frequency component
  Side frequency number (not to be confused with sidebands)
  Modulation index
As a point of interest, Bessel's functions are a solution to the following equation:

Bessel's functions occur in the theory of cylindrical and spherical waves, much like sine waves occur in the theory of plane waves.

It turns out that FM generates an infinite number of side frequencies (in both the upper and lower sidebands). Each side frequency is an integer multiple of the modulation signal frequency. The amplitude of higher order side frequencies decreases rapidly and can generally be ignored.

The amplitude of the carrier signal is also a function of the modulation index and under some conditions, its amplitude can actually go to zero. This does not mean that the signal disappears, but rather that all of the broadcast energy is redistributed to the side frequencies.

A plot of the carrier and first five side frequency amplitudes as a function of modulation index resembles:


The Bessel coefficients have several interesting properties including:


One very useful interpretation of this is:   represents the voltage amplitude of the carrier,   represents the amplitude of the 1st side frequency,   the 2nd side frequency etc. Note that the sum of the squares (power) remains constant.

FM Bandwidth edit

FM generates upper and lower sidebands, each of which contain an infinite number of side frequencies. However, the FM bandwidth is not infinite because the amplitude of the higher order side frequencies decreases rapidly. Carson's Rule is often used to calculate the bandwidth, since it contains more than 90% of the FM signal.

Carson's Rule

In commercial broadcast applications, for a purely monaural station, the maximum modulation index ( ) = 75/15 = 5, coming from: the maximum carrier deviation ( ) = 75 kHz, and maximum modulation frequency ( ) = 15 kHz. The total broadcast spectrum according to Carson's rule is 180 kHz, but an additional 20 kHz guard band is used to separate adjacent radio stations. Therefore, each FM radio station is allocated 200 kHz.

For stereo stations, the maximum modulation index is significantly reduced because the information needed to separate the channels has to be transmitted along with the mono audio signal. This increases the required bandwidth to 53 kHz. Hence, the max. modulation index is = 75/53 = 1.41509434... Radio Data System (RDS) information, increase this further to ~60 kHz, reducing the max. modulation index to 75/60 = 1.25.

How FM Stereo works edit

The mono signal is M = L + R, with the stereo difference being S = L - R. Adding both simultaneous equations together gives M + S = 2L + (R-R), recovering the left channel, while subtracting them recovers the right channel. This is transmitted as a double sideband suppressed carrier (DSBSC), which is essentially just an AM "station" going along, but without its carrier being sent when there is nothing being transmitted on it. ("Stations" sent along with the main program (usually in ultrasonic frequencies) are known as subcarriers.) A stereo "pilot" tone is used to let the receiver know that there is a stereo signal being received, and also to allow the suppressed carrier to be regenerated (by doubling the pilot tone's frequency) so the stereo difference signal can be demodulated just like a normal AM station and the resulting signals used to separate the audio into two channels.

RDS information is yet another "AM station" being sent along with the main program, but at 3× the pilot frequency (19 kHz × 3 = 57 kHz). Its contents are not audio, but analogue values meant to represent a digital signal which carries the station name and many other info like its alternate frequencies, time of the day, program info, etc.

Noise edit

In AM systems, noise easily distorts the transmitted signal however, in FM systems any added noise must create a frequency deviation in order to be perceptible.


The maximum frequency deviation due to random noise occurs when the noise is at right angles to the resultant signal. In the worst case the signal frequency has been deviated by:


This shows that the deviation due to noise increases as the modulation frequency increases. Since noise power is the square of the noise voltage, the signal to noise ratio can significantly degrade.


To prevent this, the amplitude of the modulation signal is increased to keep the S/N ratio constant over the entire broadcast band. This is called pre-emphasis.

Pre & De-emphasis edit

Increasing the amplitude of high frequency baseband signals in the FM modulator (transmitter) must be compensated for in the FM demodulator (receiver) otherwise the signal would sound quite tinny (too much treble).

The standard curves resemble:


In commercial FM broadcast, the emphasis circuits consist of a simple RC network with a time constant of 75  Sec and a corner frequency of 2125 Hz.


The magnitude of the pre-emphasis response is defined by:


FM Transmission Power edit

The equation for the transmitted power in a sinusoid is a fundamental equation. Remember it.

Since the value of the amplitude of the sine wave in FM does not change, the transmitted power is a constant. As a general rule, for a sinusoid with a constant amplitude, the transmitted power can be found as follows:


Where A is the amplitude of the sine wave, and RL is the resistance of the load. In a normalized system, we set RL to 1.

The Bessel coefficients can be used to determine the power in the carrier and any side frequency:

  is the total power and is by definition equal to the unmodulated carrier power plus the sideband power.
  is the power of the unmodulated carrier.

As the modulation index varies, the individual Bessel coefficients change and power is redistributed from the carrier to the side frequencies.

FM Receivers edit

Any angle modulation receiver needs to have several components:

  1. A limiter, to remove abnormal amplitude values
  2. bandpass filter, to separate the out-of-band noise.
  3. A Discriminator, to change a frequency back to a voltage
  4. A lowpass filter, to remove noise added by the discriminator.

A discriminator is essentially a differentiator in line with an envelope detector:

FM ---->|Differentiator|---->|Envelope Filter|----> Signal

Also, you can add in a blocking capacitor to remove any DC components of the signal, if needed. (FM)

Phase Modulation edit

Phase modulation is most commonly used to convey digital signals. All high performance modems today use phase modulation.

Similar to FM (frequency modulation), is Phase modulation. (We will show how they are the same in the next chapter.) If we alter the value of the phase according to a particular function, we will get the following generalized PM function:


It is important to note that the fact that   for all values of t. If this relationship is not satisfied, then the phase angle is said to be wrapped.

BPSK Modulator edit

The binary phase shift keyed modulator is the simplest of app PSK modulators since it has only two output phase states. It is generally a multiplier which can either be an IC (integrated circuit) or ring modulator.


The output has two phase states:


In the above illustration, the duration of each of the phase states corresponds to one signaling element or baud. The baud rate is therefor equal to the bit rate.

The spectrum of the BPSK signal will depend upon the data being transmitted, but it is very easy to sketch it for the highest data rate input.


The resultant BPSK spectrum is:


QPSK Modulators (4-PSK) edit

Quadrature modulation uses two data channels denoted I (in phase) and Q (quadrature phase) displaced by 90o with respect to each other. It may seem somewhat paradoxical, that although these two channels are combined prior to transmission, they do not interfere with each other.


The receiver is quite capable of separating them because of their quadrature or orthogonal nature.


In the most basic configuration, there are 4 possible output phases. This suggests that each output symbol correspond to 2 bits of binary information. Since several bits can be encoded into a baud, the bit rate exceeds the baud rate.


The first thing that happens in this circuit is that the incoming bits are organized into groups of 2 called dibits. They are separated into 2 data streams and kept constant over the dibit period.


Each data stream is fed to a BPSK modulator. However, orthogonal carriers feed the two modulators. The output of the I channel modulator resembles:


The output of the Q channel modulator resembles


Combining the I and Q channels has the effect of rotating the output state by 45o.


Rotating the output reference to 45o for the sake of clarity, the transmitted output for this particular data sequence is therefor:


8-PSK edit

This process of encoding more bits into each output baud or phase state can be continued. Organizing binary bits into 3 bytes corresponds to 8 different conditions.

The output constellation diagram for the 8 different phase states is:


From this diagram it is readily apparent that two different amplitudes are needed on the I and Q channels. If the A bit is used to control the polarity of the I channel and the B bit the polarity of the Q channel, then the C bit can be used to define the two different amplitudes. In order to evenly space the phase states; the amplitudes must be ± 0.38 and ± 0.92. The magnitude of the I and Q channel signals must always be different. An inverter can be used to assure this condition.

The input bit stream is organized into 3 bit bytes. Each bit is sent to a different location to control a certain aspect of the modulator. The inputs to the 2 - 4 level converter are 0’s or 1’s but the output is ± 0.38 or ± 0.92, depending on the C bit.


Wrapped/Unwrapped Phase edit

The phase angle is a circular quantity, with the restriction  . Therefore, if we wrap the phase a complete 360 degrees around, the receiver will not know the difference, and the transmission will fail. When the phase exceeds 360 degrees, the phase value is said to be wrapped. It is highly difficult to construct a communication system that can detect and decode a wrapped phase value.

PM Transmitter edit

PM signals can be transmitted using a technique very similar to FM transmitters. The only difference is that we need to add a differentiator to it:

Signal ---->|Differentiator|---->|VCO|----> PM Signal

PM Receiver edit

PM receivers have all the same parts as an FM receiver, except for the 3rd step:

  1. A limiter, to remove abnormal amplitude values
  2. bandpass filter, to separate the out-of-band noise.
  3. A Phase detector, to convert a phase back into a voltage
  4. A lowpass filter, to remove noise added by the discriminator.

Phase detectors can be created using a Phase-Locked-Loop (again, see why we discussed them first?). (PM)

Concept edit

We can see from our initial overviews that FM and PM modulation schemes have a lot in common. Both of them are altering the angle of the carrier sinusoid according to some function. It turns out that we can go so far as to generalize the two together into a single modulation scheme known as angle modulation. Note that we will never abbreviate "angle modulation" with the letters "AM", because Amplitude modulation is completely different from angle modulation.

Instantaneous Phase edit

Let us now look at some things that FM and PM have of common:


What we want to analyze is the argument of the sinusoid, and we will call it Psi. Let us show the Psi for the bare carrier, the FM case, and the PM case:


This Psi value is called the Instantaneous phase of the sinusoid.

Instantaneous Frequency edit

Using the Instantaneous phase value, we can find the Instantaneous frequency of the wave with the following formula:


We can also express the instantaneous phase in terms of the instantaneous frequency:


Where the Greek letter "lambda" is simply a dummy variable used for integration. Using these relationships, we can begin to study FM and PM signals further.

Determining FM or PM edit

If we are given the equation for the instantaneous phase of a particular angle modulated transmission, is it possible to determine if the transmission is using FM or PM? it turns out that it is possible to determine which is which, by following 2 simple rules:

  1. In PM, instantaneous phase is a linear function.
  2. In FM, instantaneous frequency minus carrier frequency is a linear function.

For a refresher course on Linearity, there is a chapter on the subject in the Signals and Systems book worth re-reading.

Bandwidth edit

In a PM system, we know that the value   can never go outside the bounds of  . Since sinusoidal functions oscillate between [-1, 1], we can use them as a general PM generating function. Now, we can combine FM and PM signals into a general equation, called angle modulation:


If we want to analyze the spectral components of this equation, we will need to take the Fourier transform of this. But, we can't integrate a sinusoid of a sinusoid, much less find the transform of it. So, what do we do?

It turns out (and the derivation will be omitted here, for now) that we can express this equation as an infinite sum, as such:


But, what is the term  ? J is the Bessel function, which is a function that exists only as an open integral (it is impossible to write it in closed form). Fortunately for us, there are extensive tables tabulating Bessle function values.

The Bessel Function edit

The definition of the Bessel function is the following equation:


The bessel function is a function of 2 variables, N and  .

Bessel Functions have the following properties:

  • If n is even:
  • If n is odd:
  •  .

The bessel function is a relatively advanced mathematical tool, and we will not analyze it further in this book.

Carson's Rule edit

If we have our generalized function:


We can find the bandwidth BW of the signal using the following formula:


where   is the maximum frequency deviation, of the transmitted signal, from the carrier frequency. It is important to note that Carson's rule is only an approximation (albeit one that is used in industry frequently).

Demodulation: First Step edit

Now, it is important to note that FM and PM signals both do the same first 2 steps during demodulation:

  1. Pass the signal through a limiter, to remove amplitude peaks
  2. Pass the signal through a bandpass filter to remove low and high frequency noise (as much as possible, without filtering out the signal).

Once we perform these two steps, we no longer have white noise, because we've passed the noise through a filter. Now, we say the noise is colored.

here is a basic diagram of our demodulator, so far:

s(t) ---------> r(t) --->|Limiter|--->|Bandpass Filter|---->z(t)

Where z(t) is the output of the bandpass filter.

Filtered Noise edit

To denote the new, filtered noise, and new filtered signal, we have the following equation:


Where we call the additive noise   because it has been filtered, and is not white noise anymore.   is known as narrow band noise, and can be denoted as such:


Now, once we have it in this form, we can use a trigonometric identity to make this equation more simple:




Here, the new noise parameter R(t) is a rayleigh random variable, and is discussed in the next chapter.

Noise Analysis edit

R(t) is a noise function that affects the amplitude of our received signal. However, our receiver passes the signal through a limiter, which will remove amplitude fluctuations from our signal. For this reason, R(t) doesnt affect our signal, and can be safely ignored for now. This means that the only random variable that is affecting our signal is the variable  , "Theta". Theta is a uniform random variable, with values between pi and -pi. Values outside this range "Wrap around" because phase is circular.

Transmission edit

This page will discuss some of the fundamental basics of EM wave propagation.

Electromagnetic Spectrum edit


Radio Waves edit

Maxwell first predicted the existence of electromagnetic waves in the 19th century. He came to this conclusion by careful examination of the equations describing electric and magnetic phenomenon. It was left up to Hertz to create these waves, and Marconi to exploit them.

In spite of one hundred years of study, exactly what radio waves are and why they exist, remain somewhat of a mystery.

Electromagnetic waves in free space, or TEM waves, consist of electric and magnetic fields, each at right angles to each other and the direction of propagation.


The relationship between wavelength and frequency is given by:

where c is the speed of light (approximately 300,000 m/s in vacuum), f is the frequency of the wave, and λ is the wavelength of the wave.

Radio waves can be reflected and refracted in a manner similar to light. They are affected by the ground terrain, atmosphere and other objects.

Maxwell’s equations state that a time varying magnetic field produces an electric field and a time varying electric field produces a magnetic field. This is kind of a chicken and egg situation.

Radio waves propagate outward from an antenna, at the speed of light. The exact nature of these waves is determined by the transmission medium. In free space, they travel in straight lines, whereas in the atmosphere, they generally travel in a curved path. In a confined or guided medium, radio waves do not propagate in the TEM mode, but rather in a TE or TM mode.

Radio waves interact with objects in three principle ways:

Reflection – A radio wave bounces off an object larger than its wavelength.
Diffraction – Waves bend around objects.
Scattering – A radiowave bounces off an object smaller than its wavelength.

Because of these complex interactions, radio wave propagation is often examined in three distinct regions in order to simplify the analysis:

Surface (or ground) waves are located very near the earth’s surface.
Space waves occur in the lower atmosphere (troposphere).
Sky waves occur in the upper atmosphere (ionosphere).

The boundaries between these regions are somewhat fuzzy. In many cases, it is not possible to examine surface waves without considering space waves.


Common RF Band Designations edit

Frequency band name Frequency Wavelength
ELF - Extremely Low Frequency 3 – 30 Hz 100000 – 10000 km
SLF - Super Low Frequency 30 – 300 Hz 10000 – 1000 km
ULF - Ultra Low Frequency 300 – 3000 Hz 1000 – 100 km
VLF - Very Low Frequency 3 – 30 kHz 100 – 10 km
LF - Low Frequency 30 – 300 kHz 10 – 1 km
MF - Medium Frequency 300 – 3000 kHz 1000 – 100 m
HF - High Frequency 3 – 30 MHz 100 – 10 m
VHF - Very High Frequency 30 – 300 MHz 10 – 1 m
UHF - Ultra High Frequency 300 – 3000 MHz 1000 – 100 mm
SHF - Super High Frequency 3 – 30 GHz 100 – 10 mm
EHF - Extremely High Frequency 30 – 300 GHz 10 – 1 mm
THF - Tremendously High Frequency 300 – 3000 GHz 1 – 0.1 mm

Surface Waves edit

These are the principle waves used in AM, FM and TV broadcast. Objects such as buildings, hills, ground conductivity, etc. have a significant impact on their strength. Surface waves are usually vertically polarized with the electric field lines in contact with the earth.


Refraction edit

Because of refraction, the radio horizon is larger than the optical horizon by about 4/3. The typical maximum direct wave transmission distance (in km) is dependent on the height of the transmitting and receiving antennas (in meters):


However, the atmospheric conditions can have a dramatic effect on the amount of refraction.

Super Refraction edit

In super refraction, the rays bend more than normal thus shortening the radio horizon. This phenomenon occurs when temperature increases but moisture decreases with height. Paradoxically, in some cases, the radio wave can travel over enormous distances. It can be reflected by the earth, rebroadcast and super refracted again.

Sub refraction edit

In sub refraction, the rays bend less than normal. This phenomenon occurs when temperature decreases but moisture increases with height. In extreme cases, the radio signal may be refracted out into space.

Space Waves edit

These waves occur within the lower 20 km of the atmosphere, and are comprised of a direct and reflected wave. The radio waves having high frequencies are basically called as space waves. These waves have the ability to propagate through atmosphere, from transmitter antenna to receiver antenna. These waves can travel directly or can travel after reflecting from earth’s surface to the troposphere surface of earth. So, it is also called as Tropospherical Propagation. In the diagram of medium wave propagation, c shows the space wave propagation. Basically the technique of space wave propagation is used in bands having very high frequencies. E.g. V.H.F. band, U.H.F band etc. At such higher frequencies the other wave propagation techniques like sky wave propagation, ground wave propagation can’t work. Only space wave propagation is left which can handle frequency waves of higher frequencies. The other name of space wave propagation is line of sight propagation. There are some limitations of space wave propagation.

  1. These waves are limited to the curvature of the earth.
  2. These waves have line of sight propagation, means their propagation is along the line of sight distance.

The line of sight distance is that exact distance at which both the sender and receiver antenna are in sight of each other. So, from the above line it is clear that if we want to increase the transmission distance then this can be done by simply extending the heights of both the sender as well as the receiver antenna. This type of propagation is used basically in radar and television communication.

The frequency range for television signals is nearly 80 to 200 MHz. These waves are not reflected by the ionosphere of the earth. The property of following the earth’s curvature is also missing in these waves. So, for the propagation of television signal, geostationary satellites are used. The satellites complete the task of reflecting television signals towards earth. If we need greater transmission then we have to build extremely tall antennas.

Direct Wave edit

This is generally a line of sight transmission, however, because of atmospheric refraction the range extends slightly beyond the horizon.

Ground Reflected Wave edit

Radio waves may strike the earth, and bounce off. The strength of the reflection depends on local conditions. The received radio signal can cancel out if the direct and reflected waves arrive with the same relative strength and 180o out of phase with each other.

Horizontally polarized waves are reflected with almost the same intensity but with a 180o phase reversal.

Vertically polarized waves generally reflect less than half of the incident energy. If the angle of incidence is greater than 10o there is very little change in phase angle.

Sky Waves edit

These waves head out to space but are reflected or refracted back by the ionosphere. The height of the ionosphere ranges from 50 to 1,000 km.[1]

Radio waves are refracted by the ionized gas created by solar radiation. The amount of ionization depends on the time of day, season and the position in the 11-year sun spot cycle. The specific radio frequency refracted is a function of electron density and launch angle.

A communication channel thousands of kilometers long can be established by successive reflections at the earth’s surface and in the upper atmosphere. This ionospheric propagation takes place mainly in the HF band.

The ionosphere is composed of several layers, which vary according to the time of day. Each layer has different propagation characteristics:

D layer – This layer occurs only during the day at altitudes of 60 to 90 km. High absorption takes place at frequencies up to 7 MHz.
E layer – This layer occurs at altitudes of 100 to 125 km. In the summer, dense ionization clouds can form for short periods. These clouds called sporadic E can refract radio signals in the VHF spectrum. This phenomenon allows amateur radio operators to communicate over enormous distances.
F layer - This single nighttime layer splits into two layers (F1 and F2) during the day. The F1 layer forms at about 200 km and F2 at about 400 km. The F2 layer propagates most HF short-wave transmissions.

Because radio signals can take many paths to the receiver, multipath fading can occur. If the signals arrive in phase, the result is a stronger signal. If they arrive out of phase with each other, they tend to cancel.

Deep fading, lasting from minutes to hours over a wide frequency range, can occur when solar flares increase the ionization in the D layer.

The useful transmission band ranges between the LUF (lowest usable frequency) and MUF (maximum usable frequency). Frequencies above the MUF are refracted into space. Below the LUF, radio frequencies suffer severe absorption. If a signal is near either of these two extremes, it may be subject to fading.

Meteors create ionization trails that reflect radio waves. Although these trails exist for only a few seconds, they have been successfully used in communications systems spanning 1500 km.

The Aurora Borealis or Northern Lights cause random reflection in the 3 - 5 MHz region. Aurora causes signal flutter at 100 Hz to 2000 Hz thus making voice transmission impossible.

Fading and Interference edit

Radio signals may vary in intensity for many reasons.

Flat Earth Reflections (Horizontal Polarization) edit

There are at least two possible paths for radio waves to travel when the antennas are near the earth: direct path and reflected path. These two signals interact in a very complex manner. However, ignoring polarization and assuming a flat earth can produce some interesting mathematical descriptions.

p1 = direct wave path length
p2 = reflected wave path length
 p = p2 - p1 difference in path lengths
d = distance

From the geometry we can observe:


But  and  


If the difference in the two paths  p, is 1/2   long, the two signals tend to cancel. If  p is equal to  , the two signals tend to reinforce. The path difference  p therefore corresponds to a phase angle change of:


The resultant received signal is the sum of the two components. The situation is unfortunately made more complex by the fact that the phase integrity of the reflected wave is not maintained at the point of reflection.

If we limit the examination of reflected waves to the horizontally polarized situation, we obtain the following geometry:


Applying the cosine rule to this diagram, we obtain a resultant signal of:


The signal strength of the direct wave is the unit distance value divided by the distance:   Therefore, the received signal can be written as:


For small angles this can be approximated by:


Multipath Fading edit

The received signal is generally a combination of many signals, each coming over a different path. The phase and amplitude of each component are related to the nature of the path. These signals combine in a very complex manner. Some multipath fading effects are characterized by delay spread, Rayleigh and Ricean fading, doppler shifting, etc. Fading is the most significant phenomenon causing signal degradation. There are several different categories of fading:

  • Flat fading: the entire pass band of interest is affected equally (also known as narrow or amplitude varying channels).
  • Frequency selective fading: certain frequency components are affected more than others (also known as wideband channels). This phenomenon tends to introduce inter-symbol interference.
  • Slow fading: the channel characteristics vary at less than the baud rate.
  • Fast fading: the channel characteristics vary faster than the baud rate.

Time Dispersion edit

Time dispersion occurs when signals arrive at different times. Signals traveling at the speed of light move about 1 foot in 1 nanosecond. This spreading tends to limit the bit rate over RF links.

Rayleigh Fading edit

The Rayleigh distribution can be used to describe the statistical variations of a flat fading channel. Generally, the strength of the received signal falls off as the inverse square of the distance between the transmitter and receiver. However, in cellular systems, the antennas are pointed slightly down and the signal falls of more quickly.


Ricean Fading edit

The Ricean distribution is used to describe the statistical variations of signals with a strong direct or line-of-sight component and numerous weaker reflected ones. This can happen in any multipath environment such as inside buildings or in an urban center.

A received signal is generally comprised of several signals, each taking a slightly different path. Since some may add constructively in-phase and others out of phase, the overall signal strength may vary by 40 dB or more if the receiver is moved even a very short distance.

Doppler Shift edit

A frequency shift is caused by the relative motion of the transmitter and receiver, or any object that reflects/refracts signal. This movement creates random frequency modulation. Doppler frequency shift is either positive or negative depending on whether the transmitter is moving towards or away from the receiver.

This Doppler frequency shift is given by:


vm is the relative motion of the transmitter with respect to the receiver, c is the speed of light and fc is the transmitted frequency. In the multipath environment, the relative movement of each path is generally different. Thus, the signal is spread over a band of frequencies. This is known as the Doppler spread.

Atmospheric Diffraction edit

Radio waves cannot penetrate very far into most objects. Consequently, there is often a shadow zone behind objects such as buildings,hills, etc.

The radio shadow zone does not have a very sharp cutoff due to spherical spreading, also called Huygens’ principle. Each point on a wavefront acts as it were a point source radiating along the propagation path. The overall wavefront is the vector sum of all the point sources or wavelets. The wavelet magnitude is proportional to   where   is measured from the direction of propagation. The amplitude is maximum in the direction of propagation and zero in the reverse direction.

Reflection edit

Reflection normally occurs due to the surface of earth or building & hills which have large dimension relative to the wavelength of the propagation waves. The reflected wave changes the incident angle.

There is similarity b/w the reflection of light by a conducting medium. In both cases, angle of reflection is equal to angle of incidence. The equality of the angles of reflection & incidence follows the second law of reflection for light.

Diffraction edit

Diffraction occurs in beams of light or waves when they become spread out as a result of passing through a narrow slit. Maximum diffraction occurs when the slit through which the wave passes through is equal to the wavelength of the wave. Diffraction will result in constructive and destructive interference.

Path Loss edit

References edit

This page is going to talk about the effect of noise on transmission systems.

Types of Noise edit

Most man made electro-magnetic noise occurs at frequencies below 500 MHz. The most significant of these include:

• Hydro lines
• Ignition systems
• Fluorescent lights
• Electric motors

Therefore deep space networks are placed out in the desert, far from these sources of interference.

There are also a wide range of natural noise sources which cannot be so easily avoided, namely:

Atmospheric noise - lighting < 20 MHz
Solar noise - sun - 11 year sunspot cycle
Cosmic noise - 8 MHz to 1.5 GHz
Thermal or Johnson noise. Due to free electrons striking vibrating ions.
White noise - white noise has a constant spectral density over a specified range of frequencies. Johnson noise is an example of white noise.
Gaussian noise - Gaussian noise is completely random in nature however, the probability of any particular amplitude value follows the normal distribution curve. Johnson noise is Gaussian in nature.
Shot noise - bipolar transistors
where q = electron charge 1.6 x 10-19 coulombs
Excess noise, flicker, 1/f, and pink noise < 1 KHz are Inversely proportional to frequency and directly proportional to temperature and dc current
Transit time noise - occurs when the electron transit time across a junction is the same period as the signal.

Of these, only Johnson noise can be readily analysed and compensated for. The noise power is given by:



k = Boltzmann's constant (1.38 x 10-23 J/K)
T = temperature in degrees Kelvin
B = bandwidth in Hz

This equation applies to copper wire wound resistors, but is close enough to be used for all resistors. Maximum power transfer occurs when the source and load impedance are equal.

Combining Noise Voltages edit

The instantaneous value of two noise voltages is simply the sum of their individual values at the same instant.


This result is readily observable on an oscilloscope. However, it is not particularly helpful, since it does not result in a single stable numerical value such as one measured by a voltmeter.

If the two voltages are coherent [K = 1], then the total rms voltage value is the sum of the individual rms voltage values.


If the two signals are completely random with respect to each other [K = 0], such as Johnson noise sources, the total power is the sum of all of the individual powers:


A Johnson noise of power P = kTB, can be thought of as a noise voltage applied through a resistor, Thevenin equivalent.


An example of such a noise source may be a cable or transmission line. The amount of noise power transferred from the source to a load, such as an amplifier input, is a function of the source and load impedances.


If the load impedance is 0  , no power is transferred to it since the voltage is zero. If the load has infinite input impedance, again no power is transferred to it since there is no current. Maximum power transfer occurs when the source and load impedances are equal.


The rms noise voltage at maximum power transfer is:


Observe what happens if the noise resistance is resolved into two components:


From this we observe that random noise resistance can be added directly, but random noise voltages add vectorially:


If the noise sources are not quite random, and there is some correlation between them [0 < K < 1], the combined result is not so easy to calculate:

K = correlation [0 < K < 1]
R0 = reference impedance

Noise Temperature edit

The amount of noise in a given transmission medium can be equated to thermal noise. Thermal noise is well-studied, so it makes good sense to reuse the same equations when possible. To this end, we can say that any amount of radiated noise can be approximated by thermal noise with a given effective temperature. Effective temperature is measured in Kelvin. Effective temperature is frequently compared to the standard temperature,  , which is 290 Kelvin.

In microwave applications, it is difficult to speak in terms of currents and voltages since the signals are more aptly described by field equations. Therefore, temperature is used to characterize noise. The total noise temperature is equal to the sum of all the individual noise temperatures.

Noise Figure edit

The terms used to quantify noise can be somewhat confusing but the key definitions are:

Signal to noise ratio: It is either unitless or specified in dB. The S/N ratio may be specified anywhere within a system.
Noise Factor (or Noise Ratio):   (unit less)
Noise Figure:   dB

This parameter is specified in all high performance amplifiers and is measure of how much noise the amplifier itself contributes to the total noise. In a perfect amplifier or system, NF = 0 dB. This discussion does not take into account any noise reduction techniques such as filtering or dynamic emphasis.


Friiss' Formula & Amplifier Cascades edit

It is interesting to examine an amplifier cascade to see how noise builds up in a large communication system.


Amplifier gain can be defined as:  

Therefore the output signal power is:  
and the noise factor (ratio) can be rewritten as:  
The output noise power can now be written:  

From this we observe that the input noise is increased by the noise ratio and amplifier gain as it passes through the amplifier. A noiseless amplifier would have a noise ratio (factor) of 1 or noise figure of 0 dB. In this case, the input noise would only be amplified by the gain since the amplifier would not contribute noise.

The minimum noise that can enter any system is the Johnson Noise:
Therefore the minimum noise that can appear at the output of any amplifier is:
The output noise of a perfect amplifier would be (F = 1):
The difference between these two values is the noised created (added) by the amplifier itself:
This is the additional (created) noise, appearing at the output.

The total noise out of the amplifier is then given by:


If a second amplifier were added in series, the total output noise would consist the first stage noise amplified by the second stage gain, plus the additional noise of the second amplifier:

If we divide both sides of this expression by the common term:  
we obtain:

This process can be extended to include more amplifiers in cascade to arrive at:

Friiss' Formula

This equation shows that the overall system noise figure is largely determined by the noise figure of the first stage in a cascade since the noise contribution of any stage is divided by the gains of the preceding stages. This is why the 1st stage in any communication system should be an LNA (low noise amplifier).

Receiver Sensitivity edit

In a given bandwidth, W, we can show that the noise power N equals:


From N, we can show that the sensitivity of the receiver is equal to


Cascaded Systems edit

This page will discuss the topic of signal propagation through physical mediums, such as wires.

Transmission Line Equation edit

Many kinds of communication systems require signals at some point to be conveyed over copper wires.

The following analysis requires two assumptions:

• A transmission line can be decomposed into small, distributed passive electrical elements
• These elements are independent of frequency (i.e. although reactance is a function of frequency, resistance, capacitance and inductance are not)

These two assumptions limit the following analysis to frequencies up to the low MHz region. The second assumption is particularly difficult to defend since it is well known that the resistance of a wire increases with frequency because the conduction cross-section decreases. This phenomenon is known as the skin effect and is not easy to evaluate.


The purpose behind the following mathematical manipulation is to obtain an expression that defines the voltage (or current) at any time (t) along any portion (x) of the transmission line. Later, this analysis will be extended to include the frequency domain.

Recall the characteristic equations for inductors and capacitors:


Kirchoff's Voltage Law edit

Kirchoff's voltage law (KVL) simply states that the sum of all voltage potentials around a closed loop equal zero. Or in other words, if you walked up a hill and back down, the net altitude change would be zero.

Applying KVL in the above circuit, we obtain:
But the LHS (left hand side) of the above equation, represents the voltage drop across the cable element  , therefor:
Dividing through by  , we obtain:
The LHS is easily recognized as a derivative. Simplifying the notation:

This expression has both current and voltage in it. It would be convenient to write the equation in terms of current or voltage as a function of distance or time.

Simplifying the Equation (trust me) edit

The first step in separating voltage and current is to take the derivative with respect to the position x (Equation 1):

The next step is to eliminate the current terms, leaving an expression with voltage only. The change in current along the line is equal to the current being shunted across the line through the capacitance C and conductance G. By applying KCL in the circuit, we obtain the necessary information (Equation 2):
Taking the derivative with respect to time, we obtain (Equation 3):
Substituting (Equation 2) and (Equation 3) into (Equation 1), we obtain the desired simplification:
Collecting the terms, we obtain:
The Transmission Line Equation for Voltage

This equation is known as the transmission line equation. Note that it has voltage at any particular location x as a function of time t.

Similarly for current, we obtain:
The Transmission Line Equation for Current

But we're not quite done yet.

Solving the Transmission Line Equation edit

Historically, a mathematician would solve the transmission line equation for v by assuming a solution for v, substituting it into the equation, and observing whether the result made any sense. An engineer would follow a similar procedure by making an “educated guess” based on some laboratory experiments, as to what the solution might be. Today there are more sophisticated techniques used to find solutions. In this respect, the engineer may lag behind the mathematician by several centuries in finding applications for mathematical tools.

To solve the transmission line equation, we shall guess that the solution for the voltage function is of the form:


The first term represents a unity vector rotating at an angular velocity of   radians per second, in other words, a sine wave of some frequency. The second term denotes the sinusoid being modified by the transmission line, namely its amplitude decaying exponentially with distance. If we let   be a complex quantity, we can also include any phase changes which occur as the signal travels down the line.

The sine wave is used as a signal source because it is easy to generate, and manipulate mathematically. Euler’s Identity shows the relationship between exponential notation and trigonometric functions:
Euler's Identity
Going back to our educated guess, we will let  , therefore:
The term   represents the exponential amplitude decay as this signal travels down the line.   is known as the attenuation coefficient and is expressed in Nepers per meter.
The term   represents the frequency of the signal at any point along the line.   component is known as the phase shift coefficient, and is expressed in radians per meter.
Substituting our educated guess
into the transmission line equation for voltage, we obtain:

This looks pretty intimidating, but if you can do basic differentials and algebra, you can do this!

Simplifying the Equation (trust me) edit

The idea now is to work through the math to see if we come up with a reasonable solution. If we arrive at a contradiction or an unreasonable result, it means that our educated guess was wrong and we have to do more experimenting and come up with a better guess as to how voltage and current travel down a transmission line.

Let's look at this equation one term at a time:

LHS = RHS Term 1 + RHS Term 2 + RHS Term 3
Starting with the left hand side (LHS) we get the following simplification:
Believe it or not, the RHS Term 1 does not need simplifying.
Simplifying the RHS Term 2, we obtain:
Simplifying the RHS Term 3, we obtain:
Let's put it all back together:
Note that each of the four terms contain the expression  .
Therefore we end up with:
And this can be further simplified to:
Attenuation and Phase Shift Coefficients

This result is not self contradictory or unreasonable. Therefore we conclude that our educated guess was right and we have successfully found an expression for attenuation and phase shift on a transmission line as a function of its distributed electrical components and frequency.

Lossless Transmission Line edit

Signal loss occurs by two basic mechanisms: signal power can be dissipated in a resistor [or conductance] or signal currents may be shunted to an AC ground via a reactance. In transmission line theory, a lossless transmission line does not dissipate power. Signals, will still gradually diminish however, as shunt reactances return the current to the source via the ground path. For the power loss to equal zero, R = G = 0. This condition occurs when the transmission line is very short. An oscilloscope probe is an example of a very short transmission line. The transmission line equation reduces to the voltage equation:

and the current equation:

To determine how sinusoidal signals are affected by this type of line, we simply substitute a sinusoidal voltage or current into the above expressions and solve as before, or we could take a much simpler approach. We could start with the solution for the general case:

Let R = G = 0, and simplify:
Equating the real and imaginary parts:

This expression tells us that a signal travelling down a lossless transmission line, experiences a phase shift directly proportional to its frequency.

Phase Velocity edit

A new parameter, known as phase velocity, can be extracted from these variables:

  meters per second

Phase velocity is the speed at which a fixed point on a wavefront, appears to move. In the case of wire transmission lines, it is also the velocity of propagation., typically: 0.24c < Vp < 0.9c .

The distance between two identical points on a wavefront is its wavelength ( ) and since one cycle is defined as 2  radians:


In free space, the phase velocity is 3 x 108 meters/sec, the speed of light. In a cable, the phase velocity is somewhat lower because the signal is carried by electrons. In a waveguide transmission line, the phase velocity exceeds the speed of light.

Distortionless Transmission Line edit

A distortionless line does not distort the signal phase, but does introduce a signal loss. Since common transmission lines are not super conductors, the signal will decrease in amplitude but retain the same shape as the input. This characteristic is essential for long cable systems.

Phase distortion does not occur if the phase velocity Vp is constant at all frequencies.

By definition, a phase shift of 2  radians occurs over one wavelength  .


This tells us that in order for phase velocity Vp to be constant, the phase shift coefficient  , must vary directly with frequency  .


The problem now is to find  . This can be done as follows:


It may seem that we have lost  , but do not give up. The 2nd and 3rd roots can be expanded by means of the Binomial Expansion.

In this instance n = 1/2. Since the contribution of successive terms diminishes rapidly,   is expanded to only 3 terms:

This may seem complex, but remember it is only algebra and it will reduce down to simple elegance. Expanding the terms we obtain:

Since  , we merely have to equate the real and imaginary terms to find  .
Note that if   then  

From this we observe that   is directly proportional to  .

Therefore the requirement for distortionless transmission is:
This is one of the essential design characteristics for a broadband coax cable network.

If we equate the real terms, we obtain:


So there is a reason to study algebra after all!

The Frequency Domain edit

Signal analysis is often performed in the frequency domain. This tells us how the transmission line affects the spectral content of the signals they are carrying.

To determine this, it is necessary to find the Fourier Transform of the transmission line equation. Recall:


and recall (hopefully) the Fourier Transform (which converts the time domain to the frequency domain):


To prevent this analysis from ‘blowing up’, we must put a stipulation on the voltage function namely, that it vanishes to zero at an infinite distance down the line. This comprises a basic boundary condition.


This stipulation is in agreement with actual laboratory experiments. It is well known that the signal magnitude diminishes as the path lengthens.

Likewise, a time boundary condition, that the signal was zero at some time in the distant past and will be zero at some time in the distant future, must be imposed.


Although engineers have no difficulty imposing these restrictions, mathematical purists, are somewhat offended. For this and other reasons, other less restrictive transforms have been developed. The most notable in this context, is the Laplace transform, which does not have the same boundary conditions.

Having made the necessary concessions in order to continue our analysis, we must find the Fourier Transform corresponding to the following terms:

Then applying the transform on the derivative, we obtain:

This equation can be solved by using integration by parts:


Applying the boundary conditions when t goes to infinity makes the 1st term disappear.


Note that the resulting integral is simply the Fourier Transform. In other words:


We can now write the transmission line equation in the frequency domain:

Rearranging the terms, we obtain: