Communications

Abstract

At a time when we are surrounded by mobile phones and Internet-based services, communication is a central component of virtually every aspect of our life. At the same time, communication itself is a rich and multifaceted notion. For example, we can consider why or how we communicate. We touch on the first question in the Game Theory chapter. In this chapter, we turn to the second question, and in particular, the fundamental concerns of what constitutes communication, what are the hard limits on telecommunications, how such limits arise, and how we can model their effects.

At a time when we are surrounded by mobile phones and Internet-based services, communication is a central component of virtually every aspect of our life. At the same time, communication itself is a rich and multifaceted notion. For example, we can consider why or how we communicate. We touch on the first question in the Game Theory chapter. In this chapter, we turn to the second question, and in particular, the fundamental concerns of what constitutes communication, what are the hard limits on telecommunications, how such limits arise, and how we can model their effects.

1 Communication , Certainty , Uncertainty , and Belief

We begin by defining some basic concepts in a manner that is as independent as possible of the details of current technologies. There are two reasons for doing so. The first is to clarify the terms used in this chapter and to reduce the chance of misunderstandings. For all the notions discussed here there are alternative interpretations, and a comprehensive treatment of these alternatives is beyond the scope of this book. The second is to settle on notions that we hope are simpler and may last a bit longer than the rapidly changing current technologies, and that may be compatible with the confluence of different technical disciplines.

To communicate is to share information. A speech by the leader of a nation shares information with interested citizens who may be present in person or watching the speech over modern telecommunication infrastructure. A server storing a digital copy of this book shares information with a smartphone or a digital device where it is downloaded over the Internet.

That communication involves sharing is straightforward. That it involves information is more interesting. Information is an abstract notion related to attaining certainty, which, in turn, is firm or absolute belief. For example, consider a value drawn from the set of Boolean s {True, False}. If we are uncertain about this value, we hold no belief about it beyond its possible values (which can be viewed as its “type”). When we are certain about the value, we either believe that it is True or believe that it is False. This is a set-based notion of uncertainty, as illustrated in Figure 8.1.

Fig. 8.1

Set-based uncertainty about a Boolean quantity

This is the notion of uncertainty, and in turn, the notion of information, typically addressed by set-based and interval analysis methods, whether applied to numerical computations or programs.

Other notions of uncertainty exist. For example, in cases where it is either impossible or unreasonable to hold absolute belief, we may associate probabilities to different values. For example, we may believe that the value may have a probability of 0.95 of being True. This is a distribution-based, probabilistic , or stochastic notion of uncertainty (See Figure 8.2). It is this notion that we typically see being used in information and coding theory and in probabilistic and statistical methods.

Fig. 8.2

Probabilistic uncertainty about a Boolean quantity

For cyber-physical systems we need to consider both notions because we are often interested not only in stochastic guarantees (as is typical in communication theory) but also in deterministic ones.

2 Messages : From Information to Representation

In contrast to information, which is a notion centered around the belief of an agent , a message is data (a string) that is used to represent and realize the transfer of information. For example, the message “Tom and Jerry are here” carries information for someone who is unaware of their presence. It also carries no information to someone who is already believes they are here. Accordingly, whether a message carries information is critically dependent on the receiver and, in particular, the receiver’s prior beliefs. This is illustrated by the following table:

Message	Belief before	Belief after	Information
Tom & Jerry are here	ϕ	Tom & Jerry are here	Tom & Jerry are here
Tom & Jerry are here	Tom & Jerry are here	Tom & Jerry are here	ϕ
They are here	They are Tom & Jerry	They are Tom & Jerry	Tom & Jerry are here
		Tom & Jerry are here
They are here	They are Tom & Jerry	They are Tom & Jerry	ϕ
	Tom & Jerry are here	Tom & Jerry are here

The table also illustrates how correct transmission and message content are different from the information that the message carries. The message “they are here!” also carries the same information for the first receiver if they know the context, but the raw data (the sequence of letters) being transmitted is clearly different.

The following two exercises, one for each notion of uncertainty, provide more examples that illustrate the difference between the message and the information it carries.

Exercise 8.1 Consider an agent that believes that x + y = z where +  is addition on the real numbers and that x ∈{1..2}, that is, x is in the set of all real numbers between 1 and 2, inclusive.

• 1. If it receives a message that y ∈{3..4}, what should it believe about z?

• 2. If it receives a message that z ∈{3..4}, what should it believe about x?

Exercise 8.2 Consider an agent that believes that (x ∧ y) = z where ∧ is conjunction (AND) on Booleans and that the probability of x being True is 0.5.

• 1. If it receives a message saying that the same probability holds for y, what should it believe about z?

• 2. If it receives a message saying that the same probability holds for z, what should it believe about x?

3 Belief , Knowledge , and Truth

Knowledge is belief that is true. This is consistent with our intuitive understanding of these notions: In everyday life we would not use the term Knowledge to describe the false belief that “The planet Earth was entirely pink on January 1st 2019.” Rather, we would describe it as a misconception or simply a false belief. Similarly, we would say that Jane knows that 1 + 1 = 2 when she believes that statement, since it is a true statement. We may also say that Jane believes that a variable x has a value of 17. When we say that we leave it open whether her belief is correct. In contrast, when we say that Jane knows that x has a value of 17, we are also asserting that her belief is true. Thus, even in everyday language we take knowledge to be the intersection of belief and truth (Figure 8.3).

Fig. 8.3

Knowledge is the intersection of belief and truth

The distinctions are important for two reasons. The first is that they can often guide us in comparing different ways to solving a problem. Consider the following exercise:

Exercise 8.3 The length x of some rope is exactly 3.14 m.

• 1. A camera-based app on a smartphone estimates it to be 3.15. Can we say that the phone has belief or knowledge about the device?

• 2. What if, instead, the phone estimates the value to be in {3..4}?

This exercise illustrates that awareness and tracking of the error in a measurement can have a significant impact on the validity of calculations based on this measurement. Now consider the following situation.

Exercise 8.4 The altitude (distance from sea level) x of a ship is varying over time t according to the equation $x(t)=\sin {}(t)$. An aircraft trying to land on the ship has a device the provides a measurement of the height of the ship. Explain for each of the following cases if the aircraft has belief and/or knowledge:

• 1. The device provides a perfect and continuous measurement y where y(t) = x(t).

• 2. The device provides a delayed continuous measurement $y(t)=\sin {}(t-d)$ where the amount of positive delay d is fixed but unknown.

• 3. The device provides a delayed continuous measurement $y(t)=\sin {}(t-d)$ and the value of the delay d.

• 4. The device provides a delayed continuous measurement $y(t)=\sin {}(t-d)$, the value of the delay constant d, and the aircraft knows that the value being measured is a periodic signal .

• 5. The device provides two discrete measurements, $y_1=\sin {}(t_1-d)$, $y_2=\sin {}(t_1+1-d)$ the value of the delay constant d, the measurement time t ₁ + 1, and the aircraft knows that the value being measured is a periodic signal.

This example illustrates the value of not only of keeping track of measurement error but also of keeping track of other effects that may arise with measurement, such as delay, and of knowledge about the nature of the signal being measured.

3.1 Broader Implications

The above distinctions help us understand what constitutes correct communication in cyber-physical systems. The distinctions are critical for two main reasons.

First, since truths in general are not affected by what we share or do not share about them, information affects primarily belief, and only affects knowledge to the extent that the change in belief overlaps with truth. Information is thus related to belief but not necessarily to knowledge. Logic and its rules are generally based on the assertion that truths need to be consistent and free of contradiction. This does not apply to beliefs. We generally work to ensure their consistency, but such efforts may well fail.

Second, whereas data processing systems are developed with some particular intent in mind, computations themselves are generally oblivious to this intent. Today, this dichotomy is touched upon quite often in the context of the so-called smart contracts, which are digital contracts that execute automatically. Our intent when writing a contract is based on our beliefs at one time, but smart contracts must run correctly for a very long time. Even if we consider such instruments to be “only” affecting money (money still has tremendous impact on people’s livelihood), with increasingly more services that take orders online to perform real-world functions (such as Amazon, Uber, or a wide range of other services), unintended actions from such systems can have tremendous undesirable impact on our life, and at a very large scale.

For these reasons, the importance of maintaining awareness of the real-world meaning and real-world veracity of the information being communicated and manipulated by automated systems cannot be understated. As innovators we have significant responsibility if not legally or socially then ethically towards the programs, controllers, and cyber-physical systems that we develop.

4 Carrier Signal, Medium , and Link

Let us now turn our attention away from information back to sharing. In particular, let us consider how sharing is carried out and the characteristics of such processes. For a transmission to occur in the real world, a message needs to be represented and physically transmitted. Transmission occurs using a carrier signal. Often, the carrier is transmitted over a medium. For example, consider the process of handing a cone of ice-cream to a friend (See Figure 8.4). Here, the carrier signal is ice-cream cones, the medium is the space in which it travels between the first and the second person. The message is whether or not we send a cone. As usual, the information being transmitted depends on the beliefs of the receiver, but one can imagine in this example the message is intended to communicate a positive sentiment. When we consider the communication between you and your friend in this situation, we can think of this entire process as a communication link.

Fig. 8.4

A simple communications channel

There is a wide variety of possible carrier types, and which we can call communication modes . The following table gives some examples.

Mode	Medium	Examples
Transportation	Optional	Postal service. Cell biology. Olfaction
Light	Optional	Lighthouses. Gestures. Sign language. LiFi.
Radio waves	Optional	Cellular. Bluetooth. WiFi.
Electric current	Mostly	Phone lines. Twisted pairs.
Electric potential	Rarely	Across capacitors
Vibration	Necessary	Sound (sonic). Ultrasonic.
Pressure	Necessary	Steering. Pressure modulation.
Temperature	Necessary	Covert communication.

The first three modes require no medium. While traditional treatments of communication will not typically include transportation as a mode of communication, in a cyber-physical setting it can be very useful to consider it as such. It is also useful to recognize that it happens both in man-made and natural systems. For example, the transport of molecules within and between living cells is essential for regulating the processes of a living organism.

Light and radio waves, which are both electromagnetic waves, can be viewed as transportation of photons. Physically, electromagnetic signals are different from what we consider as everyday transportation because photons have both wave and particle properties.

The next three modes typically require a medium, but it is useful to note that it is not always necessary. Electric signals involve the transportation of electrons, which is generally more controlled through a medium. It is possible to move electrons in free space, but managing them in this manner is quite different from designing electric circuits. In addition, traditional electric circuits also involve components such as capacitors and inductors, which involve a discontinuity in the conductive medium. Such gaps can be a medium, in which case they are called dielectrics, or it can be a vacuum. Dielectrics are more common as they would generally help avoid physical contact between the components involved.

The last three modes require a physical medium and cannot exist without it because they are, in effect, changes to the physical state of a given medium. Vibration in general and sound in particular are interesting not only because we as humans have used them since the beginning of time, but also in the cyber-physical setting, they can be related to security. For example, it is known that a malicious agent can implant a virus on computers with no traditional communication ports so as to use cooling fans to export information from such devices.

The last two are examples of more rare modes of communication, but it is useful to be aware of such examples for both cooperative and uncooperative situations. Pressure modulation, for example, has been used by oil exploration companies to send signals from deep in the ground to the surface during drilling, a situation which makes other modes of communication difficult. We are not aware of uses of temperature for communication in cooperative situations, but it is known that temperature can leak significant information about encryption keys when, for example, the temperature of a smart card chip is used to extract information that can be used to find the key.

5 Link Characteristics

The wealth of possibilities for communication modes is challenging and inspiring. At the same time, when we want to design specific systems, it is useful to have ways to compare different possible choices. The question of cost is of course always an overarching one. The next question then is how to quantify what is being communicated so that we can compare costs between solutions of similar or comparable performance.

In general, the performance of a communication link is not quantified as a single quantity, but rather, as a composite of different characteristics. Commonly considered characteristics are latency, bandwidth, and various notions of reliability. While ice-cream, of course, is not the only means of communication, it can illustrate some common physical characteristics of links. In the context of cyber-physical systems, the mobility of the communicating entities makes the situation more interesting, as all of these characteristics can depend on both the relative location of the entities as well as their environment.

5.1 Latency

Latency is the time it takes from sending a signal to receiving it. If we imagine that the two people are 50 cm apart and the ice-cream can be moved at a speed of 1 m/s, then it will take 0.5 s to transmit the ice-cream. While for interpersonal communication such delays may be acceptable, for applications such communicating a signal from car brake pedals to wheels, much shorter delays are required.

Clearly, faster transport speeds can lead to shorter latencies. This is an important reason why media such as electric current and electromagnetic waves (light and radio) are popular. Light can move much faster than our ice-cream in this example, in fact, almost 300 million times faster. But we should keep in mind that for any non-zero distance and finite speed, transmission over this distance will experience non-zero delay.

Physics limits the minimum latency more than we may realize at first. In particular, the theory of special relativity suggests that not only is there always a non-zero delay, but there may be an absolute, minimum delay between two objects with a non-zero distance between them. In particular, the theory suggests that it is impossible to travel faster than the speed of light. This means that the shortest time any transmission can take between the two people in our ice-cream example is about 1.67 ns (nanoseconds). Another way of looking at this is that no signal can travel more than about 30 cm in a nanosecond. This constraint is significant in large-scale systems such as communication via satellites or when communicating with someone on the moon. Light takes about 1.3 s to travel between Earth and the moon.

As an aside, for a historic illustration of the significance of understanding these basic constraints, and if you have not done so already, we recommend that you find and watch the two-minute YouTube video entitled “Admiral Grace Hopper Explains the Nanosecond.”

5.2 Bandwidth

Bandwidth is the number of messages that can be sent per unit time. Note that this notion cannot be meaningful unless messages can only be split into a finite number of indivisible messages. Thus, the notion of bandwidth requires that messages are discrete entities. For uniformity, a message can be taken to be one of exactly two possible values, that is, one bit. Note also that bandwidth is based on the data being transmitted rather than the information it conveys, as the latter is always a function of the beliefs of the receiver.

Considering our example above, if we assume that there are no verbal or visual hints given by the first person, the “message” can be seen as being one of two things: Either one ice-cream is handed over, or none are. If we further consider that this event can occur only once per day, then the maximum transfer rate is one message per day. Since there are only two possible events, let us consider the message to be one bit.

Assuming that the information the receiver takes from getting the ice-cream is that the sender likes them, this is a like/neutral signal. If the sender and/or the receiver would like more detailed information, such as really-like/like/neutral, then more bandwidth would be needed. This can be achieved, for example, through the use of two ice-creams. So that the information mentioned can be represented by two-ice-creams/one-ice-cream/no-ice-cream. Of course, such transmissions may cost more or require more work, but the amount of information that can be transmitted increases. As we will often see, physical resources can often limit the rate of transfer. For example, there is only so many standard sized ice-cream cones and scoops on the planet. But what is physically transmitted is only one source of limitation. In the following exercise, we consider some others.

Exercise 8.5 In the above example, using twice as many ice-creams did not double the levels of “like” that we have.

• 1. Are there ways in which a maximum of two ice-creams per day can be used to communicate four like levels, such as like-a-lot/like/like-a-bit/neutral?

• 2. What is the key idea that you are using to achieve this higher level of information transfer? In other words, is there a reason why this method can be expected to generalize to other situations?

5.3 Reliability

Under idealized conditions, for example, the universe consists only of you, your friend, and the ice-cream, the transmission of the ice-cream should be quite reliable: Once you start the process of handing over the ice-cream to your friend, the expected outcome for them should be that they receive it and recognize the message. But idealized conditions may be hard or even too difficult to provide. Instead, you and your friend could be standing outdoors on a windy day, they may be looking the other way as you get the ice-cream that you wish to give to them, and a wind might come and blow away the ice-cream before you are able to offer it to your friend. Alas, the physical evidence of the ice-cream is now gone. This kind of situation is a simplified example of the reliability issues that arise in almost all real-world communications. In general, they can also become more challenging as we try to transmit more information, over larger distances, in dynamic environments, and between mobile entities.

6 Fundamental Limits from Physics

Nature poses fundamental limits on link characteristics. These limits tend to become significant at extremes of transportation speeds or energy usage. For example, as we approach extremes of low energy, the smallest possible unit of energy transmission is one photon. If we also reduce our sampling period to a small enough period, to correctly detect a photon as it arrives would mean that we would know its speed and position, and that would lead up to other known limits posed by what is called the Heisenberg uncertainty principle.

Another fundamental physical limitations on bandwidth is to consider that the highest known frequencies for electromagnetic waves, gamma rays, are about 10²⁵. Even if we assume that we can pass one wave or skip it to encode a bit, this would be the maximum bandwidth. In practice, there are numerous reasons why even this assumption cannot be realized. But at least we have a relatively easy way to see that there are some hard limits on a single-bit communication channel.

While we can see these physics-imposed limits on latency and bandwidth as constraints on a space of possible solutions, they can also be seen as sources of inspiration for further research and innovation.

7 Limits Due to Component Dynamics

While nature can limit latency and bandwidth at a fundamental level, dynamics of the components used to build the communication link will generally pose significantly greater effects that will lead to the dominant practical concerns. To illustrate this concretely, we consider the common case of what happens when we use electric circuits to communicate.

7.1 Electrical Signal Transmission

The simplest example of an electric circuit where a signal can be transmitted is one where there is a constant current or voltage being transmitted from one entity to another. Let us focus on the case when we are transmitting a voltage (a similar analysis can be carried out if we transmit via current). As noted above, due to the laws of physics, our ability to observe any physical phenomena is always limited by some minimal quantity that can be measured. For this reason, we consider only discrete levels of voltage difference. The simplest case is to have two levels. We can after all use a series of such transmissions to represent any number of levels. For the sender to build up the voltage to be transmitted, a sufficient number of electrons must be moved from one side of the circuit to the other. Voltage difference is proportional to the amount of electrons moved. The rate at which electrons move is called current. Current generates heat and thus consumes energy, and so has to be limited in any circuits otherwise it will overheat. Limiting current means that building up the voltage takes time. This means that there is a minimum time needed to change from one voltage level to the other. This, in turn, limits the rate for data transfer (bandwidth) on this wire.

The situation described above can be modeled by a series RC circuit (shown in Figure 8.5) where a voltage source V _i is connected in series to a resistance R and then a capacitance C. We will call the current flowing the circuit I and the charge across the capacitor Q. Using the principles introduced in the chapter on physical modeling, this circuit’s dynamics is governed by the following two equations:

Fig. 8.5

A series RC circuit model of an electric signal transmission channel

$$\displaystyle \begin{aligned} V_i(t) = I(t)R + Q(t)/C, \end{aligned} $$

(8.1)

and

$$\displaystyle \begin{aligned} Q(t)=\int_{0}^{t}I(s)\,ds. \end{aligned} $$

(8.2)

The first equation reflects the fact that the input voltage must be equalized by the voltage from the rest of the circuit. The second equation models the way the voltage at the target of the signal (represented by the capacitor) is a function of the current being transmitted and the time lapsed. This equation captures the effect of the physical movement of electrons that is necessary to build up the voltage at the target, and that will make it possible for the target to measure a change in the circuit.

To make these two equations easier to recognize we will note that since Q is the integral of I (which is what the second equation states) then we also know that I is the derivative of Q. That means

$$\displaystyle \begin{aligned} Q'(t)=I(t). \end{aligned} $$

(8.3)

With this observation we can rewrite the first equation as

$$\displaystyle \begin{aligned} V_i(t) = Q'(t)R + Q(t)/C. \end{aligned} $$

(8.4)

Using basic arithmetic we can turn this equation into

$$\displaystyle \begin{aligned} Q'(t) = \frac{V_i(t) - Q(t)/C}{R}, \end{aligned} $$

(8.5)

which is an ODE. In Acumen, this would be written as

To have a full simulation model one only needs an initially section that provides some example parameters such as

The constants are selected here only for illustration and not for resemblance to any concrete circuit parameters. The key take away from running this simulation is that it takes some time for the voltage at the target, Q∕R, which is simply Q in this case, to reach the value of the source.

This simulation suggests several observations that can be confirmed through further mathematical analysis of the equations. For example, if we consider the start of the simulation, we can see that for any non-zero sensitivity to detecting voltage change there is a non-zero time needed to allow the voltage to grow to this level. But it is also important to note that this simple experiment does not tell the full picture. If V _i is to be changed to transmit both zeros and ones in sequence, then in general it may not be easy to detect voltage changes, but rather, we may want to have the voltage to reach certain specific values to consider this a reliable measurement. Such a requirement would further increase the time that we must allow for the signal at the target to reach a measurable level. The key take away here is that detecting signals requires time.

7.2 Variability in Component Parameters

Bandwidth and latency also suffer due to both capacitive and inductive effects of electrical wires. In addition, another important practical source of limitations is the variability in individual components. With any manufacturing technology, it is hard to create components that have identical characteristics, due to natural variations in the environment, materials, processes, and other factors. In poorly designed systems, the variability in individual components can be greatly magnified when we put them together. Techniques such as feedback, discretization, and quantization all provide important tools for managing this problem. For the purposes of this chapter it suffices to be aware of this issue and the need for these methods to address it.

7.3 Light and Radio Transmission

In contrast to electrical signals, light and radio wave transmission can have an advantage in terms of maximum bandwidth and latency. To give a concrete example, whereas twisted pairs can have speeds of up to 10 GHz, fiber optics transmission can go to 200 GHz and beyond. Electromagnetics in free space can have frequencies in the THz, therefore, bandwidth can in principle also approach these frequencies. However, by definition they are in general not directed, and therefore can be subjected to large dissipation effects that can limit their range. In many cases, however, there are physical obstacles on a transmission path, which stop or significantly reduce the signal.

8 Limits Due to Noise

Noise is a term that is used to describe environmental factors that can make sensing difficult. The simplest example is when we are talking to a friend in a busy gathering and find it difficult to hear each other because others are talking in the background. In this case, the medium you are using to communicate is also being used by other messages in a way that makes it hard for you to receive your friends message correctly.

Noise arises in virtually all known modes of communication. Ambient vibration, sound, light, heat, and radiation are all phenomena that can be considered to be types of noise, and that can affect a communication channel. In transportation, a message can be influenced by the many other messages that go through the system, as well as the occasional failures in the process. Electrical signals can be influenced by cross talk due to effects of varying electric potentials and electromagnetic effects from the rest of the system and the surrounding environment. Especially at high altitudes and in outer space, they can also be influenced by background radiation. Noise can also arise from contention over shared resources, which can be seen, for example, in the effect of nearby channels in radio transmission.

The inevitable presence of noise has several significant effect on signals. A particularly significant issue is that for all practical purposes there is a minimal precision for measurements. This is an important justification for why we thinking of messages as having discrete values. In essence, this decision reflects the fact that below a certain level it is impossible to make any measurement reliably. Another effect is stochasticity: Noise can be unbounded and then all we can hope for is that it follows a probabilistic distribution. Depending on this distribution and the possible magnitudes of the noise, it may be that correct measurement cannot be guaranteed with absolute certainty. This gives rise to the need for probabilistic methods in communication. This is a highly significant concern since sufficiently large levels of noise (or, alternatively, sufficiently low signal levels) lead to a situation where the probability of a correct reception of a message is indistinguishable from a guess by flipping a coin.

The situation in the last case deserves some attention, as it can be undesirable to receive a random message believe that was the message intended by the sender. A wide range of methods are used to allow the receiver to check the integrity of the received message. At the very least it allows the receiver to ignore the message, but more commonly it makes it possible to request a retransmission.

9 Limits Due to Energy Dissipation

While electrical transmission and light transmission are guided (for example, by placement of the wire or optical fiber, respectively), electromagnetic transmission in free space is unguided. Guiding is significant for preserving the energy in a signal, thus facilitating its travel to greater distances. To illustrate, you may be familiar with the experiment of creating a mechanical telephone. Such a device can be created by piercing two metal cans or plastic cups and using them to stretch a plastic fishing wire at a distance. As long as the wire is stretched, it can guide the transmission of vibrations from one end to the other rather efficiently. This allows a person holding one end to hear what the other person holding the other ends says into their side of the device. This experiment offers a hint that a signal traveling along a wire (a one-dimensional space) can be preserved quite well. The only limitations to the transmission of such a signal are dissipation due to inelastic effects in the fishing wire, which would transform the vibrations into heat, or the leakage of the mechanical vibration into the surrounding environment. Ignoring these effects, this gives us a good starting point for thinking about what happens to a wave when it is not guided.

For example, let us imagine that we have a signal propagating in two dimensions, such as what we might see if we drop a marble into the middle of a still pool. For simplicity, let us once more ignore secondary effects and assume that the energy of the wave that starts right where the marble was dropped is preserved as the wave spreads out. We know from basic geometry that the radius of the circle that represents the center of the wave as it goes out is linearly proportional to its circumference. Given the symmetry of the situation, it is reasonable to assume that the energy will be spread equally around the circumference. This means that if we observe the energy at any point on the circle (the wave), the energy of the signal at this point will go down in inverse proportion with the distance from the center. Similarly, in a three-dimensional setting where the wave is propagating spherically, the energy will go down in proportion to the square of the inverse of the distance.

10 Other Sources of Limitations

Several other practical aspects can also lead to limitations on communication. One is the clock speed of the sending and receiving systems, which occasionally need to be shared with other components of the communication system. Clocking is a discretization technique that facilitates the design and operation of large digital circuits. However, clock speeds often have to fit with the timing need of the most complex components on the system. In simple designs this may need to be matched or aligned with the clock rate of the transmitting or receiving device.

11 Chapter Highlights

• 1. Communication as transfer of information

  • (a) Information and certainty/uncertainty

  • (b) Information and knowledge, belief, and truth

• 2. Widely applicable concepts relating to communication

  • (a) Latency, the time to get a message through the channel

  • (b) Bandwidth, the maximum rate of sending messages

  • (c) Reliability, knowing that the message will come through when you send it

  • (d) Possible connections between each of these different concepts

• 3. Fundamental limitations and their sources

  • (a) Effects from physics (nature)

  • (b) Effects from physical component dynamics

  • (c) Effects due to noise (and/or “disturbance”)

  • (d) Effects due to energy limitations

12 Study Problems

The problems in this section can be investigated individually or in groups. They are larger than in the earlier chapters, so, expect them to require more time to solve.

• 1. Model a system for transmitting the bits of the binary representation for 42 on the RC channel presented in this chapter. Your model should include a vector representation of the binary representation for the message at the sender and the receiver.

  • (a) Find the shortest clocking period that would allow the correct transmission of the signal for this initial test message.

  • (b) Once this time has been determined, find another string that would not be transmitted correctly using these settings.

  • (c) Explain why the evaluation using the first string was not sufficient.

  • (d) Find a way to determine the fastest clocking period.

• 2. Amplitude modulation (AM) transmits a signal using a carrier that is itself a fixed frequency wave. It is the basis of AM radio. In essence, the signal and the carrier are multiplied to generate the transmitted signal.

  • (a) Model a source and signal generation for such a communication, assuming that the transmission channel is perfect. Assume that the carrier frequency is 100 Hz.

  • (b) Assume that the target knows the exact transmission frequency but not the phase. Model the target and explain the mechanism for determining the phase for the carrier signal.

  • (c) Use the channel to transmit the 4 bit representation of the numbers from 0 to 8.

  • (d) Determine experimentally the fastest rate with which this transmission can be done correctly using this channel.

13 To Probe Further

• Forbes article entitled The Real Reasons Quantum Entanglement Doesn’t Allow Faster-Than-Light Communication.