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## Production of Electromagnetic Waves

### Learning Objectives

By the end of this section, you will be able to:

• Describe the electric and magnetic waves as they move out from a source, such as an AC generator.
• Explain the mathematical relationship between the magnetic field strength and the electrical field strength.
• Calculate the maximum strength of the magnetic field in an electromagnetic wave, given the maximum electric field strength.

We can get a good understanding of electromagnetic waves (EM) by considering how they are produced. Whenever a current varies, associated electric and magnetic fields vary, moving out from the source like waves. Perhaps the easiest situation to visualize is a varying current in a long straight wire, produced by an AC generator at its center, as illustrated in Figure 1.

Figure 1. This long straight gray wire with an AC generator at its center becomes a broadcast antenna for electromagnetic waves. Shown here are the charge distributions at four different times. The electric field (E) propagates away from the antenna at the speed of light, forming part of an electromagnetic wave.

The electric field (E) shown surrounding the wire is produced by the charge distribution on the wire. Both the E and the charge distribution vary as the current changes. The changing field propagates outward at the speed of light.

There is an associated magnetic field (B) which propagates outward as well (see Figure 2). The electric and magnetic fields are closely related and propagate as an electromagnetic wave. This is what happens in broadcast antennae such as those in radio and TV stations.

Closer examination of the one complete cycle shown in Figure 1 reveals the periodic nature of the generator-driven charges oscillating up and down in the antenna and the electric field produced. At time t=0, there is the maximum separation of charge, with negative charges at the top and positive charges at the bottom, producing the maximum magnitude of the electric field (or E-field) in the upward direction. One-fourth of a cycle later, there is no charge separation and the field next to the antenna is zero, while the maximum E-field has moved away at speed c.

As the process continues, the charge separation reverses and the field reaches its maximum downward value, returns to zero, and rises to its maximum upward value at the end of one complete cycle. The outgoing wave has an amplitude proportional to the maximum separation of charge. Its wavelength (λ) is proportional to the period of the oscillation and, hence, is smaller for short periods or high frequencies. (As usual, wavelength and frequency(f) are inversely proportional.)

## Electric and Magnetic Waves: Moving Together

Following Ampere’s law, current in the antenna produces a magnetic field, as shown in Figure 2. The relationship between E and B is shown at one instant in Figure 2a. As the current varies, the magnetic field varies in magnitude and direction.

Figure 2. (a) The current in the antenna produces the circular magnetic field lines. The current (I) produces the separation of charge along the wire, which in turn creates the electric field as shown. (b) The electric and magnetic fields (E and B) near the wire are perpendicular; they are shown here for one point in space. (c) The magnetic field varies with current and propagates away from the antenna at the speed of light.

The magnetic field lines also propagate away from the antenna at the speed of light, forming the other part of the electromagnetic wave, as seen in Figure 2b. The magnetic part of the wave has the same period and wavelength as the electric part, since they are both produced by the same movement and separation of charges in the antenna.

The electric and magnetic waves are shown together at one instant in time in Figure 3. The electric and magnetic fields produced by a long straight wire antenna are exactly in phase. Note that they are perpendicular to one another and to the direction of propagation, making this a transverse wave

Figure 3. A part of the electromagnetic wave sent out from the antenna at one instant in time. The electric and magnetic fields (E and B) are in phase, and they are perpendicular to one another and the direction of propagation. For clarity, the waves are shown only along one direction, but they propagate out in other directions too.

Electromagnetic waves generally propagate out from a source in all directions, sometimes forming a complex radiation pattern. A linear antenna like this one will not radiate parallel to its length, for example. The wave is shown in one direction from the antenna in Figure 3 to illustrate its basic characteristics.

Instead of the AC generator, the antenna can also be driven by an AC circuit. In fact, charges radiate whenever they are accelerated. But while a current in a circuit needs a complete path, an antenna has a varying charge distribution forming a standing wave, driven by the AC. The dimensions of the antenna are critical for determining the frequency of the radiated electromagnetic waves. This is a resonant phenomenon and when we tune radios or TV, we vary electrical properties to achieve appropriate resonant conditions in the antenna.

## Receiving Electromagnetic Waves

Electromagnetic waves carry energy away from their source, similar to a sound wave carrying energy away from a standing wave on a guitar string. An antenna for receiving EM signals works in reverse. And like antennas that produce EM waves, receiver antennas are specially designed to resonate at particular frequencies.

An incoming electromagnetic wave accelerates electrons in the antenna, setting up a standing wave. If the radio or TV is switched on, electrical components pick up and amplify the signal formed by the accelerating electrons. The signal is then converted to audio and/or video format. Sometimes big receiver dishes are used to focus the signal onto an antenna.

In fact, charges radiate whenever they are accelerated. When designing circuits, we often assume that energy does not quickly escape AC circuits, and mostly this is true. A broadcast antenna is specially designed to enhance the rate of electromagnetic radiation, and shielding is necessary to keep the radiation close to zero. Some familiar phenomena are based on the production of electromagnetic waves by varying currents. Your microwave oven, for example, sends electromagnetic waves, called microwaves, from a concealed antenna that has an oscillating current imposed on it.

## Relating E-Field and B-Field Strengths

There is a relationship between the E– and B-field strengths in an electromagnetic wave. This can be understood by again considering the antenna just described. The stronger the E-field created by a separation of charge, the greater the current and, hence, the greater the B-field created.

Since current is directly proportional to voltage (Ohm’s law) and voltage is directly proportional to E-field strength, the two should be directly proportional. It can be shown that the magnitudes of the fields do have a constant ratio, equal to the speed of light. That is, $\frac{E}{B}=c\\$ is the ratio of E-field strength to B-field strength in any electromagnetic wave. This is true at all times and at all locations in space. A simple and elegant result.

### Example 1. Calculating B-Field Strength in an Electromagnetic Wave

What is the maximum strength of the B-field in an electromagnetic wave that has a maximum E-field strength of 1000 V/m?

#### Strategy

To find the B-field strength, we rearrange the above equation to solve for B, yielding

$B=\frac{E}{c}\\$.

#### Solution

We are given E, and c is the speed of light. Entering these into the expression for B yields

$\displaystyle{B}=\frac{1000\text{ V/m}}{3.00\times10^{8}\text{ m/s}}=3.33\times10^{-6}\text{ T}\\$,

Where T stands for Tesla, a measure of magnetic field strength.

#### Discussion

The B-field strength is less than a tenth of the Earth’s admittedly weak magnetic field. This means that a relatively strong electric field of 1000 V/m is accompanied by a relatively weak magnetic field. Note that as this wave spreads out, say with distance from an antenna, its field strengths become progressively weaker.

The result of this example is consistent with the statement made in the module Maxwell’s Equations: Electromagnetic Waves Predicted and Observed that changing electric fields create relatively weak magnetic fields. They can be detected in electromagnetic waves, however, by taking advantage of the phenomenon of resonance, as Hertz did. A system with the same natural frequency as the electromagnetic wave can be made to oscillate. All radio and TV receivers use this principle to pick up and then amplify weak electromagnetic waves, while rejecting all others not at their resonant frequency.

### Take-Home Experiment: Antennas

For your TV or radio at home, identify the antenna, and sketch its shape. If you don’t have cable, you might have an outdoor or indoor TV antenna. Estimate its size. If the TV signal is between 60 and 216 MHz for basic channels, then what is the wavelength of those EM waves?

Try tuning the radio and note the small range of frequencies at which a reasonable signal for that station is received. (This is easier with digital readout.) If you have a car with a radio and extendable antenna, note the quality of reception as the length of the antenna is changed.

## PhET Explorations: Radio Waves and Electromagnetic Fields

Broadcast radio waves from KPhET. Wiggle the transmitter electron manually or have it oscillate automatically. Display the field as a curve or vectors. The strip chart shows the electron positions at the transmitter and at the receiver.

## Section Summary

• Electromagnetic waves are created by oscillating charges (which radiate whenever accelerated) and have the same frequency as the oscillation.
• Since the electric and magnetic fields in most electromagnetic waves are perpendicular to the direction in which the wave moves, it is ordinarily a transverse wave.
• The strengths of the electric and magnetic parts of the wave are related by $\frac{E}{B}=\text{c}\\$,
which implies that the magnetic field B is very weak relative to the electric field E.

### Conceptual Questions

1. The direction of the electric field shown in each part of Figure 1 is that produced by the charge distribution in the wire. Justify the direction shown in each part, using the Coulomb force law and the definition of $\mathbf{E}=\frac{\mathbf{F}}{q}\\$, where q is a positive test charge.
2. Is the direction of the magnetic field shown in Figure 2a consistent with the right-hand rule for current (RHR-2) in the direction shown in the figure?
3. Why is the direction of the current shown in each part of Figure 2 opposite to the electric field produced by the wire’s charge separation?
4. In which situation shown in Figure 4 will the electromagnetic wave be more successful in inducing a current in the wire? Explain.

Figure 4. Electromagnetic waves approaching long straight wires.

5. In which situation shown in Figure 5 will the electromagnetic wave be more successful in inducing a current in the loop? Explain.

Figure 5. Electromagnetic waves approaching a wire loop.

6. Should the straight wire antenna of a radio be vertical or horizontal to best receive radio waves broadcast by a vertical transmitter antenna? How should a loop antenna be aligned to best receive the signals? (Note that the direction of the loop that produces the best reception can be used to determine the location of the source. It is used for that purpose in tracking tagged animals in nature studies, for example.)
7. Under what conditions might wires in a DC circuit emit electromagnetic waves?
8. Give an example of interference of electromagnetic waves.
9. Figure 6 shows the interference pattern of two radio antennas broadcasting the same signal. Explain how this is analogous to the interference pattern for sound produced by two speakers. Could this be used to make a directional antenna system that broadcasts preferentially in certain directions? Explain.

Figure 6. An overhead view of two radio broadcast antennas sending the same signal, and the interference pattern they produce.

### Problems & Exercises

1. What is the maximum electric field strength in an electromagnetic wave that has a maximum magnetic field strength of 5.00 × 10−4 T (about 10 times the Earth’s)?
2. The maximum magnetic field strength of an electromagnetic field is 5 × 10−6 T. Calculate the maximum electric field strength if the wave is traveling in a medium in which the speed of the wave is
0.75 c.
3. Verify the units obtained for magnetic field strength B in Example 1 (using the equation $B=\frac{E}{c}\\$) are in fact teslas (T).

## Glossary

electric field: a vector quantity (E); the lines of electric force per unit charge, moving radially outward from a positive charge and in toward a negative charge

electric field strength: the magnitude of the electric field, denoted E-field

magnetic field: a vector quantity (B); can be used to determine the magnetic force on a moving charged particle

magnetic field strength: the magnitude of the magnetic field, denoted B-field

transverse wave: a wave, such as an electromagnetic wave, which oscillates perpendicular to the axis along the line of travel

standing wave: a wave that oscillates in place, with nodes where no motion happens

wavelength: the distance from one peak to the next in a wave

amplitude: the height, or magnitude, of an electromagnetic wave

frequency: the number of complete wave cycles (up-down-up) passing a given point within one second (cycles/second)

resonant: a system that displays enhanced oscillation when subjected to a periodic disturbance of the same frequency as its natural frequency

oscillate: to fluctuate back and forth in a steady beat

1. 150 kV/m


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# Course:CPSC522/Adaptive Network Routing using ACO

## Adaptive Network Routing using ACO

In this page we discuss the application and analysis of Ant Colony Optimization (ACO) in building an efficient network protocol.

Principal Author: Kumseok Jung
Collaborators:

## Abstract

Ant Colony Optimization (ACO) is a metaheuristic for finding solutions to combinatorial optimization problems. Inspired by the behaviour of ants, it uses deliberative mobile agents to assess different parts of the problem space, and applies the concept of stigmergy - an indirect and asynchronous way of exchanging information through the environment - to propagate feedback signals for iteratively correcting the current solution until they converge to the optimal solution. The problem of finding the optimal path in a communication network is a multi-objective optimization problem in a dynamic environment, and is not trivial. AntNet applies ACO to implement a routing protocol in datagram networks with irregular topology, and experiments indicate that it outperforms existing state-of-the-art protocols such as OSPF (Open Shortest Path First) currently used in the Internet and is comparable to Dijkstra's shortest path algorithm, which is the theoretical optimal solution.

### Builds on

• Routing Protocol specifies how routers communicate with each other, distributing information that enables them to select routes between any two nodes on a computer network.
• Dijkstra's Shortest Path Algorithm is an algorithm for finding the shortest paths between nodes in a graph.
• Metaheuristic is a higher-level procedure or heuristic designed to find, generate, or select a heuristic (partial search algorithm) that may provide a sufficiently good solution to an optimization problem, especially with incomplete or imperfect information or limited computation capacity.[3]
• Swarm Intelligence is a form of collective intelligence that emerges as a result of a group of natural or artificial agents interacting with each other and the environment.
• Ant Colony Optimization is a population-based metaheuristic that can be used to find approximate solutions to difficult optimization problems.[4]
• Combinatorial Optimization is a topic that consists of finding an optimal object from a finite set of objects.[5]
• Multi-agent Systems is an area of artificial intelligence concerned with the study of multiple autonomous agents acting in an environment.
• Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.
• Temporal Difference Learning is an approach to learning how to predict a quantity that depends on future values of a given signal.
• Reinforcement Learning is learning by interacting with an environment.

## Content

### Motivation

In a computer network, not all devices have direct physical links between each other and therefore communication between any two devices are established by a series of routers redirecting data packets from the source node to the destination node. Routing involves each device maintaining a table of neighbouring nodes and the relevant statistics, so that an incoming packet can be strategically forwarded to an outgoing link selected from the table. A routing protocol describes the structure of the routing table, the rules for updating the table, and a policy for selecting a node for a given packet. The goal of a good routing protocol is to maximize the overall network performance and to minimize the cost involved; in other words, it must maximize the throughput and minimize the latency.

"the effect of good routing is to increase throughput for the same value of average delay per packet under high offered load conditions and to decrease average delay per packet under low and moderate offered load conditions." -Bertsekas & Gallager, 1992

The problem of routing is difficult because it is intrinsically distributed. A change in the system state, say a node failure, cannot be globally observed immediately, and thus it is impossible to have an up-to-date and complete model of the distributed (global) state. Additionally, although there is certainly a global optimal policy, there is no global decision system. Each node in the network has to make routing decisions based solely on its local policy, using delayed and incomplete information about the global state. It can only be hoped that the policies employed locally by individual nodes add up to the global optimal policy. Another aspect that makes the problem difficult is that the problem itself is stochastic and non-stationary. The theoretical optimal policy varies with time and depends on the traffic status; even if a complete model was available, techniques like dynamic programming cannot be used. Furthermore, the routing problem involves optimizing multiple conflicting and intertwined aspects of the system - an example being the previously mentioned throughput and latency - while considering multiple constraints imposed by the underlying network infrastructure and the demands of the service users. Besides the algorithmic complexity, at the implementation level the solution needs to be fault-tolerant and reliable, being robust against random node failures.

### Approach

#### Taxonomy of Routing Algorithms

At a higher level, routing algorithms can be characterized by the following properties:

• Centralized vs Distributed

In most cases, centralized algorithms are not fault-tolerant and are impractical due to the delays in gathering information about the network and communicating the routing decisions across the network. Static algorithms compute routes based only on the source and destination, but does not take into account the current network state, such as number of packets queued in the outgoing links. For real-world applications, it makes more practical sense to use distributed adaptive routing algorithms so that it can adapt to varying spatial and temporal traffic conditions. For the rest of the discussion in this page, we will focus on adaptive and distributed solution.

One of the issues that arise from using an adaptive algorithm is the problem of oscillating paths, where a constantly changing forwarding policy causes a packet to travel in a cycle.

From an optimization perspective, the routing paradigms can be divided into the following:

• Minimal routing vs Non-minimal routing
• Optimal routing vs Shortest path routing

Minimal routing algorithms only select paths with minimal cost, while non-minimal routers choose from all the available paths using a heuristic. Optimal routing is an approach focused on optimizing a certain function over all the individual links of the network and requires knowledge of the traffic patterns across the whole system. For instance, minimizing the sum of all the link costs is an optimal routing approach. In contrast, shortest path routing is concerned with minimizing the communication overhead between two nodes based on the link states, and without considering a global cost function and without a priori knowledge of the traffic status. This makes shortest path routing algorithms more flexible to system changes and therfore is the most widely used approach, for example in the Internet.

ARPANET, the ancestor of Internet, first used a shortest path routing algorithm called Bellman-Ford in 1969, which belongs to the class of distance-vector algorithms based on the principles of dynamic programming. In Bellman-Ford, each router keeps a routing table of a 3-tuple of the form (Destination, Estimated Distance, Next Hop) defined for all the nodes in the network. Asynchronously and iteratively, the nodes exchange with its neighbours its current estimated distance to other nodes in the network. Each node updates the entries in its routing table corresponding to the node that sent the new values. The process converges in finite time, but it introduces circular paths and therefore was replaced by Shortest Path First (SPF) algorithm in 1980, which is classified as a link-state routing algorithm. The current version in use, Open Shortest Path First (OSPF) improves upon its predecessor and was introduced in 1994. Link-state algorithms keep routing tables much larger than those in distance-vector algorithms, each table essentially representing a dynamic map of the complete network, and involves flooding the network with local state of each node to keep the information synchronized. OSPF requires static management of the topology and delegates congestion control to an external system; it is merely a "best compromise" among efficiency, stability, and robustness[7]. This brief summary of the history of routing algorithms emphasizes the difficulty of implementing a flexible and robust system, and it is worth to consider other perspectives at tackling this problem.

#### AntNet

Di Caro and Dorigo introduced a novel routing protocol named AntNet[1], based on the principles of Ant Colony Optimization (ACO). It attempts to solve the routing problem by taking a multi-agent systems approach, in which multiple mobile agents concurrently and in parallel perform an on-line Monte Carlo (MC) simulation. However, there are some key differences that distinguish the AntNet algorithm from classical Monte Carlo:

• The simulation is "on-line", where the model is updated during the execution. This makes it also similar to Temporal Difference learning.
• The updated model is shared asynchronously with other random MC instances (ants) operating in the vicinity, but not with all the instances.
• A single MC instance solves only its local optimal solution and cannot solve the global optimal solution. The global optimal of the network emerges as a result of individual MC instances concurrently and stochastically discovering partial solutions.

Some parallels with Reinforcement Learning (RL) can also be drawn, since the routing problem can be interpreted as a stochastic time-varying RL problem[1]. In AntNet, each router receives reinforcement signals from the incoming ants to iteratively update its internal model of the world and improves its routing policy towards a local optimum. However, unlike classical RL problems, the routing problem is non-stationary and non-Markovian, making usual RL approaches unsuitable.

The following is a higher level description of the algorithm:

• Each node in the network concurrently sends out mobile agents towards a randomly selected destination.
• A single mobile agent, called an ant, has one behaviour when going in the forward direction, and has a different behaviour going in the backward direction. When going forward, it is called a forward ant, and it travels in the same traffic as the data packets. In contrast, backward ants travel in a privileged traffic.
• The mobile agents act independently and communicate with each other indirectly by reading and writing information in the network nodes.
• Each agent seeks to travel along the shortest path between the source and the destination node.
• During the trip going forward, each ant collects information about time taken to reach an intermediate node and the congestion status.
• At each node, an ant chooses the next hop based on information locally available in the node and its own private information.
• After arriving at the destination, the forward ant trasfers its accumulated knowledge to the backward ant, and the backward ant begins its trip along the same route going in the opposite direction towards the source node.
• During the backward travel, the backward ant updates the routing table in each node based on the information collected during the forward trip.
• After the backward ant reaches the source node, it is destroyed.

The technical details of the algorithm are further discussed in the following sections.

##### Model of a network node
Model of a node in AntNet[2]

In a network consisting of ${\displaystyle N}$ nodes, each node ${\displaystyle k}$ maintains a routing table ${\displaystyle {\mathcal {T}}_{k}}$. The routing table stores probabilities describing the current routing policy at node ${\displaystyle k}$. Each entry in the table indicates the likelihood ${\displaystyle P_{n,d}}$ of selecting neighbour ${\displaystyle n}$ as the next node given the destination node ${\displaystyle d}$, where:

${\displaystyle P_{d}=\sum _{n\in {\mathcal {N}}_{k}}P_{n,d}=1,\qquad d\in \lbrack 1,N\rbrack ,\qquad {\mathcal {N}}_{k}={neighbours(k)}}$

The raw value ${\displaystyle P_{n,d}}$ is not used as-is for path selection, as there are some heuristic factors to be considered that are described in the following sections. Data packets use the same routing table ${\displaystyle {\mathcal {T}}_{k}}$ for selecting the next link, but use a power function ${\displaystyle g(P)=P^{\beta },\quad \beta >1}$ that amplifies the high probability values and suppresses the low probability values, because data packets only need to exploit the information and should not explore.

Each node ${\displaystyle k}$ also maintains another table ${\displaystyle {\mathcal {M}}_{k}}$ of statistics about the current locally-observed traffic. More precisely, ${\displaystyle {\mathcal {M}}_{k}}$ stores the mean travel time ${\displaystyle \mu _{k,d}}$ from current node ${\displaystyle k}$ to the destination node ${\displaystyle d}$, the variance ${\displaystyle \sigma _{k,d}^{2}}$, and an array ${\displaystyle {\mathcal {W}}_{d}}$ storing the travel times for the last ${\displaystyle w=\vert {\mathcal {W}}\vert }$ trips, used to compute the best travel time ${\displaystyle b_{k,d,W}}$ observed during the moving observation window. The mean travel time provides an estimate of the expected time to reach the destination, and the variance indicates the reliability of this estimate. Whenever there is new data available - i.e. travel time experienced by a newly arriving backward ant - the array ${\displaystyle {\mathcal {M}}_{k}}$ is updated with the following (the index ${\displaystyle k}$ will be omitted as all updates are local to node ${\displaystyle k}$):

${\displaystyle \mu _{d}\leftarrow \mu _{d}+\eta (t_{d}-\mu _{d})}$
${\displaystyle \sigma _{d}^{2}\leftarrow \sigma _{d}^{2}+\eta ((t_{d}-\mu _{d})^{2}-\sigma _{d}^{2})}$
where ${\displaystyle t_{d}}$ is the newly observed trip time of the agent going from node ${\displaystyle k}$ to destination ${\displaystyle d}$.

The term ${\displaystyle \eta }$ is analogous to the constant ${\displaystyle \alpha _{k}}$ in the Temporal Difference formula ${\displaystyle A_{k}=A_{k-1}+\alpha _{k}(v_{k}-A_{k-1}),\alpha _{k}={\frac {1}{k}}}$ for computing the running average over ${\displaystyle k}$ samples. The moving observation window ${\displaystyle {\mathcal {W}}_{d}}$ is updated by shifting it and pushing the new trip time ${\displaystyle t_{d}}$ and the best trip time ${\displaystyle b_{d,W}}$ is recomputed with ${\displaystyle b_{d,W}\leftarrow max({\mathcal {W}}_{d})}$. The moving observation window ${\displaystyle {\mathcal {W}}_{d}}$ serves as a short-term memory providing a recent empirical lower bound of the estimated travel time to the destination.

The table ${\displaystyle {\mathcal {M}}_{k}}$ provides an estimate of the travel time to different nodes, and table ${\displaystyle {\mathcal {T}}_{k}}$ describes the "goodness" of each link for a given destination.

##### Model of a mobile agent
Buffers inside a node in AntNet[2]

Given the model of the nodes, each node ${\displaystyle s}$ generates a forward ant ${\displaystyle F_{s\rightarrow d}}$ at regular intervals ${\displaystyle \Delta t}$, travelling from the current (source) node ${\displaystyle s}$ to destination node ${\displaystyle d}$. In order for a forward ant to experience the real traffic load, they are pushed into the same queue as the outgoing data packets. Since the queue is FIFO (First in, first out), if there are data packets waiting to be sent, the forward ant's departure will also be delayed proportionally to the size of the pending queue; this allows information about the congestion to be carried implicitly by the mobile agents. The destination node ${\displaystyle d}$ is selected probabilistically according to the locally observed traffic. The probability ${\displaystyle y_{d}}$ of selecting destination ${\displaystyle d}$ is proportional to the traffic towards node ${\displaystyle d}$, such that:

${\displaystyle y_{d}={\frac {f_{s,d}}{\sum _{d'=1}^{N}f_{s,d'}}}}$
where ${\displaystyle f_{s,d}}$ is some measure (e.g. size of the queue) of the data from node ${\displaystyle s}$ to ${\displaystyle d}$.

During its trip, an ant maintains a record of the traversed path, storing the identifier of a visited node ${\displaystyle k}$ and the time taken to reach that node ${\displaystyle k}$ since its departure from node ${\displaystyle s}$. This information is stored in a stack ${\displaystyle S_{s\rightarrow d}}$ and is part of the network packet representing the ant. Therefore the size of the ant increases linearly with the number of hops, but this increase has minimal effect on the performance of the algorithm.

For each node ${\displaystyle k}$ a forward ant visits along its route, it selects the next node ${\displaystyle n}$ among the neighbours it did not already visit. It picks a node based on a corrected probability ${\displaystyle P'_{n,d}}$ computed as a function of the raw probability ${\displaystyle P_{n,d}}$ stored in the routing table ${\displaystyle {\mathcal {T}}_{k}}$ and the state of the queue at the outgoing link to node ${\displaystyle n}$. The new probability ${\displaystyle P'_{n,d}}$ takes into account a heuristic correction factor ${\displaystyle l_{n}}$ that reflects the relative size of the queue of the outgoing link to node ${\displaystyle n}$:

${\displaystyle l_{n}=1-{\frac {q_{n}}{\sum _{n'=1}^{\vert {\mathcal {N}}_{k}\vert }q_{n'}}}}$
${\displaystyle P'_{n,d}={\frac {P_{n,d}+\alpha l_{n}}{1+\alpha (\vert {\mathcal {N}}_{k}\vert -1)}}}$

The heuristic correction factor ${\displaystyle l_{n}}$ indicates how congested the outgoing link is, and since ${\displaystyle l_{n}}$ is inversely proportional to the size of the queue, the corrected probability ${\displaystyle P'_{n,d}}$ becomes smaller than the raw probability ${\displaystyle P_{n,d}}$ if the corresponding outgoing link is relatively congested. This allows packets to take alternative routes in the case the theoretical best path has too much traffic. The constant ${\displaystyle \alpha }$ determines the importance of the correction factor ${\displaystyle l_{n}}$, and experiments show that setting the value of ${\displaystyle \alpha }$ between the ranges 0.2 and 0.5 yields good results. Setting higher values of ${\displaystyle \alpha }$ leads to oscillatory behaviour in the route selection, as the decision becomes too sensitive to the traffic.

If a forward ant at node ${\displaystyle k}$ has already visited all the neighbours of ${\displaystyle k}$, it has run into a circular path. In this case, records of the nodes in the cycle are popped from the ant's stack ${\displaystyle S_{s\rightarrow d}}$. If the duration of the cycle is longer than the time taken to reach the cycle (elapsed time before the ant entered the circular path), the ant is destroyed.

When a forward ant reaches the destination, it is converted to a backward ant ${\displaystyle B_{d\rightarrow s}}$, retaining the memory stack ${\displaystyle S_{s\rightarrow d}}$ (in the original paper, it is said that the forward ant transfers its memory to a new backward ant and dies[1]). The backward ant travels along the same path it (forward ant) came from, towards the source node ${\displaystyle s}$. It traces the path by popping the stack ${\displaystyle S_{s\rightarrow d}}$ at each node ${\displaystyle k}$, and travels through a priority queue at the outgoing link. The backward ants reserve this privilege so that fresh information about the route can be propagated quickly.

Backward ants implement the concept of stigmergy, an indirect way of communicating with other ants by modifying the environment. In AntNet, at each node ${\displaystyle k}$ along the path the backward ants update the values in the table ${\displaystyle {\mathcal {M}}_{k}}$ with new observations about the network status and modify the routing table ${\displaystyle {\mathcal {T}}_{k}}$ accordingly; this is analogous to ants leaving pheromone traces. The new information given by the backward ants act as reinforcement signals to push the routing policy towards a local optimum.

In the tables ${\displaystyle {\mathcal {M}}_{k}}$ and ${\displaystyle {\mathcal {T}}_{k}}$, all the entries corresponding to the destination node ${\displaystyle d}$ are updated. Additionally, the entries belonging to nodes in between node ${\displaystyle k}$ and destination ${\displaystyle d}$ - i.e. node ${\displaystyle n\in S_{k\rightarrow d},\quad n\neq d}$ - are updated, provided that the trip times for these "sub-paths" are statistically "good". Statistically "good" means that the elapsed time is less than ${\displaystyle \mu +I(\mu ,\sigma )}$, the sum of the mean trip time and the confidence interval ${\displaystyle I(\mu ,\sigma )}$. The nodes between node ${\displaystyle k}$ and destination ${\displaystyle d}$ are potential destinations for other ants and therefore updating these "sub-paths" provide information for other source-destination pairs at zero cost.

The mean trip time ${\displaystyle \mu _{k,d}}$, variance ${\displaystyle \sigma _{k,d}^{2}}$, and best trip time ${\displaystyle b_{k,d,W}}$ during observation window ${\displaystyle {\mathcal {W}}_{d}}$ are updated in table ${\displaystyle {\mathcal {M}}_{k}}$ as described above, using information stored in the ant's memory stack ${\displaystyle S_{s\rightarrow d}}$. Given that the backward ant just arrived from node ${\displaystyle f}$, the routing table ${\displaystyle {\mathcal {T}}_{k}}$ is updated by incrementing the probability ${\displaystyle P_{f,d}}$ - i.e. the probability of choosing node ${\displaystyle f}$ as the next node when destination is ${\displaystyle d}$ - by an amount proportional to a "goodness" measure ${\displaystyle r}$ of the trip time ${\displaystyle t_{k\rightarrow d}}$. The other probabilities ${\displaystyle P_{n,d},\quad n\neq f}$ are decremented proportionally as well.

${\displaystyle P_{f,d}\leftarrow P_{f,d}+r(1-P_{f,d})}$
${\displaystyle P_{n,d}\leftarrow P_{n,d}+rP_{n,d}}$

Observed trip time ${\displaystyle t_{k\rightarrow d}}$ is the only explicit feedback signal used to evaluate the "goodness" measure ${\displaystyle r}$, since it implicitly captures information about the physical properties of the path such as transmission capacity of the links, processing speed of the nodes, number of hops; as well as dynamic information about the system like the congestion level. Evaluating the "goodness" of trip time ${\displaystyle t_{k\rightarrow d}}$ is not a trivial task, since there is no objective "optimal" trip time to compare against. An apparently poor trip time ${\displaystyle t_{k\rightarrow d}}$ under low traffic could actually be a good one under heavy traffic, so we cannot compute some kind of an error signal like we do in back-propagation (in neural nets). The "goodness" measure ${\displaystyle r}$ can only be used as a reinforcement signal like a "reward" in the context of Reinforcement Learning. ${\displaystyle r\in (0,1]}$ is a dimensionless value and is critical to the AntNet algorithm and will be discussed in more detail below.

The first challenge in determining the reinforcement signal for the path from node ${\displaystyle k}$ to node ${\displaystyle f}$ is the credit assignment problem typically seen in RL. The only available metric is the elapsed time ${\displaystyle t_{k\rightarrow d}}$ between node ${\displaystyle k}$ to the destination node ${\displaystyle d}$ and we cannot know how much the sub-path between node ${\displaystyle k}$ and ${\displaystyle f}$ contributed to the elapsed time. The second challege is finding the right balance between adaptivity and stability. As mentioned in the previous paragraph, the "goodness" is relative to the current traffic conditions, so the reinforcement signal should be able to account for the network status. At the same time, it should not be overly sensitive to the fluctuations in traffic, as this can lead to oscillating routes. In addition, an important insight is gained by considering a simple reinforcement signal ${\displaystyle r=constant}$. In this case, all paths traversed by the ants are rewarded the same amount, independent of each ant's observed statistics. There is an implicit reinforcement mechanism at work, as faster links are traversed more often and accumulate rewards at a higher rate. The obvious drawback in this case is that bad paths are rewarded too. The authors' experiments show that even with a constant reinforcement signal the network performs moderately well, suggesting that this implicit reinforcement is an important component in the AntNet algorithm. To take into account the information collected by the ants, the feedback signal ${\displaystyle r}$ is computed as a function of the entries in the table ${\displaystyle {\mathcal {M}}_{k}}$ and is given by the following:

First we compute a quantity ${\displaystyle r'}$ using the equation (obvious indices are omitted for brevity)

${\displaystyle r'=c_{1}\left({\frac {t_{best}}{t}}\right)+c_{2}\left({\frac {t_{upper}-t_{best}}{(t_{upper}-t_{best})+(t-t_{best})}}\right)}$
where ${\displaystyle t_{upper}=\mu +{\frac {\sigma }{{\sqrt {1-\gamma }}{\sqrt {\vert {\mathcal {W}}\vert }}}}}$.

${\displaystyle t_{best}=b_{k,d,W}}$ is the best trip time observed in the moving window ${\displaystyle {\mathcal {W}}_{d}}$ as previously mentioned, and ${\displaystyle t_{upper}}$ is an approximation of the upper bound of the confidence interval of the average trip time ${\displaystyle \mu }$ with specified confidence level ${\displaystyle \gamma }$. ${\displaystyle \vert {\mathcal {W}}\vert }$ is the cardinality of the observation window ${\displaystyle {\mathcal {W}}_{d}}$. Using only the upper bound of the confidence interval introduces some kind of asymmetry and inaccuracy in this equation, but the authors claim that it is practical enough considering the extra CPU consumption that would be spent on computing an exact quantity. The first term in the above equation is the ratio between the best trip time and the currently observed trip time. It represents how "good" the current record is relative to the best record seen recently and it is more important than the latter term. The second term is a correction term, evaluating how different the observed trip time is from the average time, taking into account the stability of the value of the average time. In the presented implementation of AntNet, ${\displaystyle c_{1}=0.7}$ and ${\displaystyle c_{2}=0.3}$. Experiments show that setting ${\displaystyle c_{2}\in [0.15,0.35]}$ yields good results.

Then quantity ${\displaystyle r'}$ is then transformed into ${\displaystyle r}$ by using a squash function ${\displaystyle s(x)}$:

${\displaystyle s(x)=\left(1+e^{\frac {a}{x\vert {\mathcal {N}}_{k}\vert }}\right)^{-1},\qquad x\in (0,1],a\in R^{+}}$
${\displaystyle r={\frac {s(r')}{s(1)}}}$

Using the squashing function has an effect of amplifying the high values of ${\displaystyle r}$ while saturating low values of ${\displaystyle r}$, so that "rewards" play a bigger role than "punishments". Using the final value ${\displaystyle r}$, the routing table is updated as previously mentioned:

${\displaystyle P_{f,d}\leftarrow P_{f,d}+r(1-P_{f,d})}$
${\displaystyle P_{n,d}\leftarrow P_{n,d}+rP_{n,d}}$

The authors' experiments indicate that the algorithm is robust to its internal parameter settings and AntNet performs well as long as the paramaters are set to "reasonable" values.

### Analysis

Dhillon and Van Mieghem have produced a comprehensive analysis of the AntNet algorithm[2], empirically comparing its performance against existing state-of-the-art routing protocols such as OSPF and against Dijkstra's shortest path algorithm, a theoretical optimum. The study shows that AntNet's performance is comparable to Dijkstra's algorithm and is robust under varying traffic loads.

#### Computational Complexity

To calculate the complexity of the AntNet algorithm, the complexity of a single ant is first considered. At every node along its path, it needs to compare its memory of the path traversed so far with the neighbours of the node to determine if it has visited some of the nodes before; this is done in ${\displaystyle O(1)}$. After this comparison, it needs to look through the link values in the routing table to select the best neighbour to move to. Assuming a fully connected network where each node has ${\displaystyle N_{k}}$ neighbours, this takes ${\displaystyle O(N_{k})}$. Then the ant updates its memory with the identifier of the node, taking ${\displaystyle O(1)}$ and then is pushed to the outgoing link queue in ${\displaystyle O(1)}$. With ${\displaystyle M}$ being the maximum hop count of a forward ant, the ant has to perform the described computation at each of the ${\displaystyle M}$ nodes. On its way back, at each node the backward ant pops its stack, taking ${\displaystyle O(1)}$, and is pushed to the queue in ${\displaystyle O(1)}$. Updating the routing table requires computing a value for every neighbour of the node, which takes ${\displaystyle O(N_{k})}$. This operation is also done for every hop, and thus the backward ant also has complexity of ${\displaystyle O(MN_{k})}$. Therefore the complexity of a single ant is ${\displaystyle O(MN_{k})}$. For the entire system, given that there are ${\displaystyle \rho }$ ants generated, the worst-case complexity of the AntNet algorithm is ${\displaystyle O(\rho MN_{k})}$. Given that ${\displaystyle M}$ and ${\displaystyle N_{k}}$ are relatively small, this is comparable to the worst-case complexity of Dijkstra's shortest path algorithm using a Fibonacci heap, which is ${\displaystyle O(Nlog(N)+L)}$.

#### Experiments

The authors evaluate the AntNet algorithm under static and dynamic network conditions by comparing the performance against solution found by Dijkstra's shortest path algorithm. The term static here does not have the same connotation as in static analysis; it means that the traffic status and varying size of the packets are not taken into account. More precisely, in the static evaluation it is assumed that the forward ants are only affected by the propagation delays between the links - i.e. time taken for a single bit to travel from one node to another - while queueing delays, transmission delays, and link capacity are ignored. This reduces the node selection probability ${\displaystyle P'_{n,d}}$ to the raw values in the routing table ${\displaystyle P_{n,d}}$. In the dynamic evaluation, all the factors used in the original implementation of AntNet are taken into account. It is important to note that while Dijkstra's algorithm can easily be compared with the static version of AntNet, it is more tricky to compare against the dynamic AntNet. The solution given by Dijkstra's algorithm is the sum of all the link weights (propagation delays) along the path, whereas the solution given by dynamic AntNet includes the transmission and queueing delays. To make a fair comparison between the two, transmission and queueing delays are excluded from the solution found by AntNet and only link weights are considered.

The experiments are carried out in a simulated environment, on random graphs ${\displaystyle G_{p}(N)}$ with ${\displaystyle N}$ nodes and link density ${\displaystyle p}$. In each simulation, ${\displaystyle 10^{4}}$ to ${\displaystyle 10^{5}}$ graphs are assessed. The upper bound ${\displaystyle m_{max}}$ for the average number of paths between a pair of nodes in graph ${\displaystyle G_{p}(N)}$ is given by: ${\displaystyle m_{max}=\left\lbrack p^{N-1}e^{\frac {1}{p}}(N-2)!\right\rbrack }$ The propagation delays for the links are set uniformly randomly between ${\displaystyle (0,1]ms}$ and link capacity is set to 8.192 Mbit/s. Each simulation runs for ${\displaystyle 10^{4}ms=10s}$ and consists of a training period (TP) of ${\displaystyle 10^{3}ms=1s}$ during which only ant packets are generated, and a test period (TEP) of ${\displaystyle 9\times 10^{3}ms=9s}$ during which both ant and data packets are generated. Data packets are generated from nodes selected by Poisson process with mean interarrival time of 12.5ms, headed towards a randomly selected destination. The size of data packets are also randomly set, following a negative exponential distribution with mean value of 4096 bits. The size of a forward ant is initialized at 192 bits and grows by 64 bits per hop, while backward ants are assumed to be 500 bits. The default value of ${\displaystyle a}$ in the squash function ${\displaystyle s(x)}$ is set to ${\displaystyle a=pN}$. For the comparison between AntNet and Dijkstra's algorithm, source node ${\displaystyle s=1}$ and destination ${\displaystyle d=N}$ are considered. While there is just one shortest-path given by Dijkstra's algorithm, in AntNet the data packets travel probabilistically via different routes between a given source node and a destination node. Accordingly, the average end-to-end delay and hop count of the paths taken by the data packets over the whole simulation are used for the evaluation. The effects of changing other configuration parameters such as ${\displaystyle \eta }$, ${\displaystyle \vert {\mathcal {W}}\vert }$ are studied in various simulations in the experiment.

##### Static AntNet

For evaluating AntNet under static conditions - i.e. without considering traffic status - the algorithm is modified to use ${\displaystyle \epsilon -greedy}$ policy for a forward ant choosing the next node. This is because the correction term in ${\displaystyle P'_{n,d}}$ vanishes, preventing the ants from exploring the network. In this simulation the ant generation interval ${\displaystyle I_{ant}}$ is set to 4ms during TP (Training Period) and 40ms during TEP (Test Period), ${\displaystyle \eta =0.1}$, ${\displaystyle \vert {\mathcal {W}}\vert =50}$, and link density ${\displaystyle p=0.2}$. The simulation is carried out for network sizes of ${\displaystyle N=25}$ and ${\displaystyle N=50}$, each simulation consisting of ${\displaystyle 10^{4}}$ iterations of random graphs of type ${\displaystyle G_{p}(N)}$. The effect of ${\displaystyle \epsilon }$ is also observed by setting it to ${\displaystyle \epsilon =0}$ and ${\displaystyle \epsilon =0.1}$. A special case of uniform AntNet, referred to as unfNet for the rest of the discussion, is also considered, where ${\displaystyle \epsilon =1}$ (only explore).

Probability density function of average link weight and hop count for Dijkstra's shortest path, unfNet, and AntNet for varying network size and link density[2]

Static implementation of AntNet is shown to converge to a good solution, performing better when ${\displaystyle \epsilon >0}$. In fact, unfNet is shown to perform better than regular AntNet as all paths are explored with equal probabilities.

##### Dynamic AntNet

In the evaluation of dynamic implementation of AntNet, effects of different parameters are studied, such as ant generation rate, squash function, observation window, and the power function used in the forwarding policy of data packets.

###### Effect of ant generation interval ${\displaystyle I_{ant}}$ and link density ${\displaystyle p}$

In the first analysis, three different values of ant generation interval ${\displaystyle I_{ant}}$ are used: 40ms, 4ms, and 0.4ms. These values are applied only during TP, and the generation interval is fixed at 40ms during TEP. Two different values of link density ${\displaystyle p}$ are used: 0.1 and 0.2.

The results indicate that AntNet yields near optimal solution at low values of ${\displaystyle p}$, but performance degrades as ${\displaystyle p}$ is increased. Performance is slightly worse in the larger network with ${\displaystyle N=50}$ compared to the network with ${\displaystyle N=25}$. In regards to the ant generation rate, AntNet converges to a good solution even at low ant generation rates - i.e. high generation interval ${\displaystyle I_{ant}}$. When ${\displaystyle I_{ant}=40ms}$, although only 25 forward ants are sent out per node during TP, the algorithm finds a good solution. Performance improves as the ant generation rate is increased, but no difference is observed between ${\displaystyle I_{ant}=4ms}$ and ${\displaystyle I_{ant}=0.4ms}$. This can be attributed to the correlation between the size of the observation window ${\displaystyle \vert {\mathcal {W}}\vert }$ and the ant generation rate. With ${\displaystyle \vert {\mathcal {W}}\vert =50}$, the observation window is not large enough to capture the effect of changing ${\displaystyle I_{ant}=4ms}$ to ${\displaystyle I_{ant}=0.4ms}$. Additional simulations with ${\displaystyle \vert {\mathcal {W}}\vert =200}$ and ${\displaystyle \eta =0.02}$ confirm this idea. At low ant generation rate, unfNet performs worse than AntNet since there are not enough ants to discover the shortest path in the given time. At higher ant generation rates, the performance of AntNet and unfNet are comparable.

###### Effect of moving observation window ${\displaystyle {\mathcal {W}}}$ and ${\displaystyle \eta }$

It is worth reiterating that the parameter ${\displaystyle \eta }$ used in computing the running average ${\displaystyle \mu }$ of the trip time and the size of the moving observation window ${\displaystyle {\mathcal {W}}}$ are conceptually coupled. The original AntNet uses the relation ${\displaystyle \vert {\mathcal {W}}\vert ={\frac {5c}{\eta }},\quad c<1}$. It is hard to determine the optimal size of the moving observation window since its effect is correlated with the ant generation interval. Intuitively, it would make sense to keep an observation window large enough to store diverse samples of the paths, but it cannot be guaranteed that different ants would be exploring distinct and unique paths within that window. Results show that increasing the size of the observation window with a fixed ${\displaystyle \eta }$ improves the performance slightly. At light traffic loads, ${\displaystyle \eta }$ does not influence the performance much, as the mean ${\displaystyle \mu }$ converges quickly. ${\displaystyle \eta }$ would play a more significant role when there are fluctuations in the traffic loads. In order to observe the combined effect of ${\displaystyle {\mathcal {W}}}$, ${\displaystyle \mu }$, and ${\displaystyle I_{ant}}$, an additional simulation is performed with extreme values of the parameters. Ant generation interval ${\displaystyle I_{ant}}$ is set to 0.4ms during TP and 4ms during TEP, ${\displaystyle \vert {\mathcal {W}}\vert =16000}$, and ${\displaystyle \eta =0.001}$. The size of the observation window ${\displaystyle {\mathcal {W}}}$ is large enough to store the trip times of all the forward ants generated by the node as well as other incoming ants.

###### Effect of confidence interval, squash function, and power function

The confidence interval of the mean trip time ${\displaystyle \mu }$ and the squash function ${\displaystyle s(x)}$ are used in the calculation of the reinforcement signal ${\displaystyle r}$. The confidence interval term ${\displaystyle {\frac {\sigma }{{\sqrt {1-\gamma }}{\sqrt {\vert {\mathcal {W}}\vert }}}}}$ is mainly used to determine the reliability of the mean trip time ${\displaystyle \mu }$ when comparing it with the newly observed trip time. The simulations show that this term has very little effect on the performance of AntNet and can even be removed to simplify the AntNet algorithm. Using larger values of ${\displaystyle a}$ in the squash function ${\displaystyle s(x)=\left(1+e^{\frac {a}{x\vert {\mathcal {N}}_{k}\vert }}\right)^{-1},\quad x\in (0,1],a\in R^{+}}$ is shown to improve performance, demonstrating the effect of amplifying the reinforcement signals. Finally, the power function ${\displaystyle g(P)=P^{\beta }}$ is used to determine the routes for the data packets. Setting ${\displaystyle \beta =1}$ implies that the routing decision is dictated by the values in the routing table, leading to data packets taking different sub-optimal paths with a non-zero probability. Setting a very large value for ${\displaystyle \beta }$ causes the data packets to always travel through a single best path as seen locally by the node. Results show that AntNet performs better with ${\displaystyle \beta =100}$ indicating that the shortest path was correctly discovered.

###### Behaviour under varying traffic

To observe the behaviour of AntNet under varying traffic, randomly chosen nodes are congested during each iteration of the simulation by adding a queueing delay of 10ms per packet. Since it is not possible to account for traffic in Dijkstra's algorithm, one version of AntNet "src_rt" is configured to send out data packets through the shortest path found by Dijkstra's algorithm. Then the end-to-end delays measured in "src_rt" and regular AntNet are compared to see whether the theoretical shortest path is actually the shortest when there are congested nodes along its path. Results show that AntNet outperforms "src_rt" in terms of end-to-end delay, indicating that AntNet was able to find alternative routes that do not have congested nodes. When there are no congested nodes, "src_rt" and AntNet's performance are similar. This load balancing capability of AntNet is demonstrated further by running a slightly different experiment. Five randomly chosen nodes are congested 5000ms after the start of the simulation period, and average packet delay and hopcount are computed over a moving window of 250ms and plotted. AntNet's performance is constant before the 5000ms mark, until there is a sudden increase in the delay at the point of congestion. The delay is quickly reduced in AntNet, showing that it has found an alternative route, whereas "src_rt" suffers from the congestion.

End-to-end delay and hop count of AntNet versus shortest path found by Dijkstra's algorithm under varying traffic conditions[2]
Path delay and hop count over time for AntNet and "src_rt", illustrating the load balancing capabilities of AntNet[2]

#### Discussion

Various simulations were performed to examine the different features of AntNet, and results show that AntNet is able to discover near optimal solutions quickly even with a limited number of ants. There are several configuration parameters that are correlated, where changing a single parameter has little influence on the performance unless other coupled parameters are together adjusted. More importantly, it was shown that the theoretical shortest path found by Dijkstra's algorithm may be sub-optimal in the case of congestion, and that AntNet was able to quickly find alternative routes for the data packets. Thus AntNet provides inherent load balancing capabilities and is robust to changing traffic conditions, making it an attractive routing protocol. However, with larger and denser networks, AntNet's performance degraded, suggesting it may require a longer training period. The assessment on the scalability of AntNet algorithm and its applicability in ad hoc networks remains as future work.

## Annotated Bibliography

1. Di Caro, G. and Dorigo, M., 1998. AntNet: Distributed stigmergetic control for communications networks. Journal of Artificial Intelligence Research, 9, pp.317-365.
2. Dhillon, S.S. and Van Mieghem, P., 2007. Performance analysis of the AntNet algorithm. Computer Networks, 51(8), pp.2104-2125.
3. Bianchi, Leonora; Marco Dorigo; Luca Maria Gambardella; Walter J. Gutjahr (2009). "A survey on metaheuristics for stochastic combinatorial optimization". Natural Computing: an international journal. 8 (2): 239–287.
4. M. Dorigo. 2007. Ant Colony Optimization. Scholarpedia.org
5. Schrijver, Alexander (February 1, 2006). A Course in Combinatorial Optimization
6. Bertsekas, D.P., Gallager, R.G. and Humblet, P., 1992. Data networks (Vol. 2). New Jersey: Prentice-Hall International.
7. Di Caro, G. and Dorigo, M., 1997. AntNet: A mobile agents approach to adaptive routing. Technical Report IRIDIA/97-12, IRIDIA, Université Libre de Bruxelles, Belgium.

 Permission is granted to copy, distribute and/or modify this document according to the terms in Creative Commons License, Attribution-NonCommercial-ShareAlike 3.0. The full text of this license may be found here: CC by-nc-sa 3.0

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Thread: Groups and representations View Single Post
 P: 144 Ok I can show that $$F_{\mu \nu}= F_{\mu \nu}^a t^a$$ transforms under the adjoint representation, in the sense that one of the contracted indices a gets acted on by a matrix in the adjoint rep (which is kinda weird I think), but the proof relies on the fact that it is contracted with the generators. I cant figure out why $$\psi_a \psi_b$$ would transform under the adjoint because there are no generators kicking about, so the proof I did seems not to work. RedX, when I said fundamental rep of AxB I guess I assumed that this would be isomorphic to a group whose fundamental rep would be defined.


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# Problems from Another Time

Individual problems from throughout mathematics history, as well as articles that include problem sets for students.

The perimeters of two similar triangles are 45 and 135 respectively. One side of the first triangle has length 11″ and a second side, 19″. Find the lengths of the sides of the second triangle.
A set of n disjoint, congruent circles packs the surface of a sphere S so that each region of the surface exterior to the circles is bounded by arcs of three of the circles.
Suppose General [George] Washington had 800 men and was supplied with provisions to last 2 months but he needed to feed his army for 7 months.
An oracle ordered a prince to build a sacred building, whose space would be 400 cubits, the length being 6 cubits more than the width, and the width 3 cubits more than the height. Find the dimensions of the building.
Show that the curves x2 - y2 = a2 and 2 xy = b2 cross at right angles.
Now a good horse and an inferior horse set out from Chang'an to Qi. Qi is 3000 li from Chang'an.
Two men have a certain amount of money. The first says to the second, "If you give me 5 denari, I will have 7 times what you have left."
The steamer, Katie, leaves the wharf at New Orleans and runs an average speed of 15 mph. When Katie had gone 25 miles, the steamer R.E. Lee leaves the wharf and runs the average speed of 18 mph. How far will the Lee go before she overtakes the Katie?
If I were to give 7 pennies to each beggar at my door, I would have 24 pennies left in my purse. How many beggars are there and how much money do I have?
A farmer sold a team of horses for $440, but did not receive his pay for them until 1 yr, 8 mo after the sale. He had at the same time another offer of$410 for them. Did he gain or lose by the sale and by how much, money being worth 6%/yr?


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# Linear homoheneous ODE question

1. Jun 19, 2010

### gomunkul51

1. The problem statement, all variables and given/known data

y'' + p(x)y' + q(x)y = 0

p(x), q(x) continuous for x = (-1,1)

2. Relevant equations

I. prove that if y1 is an answer to the ODE and y1(0) = y'1(0) = 0.
then: y1 = 0 for all x = (-1,1).

II. if y1 is an answer to the ODE, how would you find the second non-dependent answer to the ODE y2?

III. prove that the two solutions you got, y1 and y2 are independent.

3. The attempt at a solution

Hello!

I just cant wrap my head around this question.
i know i can answer II if i knew what p(x) and q(x) were.

y(x)=A*y1+B*y2
y'(x)=A*y'1+B*y'2
y''(x)=A*y''1+B*y''2

A,B=const.

if I insert the above in the ODE i get:
y2'' + p(0)y2' + q(0)y2 = 0

also i can prove that they are linearly independent by the Wronskian(y1,y2)(x)

Last edited: Jun 19, 2010
2. Jun 19, 2010

### HallsofIvy

Since p and q are continuous on (-1, 1), this equation satifies the conditions for the "existance and uniqueness" theorem- there exist only one function satifying this equation with taking specfic values of y(0) and y'(0).

It is clear that y(x)= 0 for all x in (-1, 1) satisfies the the equation and, certainly y(0)= 0, y'(0)= 0. Since that solution is unique, y1(x)= y(x)= 0.

3. Jun 19, 2010

### gomunkul51

the part i don't understand is how you extrapolate from knowing the answer at a single point (x=0) to knowing that y1=0 for all x (-1,1) ?

of course it's known that in a linear equation always exists a trivial answer y(x)=0, but its only one answer, how can it help me to determine that y1=0 for all x=(-1,1)?

also how can i find p(0) and q(0), if i can ?

4. Jun 19, 2010

### vela

Staff Emeritus
Did you read the first paragraph of HallsofIvy's post?

In your problem, not only do you know y=0 is a solution that satisfies the initial conditions, you know it is the only solution that does.

5. Jun 20, 2010

### gomunkul51

again, I don't understand how knowing that if y=0 if a single solution to the ODE, can i know that y1=0 for all x (-1,1)?

if its so then my answer y(x) to the ODE consists only of one solution C*y2(x) because the other linearly independent solution y1(x)=0, but it couldn't be true because i need to have 2 linearly independent solutions to answer an linear ODE.

also if you see what's my problem is can you please explain it to me, clearly i don't understand what's going on, so don't assume i know the answer :)

6. Jun 20, 2010

### HallsofIvy

Do you know "unique" means? Do you know what the "existence and uniqueness theorem" is?

And no one said that "the other linearly independent solution y1(x)=0". y(x)= 0 is always a solution to a linear homogeneous differential equation. The two "linearly independent solutions" must be non-zero solutions since any set of vectors containing the 0 vector is NOT linearly independent.

7. Jun 20, 2010

### gomunkul51

This was the question:

I. prove that if y1 is an answer to the ODE and y1(0) = y'1(0) = 0.
then: y1 = 0 for all x = (-1,1).

can someone non-cryptically explain why it is so? :)

HallsofIvy thank you for your response but its a semi-theoretical question that I'm trying to understand, it's not homework. also I'm done with my ODE course

and if anyone is kind enough i would love to have a full explanation.

8. Jun 20, 2010

### vela

Staff Emeritus
You're confusing the solutions to the differential equation and the solution to the initial value problem. There is only one solution that satisfies the differential equation and the initial conditions.

9. Jun 20, 2010

### HallsofIvy

So what you are really asking for is a proof of the "existence and uniqueness" theorem! You would start by proving:

If f(x,y) is continuous in both variables and "Lipschitz" in y (differentiable with respect to y is sufficient but not necessary) in some neighborhood of $(x_0, y_0)$, then there exist a unique function y(x) satisfying dy/dx= f(x,y) with $y(x_0)= y_0$.

That's a fairly complicated theorem in its own right but any good differential equations text book should give it. Look for "existence" or "uniqueness" in the index.

To extend that to second order equations, you do this:

Let u= y' so that u'= y" and the equation y"+ p(x)y'+ q(x)y= 0 becomes u'+ p(x)u+ q(x)y= 0 or u'= -p(x)u- q(x)y. Together with y'= u that gives you two linked first order differential equations. Now Let
$$Y(x)= \begin{pmatrix}u(x) \\ y(x)\end{pmatrix}$$
so we can write
$$\begin{pmatrix}u'(x)\\ y'(x)\end{pmatrix}= Y'= \begin{pmatrix}-p(x) & -q(x) \\1 & 0\end{pmatrix}\begin{pmatrix}u \\ y\end{pmatrix}= AY$$
so the second order differential equation becomes a first derivative equation in the vector function Y(x).

As long as p(x) and q(x) are differentiable, this equation satisfies the existence and uniqueness theorem above (of course you have to prove that you can extend it to vector functions- that's not hard). The condition that y(0)= 0, y'(0)= u(0)= 0 becomes
$$Y(0)= \begin{pmatrix}u(0) \\ y(0)\end{pmatrix}= \begin{pmatrix}0 \\ 0 \end{pmatrix}$$.

Since the function y(x)= 0 for all x satisfies y''+ p(x)y'+ q(x)y= 0 as well as y(0)= 0, y'(0)= 0, it follows from the "uniquness" that it is the only function that satisfies that equation with those conditions.

Of more interest, in my opinion, is the fact that there exist a unique function, say $y_1(x)$, satisfying that equation and the conditions $y_1(0)= 1$, $y_1'(0)= 0$ and there exist a unique function, $y_2(x)$, satisfying the equation and $y_2(0)= 0$, $y_2'(0)= 1$.

Suppose y is any equation satisfying the differential equation. Then y(0) is some number, say A, and y'(0) is some number, say B. It follows that $y(x)= Ay_1(x)+ By_2(x)$! Why? Because $y'(x)= Ay_1'(x)+ By_2'(x)$ and $y"(x)= Ay_1"(x)+ By_2"(x)$ so that $y"+ p(x)y'+ q(x)y= (Ay_1"+ By_2")+ p(x)(Ay_1'+ By_2')+ q(x)(Ay_1+ By_2)$$= A(y_1"+ p(x)y_1'+ q(x)y_1)+ B(y_2"+ p(x)y_1'+ q(x)y_2)= A(0)+ B(0)= 0$ while $Ay_1(0)+ By_2(0)= A(1)+ B(0)= A$ and $Ay_1'(0)+ By_2'(0)= A(0)+ B(1)= B$.

That is, y(x) and $Ay_1(x)+ By_2(x)$ satisfy the same differential equation as well as the same "initial conditions" and so, by the uniqueness property, are equal.

Further, it is easy to show that $y_1(x)$ and $y_2(x)$ are "independent": if $Ay_1(x)+ By_2(x)= 0$ (meaning equal to 0 for all x), then, taking x= 0, A(1)+ B(0)= 0 so A= 0. Differentiating both sides, $Ay_1'(x)+ By_2'(x)= 0$ and again taking x= 0, A(0)+ B(1)= 0 so B= 0.

That tells us that $y_1(x)$ and $y_2(x)$ for a basis for the set of all solutions to that equation and so the set of all solutions is a vector space of dimension 2.

That can be extended to show that the set of all solutions to an $n^{th}$ order, linear, homogeneous differential equation is a vector space of dimension n.

Last edited by a moderator: Jun 20, 2010


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# Ghost Vapor Tactical Flashlight

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## What is the Ghost Vapor Tactical Flashlight?

The Ghost Vapor is a small, three and a half inch tactical flashlight that is perfect for wearing on the belt or keeping in toolboxes, bags, glove compartments, or drawers. However, despite its small size, the Ghost Vapor has a bright, powerful light that rivals even the most expensive, larger tactical flashlights.

The Ghost Vapor is made of aircraft grade aluminum alloy, making it extremely durable and strong. Because of its quality material, the Ghost Vapor can take all the bangs and drops that users can dish out. It also has precision optics and an attach head bezel design. Plus, the little flashlight packs quite a punch with its high and low strobe modes. With a scalable focus and a heavy-duty belt clip, the Ghost Vapor is truly one of the best tactical flashlight options available.

And, because the Ghost Vapor is available for customers free of charge now, those who were tentative about investing in a tactical flashlight can get one now without spending money on the Ghost Vapor.

### The Ghost Vapor Features

Of course, tactical flashlights are only as good as their features. This is why the Ghost Vapor has paid such close attention to creating it with features that set it apart from all the other flashlights available on the market.

Some of the key features found in the Ghost Vapor are:

• Modes: High, Medium and Strobe
• Color: Black
• Lumens: 250LM (manufacturer rated)
• Reflector: Convex Lens
• Material: Aluminum alloy
• Model of LED: J5 LED
• Switch: Tail-cap press ON/OFF
• Battery: 1 * AA / 1 * 14500 3.7v Battery (not included)
• Size: 3.6*0.98*0.98
• Weight: 65g

### What is a Tactical Flashlight?

For those who are considering getting their free Ghost Vapor, a better understanding of tactical flashlights is needed.

A tactical flashlight is a flashlight made of superior quality that can be used for specific purposes. These tactical flashlights can be handheld or can be attached to weapons, like firearms, to help with identification. In the case that a tactical flashlight is attached to a weapon, it typically moves parallel with the bore of the weapon, giving users the opportunity to shine light on and aim at their targets.

However, the Ghost Vapor is not the type of tactical flashlight that attaches to a weapon. It is a smaller, handheld device that can also be attached with a clip to the belt.

Because tactical flashlights are made with superior materials, they tend to be heavier and more durable than the typical flashlight, they can also be used as a non-lethal force option. Flashlights like the Ghost Vapor can be used to blind attackers temporarily, which can disorient them, giving users time to get away or get help.

Larger tactical flashlights can be used to ward off attackers as well. And, because they’re usually made of the highest quality material, they can be used in self-defense. For flashlights like the Ghost Vapor, which is made from an aircraft grade aluminum alloy, even when used in extreme cases like this, the flashlights are rarely damaged.

### Who Can Use Tactical Flashlights?

So many people believe that tactical flashlights are only beneficial for those who are extremely active or who need them for their jobs. However there are so many types of people who could benefit from owning a tactical flashlight, especially one of the quality of the Ghost Vapor.

Tactical flashlights are usually extremely expensive, making this deal on the Ghost Vapor a rare and exciting thing.

Some of the people who can use tactical flashlights like the Ghost Vapor are:

• Outdoors Enthusiasts – For camping, fishing, hunting, and hiking.
• Armed Forces – For weapon mounted light.
• Law Enforcement Officers – By police officers, security guards, and firefighters.
• Families – For emergency situations like natural disasters and power outages.
• City Dwellers – In the case of power outages.
• Home Owners – To scare or alert burglars or prowlers.

Plus, anyone can benefit from having a source of light beside their bed or on their office desk. And, because the Ghost Vapor is such a small and convenient tactical flashlight, it is perfect for these types of uses.

### Key Features of Tactical Flashlights

Because the Ghost Vapor tactical flashlight is available for free, a lot of people who have been considering purchasing a tactical flashlight but haven’t taken the leap yet will be considering this opportunity. For those who are new to tactical flashlights, it’s important to know what the features are that are most commonly found on these flashlights.

A brief list of some common features found in tactical flashlights are found below. These features can be compared to the features of the Ghost Vapor listed above.

Bezels: These are the metal rings found around the lenses at the front of these flashlights. Bezels can be removed so the lenses can be cleaned or replaced. Some tactical flashlights come with different colored lenses, in which case the bezels can be unscrewed to allow users to change the lenses.

Reflectors: Often made of a shined metal or plastic, reflectors are the internal surfaces that allow the light to be directed and focused through the lenses. For example, the Ghost Vapor has a convex lens reflector, which is one of the best options available.

Lenses: The glass that the tactical flashlight uses to focus the light. Many tactical flashlights allow users to change the focus of the lenses.

Bulbs: The most important part of a tactical flashlight is the quality of light it produces, which is dependent on the bulb. Usually tactical flashlights use LED bulbs, which produce a purer, stronger light.

### Purchasing the Ghost Vapor Tactical Flashlight

The best part about the Ghost Vapor flashlight is that it is currently being offered for free. There has never been an offer like this, especially for a tactical flashlight as high of quality as the Ghost Vapor. While the flashlight is usually worth $19.95, users can now get it for free, plus a small shipping and handling cost. To receive their Ghost Vapor tactical flashlight, customers need to pay a small fee for shipping and handling. The fee is$6.95. However, for those who are worried about purchasing the Ghost Vapor, it also comes with a 60 day guarantee. For those who aren’t satisfied with their purchase, they can return or replace it within 60 days.

### DangoBuds Reviews – Are Dango Buds Wireless Earbuds Legit?

Dango Buds are wireless earbuds that claim to offer crystal clear sound and active noise canceling. The earbuds are exclusively...


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# Noise figure

Last updated

Noise figure (NF) and noise factor (F) are measures of degradation of the signal-to-noise ratio (SNR), caused by components in a signal chain. It is a number by which the performance of an amplifier or a radio receiver can be specified, with lower values indicating better performance.

## Contents

The noise factor is defined as the ratio of the output noise power of a device to the portion thereof attributable to thermal noise in the input termination at standard noise temperature T0 (usually 290  K). The noise factor is thus the ratio of actual output noise to that which would remain if the device itself did not introduce noise, or the ratio of input SNR to output SNR.

The noise figure is simply the noise factor expressed in decibels (dB). [1]

## General

The noise figure is the difference in decibels (dB) between the noise output of the actual receiver to the noise output of an “ideal” receiver with the same overall gain and bandwidth when the receivers are connected to matched sources at the standard noise temperature T0 (usually 290 K). The noise power from a simple load is equal to kTB, where k is Boltzmann's constant, T is the absolute temperature of the load (for example a resistor), and B is the measurement bandwidth.

This makes the noise figure a useful figure of merit for terrestrial systems, where the antenna effective temperature is usually near the standard 290 K. In this case, one receiver with a noise figure, say 2 dB better than another, will have an output signal to noise ratio that is about 2 dB better than the other. However, in the case of satellite communications systems, where the receiver antenna is pointed out into cold space, the antenna effective temperature is often colder than 290 K. [2] In these cases a 2 dB improvement in receiver noise figure will result in more than a 2 dB improvement in the output signal to noise ratio. For this reason, the related figure of effective noise temperature is therefore often used instead of the noise figure for characterizing satellite-communication receivers and low-noise amplifiers.

In heterodyne systems, output noise power includes spurious contributions from image-frequency transformation, but the portion attributable to thermal noise in the input termination at standard noise temperature includes only that which appears in the output via the principal frequency transformation of the system and excludes that which appears via the image frequency transformation.

## Definition

The noise factorF of a system is defined as [3]

${\displaystyle F={\frac {\mathrm {SNR} _{\text{i}}}{\mathrm {SNR} _{\text{o}}}}}$

where SNRi and SNRo are the input and output signal-to-noise ratios respectively. The SNR quantities are power ratios. The noise figure NF is defined as the noise factor in dB:

${\displaystyle \mathrm {NF} =10\log _{10}(F)=10\log _{10}\left({\frac {\mathrm {SNR} _{\text{i}}}{\mathrm {SNR} _{\text{o}}}}\right)=\mathrm {SNR} _{\text{i, dB}}-\mathrm {SNR} _{\text{o, dB}}}$

where SNRi, dB and SNRo, dB are in decibels (dB). These formulae are only valid when the input termination is at standard noise temperature T0 = 290 K, although in practice small differences in temperature do not significantly affect the values.

The noise factor of a device is related to its noise temperature Te: [4]

${\displaystyle F=1+{\frac {T_{\text{e}}}{T_{0}}}.}$

Attenuators have a noise factor F equal to their attenuation ratio L when their physical temperature equals T0. More generally, for an attenuator at a physical temperature T, the noise temperature is Te = (L 1)T, giving a noise factor

${\displaystyle F=1+{\frac {(L-1)T}{T_{0}}}.}$

## Noise factor of cascaded devices

If several devices are cascaded, the total noise factor can be found with Friis' formula: [5]

${\displaystyle F=F_{1}+{\frac {F_{2}-1}{G_{1}}}+{\frac {F_{3}-1}{G_{1}G_{2}}}+{\frac {F_{4}-1}{G_{1}G_{2}G_{3}}}+\cdots +{\frac {F_{n}-1}{G_{1}G_{2}G_{3}\cdots G_{n-1}}},}$

where Fn is the noise factor for the n-th device, and Gn is the power gain (linear, not in dB) of the n-th device. The first amplifier in a chain usually has the most significant effect on the total noise figure because the noise figures of the following stages are reduced by stage gains. Consequently, the first amplifier usually has a low noise figure, and the noise figure requirements of subsequent stages is usually more relaxed.

## Noise factor as a function of additional noise

The noise factor may be expressed as a function of the additional output referred noise power ${\displaystyle N_{a}}$ and the power gain ${\displaystyle G}$ of an amplifier.

${\displaystyle F=1+{\frac {N_{a}}{N_{i}G}}}$

### Derivation

From the definition of noise factor [3]

${\displaystyle F={\frac {\mathrm {SNR} _{\text{i}}}{\mathrm {SNR} _{\text{o}}}}={\frac {\frac {S_{i}}{N_{i}}}{\frac {S_{o}}{N_{o}}}},}$

and assuming a system which has a noisy single stage amplifier. The signal to noise ratio of this amplifier would include its own output referred noise ${\displaystyle N_{a}}$, the amplified signal ${\displaystyle S_{i}G}$ and the amplified input noise ${\displaystyle N_{i}G}$,

${\displaystyle {\frac {S_{o}}{N_{o}}}={\frac {S_{i}G}{N_{a}+N_{i}G}}}$

Substituting the output SNR to the noise factor definition, [6]

${\displaystyle F={\frac {\frac {S_{i}}{N_{i}}}{\frac {S_{i}G}{N_{a}+N_{i}G}}}={\frac {N_{a}+N_{i}G}{N_{i}G}}=1+{\frac {N_{a}}{N_{i}G}}}$

## Related Research Articles

In electronics, noise temperature is one way of expressing the level of available noise power introduced by a component or source. The power spectral density of the noise is expressed in terms of the temperature that would produce that level of Johnson–Nyquist noise, thus:

Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in decibels. A ratio higher than 1:1 indicates more signal than noise.

Signal-to-noise and distortion ratio (SINAD) is a measure of the quality of a signal from a communications device, often defined as

In telecommunications, a third-order intercept point (IP3 or TOI) is a specific figure of merit associated with the more general third-order intermodulation distortion (IMD3), which is a measure for weakly nonlinear systems and devices, for example receivers, linear amplifiers and mixers. It is based on the idea that the device nonlinearity can be modeled using a low-order polynomial, derived by means of Taylor series expansion. The third-order intercept point relates nonlinear products caused by the third-order nonlinear term to the linearly amplified signal, in contrast to the second-order intercept point that uses second-order terms.

The total harmonic distortion is a measurement of the harmonic distortion present in a signal and is defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency. Distortion factor, a closely related term, is sometimes used as a synonym.

In electronics, gain is a measure of the ability of a two-port circuit to increase the power or amplitude of a signal from the input to the output port by adding energy converted from some power supply to the signal. It is usually defined as the mean ratio of the signal amplitude or power at the output port to the amplitude or power at the input port. It is often expressed using the logarithmic decibel (dB) units. A gain greater than one, that is amplification, is the defining property of an active component or circuit, while a passive circuit will have a gain of less than one.

A Negative-feedback amplifier is an electronic amplifier that subtracts a fraction of its output from its input, so that negative feedback opposes the original signal. The applied negative feedback can improve its performance and reduces sensitivity to parameter variations due to manufacturing or environment. Because of these advantages, many amplifiers and control systems use negative feedback.

The sensitivity of an electronic device, such as a communications system receiver, or detection device, such as a PIN diode, is the minimum magnitude of input signal required to produce a specified output signal having a specified signal-to-noise ratio, or other specified criteria.

In electronics, a common collector amplifier is one of three basic single-stage bipolar junction transistor (BJT) amplifier topologies, typically used as a voltage buffer.

The asymptotic gain model is a representation of the gain of negative feedback amplifiers given by the asymptotic gain relation:

In signal processing, a matched filter is obtained by correlating a known delayed signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template. The matched filter is the optimal linear filter for maximizing the signal-to-noise ratio (SNR) in the presence of additive stochastic noise.

In electronics, a common-source amplifier is one of three basic single-stage field-effect transistor (FET) amplifier topologies, typically used as a voltage or transconductance amplifier. The easiest way to tell if a FET is common source, common drain, or common gate is to examine where the signal enters and leaves. The remaining terminal is what is known as "common". In this example, the signal enters the gate, and exits the drain. The only terminal remaining is the source. This is a common-source FET circuit. The analogous bipolar junction transistor circuit may be viewed as a transconductance amplifier or as a voltage amplifier.. As a transconductance amplifier, the input voltage is seen as modulating the current going to the load. As a voltage amplifier, input voltage modulates the current flowing through the FET, changing the voltage across the output resistance according to Ohm's law. However, the FET device's output resistance typically is not high enough for a reasonable transconductance amplifier, nor low enough for a decent voltage amplifier. Another major drawback is the amplifier's limited high-frequency response. Therefore, in practice the output often is routed through either a voltage follower, or a current follower, to obtain more favorable output and frequency characteristics. The CS–CG combination is called a cascode amplifier.

Delta-sigma modulation is a method for encoding analog signals into digital signals as found in an analog-to-digital converter (ADC). It is also used to convert high bit-count, low-frequency digital signals into lower bit-count, higher-frequency digital signals as part of the process to convert digital signals into analog as part of a digital-to-analog converter (DAC).

In electronics, the common mode rejection ratio (CMRR) of a differential amplifier is a metric used to quantify the ability of the device to reject common-mode signals, i.e. those that appear simultaneously and in-phase on both inputs. An ideal differential amplifier would have infinite CMRR, however this is not achievable in practice. A high CMRR is required when a differential signal must be amplified in the presence of a possibly large common-mode input, such as strong electromagnetic interference (EMI). An example is audio transmission over balanced line in sound reinforcement or recording.

Friis formula or Friis's formula, named after Danish-American electrical engineer Harald T. Friis, is either of two formulas used in telecommunications engineering to calculate the signal-to-noise ratio of a multistage amplifier. One relates to noise factor while the other relates to noise temperature.

This article illustrates some typical operational amplifier applications. A non-ideal operational amplifier's equivalent circuit has a finite input impedance, a non-zero output impedance, and a finite gain. A real op-amp has a number of non-ideal features as shown in the diagram, but here a simplified schematic notation is used, many details such as device selection and power supply connections are not shown. Operational amplifiers are optimised for use with negative feedback, and this article discusses only negative-feedback applications. When positive feedback is required, a comparator is usually more appropriate. See Comparator applications for further information.

In telecommunications, the carrier-to-noise ratio, often written CNR or C/N, is the signal-to-noise ratio (SNR) of a modulated signal. The term is used to distinguish the CNR of the radio frequency passband signal from the SNR of an analog base band message signal after demodulation, for example an audio frequency analog message signal. If this distinction is not necessary, the term SNR is often used instead of CNR, with the same definition.

A minimum detectable signal is a signal at the input of a system whose power allows it to be detected over the background electronic noise of the detector system. It can alternately be defined as a signal that produces a signal-to-noise ratio of a given value m at the output. In practice, m is usually chosen to be greater than unity. In some literature, the name sensitivity is used for this concept.

In electronics, excess noise ratio is a characteristic of a noise generator such as a "noise diode", that is used to measure the noise performance of amplifiers. The Y-factor method is a common measurement technique for this purpose.

An RF chain is a cascade of electronic components and sub-units which may include amplifiers, filters, mixers, attenuators and detectors. It can take many forms, for example, as a wide-band receiver-detector for electronic warfare (EW) applications, as a tunable narrow-band receiver for communications purposes, as a repeater in signal distribution systems, or as an amplifier and up-converters for a transmitter-driver. In this article, the term RF covers the frequency range "Medium Frequencies" up to "Microwave Frequencies", i.e. from 100 kHz to 20 GHz.

## References

1. Agilent 2010 , p. 7
2. Agilent 2010 , p. 5.
3. Agilent 2010 , p. 7 with some rearrangement from Te = T0(F − 1).
4. Agilent 2010 , p. 8.
5. Aspen Core. Derivation of noise figure equations (DOCX), pp. 3–4
• Agilent (August 5, 2010), Fundamentals of RF and Microwave Noise Figure Measurements (PDF), Application Note, 57-1

This article incorporates  public domain material from the General Services Administration document: "Federal Standard 1037C".(in support of MIL-STD-188)


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# Physics Cup – TalTech 2023 – Problem 1

By Jaan Kalda (TalTech).

A solid cylinder of radius $R$ and height $H$ has density $\rho_c$ and is immersed in water of density $\rho_w$. The cylinder is initially kept at rest so that its axis is vertical, and the distance between its bottom face and the bottom of the water container is $h$. The water container has a flat rigid bottom, and the depth of water in it is larger than $H+h$. At a certain moment, the cylinder is released and starts falling. How long it will take for the cylinder to hit the bottom? Neglect viscosity. Assume that $\rho_c>\rho_w$, $H, and $10\rho_ch\ll \rho_wR$.

Chinese translation of problem 1 (下沉的圆柱)

Please submit the solution of this problem via e-mail to physcs.cup@gmail.com. First intermediate results will be published on 20th November, 13:00 GMT. First hints will appear here on 27th November 2022, 13:00 GMT. After the publication of the first hint, the base score is reduced to 0.9 pts. For full regulations, see the “Participate” tab.


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# Chapter 7 - Exponents and Exponential Functions - 7-4 Division Properties of Exponents - Practice and Problem-Solving Exercises - Page 444: 83

#### Work Step by Step

$\frac{m^7}{n^7}=(\frac{m}{n})^7$ Raise the numerator and denominator to the seventh power.

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