5.1 Plane Earth Reflection | unit 4 turing machines

Electromagnetic Fields

UNIT 5

Ground Wave Propagation

5.1 Plane Earth Reflection

Reflection occurs when a wave impinges on an object with large dimensions relative to a wavelength. The surface of the earth and large buildings are examples.

In considering reflections introduced by the environment, the most significant source of reflection in terrestrial links is the earth’s surface itself. Consider the link shown below:

Figure 5.1. Plane Earth Reflection

Sample E-field vectors are shown for the parallel and perpendicular polarization cases. Note that there are two paths from the transmitter (TX) to the receiver (RX): a line-of-sight (LOS) path and a reflected path.

The polarization of the LOS path, regardless of the TE/TM polarization case, remains constant; however, for the reflected path, the polarization of the reflected components are shown 180 degrees out of phase with the LOS path.

This results when θi → 90◦ (grazing incidence) at an interface with a lossy dielectric (not a PEC). The grazing angle assumption is valid since in many cases the antenna heights h1, h2 are very small compared to the TX-RX separation.

Key takeaways:

Reflection occurs when a wave impinges on an object with large dimensions relative to a wavelength.

The surface of the earth and large buildings are examples.

5.2 Space Wave

Space wave propagation is defined as the radio waves that occur within the 20km of the atmosphere i.e. troposphere, comprising of direct and reflected waves.

These waves are also known as tropospheric propagation as they can travel directly from the earth’s surface to the troposphere surface of the earth. It is also known as line of sight propagation as the signals are sent in a straight line from the transmitter to the receiver.

Figure 5.2. Space Wave Propagation

Space wave propagation depends on three components:

Direct Wave- The radio waves that are transmitted from the transmitting antenna, reach the receiving antenna directly.

Ground Reflected Wave - The radio waves that reach the receiving antenna after reflection from the ground.

Tropospheric Wave – The radio waves that reach the receiving antenna after reflection from the troposphere.

To prevent attenuation and loss of signal strength, the height of the antennas and distance between them can be given as:

Dm = (2RHt)-½ + (2RHr)-½

Where,

Dm: the distance between the two antennas

R: radius of the earth

Ht: height of transmission antenna

Hr: height of receiver antenna

5.2.1. Applications

It is used in various communication systems like

A line of sight communication and satellite communication

Radar communication

Microwave linking

5.2.2. Limitations

These waves are affected by the curvature of the earth.

The propagation of these waves happens along the line of sight distance which is defined as the distance between the transmitting antenna and the receiving antenna which is also known as the range of communication.

Key takeaways:

Space wave propagation is defined as the radio waves that occur within the 20km of the atmosphere i.e. troposphere, comprising of direct and reflected waves.

To prevent attenuation and loss of signal strength, the height of the antennas and distance between them can be given as Dm = (2RHt)-½ + (2RHr)-½)

5.3 Surface Wave

A surface wave is a mechanical wave that propagates along with the interface between differing media.

Surface waves are typically generated when the source of the earthquake is close to the Earth’s surface. As their name suggests, surface waves travel just below the surface of the ground.

Although they move even more slowly than S-waves, they can be much larger in amplitude and are often the most destructive type of seismic wave. There are several types of surface waves, but the two most common varieties are Rayleigh waves and Love waves.

5.3.1. Rayleigh waves

Rayleigh waves, also known as ground roll, spread through the ground as ripples, similar to rolling waves on the ocean. Like rolling ocean waves, Rayleigh waves move both vertically and horizontally in a vertical plane pointed in the direction in which the waves are traveling.

Figure 5.3.1. Rayleigh waves

Rayleigh waves propagate through the ground as ripples.

Eyewitnesses have claimed to observe Rayleigh waves in large open spaces, such as car parks, where they described the vehicles moving up and down like corks floating on the ocean.

Rayleigh waves are slower than body waves and typically travel at a speed that is 10% slower than S-waves.

5.3.2. Love waves

Love waves have the same motion as S-waves but without vertical displacement. They move the ground from side to side in a horizontal plane but at right angles to the direction of propagation.

Love waves are particularly damaging to the foundations of structures because of the horizontal ground motion they generate. Love waves can also cause horizontal shearing of the ground.

Figure 5.3.2. Love waves

They usually travel slightly faster than Rayleigh waves, at a speed that is usually about 10% slower than S-waves, but like S-waves, they cannot spread through water.

Love waves are particularly damaging to the foundations of structures.

Key takeaways:

Surface waves are typically generated when the source of the earthquake is close to the Earth’s surface.

As their name suggests, surface waves travel just below the surface of the ground.

Rayleigh waves propagate through the ground as ripples. Love waves are particularly damaging to the foundations of structures.)

5.4 Space Wave Propagation

5.4.1. Introduction

These waves occur within the lower 20 km of the atmosphere and are comprised of a direct and reflected wave. The radio waves having high frequencies are called space waves.

These waves can propagate through the atmosphere, from transmitter antenna to receiver antenna. These waves can travel directly or can travel after reflecting from the earth’s surface to the troposphere surface of the earth. So, it is also called Tropospheric Propagation.

The technique of space wave propagation is used in bands having very high frequencies. E.g. V.H.F. band, U.H.F band, etc. At such higher frequencies, the other wave propagation techniques like skywave propagation, ground wave propagation can’t work.

Only space wave propagation is left which can handle frequency waves of higher frequencies. The other name of space wave propagation is a line of sight propagation.

5.4.2. Principle used in space wave propagation

The space wave follows two distinct paths from the transmitting antenna to the receiving antenna - one through the air directly to the receiving antenna, the other reflected from the ground to the receiving antenna.

The primary path of the space wave is directly from the transmitting antenna to the receiving antenna. So, the receiving antenna must be located within the radio horizon of the transmitting antenna.

Because space waves are refracted slightly, even when propagated through the troposphere, the radio horizon is about one-third farther than the line-of-sight or natural horizon.

Figure 5.4.2. The principle used in space wave propagation

Although space waves suffer little ground attenuation, they nevertheless are susceptible to fading. This is because space waves actually follow two paths of different lengths (direct path and ground reflected path) to the receiving site and, therefore, may arrive in or out of phase.

If these two component waves are received in phase, the result is a reinforced or stronger signal. Likewise, if they are received out of phase, they tend to cancel one another, which results in a weak or fading signal.

5.4.3. Field strength

In principle, well removed from the earth’s surface, the field strength of the space wave is given by the inverse distance law. However, most services operate in the vicinity of the earth and propagation paths other than the direct ray from the transmitting to the receiving antennas are possible, most notably involving reflection from the ground.

As expected, the difference in path lengths between the direct and ground reflected rays will lead to a phase difference on reception; that will cause interference between the two rays.

Besides, there is a further phase change introduced into the ground reflected ray at the point of reflection that adds to the interference.

We will now analyze this situation to derive an expression for space wave field strength. Consider the geometry of Fig. below, from which we can see that the path lengths for the two rays are given by the following expressions, noting the simplifications possible since generally d >> ht, hr.

dd = d

dr = d

Figure 5.4.3. The geometry of space wave propagation

The difference in path lengths is

- =

From which the phase difference (calculated as 2π times the fractional difference in wavelength) is seen to be

This is one element of the phase delay of the reflected ray compared with the direct ray. As noted above, there is a further phase delay introduced at the point of reflection with the ground.

The electric field just after reflection compared with that just before is described by the reflection coefficient ρ = |ρ|

, a complex quantity that describes a change in amplitude and phase.

5.4.4. Effect of curvature of the earth

When the distance between the transmitting and receiving antennas is large, the curvature of the earth has a considerable effect on SWP. The field strength at the receiver becomes small as the direct ray may not be able to reach the receiving antenna.

The earth reflected rays diverge after their incidence on the earth. The curvature of the earth creates shadow zones.

5.4.5. Effect of Imperfection of earth

Earth is imperfect and electrically rough.

When a wave is reflected from perfect earth, its phase change is 180̊ . But actual earth makes the phase change different from180̊

The amplitude of the ground reflected ray is smaller than that of the direct ray.

The field at the receiving point due to space is reduced by the earth’s imperfection and roughness

5.4.6. Limitations of space wave Propagation

As a form of electromagnetic radiation, like light waves, radio waves are affected by the phenomena of reflection, refraction, diffraction, absorption, polarization, and scattering.

There are some limitations of space wave propagation: These waves are limited to the curvature of the earth. These waves have a line of sight propagation, which means their propagation is along the line of sight distance.

Key takeaways:

These waves occur within the lower 20 km of the atmosphere and are comprised of a direct and reflected wave. The radio waves having high frequencies are called space waves.

The primary path of the space wave is directly from the transmitting antenna to the receiving antenna. So, the receiving antenna must be located within the radio horizon of the transmitting antenna.

The earth reflected rays diverge after their incidence on the earth. The curvature of the earth creates shadow zones.

When a wave is reflected from perfect earth, its phase change is 180̊ . But actual earth makes the phase change different from180̊.)

References

“Antennas and Wave Propagation” by A R Harish and M Sachidananda.

G. S. N. Raju, “Antennas and Wave Propagation”, Pearson Education.

K. D. Prasad, “Antenna and Wave Propagation”, Satya Prakashan.

Unit - 4

Turing machines

4.1 The basic model for Turing machines (TM)

An information processing system (TM) may be a mathematical model that consists of

Associate infinite length tape divided into cells on which input is given. It consists of a head that reads the input tape.

A state register stores the state of the information processing system. Once reading an associated input image, it’s replaced with another image, its internal state is modified, and it moves from one cell to the proper or left. If the metal reaches the ultimate state, the input string is accepted, otherwise rejected.

A metal will be formally delineate as a 7-tuple (Q, X, ∑, δ, q0, B, F)

Where −

• Q may be a finite set of states

• X is that the tape alphabet

• ∑ is that the input alphabet

• δ may be a transition function; δ : alphabetic character × X → alphabetic character × X ×

• q0 is that the initial state

• B is that the blank image

• F is that the set of ultimate states

Basic model of turing machine

With the help of the following representation, the turning machine can be modelled.

The input tape contains an infinite number of cells, each cell containing one input symbol, so it is possible to position the input string on the tape. Blank characters fill up the empty tape.

Fig 1: Input tape

2. The finite control and the head of tape that is responsible for reading the current symbol of input. The head of the tape will switch from left to right,

3. A finite set of states through which the system has to go through.

4. Finite set of symbols that are used in the construction of the turing machine logic, called external symbols.

Fig 2: Basic model of TM

Various features of the Turing machine exist:

● It has an external memory that recalls an arbitrary, long input list.

● It has infinite capability for memory.

● The model has a method in which it is easy to read the input on the tape on the left or right.

● Depending on its input, the computer may generate a certain output. It will often be appropriate for the same input to be used to produce the output. So, the distinction between input and output has been abolished in this machine. It is therefore possible to use a standard set of alphabets for the Turing machine.

Key takeaway:

- TM may be a mathematical model that consists of associate infinite length tape divided into cells on which input is given.

- It consists of a head that reads the input tape.

- A state register stores the state of the information processing system.

4.2 Turing recognizable (Recursively enumerable) and Turing-decidable (recursive) languages and their closure properties

A language is termed Decidable or algorithmic if there’s a data processor that

Accepts and halts on each input string w. Each decidable language is Turing-

Acceptable.

Fig 3: Decidable language

A decision downside P is decidable if the language L of all affirmative instances to P

Is decidable.

For a decidable language, for every input string, the thulium halts either at the settle

For or the reject state as pictured within the following diagram -

Fig 4: shows different state

Example 1

Find out whether or not the subsequent downside is decidable or not − Is a range ‘m’ prime?

Solution

Prime numbers =

Divide the amount ‘m’ by all the numbers between ‘2’ and ‘√m’ ranging from ‘2’.

If any of those numbers turn out a remainder zero, then it goes to the “Rejected

State”, otherwise it goes to the “Accepted state”. So, here the solution may well be

Created by ‘Yes’ or ‘No’.

Hence, it’s a decidable downside.

Example 2

Given an everyday language L and string w, however will we have a tendency to

Check if w ∈ L?

Solution

Take the DFA that accepts L and check if w is accepted

Fig 5: DFA diagram

Some additional decidable issues are -

• Does DFA settle for the empty language?

• Is L1 ∩ L2 = ∅ for normal sets?

Key takeaway:

- If a language L is decidable, then its complement L’ is additionally decidable

- If a language is decidable, then there’s AN functionary for it.

4.2.1 Closure properties

Union and Intersection

Recursive

L1 x L2 and L1 x L2 are both recursive if L1 is recursive and L2 is recursive.

Using a multitape TM:

● Copy to Tape 2 and Tape 3 inputs

● Run M1 to tape 2 and M2 to tape 3 (neither will run forever; i.e.we get a result)

● They will determine whether x is in L1 and/or L2

● Test if approved by both M1 and M2 (intersection)

● Test if one is approved by M1 and M2 (union)

If L1 and L2 are recursive, the difference between L1 - L2=L1 and L2 is recursive.

Recursively Enumerable

If L1 and L2 are enumerated recursively, then L1 ∪ L2 and L1 ∩ L2 are recursively enumerated .

● Similar to the recursive case, but the case where M1 and M2 will run constantly

● Simulate simultaneously running M1 and M2-alternate one move from each machine.

A union, for reference,

● If any computer ever accepts it, accept it.

● If any machine ever refuses or fails, then the other machine continues to run.

If L is enumerated recursively and L is enumerated recursively, L is enumerated recursively.

● Let M and M be TMs, respectively, that accept L and L.

● Simultaneously, Run M and M'

● It must be agreed for any term x by any one of M or M'

● Therefore, either M or M 'will stop and agree

● If M stops and agrees, then stop and embrace.

● If M' stops and embraces, then stop and deny.

The TM that runs M and M at the same time always stops and accepts or refuses so that L and L are recursive.

If L is enumerable recursively, and L is not recursive, then L is not enumerable recursively.

Concatenation

If L1 and L2 are two recursive languages, the L1.L2 concatenation would also be recursive. For instance:

L1 = {anbncn|n>=0}

L2 = {dmemfm|m>=0}

L3 = L1.L2

= {anbncndn emfm|m>=0 and n>=0} is also recursive.

L1 states that n no. Of a's is followed by n no. Of b's, and n no. Of c's. L2 says that m no. Of d's is followed by m no. Of e's, m no. Of f's. First, their concatenation matches no. Of a's, b's, and c's, then no. Of d's, e's, and f's. It can therefore be determined by TM.

Kleene Closure

If L1 is recursive, its L1* smaller closure would also be recursive. For instance:

L1 = {anbncn|n>=0}

L1*= { anbncn||n>=0}* is also recursive.

4.3 variants of Turing machines

1. Multiple tracks Alan Turing Machine:

● A k-tack Alan Turing machine (for some k>0) has k-tracks and one R/W head

That reads and writes all of them one by one.

● A k-track information processing system may be simulated by one track information processing system.

2. Two-way infinite Tape Alan Turing Machine:

● Infinite tape of two-way infinite tape information processing system is

Boundless in each direction left and right.

● Two-way infinite tape information processing system may be simulated by

Unidirectional infinite {turing|Turing|Alan Alan Turing|Alan Mathison

Turing|mathematician} machine (standard Turing machine).

3. Multi-tape Alan Turing Machine:

● It has multiple tapes and is controlled by one head.

● The Multi-tape information processing system is completely different from the k-track information processing system however communicative power is the same.

● Multi-tape information processing system may be simulated by a single-tap information processing system.

4. Multi-tape Multi-head Alan Turing Machine:

● The multi-tape information processing system has multiple tapes and multiple Heads.

● Each tape is controlled by a separate head.

● Multi-Tape Multi-head information processing systems may be simulated by normal information processing systems.

5. Multi-dimensional Tape Alan Turing Machine:

● It has multi-dimensional tape wherever the head will move any direction that’s left, right, up or down.

● Multi-dimensional tape information processing system may be simulated by a one-dimensional information processing system.

6. Multi-head Alan Turing Machine:

● A multi-head information processing system contains 2 or additional heads to scan the symbols on constant tape.

● In one step all the heads sense the scanned symbols and move or write severally.

● Multi-head information processing system may be simulated by single head information processing system.

7. Non-deterministic Alan Turing Machine:

● A non-deterministic information processing system encompasses a single, a method infinite tape.

● For a given state and input image has at least one option to move (finite range of decisions for consecutive move), every selection has many decisions of path that it’d follow for a given input string.

● A non-deterministic information processing system is an adored settled information processing system.

4.4 Nondeterministic TMs and equivalence with deterministic TMs

In a Non-Deterministic computing device, for each state and image, there area unit a

Bunch of actions the thulium will have. So, here the transitions don’t seem to be

Settled. The computation of a non-deterministic computing device could be a tree of

Configurations which will be reached from the beginning configuration.

An input is settled fored if there’s a minimum of one node of the tree that is associate degree accept configuration, otherwise it’s not accepted. If all branches of the machine tree halt on all inputs, the non-deterministic computing device is named a Decider and if for a few inputs, all branches area unit rejected, the input is additionally rejected.

A non-deterministic computing device are often formally outlined as a 6-tuple

(Q, X,∑, δ, q0, B, F) where −

● Q could be a finite set of states

● X is that the tape alphabet

● ∑ is that the input alphabet

● δ could be a transition function;

δ: letter × X → P (Q × X ×).

● q0 is that the initial state

● B is that the blank image

● F is that the set of ultimate states

Key takeaway:

- In a Non-Deterministic Turing Machine, there is a group of acts the TM can have for every state and symbol. So, the transformations aren't deterministic here.

- A non-deterministic Turing Machine calculation is a tree of configurations that can be accessed from the configuration at the start.

4.5 Unrestricted grammars and equivalence with Turing machines

Unrestricted Grammars

Recall, a descriptive linguistics is Associate in Nursing abstract entity that makes an

Attempt to explain the “strings” [aka sentences] of a language. As a proper extension

Of a context-free grammar:

Definition: Associate in Nursing unrestricted descriptive linguistics could be a 4-tuple (T,N,P,S), consisting of :

1. T = set of terminals (the legal “tokens” of the language)

2. N = set of nonterminals (aka variables)

3. P = as set of productions, every of the form:

v -> w (what will this mean?)

Where v and w area unit strings consisting of nonterminals and terminals.

4. S = a special nonterminal referred to as the beginning image.

This type of descriptive linguistics is additionally referred to as a sort zero descriptive linguistics. It’s such a computing device. [What will this mean? Given a sort zero descriptive linguistics, there exists a metallic element which will “recognize” identical language as that descriptive linguistics. And, conversely, a sort zero descriptive linguistics will continuously be found for the language recognized or generated by any metallic element.]

It is conjointly such as a recursively calculable language. [What will this mean?]

E.g.: G with set of productions:

P =

L(G) (language generated or recognized by G) =

w = ai {and i = 2k, k > 0}

Note: traditional methodology of specifying repetition of strings.

Definition: context-sensitive descriptive linguistics [aka kind one grammar]

All productions area unit of kind

v -> w wherever |v| < |w|

Note: one special traditional kind [what will this mean?] of this sort of descriptive

Linguistics needs productions of form:

UAv -> uwv with w != letter of the alphabet,

I.e., A -> w however solely within the context of u nine v.

A context-sensitive descriptive linguistics is such as a linear finite automaton (LBA)

And to a context-sensitive language.

Almost all languages area unit context-sensitive. [What will this mean exactly? -

They’ll really be less complicated than context-sensitive.]

Definition: context-free descriptive linguistics [aka kind a pair of grammar]

All productions area unit of the form:

A -> x — wherever A is nonterminal, x could be a string of nonterminals and Terminals.

A context-free descriptive linguistics is such as a pushdown automaton (PDA) and to

Context-free languages.

Definition: Regular descriptive linguistics [aka kind three descriptive linguistics, aka

Linear grammar]

All productions area unit of the form:

A -> wB

A -> w wherever A, B area unit nonterminals,

w could be a string of terminals

Regular grammars area unit such as regular sets [what will such as mean here?]

And they are such as finite automata.

Key takeaway:

- Once this constraint is removed, it is interesting to explore the possibilities.

- In fact, such grammars are called unrestricted grammars, where any combination of variables and terminals will appear on the left side.

4.6 TMs as enumerators

Fig 5: Enumerator

Enumerator

(Q , Σ ,「,δ , qstart , qhalt )

δ : Q ｘ「 → Q ｘ「 {L , R}

ｘ (Σ ｘ{R} ) ∪ {(ε , S)}

Description

Use Enumeration attribute if you have got fields or properties of enumerated sorts

And you wish to save lots of them within the info. Victimization functionary you outline

However the enumerated values are saved and loaded from the info. The functionary

Attribute should be declared right higher than the enumerated sort.

Creator

Constructor Create(MappedType: TEnumMappingType); overload;

Constructor Create(MappedType: TEnumMappingType; MappedValues: string);

Overload;

Parameters

MappedType Indicated the kind of the enumerated worth within the info. Valid values square measure (prefixed by TEnumMappingType):

EmChar Enumerated values are saved as single-chars within the info

EmInteger Enumerated values are saved as number values. {the worth|the worth}

Used is that the ordinal value of the enumerated sort, i.e, the primary worth within the functionary are saved as zero, the second as one, etc..

EmString Enumerated values are saved as strings within the info

MappedValues If MappedType is char or string, then you need to use this

Parameter to specify the char/string worths reminiscent of every enumerated value.

The values should be comma-separated and should be within the same order

Because the values within the enumerated sort.

References:

John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman, Introduction to

Automata Theory, Languages, and Computation, Pearson Education Asia.

2. John Martin, Introduction to Languages and the Theory of Computation, Tata McGraw Hill.

3. Harry R. Lewis and Christos H. Papadimitriou, Elements of the Theory of Computation, Pearson EducationAsia.

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