Dynamic factor

In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL₂(R) on the time–frequency plane (domain).

The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL₂(R) can also be interpreted as the symplectic group Sp₂, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form.

Definition

The LCT can be represented in several ways; most easily,^[1] it can be viewed as a 2×2 matrix with determinant 1, i.e., an element of the special linear group SL₂(R). Taking a matrix $(\begin{matrix} a & b \\ c & d \end{matrix}),$ with ad − bc = 1, the corresponding integral transform is:

$X_{(a, b, c, d)} (u) = \sqrt{- i} \cdot e^{i π \frac{d}{b} u^{2}} \int_{- \infty}^{\infty} e^{- i 2 π \frac{1}{b} u t} e^{i π \frac{a}{b} t^{2}} x (t) d t,$ Template:Pad	when b ≠ 0,
$X_{(a, 0, c, d)} (u) = \sqrt{d} \cdot e^{i π c d u^{2}} x (d u),$	when b = 0.

Special cases

Many classical transforms are special cases of the linear canonical transform:

The Fourier transform corresponds to rotation by 90°, represented by the matrix:

[\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}] .

The fractional Fourier transform corresponds to rotation by an arbitrary angle; they are the elliptic elements of SL₂(R), represented by the matrices:

[\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix}] .

The Fresnel transform corresponds to shearing, and are a family of parabolic elements, represented by the matrices:

[\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 1 & λ z \\ 0 & 1 \end{matrix}] .

where z is distance and λ is wave length.

The Laplace transform corresponds to rotation by 90° into the complex domain, and can be represented by the matrix:

[\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 0 & i \\ i & 0 \end{matrix}] .

The Fractional Laplace transform corresponds to rotation by an arbitrary angle into the complex domain, and can be represented by the matrix:^[2]

[\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} i \cos θ & i \sin θ \\ i \sin θ & - i \cos θ \end{matrix}] .

Composition

Composition of LCTs corresponds to multiplication of the corresponding matrices; this is also known as the "additivity property of the WDF".

In detail, if the LCT is denoted by O_F^(a,b,c,d), i.e.

X_{(a, b, c, d)} (u) = O_{F}^{(a, b, c, d)} [x (t)]

then

O_{F}^{(a 2, b 2, c 2, d 2)} {O_{F}^{(a 1, b 1, c 1, d 1)} [x (t)]} = O_{F}^{(a 3, b 3, c 3, d 3)} [x (t)],

where

[\begin{matrix} a 3 & b 3 \\ c 3 & d 3 \end{matrix}] = [\begin{matrix} a 2 & b 2 \\ c 2 & d 2 \end{matrix}] [\begin{matrix} a 1 & b 1 \\ c 1 & d 1 \end{matrix}] .

In optics and quantum mechanics

Paraxial optical systems implemented entirely with thin lenses and propagation through free space and/or graded index (GRIN) media, are quadratic phase systems (QPS); these were known before Moshinsky and Quesne (1974) called attention to their significance in connection with canonical transformations in quantum mechanics. The effect of any arbitrary QPS on an input wavefield can be described using the linear canonical transform, a particular case of which was developed by Segal (1963) and Bargmann (1961) in order to formalize Fok's (1928) boson calculus.^[3]

Applications

Canonical transforms are used to analyze differential equations. These include diffusion, the Schrödinger free particle, the linear potential (free-fall), and the attractive and repulsive oscillator equations. It also includes a few others such as the Fokker–Planck equation. Although this class is far from universal, the ease with which solutions and properties are found makes canonical transforms an attractive tool for problems such as these.^[4]

Wave propagation through air, a lens, and between satellite dishes are discussed here. All of the computations can be reduced to 2×2 matrix algebra. This is the spirit of LCT.

Electromagnetic wave propagation

Assuming the system looks like as depicted in the figure, the wave travels from plane x_i, y_i to the plane of x and y. The Fresnel transform is used to describe electromagnetic wave propagation in air:

U_{0} (x, y) = - \frac{j}{λ} \frac{e^{j k z}}{z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{j \frac{k}{2 z} [(x - x_{i})^{2} + (y - y_{i})^{2}]} U_{i} (x_{i}, y_{i}) d x_{i} d y_{i},

with

k = 2 π / λ Template:Pad	: wave number;
λ	: wavelength;
z	: distance of propagation;
j	: imaginary unit.

This is equivalent to LCT (shearing), when

[\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 1 & λ z \\ 0 & 1 \end{matrix}] .

When the travel distance (z) is larger, the shearing effect is larger.

Spherical lens

With the lens as depicted in the figure, and the refractive index denoted as n, the result is:^[5]

U_{0} (x, y) = e^{j k n Δ} e^{- j \frac{k}{2 f} [x^{2} + y^{2}]} U_{i} (x, y)

with f the focal length and Δ the thickness of the lens.

The distortion passing through the lens is similar to LCT, when

[\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 1 & 0 \\ \frac{- 1}{λ f} & 1 \end{matrix}] .

This is also a shearing effect: when the focal length is smaller, the shearing effect is larger.

Spherical Mirror

The spherical mirror—e.g., a satellite dish—can be described as a LCT, with

[\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 1 & 0 \\ \frac{- 1}{λ R} & 1 \end{matrix}] .

This is very similar to lens, except focal length is replaced by the radius of the dish. Therefore, if the radius is smaller, the shearing effect is larger.

Example

The system considered is depicted in the figure to the right: two dishes – one being the emitter and the other one the receiver – and a signal travelling between them over a distance D. First, for dish A (emitter), the LCT matrix looks like this:

[\begin{matrix} 1 & 0 \\ \frac{- 1}{λ R_{A}} & 1 \end{matrix}] .

Then, for dish B (receiver), the LCT matrix similarly becomes:

[\begin{matrix} 1 & 0 \\ \frac{- 1}{λ R_{B}} & 1 \end{matrix}] .

Last, for the propagation of the signal in air, the LCT matrix is:

[\begin{matrix} 1 & λ D \\ 0 & 1 \end{matrix}] .

Putting all three components together, the LCT of the system is:

[\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} 1 & 0 \\ \frac{- 1}{λ R_{B}} & 1 \end{matrix}] [\begin{matrix} 1 & λ D \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ \frac{- 1}{λ R_{A}} & 1 \end{matrix}] = [\begin{matrix} 1 - \frac{D}{R_{A}} & - λ D \\ \frac{1}{λ} (R_{A}^{- 1} + R_{B}^{- 1} - R_{A}^{- 1} R_{B}^{- 1} D) & 1 - \frac{D}{R_{B}} \end{matrix}] .

Notes

↑ de Bruijn, N. G. (1973). "A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence", Nieuw Arch. Wiskd., III. Ser., 21 205-280.
↑ P.R. Deshmukh & A.S. Gudadhe (2011) Convolution structure for two version of fractional Laplace transform. Journal of Science and Arts, 2(15):143-150. [1]
↑ K.B. Wolf (1979) Ch. 9:Canonical transforms.
↑ K.B. Wolf (1979) Ch. 9 & 10.
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, §5.1.3, pp. 100–102.

References

J.J. Ding, "Time–frequency analysis and wavelet transform course note", the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
K.B. Wolf, "Integral Transforms in Science and Engineering", Ch. 9&10, New York, Plenum Press, 1979.
S.A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Amer. 60, 1168–1177 (1970).
M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 8, 1772–1783, (1971).
B.M. Hennelly and J.T. Sheridan, "Fast Numerical Algorithm for the Linear Canonical Transform", J. Opt. Soc. Am. A 22, 5, 928–937 (2005).
H.M. Ozaktas, A. Koç, I. Sari, and M.A. Kutay, "Efficient computation of quadratic-phase integrals in optics", Opt. Let. 31, 35–37, (2006).
Bing-Zhao Li, Ran Tao, Yue Wang, "New sampling formulae related to the linear canonical transform", Signal Processing '87', 983–990, (2007).
A. Koç, H.M. Ozaktas, C. Candan, and M.A. Kutay, "Digital computation of linear canonical transforms", IEEE Trans. Signal Process., vol. 56, no. 6, 2383–2394, (2008).
Ran Tao, Bing-Zhao Li, Yue Wang, "On sampling of bandlimited signals associated with the linear canonical transform", IEEE Transactions on Signal Processing, vol. 56, no. 11, 5454–5464, (2008).

[1] Bruijn, N. G. (1973). "A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence", Nieuw Arch. Wiskd., III. Ser., 21 205-280.

[2] P.R. Deshmukh & A.S. Gudadhe (2011) Convolution structure for two version of fractional Laplace transform. Journal of Science and Arts, 2(15):143-150. [1]

[3] K.B. Wolf (1979) Ch. 9:Canonical transforms.

[4] K.B. Wolf (1979) Ch. 9 & 10.

[5] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, §5.1.3, pp. 100–102.

[1]

[2]

[3]

[4]

[5]

Dynamic factor

Contents

Definition

Special cases

Composition

In optics and quantum mechanics