Relation between Laplace and Fourier Transform

The Fourier transform  of a signal x(t) is given as 

X(j\omega ) = \int_{-\infty }^{\infty } x(t) e^{-j\omega t}dt----EQN(I)

Fourier Transform exists only if \int_{-\infty }^{\infty } \left | x(t) \right |dt< \infty 

we know that s=\sigma + j\omega 

X(S) = \int_{-\infty }^{\infty } x(t) e^{-s t}dt

X(S) = \int_{-\infty }^{\infty } \left | x(t)e^{-\sigma t} \right | e^{-j\omega t}dt----EQN(II)

if we compare Equations (I) and (II) both are equal when  \sigma =0.

i.e, X(S) =X(j\omega)| \right |_{s=j\omega }.

This means that Laplace Transform is same as Fourier transform when s=j\omega.

Fourier Transform is nothing but the special case of Laplace transform where  s=j\omegaindicates the imaginary axis in complex-s-plane.

Thus Laplace transform is basically Fourier Transform on imaginary axis in the s-plane.

Normal incidence on a perfect conductor

Normal incidence on a perfect conductor

whenever an EM Wave travelling in one medium impinges second medium the wave gets partially transmitted and partially reflected depending up on the type of the second medium.

Assume the first case in Normal incidence that is Normal incidence on a Perfect conductor.

i.e an EM wave propagating in free space strikes suddenly a conducting Boundary which means the other medium is a conductor.

The figure shows a plane Wave which is incident normally upon a boundary between free space and a perfect conductor.

assume the wave is propagating in positive z-axis and the boundary is z=0 plane.

The transmitted wave  since the electric field intensity inside a perfect conductor is zero.

The incident   and reflected   waves are in the medium 1  that is free space.

The energy transmitted is zero so the energy absorbed by the conductor is zero and entire wave is reflected to the same medium

now incident wave is 

  in free space  for medium 1

 ( a wave propagating in positive z-direction) and the reflected wave is  ( a wave propagating in positive z-direction).

 .

by using tangential components  .

The resultant wave is   .

 .

the above equation is in phasor notation , converting the above equation into time-harmonic (or) sinusoidal variations

 .

This is the wave equation which represents standing wave , which is the contribution of incident and reflected waves. as this wave is stationary it does not progress.

it has maximum amplitude at odd multiples of   and minimum amplitude at multiples of  .

Similarly The resultant Magnetic field is

The resultant wave is   .

 .

the above equation is in phasor notation , converting the above equation into time-harmonic (or) sinusoidal variations

 .

this wave is  a stationary wave  it has minimum amplitude at odd multiples of   and maximum amplitude at multiples of  .

 

Nature of Magnetic materials

In order to find out the various types of materials in magnetic fields and their behaviour we use the knowledge of the action of magnetic field on a current loop with a simple model of an atom

Magnetic materials are classified on the basis of presence of magnetic dipole moments in the materials.

a charged particle with angular momentum always contributes to the permanent contributions to the angular moment of an atom

1. orbital magnetic dipole moment.

2. electron spin moment.

3. Nuclear spin magnetic moment.

Orbital Magnetic dipole Moment:-

The simple atomic model is one which assumes that there is a central positive nucleus surrounded by electrons in various circular orbits.

an electron in an orbit is analogous to a small current loop and as such experiences a torque in an external magnetic field, the torque tending to align the magnetic field produced by the orbiting electron with the external magnetic field.

Thus the resulting magnetic field at any point in the material would be greater than it would be at that point when the other moments were not considered.

so there are Quantum numbers which describes the orbital state of notion of electron in an atom there are n,l and ml

n-principal Quantum number, which determines the energy of an electron.

l-Orbital Quantum number which determines the angular momentum of orbit.

ml-magnetic Quantum number which determines the component of magnetic moment along the direction of an electric field.

electron spin Magnetic Moment:-

The angular momentum of an electron is called spin of the electron. as electron is a charged particle the spin of the electron produces magnetic dipole moment because electron is spinning about it’s own axis and thus generates a magnetic dipole moment.

\pm 9X 10^{-24} A-m^{2} is the value of electron spin when we consider an atom those electrons which are in shells which are not completely filled with contribute to a magnetic moment for the atom.

Nuclear spin Magnetic Moment:-

a third contribution of the moment of an atom is caused by nuclear spin this provides a negligible effect on the overall magnetic properties of material

That is the mass of the nucleus is much larger than an electron thus the dipole moments due to nuclear spin are very small.

so the total magnetic dipole moment of an atom is nothing but the summation of all the above mentioned .

Compression laws (A-law and u-law)

The laws used in compressor of a non-uniform Quantizer are known as compression laws.

two laws are available

  1. \mu– law.
  2. A-law.

\mu-law:-  A particular form of compression law that is used in practice is the so called  \mu– law which is defined by 

\left | v \right |= sgn(x){\frac{\ln (1+\mu \left | x \right |)}{(1+\mu )}}, for\ 0\leq \left | x \right |\leq 1.

where \left | v \right |  –  Normalized compressed output voltage.

             \left | x \right |   – Normalized input voltage to the compressor.

and  \mu  is a positive constant and its  value decides the curvature of \left | v \right |.

\mu =0   corresponds to no compression, which is the case of uniform quanization, the curve is almost linear as the value of \muincreases signal compression increases.

\mu =255  is the north american standard for PCM voice telephony.

for a given value of \mu, the reciprocal slope of the compression curve, which defines the quantum steps is given by the derivative of  \left | x \right |  with respect to \left | v \right | 

\frac{dx}{dv} = \frac{\ln (1+\mu )}{\mu } (1+\mu \left | x \right |) 

we see that therefore \mulaw is neither strictly linear nor strictly logarithmic.

i.e,

It is approximately linear at low levels of input corresponding to \mu \left | x \right |<<1. and  is approximately logarithmic at high input levels corresponding to \mu \left | x \right |>>1.

typical value of \mu =255 

A-law :- another compression law that is used in practice is the so called A- law defined by

\left | v \right | =\left\{\begin{matrix} sgn(x) (\frac{A\left | x \right |}{1+\ln A}), 0\leq \left | x \right |\leq \frac{1}{A}.\\ sgn(x) (\frac{1+ \ln A\left | x \right |}{1+\ln A}),\frac{1}{A} \leq \left | x \right |\leq 1. \end{matrix}\right.

A=1 corresponds to the case of uniform Quantization. A-law has been plotted for three different values of A. A=1, A=2, A=87.56.

typical value of A is 87.56 in European Commercial PCM standard which is being followed in India.

for both the \mu-law and A-law, the dynamic range capability of the compander improves with increasing \muand A respectively. The SNR for low-level signals increases at the expense of the SNR for high level signals.

to accommodate these two conflicting requirements (i.e, a reasonable SNR for  both low and high-level signals), a compromise is usually made in choosing the value of parameter \mufor the \mu-law and parameter A for the A-law. The typical values used in practice are \mu=255and A=87.56.

It is also interested to note that in actual PCM systems, the companding circuitry does not produce an exact replica of the non-linear compression curves shown in the compression law characteristics.

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