*:*

__Solution__For regular Stokes' parameters we have:

d =

**Ö (**Q

^{2}+ U

^{2}+ V

^{2}

**) / I**

(For 0

__<__d__<__1.)
We have for the

*normalized Stokes parameters*:_{}^{}
s

_{0}= I / S = 1
s

_{1}= Q / S
s

_{2}= U / S
s

_{3}= V / S
So through the use of algebra we obtain:

d =

**Ö**(s_{1}^{2}+ s_{2}^{2}+ s_{3 }^{2}) / s_{0}=**Ö**(s

_{1}

^{2}+ s

_{2}

^{2}+ s

_{3 }

^{2})

2) Write a matrix equation for a completely unpolarized radio wave using a left circularly polarized wave and a right circularly polarized wave.

*:*

__Solution__
A completely unpolarized wave requires the result: C =

½ [1…..0]

[0…..1]

Left circularly polarized wave: has:

½ [1…..j]

[-j…..1]

And: Right circularly polarized wave: has:

½ [1…..-j]

[j…..1]

_{}

^{}

Then to get C we need:

A + B = C

Where: A =

1/4 [1…..j]

[-j…..1]

B =

1/4 [1…..-j]

[j…..1]

.

Check by using matrix addition to add and you obtain matrix C

3) The coherency matrix of some individual radio wave is given by:

½ [1…..0]

[0…..0]

Show how the resultant unpolarized wave's coherency matrix may be obtained by showing the C-matrix for the other wave needed to combine with the individual wave above.

*:*

__Solution__
The resultant unpolarized wave's coherency matrix would be: C=

½ [1…..0]

[0…..1]

The given matrix which needs a complementary matrix to obtain the above is: A =

½ [1…..0]

[0…..0]

Then matrix subtraction, e.g. C - A yields B =

_{}

^{}

½ [0…..0]

[0…..1]

Which is the matrix needed to obtain C.

4) Four radio waves are detected and analyzed and found to have the characteristics shown below:

a) d = 0

b) d = ½ AR = 4 and t = 135 deg

c) d = ½ AR = 4 and t = - 135 deg

d) d = ½ AR = 4 and t = 45 deg

Find the normalized Stokes parameters and the coherency matrices for these waves.

__:__

*Solutions*
a) d = 0 denotes a completely unpolarized wave so

Stokes parameters: [1..0..0..0]

Coherency matrix: C =

Coherency matrix: C =

½ [1…..0]

[0…..1]

b) d = ½ AR = 4 and t = 135 deg

We have for the normalized Stokes components:

s

st cos 2 e

st cos 2 e

s

_{0 }= 1s

_{1}= d cos 2s

_{2}= d sin2s

_{3}= d sin2e_{}^{}
And: cos 2 e = (AR

^{2}- 1) / (AR^{2}+ 1)
= (4

^{2}- 1) / (4^{2}+ 1) = 15/ 17
And: cos 2t = cos (2 x 135) = cos 270 = 0

sin2t = sin (2 x 135) = sin 270 = -1

_{}

^{}

Then Stokes parameters for these characteristics:

[1..0..0....

**-**1./2]
Coherency matrix: C =

½ [1…..-j]

[j…..1/2]

(c) is analogous in solution to (b) except t= - 135 deg

Then Stokes parameters for these characteristics:

[1..0..0.... ½]

Coherency matrix: C =

Coherency matrix: C =

½ [1…..j]

[-j…..½]

d) d = ½ AR = 4 and t = 45 deg

cos 2 e = (AR

^{2}- 1) / (AR^{2}+ 1)
= (4

^{2}- 1) / (4^{2}+ 1) = 15/ 17
And: cos 2t = cos (2 x 45) = cos 90 = 0

sin2t = sin (2 x 45) = sin 90 = 1

Then Stokes parameters for these characteristics:

[1..0..0.... ½]

Coherency matrix: C =

Coherency matrix: C =

½ [1… ..j]

[-j.....½]

_{}

^{}

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