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19th December 2014, 21:11 #1
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Sanity check: Nearfield to farfield transformation
Dear forum,
I would like to do a sanity check on a nearfield to farfield transform problem that I am struggling with. I have some planar recorded nearfield data and would like to transform these data to farfield data and subsequently depict the field in spherical coordinates as E(r, theta, phi) where r is the radius, theta is the spaceangle (?) and phi is the planar angle. All this should be done using Matlab. My coordinate system is such that the planar data is the xy plane and the direction of propagation is in the zdirection, hence the planar angle is measured from the positive xaxis and counter clockwise and the spaceangle is measured from the positive zaxis.
Now, the data I have at hand is recorded (actually simulated) in front of a pyramidal horn excited in the TE_10 dominant mode. Simply, this emulates a planar gridsampling of the nearfield data.
The transformation should be done using the least amount of approximations possible and I assume that this would be by going about it the following way:
The recorded samples should be converted to a Plane Wave Spectrum (PWS) using the following expression:
 f_y(k_x, k_y) = doubleintegralsum( E_ya(x,y,z=0)*exp(j*k_x*x+k_y*y))dxdy
 "E_ya" is the sampled data and the _ya indicates that the data is linearly polarized in the ydirection. A similar double integral sum could be put in place if there was an xpolarized component present.
 "k_x" is the wavenumber in the xdirection given by k_x = sin(theta)*cos(phi)
 "k_y" is the wavenumber in the ydirection given by k_y = sin(theta)*sin(phi)
 The angles defined by theta and phi can be limited to the hemisphere of propagation, since this is a planar measurement and hence no data is recorded except in front of the antenna
 The double integral sum is taken over the limits or size of the sampling grid/plane
The next operation to perform is to convert or map these data points from the PWS to spherical coordinates by using:
 r = x*sin(theta)*cos(phi) + y*sin(theta)*sin(phi) + z*cos(phi)
theta = x*cos(theta)*cos(phi) + y*cos(theta)*sin(phi)  z*sin(theta)
phi = x*sin(phi) + y*cos(phi)
 r = y*sin(theta)*sin(phi)
theta = y*cos(theta)*sin(phi)
phi = y*cos(phi)
The coordinate mapping originates from this definition:
The final operation to be done is to integrate to the farfield in spherical coordinates:
 E(r, theta, phi) = (1/(4*pi^2))*doubleintegralsum(PWS*exp(j*k*r))dk_xdk_y
 k*r = r*[k_x*sin(theta)*cos(phi) + k_y*sin(theta)*sin(phi) + kz*cos(theta)]
which can be rewritten as
k*r = r*[k_x*sin(theta)*cos(phi) + k_y*sin(theta)*sin(phi) + sqrt(k^2  k_x^2  k_y^2)*cos(theta)]
and PWS is f_y(k_x,k_y) mapped to spherical coordinates. The double integral sum is taken over the angular limits of the hemisphere in the direction of propagation.
Where am I getting it right and where am I getting it wrong here?
Any help and/or comments is deeply appreciated!
Best Regards,

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