# Chapter 19: The Fourier Transform Relation between Aperture Distribution and Far-field Pattern

### Example 19-8.1, Page number: 690

In [17]:
from math import sin, pi

#Variable declaration
gal_ext = 400000        #Extent of galaxy (light-years)
alpha = 0.032           #Extent of galaxy (degrees)
f = 5e9                 #Frequency (Hz)
a = 36e3                #Maximum VLA Spacing (m)
c = 3e8                 #Speed of light (m/s)
wid = 0.03             #Width of image (degrees)
hei = 0.008              #Height of image (degrees)
flux_den = 2.5e-23      #Average flux density (W/m^2)
bw = 1e9                #Bandwidth (Hz)

#Calculations
dist = gal_ext/sin(alpha*pi/180)    #Distance to the galaxy (light-years)
dist_m = dist*(365*24*3600*c)

wave_lt = c/f       #Wavelength (m)
a_lambda = a/wave_lt    #Spacing in wavelength  (unitless)
pix_size = 51/a_lambda  #Resolution or Pixel size (degrees)
pix_size_arc = pix_size*3600    #Pixel size (arc seconds)

area = wid*hei      #Area of image (square degrees)
area_arc = area*(3600**2)    #Area of image (arc seconds)
num_pix = area_arc/pix_size_arc**2  #Number of pixels

#Result
print "The distance to the galaxy is", round(dist,-8), "light years"
print "The resolution or pixel size is", round(pix_size_arc,2), "arc seconds"
print "The number of pixels is", round(num_pix)
print "The radio power of the galaxy is %.1e W" % rad_pow

The distance to the galaxy is 700000000.0 light years
The resolution or pixel size is 0.31 arc seconds
The number of pixels is 33218.0
The radio power of the galaxy is 1.4e+37 W


### Example 19-8.2, Page number:691

In [19]:
from math import pi, log10

#Variable declaration
f = 10e9    #Frequency (Hz)
c = 3e8     #Speed of light (m/s)
dia = 100   #Dish diameter (m)
aper_eff = 0.725    #Aperture efficiency (unitless)

#Calculation
wave_lt = c/f       #Wavelength (m)
hpbw = 66/(dia/wave_lt) #Half power beam width (degrees)

gain = 41000/(hpbw**2)  #Gain from beamwidth (unitless)
gain_db = 10*log10(gain)    #Gain from beamwidth (dBi)

gain_ap = 4*(pi**2)*(dia/2)**2*(aper_eff)/(wave_lt**2)
#Gain from effective aperture(unitless)
gain_ap_db = 10*log10(gain_ap)  #Gain from effective aperture (dBi)

side_lobe = -23     #First side lobe level from table (dB)

#Result
print "The Half Power Beamwidth is", round(hpbw,3), "degrees"
print "The gain from beamwidth is", round(gain_db), "dBi"
print "The gain from effective aperture is", round(gain_ap_db), "dBi"
print "The first side-lobe level is", side_lobe,"dB"

The Half Power Beamwidth is 0.02 degrees
The gain from beamwidth is 80.0 dBi
The gain from effective aperture is 79.0 dBi
The first side-lobe level is -23 dB