a Matlab package for phased array beam shape inspection

4.6 Parseval’s theorem 19
closed form as a product of two Direchlet kernels:
AF =
M∑
x−1
m x=0
M
∑ y−1
e iΨxmx
m y=0
e iΨymy (50)
= e iΦ M x M y · diric(Ψ x , M x ) · diric(Ψ y , M y ) , (51)
where Φ is a non-observable phase factor, which we ignore,
and the Dirichlet kernel is defined by
Ψ x = 2πu x D x − δ x = 2πD x ( u x − u 0 x) (52)
Ψ y = 2πu y D y − δ y = 2πD y ( u y − u 0 y)
diric(Ψ, M) = sin(MΨ/2)
M sin(Ψ/2) . (53)
The function diric in Matlab signal processing toolbox implements Eq. (53). The magnitude
of the Dirichlet kernel is periodic by 2π. One period is plotted in Fig. 10 for a
few values of M, using plot-diric.m.
4.6. Parseval’s theorem
We will need the Parseval’s theorem for two-dimensional sequencies. Recall that for
the 1-dimensional sequence, Parseval’s theorem—spectral domain and power domain
representation of signal energy are equal—is
∫ 1
0
|ã(ν)| 2 dν = ∑ m
Moreover, due to ã(ν) periodicity by 1, it follows from Eq. (54) that
∫ 1/D
0
|a m | 2 . (54)
|ã((ν − ν 0 )D)| 2 dν = 1 ∑
|a m | 2 , (55)
D
for any constants ν 0 and D. The 2D-analogue of Eq. (55) holds,
∫
|ã((u − u 0 )D)| 2 d 2 u = 1 ∑
|a m | 2 . (56)
D x D y
4.7. Antenna gain and the power integral
The directive gain (which we will also call power gain in this note) of an antenna in the
direction û is the ratio of the power radiated per unit solid angle into that direction,
divided by the power density of the same total power radiated isotropically,
m
m
G(û) = dP
dΩ / P 4π . (57)
The power density is proportional to the squared modulus of the total field, so is proportional
to |AF| 2 . If the antenna elements themselves have non-isotropic gain, all equal
to G E (û), the power density dP/dΩ is proportional to G E (û)AF(û) 2 ,
G = 1 PI · G E |AF| 2 . (58)