To simplify the discussion, we shall discuss a onedimensional line source or length Lw. Earlier in Eqs. (6) and (7) a Fourier
CLS
‘Program to compute rectangular aperture radiation patterns REDIM A(101, 101), pattern(2 0 0)
PI = 3.14159265358#
‘Results stored in file Circ. dat OPEN »Circ. data» FOR OUTPUT AS #1
‘———— input data
frequency = 10: Dw = 100
wavelength = 30 / frequency: Dwave = Dw / wavelength ‘ Get reference power for uniformly illuminated circular array: reference = 0 FOR I = 1 TO 101 FOR J = 1 TO 101
X = Dw * (51 — I) / 100: Y = Dw * (51 — J) / 100 radius = SQR(X л 2 + Y л 2)
IF radius > (Dw / 2) THEN
A(I, J) = 0
ELSE
A(I, J) = 1 END IF
reference = reference + A(I, J) л 2
NEXT J NEXT I
‘Load aperture distribution: all dimensions in cm
FOR I = 1 TO 101
FOR J = 1 TO 101
X = Dw * (51 — I) / 100: Y = Dw * (51 — J) / 100
radius = SQR(X л 2 + Y л 2)
IF radius > (Dw / 2) THEN
A(I, J) = 0
ELSE
A(I, J) = COS(PI * radius / Dw)
‘ Note: A(I, J) is aperture distribution in the X and Y plane
END IF NEXT J NEXT I
‘Normalize power over aperture to that in a uniform distribution power = 0 FOR I = 1 TO 101
FOR J = 1 TO 101
power = power + A(I, J) л 2 NEXT J NEXT I
power = power / reference: power = SQR(power)
Cp = 1 / power
‘ Cp is the desired normalization constant phi = 0
‘Note: Phi=0 results in principal Xplane pattern ‘ Phi=45 results in intercardinal plane pattern
‘ Phi=90 results in principal Yplane pattern
‘ For a symmetrical circular distribution, all these are the same
K = 0: G0DB = 0
FOR theta = 0 TO 10 STEP .1
PRINT »Computing for theta=»; theta
RE = 0: IM = 0
K = K + 1
‘Integration over aperture FOR I = 1 TO 101 FOR J = 1 TO 101
X = Cw * (51 — I) / 100: Y = Dw * (51 — J) / 100
psi = (2 * PI * X / wavelength) * SIN(theta * PI / 180) * COS(phi * PI / 180)
psi = psi + (2 * PI * Y / wavelength) * SIN(theta * PI / 180) * SIN(phi * PI / 180)
RE = RE + A(I, J) * COS(psi)
IM = IM + A(I, J) * SIN(psi)
NEXT J
NEXT I
TMM = Cp * SQR(RE л 2 + IM л 2) / reference
IF TMM = 0 THEN TMM = 10 л 6
IF theta = 0 THEN
G0DB = 20 * LOG(TMM) / LOG(10): GV = TMM
END IF
pattern(K) = 20 * LOG(TMM) / LOG(10)
NEXT theta K = 0
FOR theta = 0 TO 10 STEP.1 K = K + 1
PRINT #1, theta, pattern(K)
PRINT »Angl=»; theta; »E = »; pattern(K)
NEXT theta Print »»
‘; GV; » G0(power) =»; GV л 2 
PRINT »Peak Gain G0 (dB)=»; G0DB; » G0(voltage) =’
CLOSE #1
END
Figure 14 (Continued)
Lw N E (в) = j V’ Cm. ej(k sin ° +фт )xdx w 
(34) 
E (в) = £ C, 
(35) 
(33) 
m=1 
transform pair was defined for a line source relating the aperture distribution and the farfield radiation pattern, and vice versa. In the synthesis process, we wish to determine the aperture distribution for a desired radiation pattern. To do this we first express the illumination function as a sum of N uniform distributions: 
E (x) = < 
m=1 
/Lw L E w m=1 yielding for a line source of length Lw I 7 ■ Lw 31П Sin в+фгп) — 
Lw Sin в +фт ) — 
N sin I r. m 
The Fourier transform can then be written as 
Thus each coefficient Cm is responsible for a (sin x)/x type of
Table 2. Radiation Pattern Characteristics Produced by Various Circular Aperture Distributions

Figure 15. Form of beam and aperture efficiencies for an aperture as a function of taper. 
beam, and there are N beams. The coefficients may be obtained manually by estimating the number of independent beams and their relative magnitudes and positions to approximate the desired radiation pattern. Alternatively they may be obtained mathematically via a Fourierseries representation. Also, the results may be extended by the reader to a twodimensional aperture.
The preceding equations form the basis for Woodward’s aperture synthesis technique (6,29) that enables the aperture illumination required to produce a given beam shape to be approximated. A new array antenna synthesis method, called the virtual array synthesis method, was recently published by Vaskelainen (23). In this method, the excitation values of a virtual array are synthesized using some known synthesis method. The geometry of the virtual array can be chosen so that there will be a suitable synthesis method for that geometry, and the synthesis of the virtual array can be done accurately enough. In the synthesis method, the excitation values of the virtual array are transformed into the excitation values of the actual array geometry. Matrix operations are simple and large arrays can be easily synthesized. Further references on recent synthesis techniques appear in Refs. 24 to 28.