APERTURE SYNTHESIS

To simplify the discussion, we shall discuss a one-dimensional 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 X-plane pattern ‘ Phi=45 results in intercardinal plane pattern

‘ Phi=90 results in principal Y-plane 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 far-field 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 Distribution Comments Normalized Half-Power Beam Width (deg) HPbw/K Normalized Null-to-Null Beam Width (deg) NULLbw*/K Side-Lobe Level (dB): SLL dB Normalized Side-Lobe Angle (deg) SLpos/K Gain Relative to Uniform (dB) G0 dB Power Gain Factor Relative to Uniform G0 power Voltage Gain Factor Relative to Uniform G0 volts Uniform 59.33 140.00 -17.66 93.67 0.00 1.000 1.000 Cosine raised n = 1 74.67 194.67 -26.07 119.33 -1.42 0.721 0.849 to power n n = 2 88.00 250.00 -33.90 145.50 -2.89 0.514 0.717 n = 3 99.33 306.67 -41.34 173.00 -4.04 0.394 0.628 n = 4 110.00 362.67 -48.51 200.30 -4.96 0.319 0.564 n = 5 120.00 420.00 -55.50 228.17 -5.73 0.267 0.517 Cosine on a p = 0.0 74.67 194.67 -26.07 119.33 -1.42 1.000 1.000 pedestal p p = 0.1 70.67 183.33 -25.61 112.67 -0.98 0.799 0.894 p = 0.2 68.67 174.00 -24.44 107.83 -0.66 0.859 0.927 p = 0.3 66.00 166.00 -23.12 104.17 -0.43 0.905 0.951 p = 0.4 64.67 159.60 -21.91 101.33 -0.27 0.9388 0.9689 p = 0.5 63.33 154.67 -20.85 99.47 -0.17 0.963 0.981 p = 0.6 62.00 150.67 -19.95 97.83 -0.09 0.9789 0.9894 p = 0.7 61.33 147.33 -19.18 96.47 -0.05 0.9895 0.9947 p = 0.8 60.67 144.00 -18.52 95.27 -0.02 0.9958 0.9979 p = 0.9 60.00 142.00 -17.96 94.57 -0.00 0.9991 0.9995 p = 1.0 59.33 140.00 -17.66 93.67 0.00 1.000 1.000 Parabolic n = 0 59.33 140 -17.66 93.67 0.00 1.00 1.00 raised to n = 1 72.67 187.33 -24.64 116.33 -1.244 0.701 0.866 power n n = 2 84.67 232.67 -30.61 138.67 -2.547 0.556 0.746 n = 3 94.67 277.2 -35.96 160.17 -3.585 0.438 0.662 n = 4 104 320.33 -40.91 181.33 -4.432 0.36 0.6

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 Fourier-series 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.