Antennas
5.5 CREATING SHAPED BEAMS
5.5.2 The Woodward-Lawson method
Another method of finding the illumination pattern for a chosen beam shape is the Woodward-Lawson sample value method [2, p. 526; 5, p. 419]. This method is illustrated here in the example for a cosecant squared pattern.
The desired pattern is defined first in (elevation) angle as (see Figure 5.51):
-10 -8 -6 -4 -2 0
5 10 15 20 25 30
Elevation angle, degrees
Sample line source radiation pattern, Fk(2) ' ak
sin Bw
8 sin(2 & k*) Bw
8 sin(2 & k*)
0.6132
I
&0.6132
sin Bq
Bq dq ' 1
&
I
0.6132&4
sin Bq
Bq dq '
I
40.6132
sin Bq
Bq dq ' 0 Figure 5.51 The specified cosecant squared pattern.
(5.85)
(5.86)
• Elevation angle 2.87 degrees (arcsin 0.05): Gain 0 dB;
• Elevation angle 10 degrees: Gain 0 dB;
• Elevation angle 30 degrees : Gain -9.1 dB; following a cosecant squared characteristic from 10 degrees.
Equally spaced samples in sine space are chosen. The spacing is normally chosen to be around the standard beamwidth, 8/w, to avoid possible grating lobes at wide angles.
The antenna pattern is built up from these samples as the sum of displaced patterns of linear apertures of the same length. The radiation pattern of a uniform line source is
where ak is the scaling factor taken from Figure 5.51 at a point 2 = k*;
k is an integer between kmax and kmin; kmax - kmin = K the number of samples;
* is the spacing between the sampled patterns.
One such sample pattern is shown in Figure 5.52. The sin x/x function has two interesting properties:
where q = (2 - *)/8.
-0.2 -0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-10 -8 -6 -4 -2 2 4 6 8 10
2 w/8 2 w/8 = 0.6132 Inside this region
the integral is unity
Outside this region the integral is zero
y = sin B 2 w/8 B 2 w/8
Aperture illumination function, g(x) '
E
kmax
k'kmin
gk(x) '
E
kmax
k'kmin
E
%N
2
n'&N 2
Fk(2n) exp j2Bn8 2 sin 2n
Amplitude ' /0 Aperture function /0
Phase ' arctan Imaginary part of aperture function Real part of aperture function Figure 5.52 The sinc function showing the regions inside and outside
±2w/8 = 0.6132.
(5.87)
(5.88) Inside the bounds ±2w/8 = 0.6132 the integral is unity, and outside these bounds there is no contribution to the inverse Fourier transform. The radiation patterns of these samples are shown in Figure 5.53, where it is easy to see that the contributions away from the peaks tend to zero. The synthesized antenna pattern is the sum of these elemental patterns and is shown in the traditional angle versus decibel form in Figure 5.54. The polar form is shown in Figure 5.55.
The aperture function, illumination function, or current distribution is the inverse Fourier transform of the antenna pattern [2, p. 534; 8, p. 24]. This is calculated from the inverse Fourier transforms of each sample, one of which is shown in Figure 5.57. The inverse transform of a displaced sin x/x function has two component rectangular functions equally spaced from the origin, and the inverse transform of all the samples is the envelope. This is shown as the illumination pattern in Figure 5.57. Sin x/x functions extend to infinity, so that the illumination function taken for a radiation pattern over a finite angle is
where 2n is one of N equally spaced pattern samples within ±0.6132 8/w of the angle k*. Note that for an angle sample spacing of a standard beamwidth (8/w), the aperture illumination function samples are spaced 8/2 apart.
The antenna pattern is not symmetrical. In the Woodward-Lawson synthesis, the amplitude function is an even function. The aiming and unbalance are provided by the phase characteristic and are given by
The aperture functions are shown in complex form in Figure 5.57, amplitude in Figure 5.59, and phase in Figure 5.60.
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Sin 2
-40 -30 -20 -10
0 10 20 30 40
Elevation angle, degrees Figure 5.53 The construction of the synthesized pattern using sinc curves spaced at 8/2
along a length of 11.58.
Figure 5.54 The desired pattern and its approximation in decibels using sinc curves spaced at 8/2 along a length of 11.58.
-15
-10
-20 0
-20
-25
-30
-35 Desired pattern
Synthesized pattern
Antenna characteristic, decibels -40
-30 Sidelobes above the beam collect
echoes from nearby aircraft flying over the radar and rain
clutter
Sidelobes below the beam collect clutter
Desired pattern
0 35
90 degrees
5 10 15 20 30
Elevation degrees
25
Length along antenna
Figure 5.55 The polar diagram of the synthesized pattern for samples spaced at 8/2 along a length of 11.58.
Figure 5.56 Inverse Fourier transforms of the individual sinc functions.
The illumination function is the sum of the Fourier transforms of the individual sinc functions in Figure 5.53. The Fourier transform of a sinc function is a helix of limited length with a phase depending on the position of the sinc function. The cluster of sinc functions before summing is shown in Figure 5.56, their vector sum in Figure 5.57, the modulus in Figure 5.58, and the phase angle along the aperture in Figure 5.59.
0 0.5
1
-0.5
0.5
1 -0.5
0.5
One sample of the radiation pattern
Radiation pattern
axis
Sample component in the positive side of the illumination function Illumination
pattern
Phase
Illumination pattern axis Sample component in
the negative side of the illumination function
-20 -15 -10 -5 0
-0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5
Length along the aperture from -0.5 to 0.5
Figure 5.57 The complex aperture function for the cosecant squared antenna of length 11.58.
Figure 5.58 The amplitude of the aperture function for the cosecant squared antenna of length 11.58.
-200 -100
0 100 200
-0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5
Distance along the aperture from -0.5 to 0.5
Figure 5.59 The phase angle of the aperture function for the cosecant squared antenna of length 11.58.
There are a number of points to note:
• The samples must be independent or orthogonal [5, p. 739].
• The pattern is the same as the desired pattern at the K sample points but is not controlled between them. If there are problems caused, more sample points must be used. The distance between the sampling points is normally one half a wavelength to avoid grating lobes. Thus more sampling points can be achieved only by increasing the size of the antenna [8, p. 24].
• Aperture function is symmetrical in amplitude and conjugate in phase [8, p. 24].
• The phase is tilted in the direction of the bulk of the beam [5, p. 739].
• This synthesis applies to linear antennas and planar arrays where the dimensions are separable [8, p. 24].
The pattern that has been formed will give a number of problems. The beam has ripple, which, in this case, will slightly alter the maximum ranges. Above and below the beam a number of sidelobes are present. With a surface radar, the sidelobe at 1.24 degrees 16.7 dB down and those below it will collect nearby clutter. The sidelobes above the beam, for example, at 30.3 degrees, 25.5 dB down, will give echoes from aircraft flying over the radar at short ranges.
After the basic design a number of numerical optimization techniques [6, p. 739] may be used to decrease the ripple and the sidelobes.
The aperture function can be approximated by specially developed horns. The phase characteristic may be controlled for a reflector antenna by changing the distance of the virtual directrix from the radiating horn. The shape of a parabolic reflector is often modified to simulate this shaped directrix.
Often, a reflector is fed by a number of horns (or similar feeds) stacked vertically to produce a cosecant-squared beam. In Figure 5.60, the transmit-receive devices are positioned between the individual horns and the power divider fed by the transmitter. The power divider gives the illumination function for a cosecant-squared transmitted beam. The returning echoes are switched to separate receivers which allow the measurement of elevation angle using monopulse evaluation in three-dimensional surveillance radars. It must be noted that the transmitting and receiving patterns of such antennas are not reciprocal.
T-R
T-R
T-R
T-R
Power divider to give cosecant squared transmitted beam
From the transmitter
Receiving beam 1 to receiver 1 Receiving beam 2 to receiver 2 Receiving beam 3 to receiver 3 Receiving beam 4 to receiver 4 Stack of
horns
Figure 5.60 The arrangement of horns and transmit-receive switches to give a single transmitted beam and a number of stacked receiving beams.