Antennas
5.3 MONOPULSE RADAR ANTENNAS
5.3.1 Tapering functions for monopulse antennas with low sidelobes
Low sidelobes for monopulse antennas give the same advantages as with simple antennas, namely reduction of clutter in the sidelobes and jamming entering through the sidelobes. Figure 5.26 shows a number of monopulse antenna patterns starting with the odd uniform distribution, and first attempts were made to achieve low sidelobes using the Taylor derivative pattern. Figure 5.26 shows what happens when the aim is 30 dB sidelobes. Bayliss modified the positions of the zeroes in the Taylor derivative pattern to give a difference pattern with really low sidelobes and the 30 dB aim is met. Later Zolotarëv polynomials were suggested as an analogue for antennas that have a Chebyshëv pattern.
5.3.1.1 Taylor derivative distribution
Many of the derivatives of sum patterns may be used to form difference patterns and the Taylor derivative [4, p. 300]
looked interesting but does not live up to its promise and the aim of 30 dB sidelobes is not met as shown in Figures 5.27, 5.28, and 5.29.
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 1
2 4 6 8 10
u
q´ = L sin 2 8
Sidelobe level 30 dB 0.6
0.5
-40 -20 -10 0
2 4 6 8 10
q´ = L sin
2
8
Sidelobe levels
30 dB -30
Figure 5.27 Taylor derivative characteristic.
Figure 5.28 Taylor derivative antenna characteristic in decibels.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1 0.2 0.3 0.4 0.5
p´= Distance from the center L
Sidelobe level dB 30 dB
Z2n%1(x) ' cosh (n%1
2) ln /(M%<, k) /(M&<, k)
x ' sn(M, k) cn(<, k) sn2(M, k) & sn2(<, k)
M ' & K(k) 2n % 1
1 & k2
Figure 5.29 The illumination function for a Taylor derivative characteristic.
(5.53) Bayliss [11] modified the Taylor derivative to give reduced near sidelobes of equal height and decaying far sidelobes.
5.3.1.2 Zolotarëv distribution
Yegor Zolotarëv was a pupil of Chebyshëv at the university of St. Petersburg and later took over from him. He developed polynomials that give a monopulse antenna pattern with low sidelobes for discrete elements described in [12, 13]. The function, like the Chebyshëv for discrete numbers of elements, has a single peak and sidelobes that alternate between ±1 and is defined parametrically and is of the form
where the number of elements is 2n + 1;
/(<, k) is the Jacobi Eta function with modulus k [13, p. 577];
;
sn(M, k), cn(M, k), and dn(M, k) are the Jacobi elliptic functions [6, p. 570; 14, p. 942];
K(k) is the complete elliptic integral of the first kind [6, p. 590; 14, p. 537];
k is the complementary modulus 1 .
Note: In the literature k´, K´, and q´ are used for complementary functions that has lead to misprints, so that k , K , and1 1 q are used here for clarity. The full procedure for calculating a Zolotarëv distribution, from [12], is given in the appendix1 at the end of this chapter.
The sidelobe ratio is determined by the value at x = x and depends on n, and Figure 5.30 gives the values of k to2 achieve this.
Plot for k against SLL dB for 7, 9, 11, 13, 15, 17 and 21 elements
15 20 25 30 35 40 45 50
SLL dB
0.9999 0.99992 0.99994 0.99996 0.99998
k
7 9 11 13 21
Number of elements
0 -1
10 15 20 25 30
0.2 0.4 0.6 0.8
1
x1 x2 x3
5
+1
Figure 5.30 Values of k for sidelobe levels and numbers of elements.
Figure 5.31 Zolotarëv characteristic with 30 dB sidelobes.
Figures 5.31and 5.32 show the characteristics of an array of 20 elements (the center element is not driven) as the Zolatarëv polynomial and the plot of one side of the antenna difference pattern in decibels.
-20 -10 0 10 20 30
0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1 0.2 0.3 0.4 0.5
p´= Distance from the center L
Sidelobe level dB 30 dB
Figure 5.32 Zolotarëv antenna characteristic in decibels with 30 dB sidelobes.
Figure 5.33 Zolotarëv antenna illumination function for the example.
Figure 5.33 shows the excitation derived numerically from the discrete Fourier transform. The left-hand 10 elements are driven with the same amplitude but in antiphase.
5.3.1.3 Bayliss distribution
Bayliss [11] found the way to reduce the first four sidelobes of the Taylor derivative pattern by moving the first nulls.
No closed function is available for the positions of the nulls and an iterative procedure was used to find them. The modified or corrected table [15] for finding the positions of the zeroes is reproduced in Appendix B, Tapering functions.
F(u) ' u cosBu
J
n&1n'1
1 & u2 F2 zn2
J
n&1n'0
1 & u2
(n % ½)2
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-10 -8 -6 -4 -2 2 4 6 8 10
u
-50 -40 -30 -20 -10 0
2 4 6 8 10
u
(a) 30 dB Bayliss difference pattern (b) Bayliss difference patterns with 20 dB to 45 dB sidelobes 20 dB 30 dB 40 dB 25 dB 35 dB 45 dB
g(p) '
3
n&1
n'0
Bn sin Bp n % ½
Bm ' &(&1)m (m % ½)2
J
n&1
n'1
1 & (m % ½)2 F2zn2
J
n&1
n'0, n†m
1 & (m % ½)2 (n % ½)2
n ± A2 % n2 F ' n % ½
zn
(5.54)
Figure 5.34 A linear Bayliss pattern with 30 dB sidelobes and a family of patterns in decibels.
(5.55)
(5.56) Similarly to the Taylor distribution the pattern is defined and the distribution is derived from the pattern. First for a given sidelobe level, SLL db, the factor A is found from the polynomial approximation given in Appendix 5A.
The pattern is given by
where z is calculated from the polynomials in the Appendix for n less than or n for n outside the center region and the dilation factor F is given by .
A typical pattern with 30 dB sidelobes is illustrated in Figure 5.34(a) or in decibels in Figure 5.34(b).
The excitation with p as the distance from the center (values from 0 to 1/2) is given by
where
Examples are shown in Figure 5.35.
An alternative expression for the difference pattern using B is given byn
45 dB 40 dB 35 dB 30 dB 25 dB 20 dB
0 0.1 0.2 0.3 0.4
0.1 0.2 0.3 0.4 0.5
x
F(u) ' u cosBu
3
n&1n'0
(&1)n Bn n % 1
2
2
& u2
F(u) ' u J´1(Bu)
J
n&1
n'1
1 & u2 F2 z2n
J
n&1
n'0
1 & u2 µ2n
F(u) ' u J´1(Bu)
3
n&1n'0
Bn J1(Bµn) µ2n & u2 µn
A2 % n2
Figure 5.35 Examples of the illumination function for Bayliss antenna patterns.
(5.57)
(5.58)
(5.59) Bayliss’ article [11] started with his difference pattern for a circular aperture in the plane of the cross-section of the difference pattern. The calculation for A and the zeroes using the data from Table 5A.1 is the same and the pattern is given by (the cosN term for the other dimension has been omitted)
where J ´(x) is the differential of the Bessel function of the first kind, order one J (x); 1 1 µ is the nth zero of J (B x); n 1´
the dilation factor, F, is given by .
Typical Bayliss circular patterns are shown in Figure 5.36.
Alternately the pattern may be calculated from the expression
where
Bm ' 2µ2m
J
n&1n'1
1 & µ2m F2 zn2
J1(Bµm)
J
n&1n'0, n†m
1 & µ2m µ2n
m ' 0, 1, 2, .. n & 1
-0.4 -0.2 0 0.2 0.4
-10 -8 -6 -4 -2 2 4 6 8 10
u
-50 -40 -30 -20 -10 0
2 4 6 8 10
u
20 dB 25 dB 30 dB 35 dB 40 dB 45 dB (a) Circular Bayliss patterns with sidelobe levels from 20 dB to 45 dB (b) Circular Bayliss patterns in dB w ith sidelobe levels from 20 dB to 45 dB
g(p) '
3
n&1n'0
J1(µn p)
45 dB 40 dB 35 dB 30 dB 25 dB 20 dB
0 0.1 0.2 0.3 0.4
0.1 0.2 0.3 0.4 0.5
x
(5.60)
Figure 5.36 A family of circular Bayliss antenna patterns in the plane of the difference pattern.
(5.61)
Figure 5.37 Aperture distribution for a number of linear Bayliss patterns.
The aperture distribution is given by
Typical aperture distributions for circular apertures with Bayliss patterns are shown in Figure 5.37.
Broadside wavefront field ' K
3
N&1
n'0
An exp j Rn 2B 8
d
Distant wavefront NORMAL TO THE LINE OF RADIATORS OR BROADSIDE
R
d
2
x = (N - 1)d sin 2 m x = (N - 1)d 2B/8 sin 2 radians
x
Distant wavefront OTHER ANGLES, here 2222
Rn
2B 8
(5.62)
Figure 5.38 The contribution of individual antenna elements to a distant wavefront.
Figure 5.39 The contribution of individual antenna elements to a distant wavefront.