Antennas
5.1 LINEAR AND RECTANGULAR RADIATORS
5.1.4 Low-sidelobe tapering functions
Dolph used Chebyshëv polynomials to create an antenna pattern that were used in the late 1940s to produce antennas with much lower sidelobe levels, mainly with cylindrical reflector antennas fed by slotted waveguides using the slots as discrete radiating elements. The illumination function was derived from the antenna pattern and the sidelobe level attainable depended on how accurately the radiating slots could be cut into the waveguide.
Taylor used the Dolph-Chebyshëv pattern for use with continuous radiators and examples of uniform, Dolph- Chebyshëv, and Taylor patterns are shown in Figure 5.15.
-40 dB -30 -20 -10
q´
-8 -6 -4 -2 0 2 4 6 8
sin Bq´
Bq´
Chebyshev with sidelobe level 30 dB
Taylor, 30 dB
n 2 n 2
Taylor sidelobes nearly equal
Taylor sidelobes " 1/q´ Taylor sidelobes " 1/q´
-3 -2 -1 0 1 2 3
-1.5 -1 -0.5 0.5 1 1.5
x
1
1 2 2
3
3 4 6
5 7
7-4
Sidelobes are mapped
into this region Main lobe is mapped into this region
&1n cosh(n arccosh|x|) x < &1 Tn(x) ' cos(n arccos x) &1 < x < 1
cosh(n arccosh x) x > 1
Figure 5.15 Comparison of antenna patterns with uniform, Chebyshëv, and Taylor distributions. [Source: Meikle, H. D., A New Twist to Fourier Transforms, Wiley-VCH, 2004.]
Figure 5.16 Plots of the first seven Chebyshëv polynomials.
(5.24) 5.1.4.1 Dolph-Chebyshëv tapering function
The Dolph-Chebyshëv pattern has equal sidelobes outside the main beam together with minimum main beam spreading for a given sidelobe level. This function is defined for a line of discrete radiators, so strictly this topic belongs in Section 5.4 [5, p. 714]. Chebyshëv polynomials are used for economizing approximations in the domain ±1, in which the error, or ripple, oscillates between ±1. The first seven polynomials are shown in Figure 5.16.
Chebyshëv polynomials of degree n are given by [5, p. 775; 6, p. 774]
G(u)) ' TN&1(x0 cos Bu)) where u) ' w 8 sin 2 ' cos (N & 1) arccos(x0 cos Bu))
Tn(x) ' cosh(n arccosh x) x > 1
Sidelobe ratio, SLR ' 10
SLL
20 in the above example 10, 31.62, 100, voltage ratio
x0 ' cosh arccosh SLR N&1
weight[K] ' N & 1 N & K K
3
&2S'0
(K & 2)! (N & K)! "S%1
S! (K & 2& S)! (S % 1)! (N & K & S & 1)!
where " ' tanh arccosh(SLR) N & 1
2
and weight[1] ' weight[N] ' 1
(5.25)
(5.26)
(5.27)
(5.28)
(5.29) Dolph [7] saw that if the center (cosine) region could be mapped onto the sidelobes and the (hyperbolic cosine) region beyond +1 onto the main lobe, then an illumination function for a pattern with constant sidelobes could be calculated.
An N element array has N - 1 zeroes in its antenna pattern. All curves pass through the point 1, 1. The Chebyshëv polynomial has n zeroes, so a polynomial of order n-1 is used for mapping.
The antenna pattern is of the type [5, p. 715]
The antenna pattern G(u) must be normalized by dividing it by the maximum value TN-1 (x ). The value u´ has a range0
±½ so that Bu´ radians varies between ±90 degrees. The value N is the number of elements in the antenna, and x is a0 scaling factor that defines the sidelobe levels. This is calculated from the Chebyshëv polynomial of degree n beyond +1.
T (x) is equal to the desired voltage sidelobe level, SLR. Commonly, the sidelobe level is given in decibels, SLL dB, forn example 20, 30, and 40 dB, so that
Solving for x , the value of x giving the sidelobe ratio, SLR0
The values of x for a seven element array in the example are 1.127, 1.248, and 1.416.0
Figure 5.17 shows the normalized voltage field antenna pattern for a seven element antenna. This has six zeroes.
The Chebyshëv polynomial with six zeroes is the T (x) polynomial. The antenna diagrams using the T (x) polynomials6 6 have been plotted for 20, 30, or 40 dB sidelobes on the same diagram. The customary decibel plot is shown in Figure 5.18.
The weighting coefficients are obtained from the inverse Fourier transform of (5.25). A number of methods for the calculation have been developed, one of which uses a large number of whole numbers. They are suitable for calculation by mathematical programs [8, p. 260]. For the Kth weight for an antenna of N elements,
These weights are all greater than 1 and must be divided by the center weight to give the normalized values which are less than unity.
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.5 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5
20 dB 30 dB 40 dB Chebyshev
sidelobe levels
Sin 7Bu 7Bu Sin 7Bu
7Bu
-50 -40 -30 -20 -10 0
-0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5
Sin 7Bu 7Bu
20 dB
30 dB
40 dB Chebyshev
sidelobe levels
Figure 5.17 Antenna pattern with scaled Chebyshëv polynomials for a seven element array.
Figure 5.18 Decibel plot of the antenna pattern with scaled Chebyshëv polynomials for a seven element array.
R ' 10
SLL
20 voltage ratio
At ' arccosh(R) B
G(u) ' coshB At2 & u2 for u < A main beam
or cos B u2 & At2 for u > A sidelobe
minimum value of ¯n $ 2At2 % 0.5
F ' ¯n
At2 % ( ¯n & 0.5)2
z(n) ' ±F At2% n&1 2
2
for 1 # n # ¯n
' ±n for ¯n # n # 4
F(u)) ' sin(Bu)) B u)
J
¯n&1n'1
1 & u)² z(n)² 1 & u)²
n²
g(x)) ' 1 % 2
3
¯n&1n'1
Fn cos(2 B n x)) At2 % (n & 1/2)2
(5.30)
(5.31)
(5.32)
(5.33)
(5.34)
(5.35)
(5.36)
(5.37) 5.1.4.2 Taylor tapering function
Taylor modified the Dolph-Chebyshëv illumination function for continuous illumination [5, p. 719]. As with the Dolph- Chebyshëv taper, first choose the level of the first sidelobe SLL dB. Then,
Calculate the value A :t
The antenna pattern for the center region is defined as [5, p. 719]
Equation (5.32) describes a central beam of amplitude A and sidelobes of amplitude 1. The zeroes of the sidelobes are at . Taylor chose to dilate the main and sidelobe widths and null positions inside the n2th sidelobe so that the amplitudes decay with a rate of 1/u. A value of n2 for the transition point is chosen such that
The dilation factor, F, is
which gives the zero locations for the pattern
Using the unit circle approach, the radiation pattern is given by
The illumination function is found from the radiation pattern. Typically,
Fn ' (n & 1)!2
(n & 1 % n)! (n & 1 & n)!
J
n&1m'1
1 & n2 zm2
F(u), R) ' 2 B R2
I
10
f(r) r) J0(u) r)) dr)
I
10
x<%1 J<(a x) dx ' J<%1(a) a
F(u), R) ' 2 B R2 J1(u)) u) u) ' 2BR
8 sin 2 ' BD 8sin 2
(5.38)
(5.39)
(5.40)
(5.41) where the coefficients F are given byn
A number of Taylor illumination functions and patterns are given in Appendix B. This is not the end of the design and [5, p. 739] lists a number of references for optimization procedures to improve the design.