CHAPTER 2. RESEARCH BACKGROUND 13
3.4 Antenna Arrays Analysis and Synthesis 57
4.2.2 Dolph-Chebyshev arrays method for the proposed model 68
Chebyshev array is one of the uniform space linear arrays that has correlation with the conventional antenna arrays with respect to directivity and half-power beam width in an innovative manner [138]. Chebyshev is used to obtain sidelobe level. For the analysis of Chebyshev arrays, assuming we have an array factor of f(θ), N antenna element, where θ is the angle of elevation. For p basis Chebyshev arrays, and f(θ) as the array factor of the nth basis Chebyshev array (1≤ n ≤ p), then:
1
( ) ( )
p n n
f f
(4.1) If f1(θ) = … = fp(θ), hence f( )
fop( )
, this represent the modified Chebyshev array.For series of fn(θ), we can write f(θ) as [138]:
1 , 10 0
1
( ) 2 cos cos exp( )
Nn
p N
k q
o n k q
k q
n
f f u d I ju I z
(4.2) where d is the inter-element spacing, λ is the wavelength, θo is the angle of elevation of the maximum radiation, Nn is the number of elements of the nth basis array, In,k is the excitation of the kth element of the nth basis array, and Iq is the excitation of the qth element of the generalized Chebyshev array. From Equation (4.2), it can be deduced that:
1
1 1 1
p p
n n
n n
N N N N p
(4.3) So, to design an N element generalized Chebyshev array, the basis Chebyshev arrays should be selected such that the sum of their number of element is:
1
1
p n n
N N p
(4.4) and their sidelobe levels should be such that:
1 p
n n
R R
(4.5) where R is the desired upper bound on the sidelobe level, and Rn is the sidelobe level ratio of the nth basis Chebyshev array.Dolph-Chebyshev distribution of series of excitation coefficients for an equi-spaced linear array antenna such that the array factor can be stated as a Chebyshev polynomial. The Dolph-Chebyshev pattern is conceptually simple, as it consists of a main pencil beam, plus side lobes of equal level [137, 138]. Dolph-Chebychev linear arrays have ideal radiation properties in that all side lobes in their radiation pattern are of equal magnitude, they provide the narrowest
69 first null beamwidth possible for a given sidelobe ratio [75, 139]. In addition, the connection between the directivity and sidelobe level for these arrays is optimal because for a given sidelobe level the beam width is the lowest, and, otherwise for a specified beam width the sidelobe level is the lowest. These acceptable radiation characteristics, conversely, put a constraint on the flexibility of putting nulls in the sense that once the sidelobe level or directivity is fixed, the nulls have directions dictated by the Dolph-Chebychev excitations [140]. The Dolph-Chebyshev pattern is given by [141]:
( ) 1 cos
N o 2
F u T x u (4.6) The excitation An of the nth element is obtained by writing the pattern with the array center as phase reference:
1
( ) exp 2 1
2
N n
F u
A j n N u (4.7) This is a finite Fourier series and the inverse gives the coefficients. The Dolph- Chebyshev polynomial of order m, is defined by Equation (4.8), where m is an integer. The array polynomial method is used to synthesize the equispaced linear array pattern with null steering [76, 141, 142]:
1 1
1
cos( cos ) 1 1( )
( ) cosh( cosh ) 1 ( )
1 ( ) ( 1) cosh cosh
m
m
m x x a
T x m x x b
x c
m x
(4.8)
It can be confirmed by means of simple trigonometry that Tm(x) is a polynomial of x of order m and the subsequent iteration holds:
Tm+1(x) = 2xTm(x)-Tm-1(x) (4.9) Here m is integer constant with range from 0 to ∞, let us now obtain Tchebyscheff polynomials for different values of m. The Tchebyscheff polynomials are summarized in Table 4.1 using Equation (4.8).
Now the polynomials with higher values of m (5-7) can be obtained by using recursive formula given by:
T x
m( ) 2 xT x
m( ) T
m1( ) x
(4.10) From (4.8) we see that the magnitude of a Chebyshev polynomial lies between zero and unity if x 1and exceeds unity if x 1. This property of the polynomial is utilized to construct the array factor.70 Table 4.1. Tchebyscheff Polynomials for the proposed model.
S/N m Tm(x)
0 0 1
1 1 x
2 2 2x2-1
3 3 4x3-3x
4 4 8x4-8x2+1
5 5 16x5-20x3+5x
6 6 32x6-48x4+18x2-1 7 7 64x7-112x5+56x3-7
Figure 4.2. Dolph-Chebyshev polynomials for m = 2, 3, and 4.
Figure 4.2 to 4.4 shows the graph of the first six Chebyshev polynomials for 0 m 6. Chebyshev polynomial can be used to obtain sidelobes level. The array of Chebyshev are non- unform amplitude, when fed with optimum source of amplitude distribution for a specified side lobe level, it will produce same level for all side lobes [143]. Given a side lobe level for Chebyshev polynomial method, it is certain to have the narrowest main-lobe; if the width of the main lobe is given, we can gain the lowest side-lobe on the same level, as it shown in Figure 4.5.
71 Figure 4.3. Dolph-Chebychev polynomial for m = 5.
Figure 4.4. Chebyshev polynomial of the seventh degree for m = 6.
72 Figure 4.5. Patterns of Dolph-Chebyshev arrays with eight elements.
If the sidelobes Sll (dB) is below the peak of the main beam, the value of the Chebyshev polynomial at the peak of the main beam must be equal to [42]:
1( ) 1020
sll
N mb
T x (4.11) Setting Equation (4.8) to (4.11), then we obtain of the main beam at
cosh
mb 1 x A
N
(4.12) where
1cosh 1 1020
sll
A
(4.13) From Equation (4.12) we observed that the main beam maps the Chebyshev polynomial, while the array factor zeros (nulls) map the zeros of the Chebyshev polynomial. The location of the zeros of the Chebyshev polynomial are at:
( 0.5)
cos 1
n
x n
N
(4.14) The zeros of the array factor mapping to the zeros of the Chebyshev polynomial can be obtained using:
73 cos
2
n
n mb
x x (4.15) The zeros of the array factor that match up with a sidelobe level (Sll) can be given as:
1
( 0.5)
cos 1
2 cos
cosh 1
n
n N
A N
(4.16)
For a specified number of elements and sidelobe level, we can simply get the null locations on the unit circle. Equation (4.16) can be used to design an 8-element array (d = 0.5λ) with -20-dB sidelobes. It can also be repeated for -30-sidelobe, and -40-sidelobes levels as seen in Figure 4.5. From Equation (4.16),
1 ( 0.5) 0 0 0
2 cos 73.2 , 120.5 ,180
cosh(0.1)
m
m
The factored polynomial in factored can be expressed as:
73.2 73.2 120.5 120.5 180
(
j)(
j)(
j)(
j)(
j)
AF z e z e
z e z e
z e
The factors multiplication gives
5 4 3 2
1.44 1.85 1.85 1.44 1
AFz z z z z
The coefficients of AF are the amplitude weights for the 8-element array pattern in Figure 4.6. Figure 4.5 are the normalized weights for an 8-element Chebyshev array as a function of sidelobe level. The end element does not have the smallest amplitude for sidelobe levels above -22 dB. At -22 dB and below, the weights monotonically decrease from the centre to the edge.