CHAPTER 2. RESEARCH BACKGROUND 13
3.4 Antenna Arrays Analysis and Synthesis 57
3.4.2 Uniform circular array radiation pattern synthesis 62
62
63 ( ) s( ) j xu2 a
A x E u e du
1 2
1
( )
n j xu
a s
n n
E u e du
(3.28) That is, A(x) is a weighted sum of integrals of the form,sin( ) 2
( )
j xu
u n a
e du
u n
Let u! = u-n results in! !
2 2
!
!
sin
j n j xu
a u a
e e du
u
Since the imaginary part of the integrand is odd, this becomes
! !
2 !
!
sin cos 2
j n x a
u xu
e a du
u
! !
2 !
!
2 2
sin 1 sin 1
1 2 2
2
j n x a
x x
u u
e du
u
(3.29)A standard definite integral is
sinbzdz z
0
0 0
0 for b for b for b
Application of this integral to Equation (3.29) and hence to Equation (3.28) yields
1 2
1
( ) ( )
n j nx
a s
n n
A x E n e
1 1
(0) ( ) cos2
n
s s
n n
E E n nx for x
a
≪a2 (3.30)= 0 for |x|>a/2
The continuous aperture distribution given by Equation (3.30) is sampled to give the element excitation values for a discrete array. This last step is approximate, and the pattern function of the array is obviously different from Es(u). This approximation is acceptable provided that the number of elements in the array is much greater than 𝑛̅ and the sidelobe level is not extremely low.
64 Those equations will determine the aperture illumination coefficients for a linear array of N elements to produce a Taylor-type pattern function with n sidelobes on each side of the main beam at a level (L) dB. The procedure involves three steps. The first n -1 zeros of the pattern are determined. Then the appropriate pattern function samples are determined. Finally, the array element illumination coefficients are determined by a harmonic analysis of the pattern function samples.
A particular advantage of this synthesis is that the knowledge of all of the pattern function zeros allows the computation of the pattern function as a product rather than as a polynomial.
The product computation involves only one trigonometric function evaluation for each pattern function value. All other constants need to be evaluated only once for each array. The pattern function zeros are given by:
2 2
2 2
2 1
2 1 1
1 2
n
A n
z for n to n
N A n
(3.31)
2 n ,
for n n to M N
where
int 1 2 M N
and A is given by:
1 20
1cosh 10
L
A
6.02
27.29 L
(3.32) where L is the sidelobe level (positive) in dB. Equation (3.32) is an excellent approximation, especially for large L. The pattern function is given by
1
cos cos ( ) cos
2 1 cos
M
n
n n
z z
E z z
z
N even
1
cos cos 1 cos
M
n
n n
z z
z
N odd (3.33) The pattern samples to be used to find the array element illumination coefficients are given by:65
2 1 1.
m
a E m for m to n
N
The element excitation coefficients are given by
1 1
(2 1)
1 2 cos
n
p m
m
m p
e a
N
N even, p = 1 to M + 11 1
1 2 cos2
n m m
a mp
N
N odd, p = 0 to M (3.34) where p is an index or element number staring at the center and moving to either end of the array.3.5. Conclusions
Smart antenna systems consist of four assemblages: the physical antenna, radio unit, beamforming, and the Digital Signal Processor (DSP). In this chapter, circular pin-fed linearly patch antenna and waveguide-fed pyramidal horn as antenna elements have been considered.
A circular pin-fed linearly polarized patch antenna is a directional antenna adapted for determining and transmitting signals in a specified direction, especially for radio broadcast and wireless communication systems due to the unique property of its radiation.
Wave-guide pyramidal horn antenna is a microwave horn antenna that has a flickering metal waveguide conFigured to optimize radio waves in a beam. The waves then radiate out the horn end in a narrow beam. Wave-guide pyramidal horn antenna has been considered because its popularity at UHF (300 MHz – 3 GHz) and higher frequencies it is somewhat intuitive and relatively simple to manufacture. The design of smart antenna using waveguide- fed pyramidal horn antenna gives a better system performance of directional radiation beam pattern with a high gain and wide impedance bandwidth.
66
CHAPTER 4
Performance Analysis of Smart Antenna Arrays
Array of antenna consist of number of antenna elements. The antenna element has spacing between them, and the phase and excitation parameters of these array of antenna determines its characteristics which will subsequently determines the directivity factor, main lobe, and sidelobe levels. Due to changes in characteristic of an array antenna by controlling its parameters give rise to synthesis problem [135].
4.1. Introduction
In antenna design, pattern synthesis is one of the most important analysis that must be put into consideration [77]. Antenna arrays can be synthesized with a few number of antenna elements. There are a lot of benefits attach to the synthesis of antenna arrays. The benefits varies from weight reduction to simplification in feeding network [136].
Until now, we have a lot of approaches to synthesize antenna arrays so as to get the radiation pattern. For a linear array to be synthesized, Woodward–Lawson and Dolph–
Chebyshev techniques are normally applied for the synthesis. Each of these techniques creates their own special radiation pattern. The radiation pattern created by Dolph-Chebyshev is different from the one created by Woodward-Lawson technique. With Woodward-Lawson technique, it has a favourite radiation pattern in its sampling point position. Its disadvantage is that it cannot control its radiation ripples generated at a specified direction and the sidelobes level. Chebyshev technique is normally use to realize the narrowest main lobe for a specified sidelobe level. Taylor method current distribution technique more gradual in comparison to Chebyshev approach. Hence, it is extensively applied in aperture antennas and array antennas [135].
4.2. Analysis of the Proposed Model’s Weighting Methods and Optimization of Radiation Pattern
Antenna array distribution and their associated patterns are now designed on physical principles, based on placement of zeros of the array polynomial. Distribution discussed in this
67 section are Dolph-Chebyshev, phase-tapered weights, and those that allow side lobe envelope shaping. All distributions have constant phase [137].