[1]
For the generation of narrow beams and other desired beam shapes, the antenna arrays are widely used. Mostly in high resolution radars and for point to point communication, narrow beams are used. There will be one major lobe followed by a number of minor lobes for any pattern produced by array antenna. For the purpose of covering wide angular regions, flat or sector beams are used. For ground mapping and also for airport surveillance, cosecant beams are used. The width of main beam becomes very small and is mostly a useful point to point communication, when the selected antenna array is sufficiently large enough. Generally, stair- step patterns are employed to identify when there is more than one target moving with both different altitudes and angular regions. However, it is required to design the sufficiently large enough arrays to produce optimized patterns. In this paper, Fourier transform technique is extended to design an aperture distribution for the generation of stair- step patterns and also an amplitude control technique for the generation sector beams to produce the ramp patterns.
Synthesis methods are divided based on the antenna or type of pattern. Some methods can be applied to a number of antenna and pattern types. Antenna arrays can be synthesized for low side lobes, narrow main beam, to produce nulls in the desired direction and generate desired shaped beams. Synthesis techniques for the generation of shaped beams are totally different from those used for the generation of radiation patterns with low side lobes, narrow main beam patterns and nulls in the desired directions [1]. Schelkunoff introduced synthesis of antenna arrays to give nulls or zero radiation in the prescribed directions. Both continuous line sources and discrete antenna arrays can be synthesized using antenna synthesis methods. Discrete linear arrays with reasonable number of elements can be synthesized practically, but those with large number of elements, occupy more space, requires a complex feed network, costly and not practical. In such cases antennas with continuous line distributions are suitable, but these antennas produce large number of side lobes and difficult to scan at fast rate. Any continuous source can be approximated to a closely spaced discrete array, hence synthesis of continuous source can be extended to discrete linear array of antenna elements. A continuous source distribution is sampled at regular specified points and used as excitation of discrete elements in the array at that position. A linear discrete antenna array can be excited for uniform amplitude and non-uniform amplitudes. Uniform excitation of a linear antenna array radiates broad side with narrow main beam and many side lobes. Standard non-uniform excitations are Binomial and Chebyshev methods. The binomial excitation generates radiation pattern with wide beam width and no side lobes, whereas Chebyshev excitation radiates a main beam of moderate width and all the side lobes of same prescribed level. Non-uniform excitation methods are used to synthesize both continuous and discrete antenna arrays [2-6] .
Synthesis is one of determining the required amplitude distributions to generate the desired shaped radiation pattern over a specified angular region or between specified points. Sector and cosecant shaped radiation patterns have been synthesized for various array lengths and over different angular regions. This technique is suitable to implement in arrays of extremely large dimensions [1-3]. Synthesis of unequally-spaced arrays to generate the desired radiation pattern was done using Poisson's sum formula. The synthesis also extended for equally spaced arrays to obtain reduced side lobe levels with applications. Antenna arrays are synthesized for minimum number of side lobes with low amplitude levels by changing the position of the elements in the array without changing the excitations in non-linear minimax problem. Radiation patterns with one main beam from unequally spaced arrays with even and odd number of elements have been synthesized [4-14].
In the present conventional radar system, antenna is one of the most important radiating elements, as the radar range varies with respect to the square root of the antenna gain wavelength product. In order to double the radar range, a fourfold increase in antenna is done, and for the same type of effect, a sixteen-fold increase in transmitter power or receiver sensitivity is done. In radar system, planning is one of the important conservations in free space radiation pattern of the antenna. Many times, certain beam shaping of the pattern is desirable. Gain and beam angle requirements are weighed against other system only after setting the Minor-lobe levels on a compromise bases. In general, when radar is used in electronic counter measure applications in military, the radar antenna never had sufficiently high gain or low enough side lobe levels. As the access to the radar by the jammer is not restricted to the main antenna beam, but can also be detected through the side lobes because this signal to jamming ration is directly proportional to radar antenna gain. Hence in anti-jamming applications, minimum side lobe level pattern has to be synthesized. Maximum range high gain over side lobe suppression is required by a Search radar. High gain and low-side lobes are very important in many applications.
For the generation of shaped beams, Antenna size, weight and conformability of the elements in the array plays an important role in the design of antenna arrays [7- 8].
In amplitude control method of synthesis of discrete linear antenna arrays, excitation phase is not varied. The commonly used non-uniform excitation amplitude distributions such as uniform, sinusoidal, cosinusoidal are not suitable to produce shaped beams. Hence, Fourier transform method is used to generate desired shaped beams such as sector, stair-step, ramp, trapezoidal, and cosecant beams over specified regions from continuous line sources. But the continuous line source is not in practice.
Identical discrete antenna elements placed at equal distance are synthesized to generate complex beam shapes such as sector, positive ramp, negative ramp, trapezoidal, and cosecant radiation patterns using amplitude control method. A suitable synthesis method of amplitude control technique is Fourier series for discrete antenna arrays [9-10].
Array factor of uniformly spaced linear array of identical elements is expressed as Fourier series summation of finite number of terms proportional to the finite number of elements in the discrete array. Excitation amplitude of each element in the array is equal to the corresponding coefficients in the Fourier series summation. Half wave length separation between the elements of the array is preferred to reduce mutual coupling between the elements of the array [5]. An array may contain odd or even number of elements. The array factor or angular variation of far field of a linear uniformly placed array of 'N' antennas can be expressed as,
In (t) = complex excitation coefficients of the nth element.
tn = location of nth element of the array.
Substituting tn = L xn and let 2L – be the length of the array.
xn be the position of the nth element of the array.
The above expression for the far field becomes
The complex amplitude distribution can be expressed as,
Where b(xn) is the amplitude of the current in the nth element.
Substituting, 
and 
in the above equation
Far field radiation for continuous source is determined using radiation integrals. But for discrete array considered here, fields from individual elements of the array are summed by including excitation phase levels. A uniform linear antenna array may consist of even or odd number of elements placed at equal intervals. In an array of odd number of elements, the position of each element is,
N= 2M+1 is the total number of elements of the array, M is an integer.
Antenna arrays consisting of even number of elements, N = 2M and the position of each element is expressed as,
But one expression that can be used for computing position of each element for arrays consisting of even or odd number of elements is,
When using amplitude control to the array factor, the far field is established by substituting the phase variation as zero.
Excitation amplitude coefficients of the individual elements in the array are determined by the formula,
The array extends from -1 to 1 and 'xn ' is the position of the elements in the array. E(u) is the desired radiation pattern. Here the element excitation amplitude coefficients are equal to the coefficients of the Fourier series and are determined from the desired radiation pattern [11-13].
Generation of sector, positive ramp, negative ramp, trapezoidal, and cosecant radiation patterns are considered. Expressions for the desired radiation patterns are as follows [14],
Array antennas are synthesized to generate the shaped radiation patterns using amplitude control method. Fourier series method of amplitude control techniques are used to generate the complex beam shapes and the amplitude coefficients necessary to generate the respective shaped beams using Fourier series.
Figure 1. Normalized Sector Pattern 2L/λ =25, N=50, uo =0.8
Figure 2. Normalized Sector Pattern 2L/λ =50, N=100, uo =0.8
Figure 3. Normalized Sector Pattern 2L/λ =100, N=200, uo =0.8
Figure 4. Normalized Sector Pattern 2L/λ =200, N=400, uo =0.8
Figure 5. Normalized Step Radiation Pattern 2L/λ =25
Figure 6. Normalized Step Radiation Pattern 2L/λ =50
Figure 7. Normalized Step Radiation Pattern 2L/λ =100
Figure 8. Normalized Step Radiation Pattern 2L/λ =200
Figure 9. Normalized Ramp Radiation Pattern 2L/λ = 50
Figure 10. Normalized Ramp Radiation Pattern 2L/λ =100
Figure 11. Normalized Ramp Radiation Pattern 2L/λ =200
Figure 12. Normalized Ramp Radiation Pattern 2L/λ =200
Figure 13. Trapezoidal Radiation Pattern 2L/λ =25, N=50
Figure 14. Trapezoidal Radiation Pattern 2L/λ =50, N=100
Figure 15. Trapezoidal Radiation Pattern 2L/λ =100, N=200
Figure 16. Trapezoidal Radiation Pattern 2L/λ =200, N=400
Figure 17. Normalized Cosecant Pattern 2L/λ =25, N=50
Figure 18. Normalized Cosecant Pattern 2L/λ 50, N=100
Figure 19. Normalized Cosecant Pattern 2L/λ 100, N=200
Figure 20. Normalized Cosecant Pattern 2L/λ 200, N=400
Figures 1 to 20 show the synthesized shaped beams using Fourier Series method.
The shaped radiation patterns necessary for various applications are generated by synthesizing linear antenna arrays using amplitude control methods. The amplitude control method of the synthesis of discrete linear array is a non-uniform amplitude excitation method. Fourier series and Woodward-Lawson methods of synthesis of linear antenna array are used to generate desired beam shapes. The generated patterns are useful for non-scan applications. Excitation coefficients required to generate each shaped beam are tabulated. Excitation coefficients are different for different angular regions or bounds for the same shaped beam. Side lobes and ripples of the main beam are more in small arrays with less number of elements which is useful for low precision and low cost applications. The results show that the synthesized beam is very close to the desired pattern for an array of large number of elements. In amplitude control method, the excitation phase is zero for all the elements of the array.
The maximum radiation is directly perpendicular to the axis of the array for all the beams. There are no side lobes immediately surrounding the main beam as in conventional methods instead the side lobes are along the axis of the array. Another advantage of this synthesis is that there are no grating lobes as well. The boundaries of the covered and uncovered regions are clearly separated. Ripples of the main beam are less in arrays with large number of elements. Coverage or radiated area can be increased or reduced by changing the excitation of the elements in the array. A sector or step or flat-top pattern is symmetrical around the center of the array covering equal areas on either side. The stair-step pattern is also symmetrical around the center of the array, where its area reduces in vertical direction in gradual steps. Ramp, Trapezoidal, and Cosecant beams are nonsymmetrical about the center of the array. Each beam exhibits a particular coverage on one side of the beam.