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Keywords:

  • Bessel planar array;
  • directivity

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Results and Comments
  6. 4. Conclusion
  7. References
  8. Supporting Information

[1] In this paper, we introduce a new class of planar arrays that we call the Bessel planar arrays. A formula for the current distribution in the elements of these arrays is presented, which is related to Bessel functions. For the Bessel planar arrays, the maximal sidelobe level is controllable, the directivity is very high, and the half-power beam width is slightly larger compared to the optimal Chebyshev planar arrays. Methods to set the maximal sidelobe level and compute the directivity and the half-power beam width are described, and numerical examples are given to illustrate the features of the proposed arrays.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Results and Comments
  6. 4. Conclusion
  7. References
  8. Supporting Information

[2] In Digital Filter Design, different windowing techniques are used to reduce the oscillatory behavior in the causal FIR filter obtained by truncating the impulse response coefficients of the corresponding ideal filter. This behavior is referred to as the Gibbs phenomenon [Mitra, 1998]. The most commonly used windows are the Hanning, Hamming, Blackman, Chebyshev, and Kaiser windows. Among these, only Chebyshev and Kaiser windows provide control over the filter's ripple, since the sidelobe levels of their Fourier transform are adjustable. The coefficients of Chebyshev windows are related to Chebyshev polynomials, and those of Kaiser windows to modified Bessel functions of order zero.

[3] The linear antenna array with excitation equal to the coefficients of a Chebyshev window is the famous Chebyshev array. Chebyshev arrays have the important property of providing sidelobes of equal magnitude in their radiation patterns. They are optimum in the sense that for a specified sidelobe level they have the smallest beam width, and for a specified beam width they produce the lowest sidelobe level. However, they suffer from directivity saturation when the number of elements becomes large [Mailloux, 1994]. These properties are also shared by the Chebyshev planar arrays, which are described in Tseng and Cheng [1968].

[4] If the Kaiser window coefficients are taken as the excitations of a linear array's elements, the resulting array has a much higher directivity than the corresponding Chebyshev array at the cost of a slightly broader beam width, and a radiation pattern that does not have a constant sidelobe level, but a controllable maximal sidelobe level. In our paper, we exploit this observation and extend it to the case of planar arrays. The proposed arrays, the Bessel planar arrays, have radiation patterns close in shape to those of uniform planar arrays, but the maximal sidelobe levels are adjustable. In fact, uniform planar arrays are a special case of the proposed arrays, as will be clarified in later sections. Moreover, the Bessel planar arrays provide directivities much higher than those of Chebyshev planar arrays with the same numbers of elements, same size, and same sidelobe level, when the number of elements is high. The beam width is slightly broader than that of the optimal Chebyshev array.

[5] In Section 2, planar arrays are reviewed briefly, the Bessel planar arrays are formulated, equations of their current excitations are given, and methods for controlling the sidelobe level and estimating the half-power beam width of the proposed arrays are described. Examples are given in Section 3 to illustrate the features of the Bessel planar arrays and to compare them to the Chebyshev planar arrays. They will also be compared to some numerically synthesized high-performance planar arrays. Summarizing and concluding remarks are given in Section 4.

2. Problem Formulation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Results and Comments
  6. 4. Conclusion
  7. References
  8. Supporting Information

[6] In addition to placing elements along a line to form a linear array, individual radiators can be positioned along a rectangular grid to form a rectangular or planar array. Compared to linear arrays, planar arrays provide additional variables that can be used to control and shape the pattern of the array. They are more versatile and can provide more symmetrical patterns with lower sidelobes. In addition, they can be used to scan the main beam of the antenna toward any point in space. Applications include tracking radar, search radar, remote sensing, communications, and many others [Balanis, 1997].

[7] Consider a planar array of L × L identical elements placed in the xy plane. The z axis crosses at the center of the array, and the current amplitudes in the array elements are symmetrical about the x and y axes. The array factor, when L = 2N, is then given by [Tseng and Cheng, 1968]:

  • equation image

In (1), the Imn's denote the current magnitudes (I11 is the excitation of the 4 center elements),

  • equation image

and

  • equation image

Moreover, dx is the inter-element spacing in the x-direction, dy is the inter-element spacing in the y-direction, λ is the wavelength, θ is the elevation angle, ϕ the azimuth angle, and θ0 and ϕ0 correspond to the direction of maximum radiation. For odd values of L, L = 2N + 1, the array factor is given by

  • equation image

In (4), εm, εn are equal to 1 for m = n = 1, and equal to 2 for m, n ≠ 1.

[8] The directivity of planar arrays of isotropic elements has been derived in several places. In Kabalan et al. [2002], the formula for the directivity of an L × L planar array was given for even and odd values of L and any direction of maximum radiation. For L = 2N,

  • equation image

where

  • equation image
  • equation image

For L = 2N + 1, the directivity is given by

  • equation image

where S is as given in (6), but

  • equation image

In deriving equations (5) and (8), it was assumed that the radiation exists in both sides of the array. In this paper, we assume that the radiation occurs only into the half-space on one side of the array, as is the case in most planar array directivity formulas [Lee et al., 2000; Hansen, 1983]. As a result, the directivity values given in this paper are twice those obtained from equations (5) and (8).

2.1. Current Distribution in Bessel Planar Arrays

[9] The Kaiser window used in the design of FIR filters is defined as [Mitra, 1998]

  • equation image

where (M + 1) is the length of the window, α = M/2, I0(·) represents the zeroth-order modified Bessel function of the first kind, and β is the parameter that affects the sidelobe level of the Fourier transform of the window. From (10), it is clear that w[i] = w[Mi] for 0 ≤ iM.

[10] Adopting the N-point Kaiser window coefficients and using them as the current magnitudes of an N-element linear array with symmetric excitation, the equation for these currents will be

  • equation image

where 1 ≤ mM, a1 is the excitation of the array's center element(s), and aM is that of the two edge elements. Equation (11) is written such that aM = I0(0) = 1.

[11] An observation of the linear case enables us to define the excitation currents of an L × L Bessel planar array as follows

  • equation image

where 1 ≤ m, nM, β is the parameter that controls the sidelobe level, and I0(·) is as defined before.

[12] The current magnitudes in (12) are normalized to the currents of the edge elements. That is, IMn = ImM = IMM = 1. In addition to being symmetrical with respect to both the x and y axes, the excitation of a Bessel planar array is also symmetrical with respect to the line y = x, just like Chebyshev planar arrays. This is because Imn = Inm. Thus one needs to determine only N(N + 1)/2 excitation currents when L = 2N, and (N + 1)(N + 2)/2 excitation currents for L = 2N + 1.

[13] For β = 0, all currents Imn, as defined in (12), have a value of 1. Thus, uniform planar arrays are a special case of the Bessel planar arrays.

2.2. Sidelobe Level Control

[14] The array factor of the Bessel planar arrays is given in (1) and (4) for even and odd numbers of elements, respectively, where the excitation currents are as defined in (12). For θ0 = ϕ0 = 0, the highest sidelobe in the array pattern is in the ϕ = 0 plane, and due to the symmetry of the pattern, it also occurs in the ϕ = π/, ϕ = π, and ϕ = 3π/2 planes. The equation of the array factor for ϕ = 0 becomes

  • equation image

Using the properties of Chebyshev polynomials [Balanis, 1997], equation (13) can be written as

  • equation image

where z = cos[πdx/λsinθ], and Tp(·) denotes a Chebyshev polynomial of order p. Clearly, F(z) is a polynomial in z. The normalized version of F(z) is

  • equation image

The top points of the sidelobes in the ϕ = 0 planes correspond to real values of θ, and consequently z, at which the derivative of Fn(z) with respect to z vanishes. Since the highest sidelobe in a Bessel planar array is one that is directly next to the main lobe, the z value at which the maximal sidelobe level occurs is the maximum real root of dFn(z)/dz. Calling it zm, then

  • equation image

The corresponding θ value is

  • equation image

The dB value of the maximal sidelobe level (MSLL) is then

  • equation image

For every β value, a set of Imn values is obtained, and the corresponding MSLL is computed from (14), (15), (16), and (18). This makes it possible to plot the MSLL versus β, and from the obtained plots, to find the β value needed to set a desired maximal sidelobe level. Through experimental results, it was found out that this value is independent of dx and dy, and from θ0 and ϕ0, which were set to zero for simplification, and depends solely on L. A closed-form expression for β as a function of the MSLL is not feasible. In the Kaiser window design of FIR filters, only an empirical formula exists that relates β to the filter's ripple, and not even to the sidelobe levels of the Fourier transform of the window.

2.3. Half-Power Beam Width

[15] The half-power beam width is the elevation angle difference between the two directions, in the ϕ = ϕ0 plane, in which the radiation intensity is one-half the maximum value beam. For ϕ0 = 0, these two directions correspond to the two real roots of

  • equation image

Let zh1 and zh2 be these two roots, and θh1 and θh2 the two directions of half-power, such that θh2 > θh1. The half-power beam width (HPBW) is then

  • equation image

3. Results and Comments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Results and Comments
  6. 4. Conclusion
  7. References
  8. Supporting Information

[16] A planar array with 13 × 13 elements uniformly spaced in the x and y directions is first considered. We choose dx = dy = 0.5λ and θ0 = ϕ0 = 0. Figure 1 shows the curves of the MSLL of the corresponding Bessel planar arrays plotted against β for some values of L. The parameter β is varied from 0 to 6. For a desired MSLL of −20 dB, β should take the value 2.542. At this value, the excitation currents of the Bessel planar array are calculated and are given in Table 1. Note that, due to their symmetry with respect to the x and y axes, only 49 out of the 169 currents are given. Furthermore, the symmetry around the line y = x is evident in the table. Also evident is the fact that in a Bessel planar array all edge elements have the same excitation, which makes the calculation of the currents in a Bessel planar array easier and faster. In Table 1, the currents are normalized so that the array's edge elements have a current magnitude of 1. The directivity of this Bessel planar array is DB = 448.174 or 26.514 dB and its half-power beam width is HPBWB = 9.0150. Its 3-D array factor plot is given in Figure 2.

image

Figure 1. Maximal Sidelobe Level versus β of a 13 × 13-element Bessel planar array for 0 ≤ β ≤ 6.

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image

Figure 2. Three-dimensional antenna pattern of a Bessel planar array with 13 × 13 isotropic elements, MSLL = −20 dB, dx = dy = 0.5λ, and θ0 = ϕ0 = 0.

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Table 1. Excitation Currents of a Bessel Planar Array With 13 × 13 Elements, and MSLL = −20 dB
Imnn = 1n = 2n = 3n = 4n = 5n = 6n = 7
m = 13.39763.30613.04152.63192.12011.55801
m = 23.30613.21882.96592.57372.08241.54061
m = 33.04152.96592.74632.40411.97171.48931
m = 42.63192.57372.40412.13721.79521.40591
m = 52.12012.08241.97171.79521.56431.29361
m = 61.55801.54061.48931.40591.29361.15661
m = 71111111

[17] The corresponding 13 × 13-element Chebyshev planar array with a −20dB sidelobe level has a directivity DC = 23.734 dB and a half-power beam width HPBWC = 8.4350. Its current magnitudes are listed in Table 2. They are normalized to the currents of the corner elements. Its array factor is plotted in Figure 3. The directivity of the Bessel planar array is 89.7% larger than the Chebyshev one, but its half-power beam width is also larger by 6.88%. The improvement in directivity the Bessel planar array has over its Chebyshev counterpart is due to the shape of its sidelobes. They decrease in level as θ increases, and they are not present for all ϕ values, as is the case of Chebyshev planar arrays. This results in less radiation power being lost in the direction of the sidelobes for the Bessel planar array. One remark is that the variations in current magnitudes in the Bessel planar array are much less than those of the Chebyshev planar array (maximum to minimum current ratio is 3.3976 for the Bessel case compared to 924 for the Chebyshev case).

image

Figure 3. Three-dimensional antenna pattern of a Chebyshev planar array with 13 × 13 isotropic elements, SLL = −20 dB, dx = dy = 0.5λ, and θ0 = ϕ0 = 0.

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Table 2. Excitation Currents of a Chebyshev Planar Array With 13 × 13 Elements, and SLL = −20 dB
Imnn = 1n = 2n = 3n = 4n = 5n = 6n = 7
m = 1778.2174179.5994755.518263.6650719.7393285.5622924
m = 2179.5994750.1243190.3872565.4350269.404026.0315792
m = 3755.5182190.3872479.0098536.0882101.2755524.0180495
m = 463.6650565.4350536.088265.4408321.0802609.0068220
m = 5719.7393269.4040101.2755321.0802606.5583340.668266
m = 6285.562226.0315524.0180609.0068340.668298.866812
m = 792479249522066121

[18] For a β value of 0, the MSLL of the Bessel planar array is −13.087 dB, which is the MSLL of the 13 × 13 uniform planar array. Since I0(·) is an even function, higher MSLLs cannot be set using negative values of β. To work it around, β should take imaginary values. In Figure 4, we plot the maximal sidelobe level versus β/i, where i is the imaginary number. It is noted that, when β takes on imaginary values, the edge elements of the resulting Bessel planar array acquire the largest current magnitudes, whereas the lowest excitation is that of the center element(s), a situation totally opposite to that when β has real values.

image

Figure 4. Maximal Sidelobe Level versus β/i of a 13 × 13-element Bessel planar array for 0 ≤ β/i ≤ 3.

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[19] In Figures 5 and 6, respectively, the variations of directivity and HPBW of Bessel and Chebyshev planar arrays with MSLL = SLL = −20dB are plotted against L. It is noticed that, although the directivity of the Chebyshev planar arrays saturates, the corresponding Bessel planar arrays shows always-increasing directivities. The realized improvements in directivity are very clear. As for the HPBW, it is always smaller for the Chebyshev planar arrays, which are optimal with respect to beam width, but the improvement in directivity justifies, for most applications, the slightly broader beam width the Bessel planar arrays have.

image

Figure 5. Directivity versus L for Bessel and Chebyshev planar arrays with MSSL = SLL = −20dB, dx = dy = 0.5λ, and θ0 = ϕ0 = 0.

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image

Figure 6. HPBW versus L for Bessel and Chebyshev planar arrays with MSSL = SLL = −20dB, dx = dy = 0.5λ, and θ0 = ϕ0 = 0.

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[20] For lower sidelobe levels (<−30dB), most power is contained in the main lobe and less is radiated in the direction of the sidelobes. Thus, the suppression of the sidelobes, as done by the Bessel planar arrays, does not contribute to a significant improvement in directivity. Actually, for low sidelobes, the Bessel planar arrays achieve directivities higher than their Chebyshev counterparts only when L is larger than a certain minimum for the given sidelobe level and element spacing, or when the element spacing is larger than a certain minimum for the given sidelobe level and number of elements. For MSLL = SLL = −30dB, dx = dy = 0.5λ and θ0 = ϕ0 = 0, the directivity of Bessel planar arrays surpasses that of Chebyshev planar arrays only when L ≥ 20, as indicated in Figure 7. But as L is increased beyond 20, the directivity of the Chebyshev array saturates whereas that of the Bessel array keeps on increasing. For L = 13, MSLL = SLL = −30dB and θ0 = ϕ0 = 0, the directivity of Bessel planar arrays exceeds that of Chebyshev planar arrays when dx = dy ≥ 0.666λ.

image

Figure 7. Directivity versus L for Bessel and Chebyshev planar arrays with MSSL = SLL = −30dB, dx = dy = 0.5λ, and θ0 = ϕ0 = 0.

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[21] To analyze the performance of Bessel planar arrays for off-broadside scan angles, we take θ0 = 5π/12, ϕ0 = π/2 for L = 13 and MSLL = SLL = −20dB. The spacing value dx = dy = 0.5λ causes a grating lobe to appear in the pattern of the Bessel planar array, and also in that of the corresponding Chebyshev planar array. The directivity of the Bessel array is 12.56% larger than that of the Chebyshev array. For an inter-element spacing that prevents grating lobes, the improvement in directivity achieved by the Bessel arrays will be more significant since, in this case, more power percentage is contained in the sidelobes that the Bessel array design suppresses. For dx = dy = 0.4λ, the gain in directivity is 17.49%. The array factor of the Bessel array for this spacing is shown in Figure 8.

image

Figure 8. Three-dimensional antenna pattern of a Bessel planar array with 13 × 13 isotropic elements, MSLL = −20dB, dx = dy = 0.25λ, θ0 = 5π/12 and ϕ0 = π/2.

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[22] Numerical techniques exist and help synthesize planar arrays with high performance. In Rivas et al. [2001] and Bucci et al. [2002], the numerically designed arrays are compared to Chebyshev (Tseng-Cheng) planar arrays. Optimized from a 12 × 12 Chebyshev planar array with a −20dB SLL, the array designed in Rivas et al. [2001] has a 23.02dB directivity, −19.1dB MSLL, a HPBW of 10.8°, and a maximum to minimum current ratio of 3.26. The 12 × 12 Bessel planar array with MSLL = −20dB has a directivity DB = 25.82dB and a half-power beam width HPBWB = 9.770. The max-to-min current ratio is 3.31. The analytically designed Bessel planar array outperforms in both the directivity and the beam width. Even the max-to-min current ratio is smaller (2.94) if the MSLL is set to −19.1dB. In Bucci et al. [2002], the array considered has 15 × 15 elements and a MSLL = −20dB. The ratio of the directivity of the synthesized array to that of Chebyshev (Tseng-Cheng) array is 1.93. For a 15 × 15 Bessel planar array with MSLL = −20dB, this ratio is 2.32, which signifies a better directivity performance for the Bessel planar array.

[23] To analyze the immunity of the directivity of Bessel planar arrays to slight changes in excitations, these excitations were allowed to change by up to 10% of the maximum excitation value through the introduction of uniformly distributed noise. The maximum recorded deviation from the true directivity value did not exceed 4%, which indicates that the directivity of Bessel arrays is stationary.

4. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Results and Comments
  6. 4. Conclusion
  7. References
  8. Supporting Information

[24] In this paper, the Bessel planar arrays were proposed. Their idea was incited by Kaiser windows used in FIR filter design. Analytical expressions for their excitation currents were presented, and methods for calculating their half-power beam width and controlling their maximal sidelobe level were explained. These arrays have stable performance and result in much higher directivities than the Chebyshev planar arrays for large number of elements. These improvements and other properties of the proposed arrays were illustrated through numerical examples.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Results and Comments
  6. 4. Conclusion
  7. References
  8. Supporting Information
  • Balanis, C. A. (1997), Antenna Theory, John Wiley, Hoboken, N. J.
  • Bucci, O. M., L. Caccavale, and T. Isernia (2002), Optimal far-field focusing of uniformly spaced arrays subject to arbitrary upper bounds in nontarget directions, IEEE Trans. Antennas Propag., 50(11), 15391554.
  • Hansen, R. C. (1983), Planar arrays, in The Handbook of Antenna Design, vol. 2, edited by A. W. Rudge, and K. Milne, p. 154, Peter Peregrinus, London.
  • Kabalan, K. Y., A. El-Hajj, and M. Al-Husseini (2002), The modified Chebyshev planar arrays, Radio Sci., 37(5), 1082, doi:10.1029/2002RS002619.
  • Lee, M. J., I. Song, S. Yoon, and S. R. Park (2000), Evaluation of directivity for planar antenna arrays, IEEE Antennas Propag. Mag., 42(3), 6467.
  • Mailloux, R. J. (1994), Phased Array Antenna Handbook, Artech House, Norwood, Mass.
  • Mitra, S. K. (1998), Digital Signal Processing: A Computer-Based Approach, McGraw-Hill, New York.
  • Rivas, A., J. A. Rodriguez, F. Ares, and E. Moreno (2001), Planar arrays with square lattices and circular boundaries: Sum patterns from distributions with uniform amplitude or very low dynamic-range ratio, IEEE Antennas Propag. Mag., 43(5), 9093.
  • Tseng, F. I., and D. K. Cheng (1968), Optimum scannable planar arrays with an invariant sidelobe level, Proc. IEEE, 56(11), 17711778.

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Results and Comments
  6. 4. Conclusion
  7. References
  8. Supporting Information
FilenameFormatSizeDescription
rds4994-sup-0001tab01.txtplain text document1KTab-delimited Table 1.
rds4994-sup-0002tab02.txtplain text document1KTab-delimited Table 2.

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