In this paper, a design method for wideband smart antennas with independently controllable beam width and sidelobe level in their patterns is demonstrated. Using a frequency domain frequency-invariant beam former, the phases of the complex weights can be adjusted to obtain a beam pattern that is invariant over the frequency range of interest, and a modified Chebyshev approach is used to calculate the amplitudes of these complex beam-forming weights, ensuring a relatively independent control of the beam width and the sidelobe level. The paper also briefly discusses different approaches for selecting the amplitudes leading to patterns with high directivities.
 In mobile communications, smart antennas have received huge interest worldwide as they result in larger system capacity and increased range and provide the potential to introduce new services. Antenna arrays are the technology that is almost exclusively suggested for land-based mobile and personal communications systems [Lehne and Pettersen, 1999].
 In previous work, novel classes of linear and planar arrays were introduced. The generalized Chebyshev arrays [El-Hajj et al., 2005], the modified Chebyshev planar arrays [Kabalan et al., 2002], and the Bessel planar arrays [Kabalan et al., 2004] were designed to provide highly directive patterns with controllable sidelobe levels while keeping the beam width the smallest possible. Arrays with the narrowest possible beams are ideal in some applications such as in radars, but other applications, like wireless applications, require a flexible relationship between beam width and sidelobe level. M. Al-Husseini et al. (On the design of planar arrays with adjustable sidelobe level and beamwidth using a modified Chebyshev approach, paper to be presented at 2006 International Conference on Computer Engineering and Systems (ICCES06), Inst. of Electr. and Electron. Eng., Cairo, 5–7 Nov., hereinafter referred to as Al-Husseini et al., paper in preparation, 2006) devised planar arrays with independently controllable beam width and sidelobe level on the basis of a method proposed by Freitas de Abreu and Kohno  for uniform linear and circular antenna arrays. These arrays were narrowband beam formers and the problem was to select the appropriate weights to achieve the desired beams.
 It is well known that narrowband antenna arrays are not suitable for the ubiquitous applications that require operation over a wide band of frequencies. When the frequency decreases, the beam width increases leading to reduced spatial resolution, and when the frequency increases, the beam width decreases and grating lobes may appear in the array beam pattern [Ward et al., 1994]. Conventional wideband antenna arrays are characterized by a phase response that varies with frequency. They employ a combination of spatial and temporal filtering and use tapped delay lines (TDLs) [Godara, 1997] or recursive filters [Ghavami and Kohno, 2000] to equalize the phase shifts due to higher and lower frequencies.
Do-Hong and Russer  proposed a new method for wideband beam forming in frequency domain where a FFT is used at each antenna element to transform the wideband signals into frequency domain and then the signals at each frequency bin are weighted by complex weights whose phases are selected appropriately so that the beam pattern is constant over the frequency band. The amplitudes of the complex weights are used to control the characteristics of the beam pattern, including the beam width and the sidelobe level. One of the advantages this method has over time domain methods is that frequency invariance depends solely on the phases of the complex weights, which enables the control of the beam width and sidelobe level over the whole frequency band.
 In this paper, a design method for wideband antenna arrays is described. The phases of the complex weights are found using the method presented by Do-Hong and Russer , whereas their amplitudes are calculated using a modified Chebyshev approach that enables independent control of the beam width and the sidelobe level in the array pattern. The formulation is given in section 2. In section 3, methods to calculate the amplitudes of the complex weights with the goal of producing highly directive patterns are briefly discussed. Numerical examples and simulation results are given in section 4, and finally section 5 concludes the paper.
2. Independent Control of Beam Width and Sidelobe Level
 An arbitrary antenna array with M identical elements is considered. Ignoring mutual coupling, the far-field beam pattern at frequency fk is given by
Vector dm = T denotes the coordinates of the mth element within the array, m = 1…M, and vector Ωs = [cos ϕs sin θs, sin ϕs sin θs, cos θs]T. The angles ϕs and θs are respectively the steered azimuth and elevation angles, and c is the propagation speed. In (1), H denotes conjugate transpose and w(fk) is the vector of the complex weights given by
where wm(fk) is the complex weight applied to the mth element at frequency fk.
 To obtain a frequency-invariant beam pattern in the frequency range from fl to fh, the phases of the weights at each frequency should be selected such that F(fk) = F(f0), ∀ fk ∈ [fl, fh] where f0 is a focusing frequency in [fl, fh]. As a result, the complex weights at frequency fk should be determined as Do-Hong and Russer 
where am denotes the amplitude part of the weight wm and is frequency independent.
 The choice of am affects the characteristics of the beam pattern especially the beam width and sidelobe level.
 In the case of linear arrays with equal interelement spacing, the classic Dolph-Chebyshev design, which is based on Chebyshev polynomials, results in equiripple pattern and is optimum in the sense that for a desired sidelobe level, the beam is narrowest and for a prescribed beam width, the sidelobe level is lowest. In the work by Freitas de Abreu and Kohno , a Chebyshev-like function is given by
where N is the order of the function, and α and β are real parameters to be determined. Clearly, for α = 0 and β = 2, (4) reduces to an Nth-order Chebyshev polynomial. Using (4) and by choosing appropriate values for α and β, Freitas de Abreu and Kohno  were able to design linear arrays with independently controllable beam width and sidelobe level.
 In a parallel work, the Chebyshev-like function given by (4) was also used to design planar arrays with independently adjustable beam width and sidelobe level. Suppose the array has L × L identical elements, then α is iteratively obtained from
and β from the closed-form expression
Here xρ is a parameter that depends on the specified beam width and the specified sidelobe level ratio R. An algorithm for calculating xρ is given by Al-Husseini et al. (paper in preparation, 2006). Upon determining the parameters xρ, α and β, the amplitudes of the weights applied to the array are calculated from
when L is even and
for odd L. The amplitude a1,1 corresponds to the center element when L is odd, and to the four center elements when L is even since the amplitudes are symmetrical with respect to two principal axes in the plane of the array. In (9), ɛp and ɛq are equal to 1 for p = q = 1, and equal to 2 for p,q ≠ 1. When N(β − eα∣x∣) is not an integer, Equation (4) does not correspond to a polynomial, and as a result, the sidelobes will not necessary be of equal magnitude.
3. Highly Directive Patterns
 The amplitudes can also be chosen to produce patterns that are highly directive, and in this case either the beam width or the sidelobe level can be adjusted. The conventional Dolph-Chebyshev method used in the design of linear and planar arrays provides means to control either the beam width or the sidelobe level, but the resulting arrays suffer from directivity saturation in the sense that, beyond a certain number of elements, the directivity ceases to increase even when the number of elements is increased [Safaai-Jazi, 1998; Kabalan et al., 2002]. Several methods try to overcome this limitation. In the linear array case, the modified Chebyshev array [Safaai-Jazi, 1998] is obtained by raising the pattern of a basis linear array to an integer power. Recently, a new linear array design method consisted of multiplying the pattern of several different basis linear arrays [El-Hajj et al., 2005]. For the resulting Generalized Chebyshev array, the amplitudes are given by
where M is the number of elements, R is the desired sidelobe level ratio, and
The amplitudes are symmetrical with respect to the center of the array, a1 being the amplitude corresponding to the center element(s). In (11), b is the number of basis arrays, Mk and Rk are respectively the number of elements and the sidelobe level ratio of the kth basis array, TN denotes a Chebyshev polynomial of order N, and γk = cosh[cosh−1(Rk)/(Mk − 1)]. Both the conventional Dolph-Chebyshev and the modified Chebyshev linear arrays are special cases of the generalized Chebyshev arrays.
 The modified Chebyshev design of Safaai-Jazi  was extended to the planar array case of Kabalan et al. . For an array with L × L identical elements and a desired sidelobe level ratio R, the amplitudes of a modified Chebyshev planar array are given by
when L is even and
for odd L. The amplitude a1,1 corresponds to the center element when L is odd, and to the four center elements when L is even. Also, γ0 = cosh[b cosh−1(R1/b)/(L − 1)] and L0 = (L − 1)/b + 1. When b = 1, the resulting array is clearly a conventional Chebyshev planar array.
 The idea of the Bessel planar arrays [Kabalan et al., 2004] was “borrowed” from digital filter design and more specifically from Kaiser windows and extended to the planar case. The amplitudes of a Bessel planar array are given by
where I0(·) represents the zeroth-order modified Bessel function of the first kind, and γ is the parameter that controls that sidelobe level. A method for finding γ given a desired sidelobe level is described by Kabalan et al. . For γ = 0, all amplitudes are equal, and hence the uniform planar array, which provides the highest directivity, is a special case of Bessel planar arrays.
 The designs discussed above surpass the Dolph-Chebyshev design in directivity when the number of elements is large enough or when the sidelobes are not too low. In the pattern of a generalized Chebyshev array, most of the sidelobes are below the desired level. A modified Chebyshev planar array has less sidelobes than its conventional Chebyshev counterpart, and the Bessel planar array has fast decaying sidelobes. These properties of the sidelobes lead to these designs having higher no-saturating directivities at the cost of slightly broader main lobes, as compared to the Dolph-Chebyshev design.
4. Simulation Results
 To illustrate the aforementioned design methods, we take as a first example a 15-element linear broadside array. The interelement spacing is d = 0.5λ, where λ is the wavelength at the focusing frequency f0. We take f0/c = 0.5, where c is the propagation speed, and set the sidelobe level to −20 dB. The conventional Dolph-Chebyshev array has a beam width of 15.53°. The beam width is taken as the angle difference between the two points on the main lobe that have the same level as the sidelobes. For a desired beam width of 25°, the frequency-variant beam pattern of the resulting array is shown in Figure 1 plotted in the normalized band [0.3, 0.5] against f/c. As evident from the plot, the main lobe and the sidelobes broaden with decreasing frequency, and vice versa. The pattern of the corresponding wideband array is depicted in Figure 2. Frequency invariance is achieved over the whole band.
 As a second example, a planar array with 9 × 9 elements is taken. The interelement spacing is d = 0.5λ calculated at the focusing frequency f0 = 0.5c. The beam is steered in the direction ϕs = 0 and θs = 30°. For a sidelobe level of −20 dB, the conventional Chebyshev design results in a beam width of 26.93° in the plane ϕ = ϕs. The modified design is carried out for a prescribed beam width of 35° and the same sidelobe level. The parameters used in (9) are found to be xρ = 1.123, α = 0.197 and β = 2.01. A cross section of the resulting array's pattern, in the plane ϕ = ϕs, is compared to that of the conventional Dolph-Chebyshev design in Figure 3. For the wideband case, these patterns are frequency independent over the band [0.3, 0.5]. The beam width was enlarged to the desired extent, and the sidelobes, which are not exactly equiripple, are at or below the −20 dB level.
 A 19-element linear array with a sidelobe level of −10 dB is then considered. The interelement spacing is d = 0.5λ, where λ is calculated at f0 = 0.5c. The aim is to obtain wideband linear arrays with high directivities. Two basis arrays with equal sidelobe levels are used in the modified Chebyshev and the generalized Chebyshev designs. For the first design, each basis array has 8 elements, whereas for the second, the first basis array has 9 elements and the second has 7 elements. Comparing them to their Dolph-Chebyshev counterpart, the modified Chebyshev array has less sidelobes in its pattern, and many of the sidelobes in the pattern of the generalized Chebyshev array are even below −10 dB, as revealed in Figure 4. The directivity is 10.7 dB for the Dolph-Chebyshev design, 10.8 dB for the modified Chebyshev design, and 11.6 dB for the generalized Chebyshev array. For lower sidelobe levels (≤−20 dB), the modified and the generalized Chebyshev arrays surpass the conventional one in directivity only when the number of elements or the spacing is large enough.
 As a last example, a planar array with 13 × 13 elements is taken. In this example, the sidelobe level is −20 dB, ϕs = 0 and θs = 0. At the focusing frequency, the corresponding Dolph-Chebyshev array has a directivity of 23.7 dB, whereas the directivity of the Bessel planar array is 26.5 dB and that of the modified Chebyshev one is 24.1 dB. The frequency-invariant beam pattern for the Bessel planar array is plotted in Figure 5. Its higher directivity results from the decaying sidelobes in its pattern, meaning that less power is lost in the direction of the sidelobes.
 Smart antennas, which are implemented as antenna arrays, are of enormous interest in mobile communications because they result in larger system capacity, increased range and more potential for new services. Many of their applications require wideband properties and also the ability to independently control the beam width and the sidelobe level in their beam patterns. In this paper, a technique was demonstrated for the design of antenna arrays with beam patterns that are frequency invariant over a specified frequency range and have independently adjustable beam width and sidelobe level. The amplitudes of the complex weights of a frequency domain frequency-invariant beam former were chosen according to a modified Chebyshev method, whereas the phases of these weights were determined at several frequencies to achieve the frequency-invariance property. Other methods for amplitude selection were briefly discussed for desired high-directivity patterns. Simulation results were given to illustrate the presented designs.
 The authors would like to acknowledge the University Research Board (URB) of the American University of Beirut.