The authors propose to use planar antennas of waveguide slots with the simple structure of postwall waveguide for base station antennas in Local Multipoint Distribution Systems (LMDS). The array has a cosecant radiation pattern with null filling in the vertical plane for uniform illumination in a sector area. The slots are paired as an element to achieve traveling wave excitation because the phase, as well as the amplitude, has to be controlled for null filling in a cosecant pattern. It is the first time to design an array of reflection-canceling slot pairs to get a tapered distribution both in amplitude and phase. This paper extends the designability of the reflection-canceling slot arrays. A 16-element array is designed at 25–26 GHz band. The model array has a cosecant pattern with 3-dB deviations and a peak gain of 17.1 dBi in measurements.
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 Various types of base station antennas with a sector beam for Local Multipoint Distribution Systems (LMDS) at 25–26 GHz band are now developing in Japan [Wako et al., 1999; Omuro and Kuzuya, 2000]. The antennas should illuminate uniformly in the radial direction in a sectoral area by adopting a cosecant radiation pattern in the vertical plane as well as in the horizontal plane. Nulls in the cosecant radiation pattern should be filled to communicate everywhere in the area. Transmission loss of a millimeter wave is larger in air so that many base station antennas with high gain should be installed. Antenna fabrication should be simplified for reducing the cost. At high frequencies such as 25–26 GHz band, a low-loss feed line has to be used to obtain high efficiency even in a base station antenna with 15–20 dBi gain.
 Conventionally, reflector antennas were used for this purpose. They have a wide bandwidth based on optics. They are suitable especially in Frequency Division Duplexing (FDD) systems to cover wide frequency bands. It is possible to shape a reflector for any coverage to be illuminated uniformly. However, a reflector antenna is large in size in comparison with a linear array of microstrip patches. Transmission loss of a microstrip line as feeder is so large that the required gain can not be achieved. On the other hand, a waveguide has a small transmission loss. It has no undesired radiation from the transmission line itself because of the closed structure. Furthermore a waveguide can take a large height in order to reduce the conductor loss in comparison with a microstrip line. The authors propose a planar antenna of reflection-canceling slot pairs on a postwall waveguide [Hirokawa and Ando, 1998] as shown in Figure 1 for the simple fabrication. The postwall waveguide consists of metal-plated via-holes densely arrayed in a metal-grounded dielectric substrate. The slots are paired for reflection canceling to be excited by a traveling wave [Sakakibara et al., 1994]. The antenna is easily realized at low cost for mass production by conventional printed-circuit-board fabrication techniques such as via-holing and metal plating for the postwall waveguide and etching for the slots. It is thin and has a small volume itself. It is a series-fed array excited by a traveling wave so that the bandwidth is narrow depending on the array size, and the main beam is squinted by the frequency change. The antenna is preferable to be used in Time Division Duplexing (TDD) system because a wide bandwidth is not required in the system. A 16-element array is designed at 25–26 GHz band on a postwall waveguide to get a cosecant radiation pattern with null filling in the vertical plane. The theory for the pattern synthesis used in this paper is well known in the work of Elliot and Stern . The design procedure for the slot array in this paper was published in the work of Takahashi et al. . In the work of Sakakibara et al. , the authors proposed reflection-canceling slot pairs. However, in that reference, an array of the slot pairs on the broad wall of a rectangular waveguide was designed to be excited uniformly both in amplitude and phase. In this paper, it is the first time to design another array of the slot pairs to get a tapered distribution both in amplitude and phase. This paper extends the designability of the reflection-canceling slot arrays. Furthermore, from practical point of view, this paper proposes to use planar antennas of waveguide slots with the simple structure of postwall waveguide for base station antennas in LMDS to get high efficiency suitable for mass production. The antenna structure is described in section 2. The design procedure of the slot array with a cosecant radiation pattern with null filling is presented in section 3. Then, the measured results of model antennas are demonstrated in section 4. Finally the conclusions are summarized in section 5.
2. Antenna Structure
Figure 1 shows the antenna structure. Via-holes are arrayed densely on a metal-grounded dielectric substrate and are plated by metal on the surface to compose a postwall waveguide. The spacing of the via-holes is determined to prevent from leakage by solving an eigenvalue problem as a function of a guide wavelength, the spacing of the via-holes and dielectric constant of the substrate [Hirokawa and Ando, 1998]. The postwall waveguide is fed through an aperture at the bottom. The size and the position of the aperture from a shorted plate are determined to minimize the reflection. Slots are etched transversely to the waveguide axis. They are paired as one element to suppress the reflection and to be excited by a traveling wave. The slot pairs are arrayed to get a cosecant beam with null-filling in the vertical plane. The design procedure of the slot array is explained in detail in section 3.
3. Design Procedure of the Slot Array
3.1. Excitation Coefficient of Elements
 At the first step, the excitation coefficient of each element should be determined. It is found by the technique for shaped beam pattern synthesis [Elliot and Stern, 1984]. In this synthesis, the element spacing is assumed to be equal as the first step of the array design. The method directly treats a linear array and begins with the synthesis of a sum pattern in which the height of every sidelobe is individually specified. In the cosecant region, the peak of the main beam and the sidelobes are adjusted to follow a cosecant shaped envelope. In the remainder of visible space, the sidelobes are adjusted to satisfy Taylor pattern to suppress undesired radiation. After this, the roots of the Schelkunoff unit circle which lie in the cosecant region are displaced radially the proper amount to give null-filling to the level of the envelope. An N-element array equally spaced by d has a polynomial representation of the pattern given by the factored form such as with w = exp(jΨ) and , where θp is the shifting direction of the main beam. k0 and λ0 are the wave number and the wavelength in free space, respectively. wn is one of the roots corresponding to the null directions. The m-th root in the cosecant region is moved radially off the unit circle a distance α (either inward or outward). The new position of the m-th root is (1 − α)wm, where α is a pure real constant (positive or negative). α should be chosen so that the m-th null in the original pattern has been filled to the proper level. By using this technique, the excitation coefficients of the elements are determined. They are complex and both the amplitude and the phase of the radiation should be controlled.
3.2. Element Design
 In the second step, an element consisting of two parallel slots should be designed to get desired coupling and to suppress reflection. The two slots are offset from the center of the waveguide axis in the opposite direction in each other to reduce the mutual coupling and to increase the radiation of the pair. One element placed on a waveguide is analyzed by method of moments [Sakakibara et al., 1994]. In the analysis, postwalls are replaced with metal walls equivalent to have equal guide wavelength by solving an eigenvalue problem including the periodicity of postarrangement [Hirokawa and Ando, 1998]. An infinite ground plane is embedded in the external region as half free space. In a slot pair, the length of one slot is used to control the amplitude of the radiation, and the length of the other slot and the slot spacing are determined to suppress the reflection of the pair. In principle, the slot spacing is a quarter of the guide wavelength because the reflected waves of the two slots have the path length difference of a half guide wavelength and are canceled out to each other. However, the actual value is smaller than a quarter of the guide wavelength depending on the mutual coupling. By using this analysis, the length of the slots and the spacing are derived as functions of the coupling, i.e. the ratio of the radiated power to the input power, for the initial design. When the coupling is stronger, the length of the slots increases and the spacing decreases in a pair. The phase of the transmitting wave and the radiating wave are also presented as a function of the coupling. It becomes large for stronger coupling. The spacing between adjacent pairs gets small as the coupling is larger.
3.3. Initial Design of Array Arrangement
 In the third step, the coupling both in amplitude and phase of each slot pair should be determined. The amplitude can initially be designed in a deterministic way based on the power conservation from the termination side to the input side because the pair is designed to suppress the reflection [Takahashi et al., 1991]. When N pairs are cascaded, the coupling C(n) in the n-th pair (1 ≤ n ≤ N) is given as:
where A(n) was the excitation coefficient determined in the first step. Then the length of the slots and the spacing were presented as functions of the coupling derived in the second step. The phase of the radiating wave between adjacent slot pairs can be controlled by the spacing of the pairs because all the pairs are excited by a traveling wave. The spacing S(n) between the n-th and n + 1-th pairs is solved by the following equation:
where ∠St and ∠Sr are the transmitting and radiating phase, respectively, all of which were presented as functions of the coupling given in the second step. In equation (2), λg is the guide wavelength. The size and the location of all the slot pairs are initially specified.
3.4. Modification by Including the Actual Mutual Coupling
 In the final step, the location of the slot pairs is modified by including the mutual coupling in an actual array of the slot pairs. The method of moments is used in the analysis, where an infinite ground plane is embedded in the external region as well as in the element design [Sakakibara et al., 1994]. In the second step, only the mutual coupling between the two slots in a pair was considered. The mutual coupling among the slot pairs should be included because all the slots are parallel to each other. The phase of the excitation between adjacent slot pairs should be controlled by the pair spacing. The amplitude of the excitation should be controlled by the length of one slot in each pair. The length of the other slot and the slot spacing in a pair should also be modified according to the relation in the pair to keep suppressing the reflection.
4. Measured Results
4.1. Model Antennas
 16-element arrays are designed at 25.6 GHz. PTFE substrate is used. The dielectric constant is 2.17. The waveguide height b in Figure 1 is 3.2 mm, which is about 0.4 wavelength including the dielectric constant. The waveguide is a closed waveguide so that it has no radiation loss and it can take a large waveguide height to reduce the conductor loss. The diameter dp of the post is 1.2 mm (about 0.15λ) and the spacing sp is 2.4 mm (about 0.3λ). We confirm in an analysis that the leakage from the posts is negligible. The width a of the postwall waveguide in the Figure 1 is 7.92 mm. The length of the slot array is about 130 mm. The size of the input aperture is 6.43 mm (la) × 4.32 mm (wa). The position sa of the input aperture is 3.48 mm. The slot width w is 0.40 mm common in all the pairs. No offset (p = 0.00 mm) is used in all the pairs.
4.2. Amplitude Distributions
Figure 2 shows the designed aperture field distribution at 25.6 GHz. The dotted lines are for no null-filling in the cosecant region for when the roots for α = 0 nulls are on the unit circle in the first step of the design. The solid lines are for null-fillings for α = 0.1 when the roots are moved off the unit circle inwardly. The input side is located at the left and the terminated one is at the right in the figure. In the first step of the design, the average element spacing d is chosen as 0.725 λ0 and the shifting direction θp of the main beam is set to be 2.0 degrees. The phase taper for the beam shifting is eliminated in the phases presented in Figure 2. The sidelobe level in Taylor region is designed to be −15 dB. In the no null-filling case, both the amplitude and the phase are almost constant. However, the amplitude decreases from the input to the termination by 8.7 dB. The phase has a difference of 48 degrees between the center and the edges. This means that the element spacing around the center is wider than that around the edges.
 The length l1, l2 of two slots and the slot spacing d is varied in each pair. They are shown in Figure 3. The slot length is larger in null-filling than that in no null-filling. This means that the slot coupling is stronger in null-filling because the desired amplitude is strong around the middle of the array and it is weak around the ends. The slot spacing is smaller in null-filling for stronger coupling. The pair spacing s is changed in each pair. It is presented in Figure 4. The n-th spacing number is related to the spacing between the n-th and n + 1-th pairs. The element spacing is smaller in the null-filling for stronger coupling. Furthermore it is changed more widely around the ends of the array in null-filling because it should be compensated the phase taper of about 50 degrees around there.
Figure 5 shows the measured distribution at 25.6 GHz. A probe of rectangular waveguide is scanned over the waveguide array at 50 mm height by 5 mm pitch. Ripples are observed in the near field measurement. In the no null-filling, the amplitude is tapered by about 2 dB and the phase is deviated by about 10 degrees. In the null-filling, the amplitude has a taper of about 8 dB and the phase has a difference of about 40 degrees between the center and the edges. The measured values are slightly smaller than the designed ones.
4.3. Radiation Patterns
Figure 6 shows the designed radiation patterns on the vertical or E plane at 25.6 GHz. The dashed line is a cosecant pattern for reference. In the cosecant region, the pattern for no null-filling has deep nulls and that for null-filling has ripples of about 5 dB. Null-filling results in gain reduction at the main-beam direction. The no null-filling design has 19.5 dBi gain and the null-filling design has 19.0 dB in the main beam direction. The sidelobe level in Taylor region is suppressed below −14 dB in both designs.
Figure 7 shows the measured patterns at 25.6 GHz. The null-filling pattern fits well with a cosecant one in an angle range up to about 55 degrees. It is almost smooth in a range to 20 degrees. The measured ripples in the cosecant region are smaller than the designed ones. It is about 3 dB in the maximum. The measured peak gain of the main beam is 17.9 dBi for the no null-filling and 17.1 dBi for the null-filling. The gain reduction is 0.8 dB. The measured reflection at the input aperture is around −13 dB at 25.6 GHz. The 3 dB beam width on the horizontal or H plane is 84 degrees in the measurements at 25.6 GHz. The slot length on a dielectric-filled waveguide is shorter than that on an empty waveguide. So the 3 dB beam width on the H plane of the slot on the dielectric-filled waveguide is slightly wider.
 When the array is applied to a base station antenna in FDD system, discussions on the frequency dependence are needed. Figure 8 shows the calculated frequency dependence of the radiation pattern and the peak gain. The array is excited by a traveling wave so that the main beam direction is squinted by frequency change. The tilting angle of the main beam is −2.2 deg at 25.4 GHz, 0.3 deg at 25.6 GHz and 2.3 deg at 25.8 GHz. As the frequency is lower, the ripple width in the cosecant region is wider and the sidelobe level in Taylor region increases. The calculated peak gain is 18.0 dBi at 25.4 GHz, 19.0 dBi at 25.6 GHz and 18.7 dBi at 25.8 GHz.
 We have presented a reflection-canceling slot array with a cosecant pattern on a postwall waveguide. We have designed a 16-element array at 25.6 GHz. A cosecant pattern with 3 dB deviation is obtained with a 17.1 dBi gain in the model antenna.